DOI: 10.6084/m9.figshare.9980744
4th Edition JASP v0.14 2020
Copyright © 2020 by Mark A Goss-Sampson.
Licenced as CC BY 4.0
All rights reserved. This book or any portion thereof may not be reproduced or used in any manner
whatsoever without the express written permission of the author except for research, education or
private study.
CONTENTS
PREFACE .................................................................................................................................................. 1
USING THE JASP ENVIRONMENT ............................................................................................................ 2
DATA HANDLING IN JASP ........................................................................................................................ 8
JASP ANALYSIS MENU ........................................................................................................................... 11
DESCRIPTIVE STATISTICS ....................................................................................................................... 14
DESCRIPTIVE PLOTS IN JASP .............................................................................................................. 19
SPLITTING DATA FILES ....................................................................................................................... 23
EXPLORING DATA INTEGRITY ................................................................................................................ 25
DATA TRANSFORMATION ..................................................................................................................... 34
EFFECT SIZE ........................................................................................................................................... 38
ONE SAMPLE T-TEST ............................................................................................................................. 40
BINOMIAL TEST ..................................................................................................................................... 43
MULTINOMIAL TEST .............................................................................................................................. 46
CHI-SQUARE ‘GOODNESS-OF-FIT’ TEST............................................................................................. 48
MULTINOMIAL AND Χ2 ‘GOODNESS-OF-FIT’ TEST. ........................................................................... 49
COMPARING TWO INDEPENDENT GROUPS .......................................................................................... 50
INDEPENDENT T-TEST ....................................................................................................................... 50
MANN-WITNEY U TEST ..................................................................................................................... 54
COMPARING TWO RELATED GROUPS ................................................................................................... 56
PAIRED SAMPLES T-TEST ................................................................................................................... 56
WILCOXON’S SIGNED RANK TEST...................................................................................................... 59
CORRELATION ANALYSIS ....................................................................................................................... 61
REGRESSION .......................................................................................................................................... 67
SIMPLE REGRESSION ......................................................................................................................... 70
MULTIPLE REGRESSION ..................................................................................................................... 73
LOGISTIC REGRESSION .......................................................................................................................... 80
COMPARING MORE THAN TWO INDEPENDENT GROUPS .................................................................... 85
ANOVA .............................................................................................................................................. 85
KRUSKAL-WALLIS .............................................................................................................................. 92
COMPARING MORE THAN TWO RELATED GROUPS ............................................................................. 95
RMANOVA ......................................................................................................................................... 95
FRIEDMAN’S REPEATED MEASURES ANOVA .................................................................................. 100
COMPARING INDEPENDENT GROUPS AND THE EFFECTS OF COVARIATES ........................................ 103
ANCOVA .......................................................................................................................................... 103
TWO-WAY INDEPENDENT ANOVA ...................................................................................................... 111
TWO-WAY REPEATED MEASURES ANOVA ........................................................................................ 119
MIXED FACTOR ANOVA ....................................................................................................................... 127
CHI-SQUARE TEST FOR ASSOCIATION ................................................................................................. 135
META-ANALYSIS .................................................................................................................................. 142
EXPERIMENTAL DESIGN AND DATA LAYOUT IN EXCEL FOR JASP IMPORT. ........................................ 150
Independent t-test .......................................................................................................................... 150
Paired samples t-test ...................................................................................................................... 151
Correlation ...................................................................................................................................... 152
Logistic Regression .......................................................................................................................... 154
One-way Independent ANOVA ....................................................................................................... 155
One-way repeated measures ANOVA ............................................................................................. 156
Two-way Independent ANOVA ....................................................................................................... 157
Two-way Repeated measures ANOVA ............................................................................................ 158
Two-way Mixed Factor ANOVA ....................................................................................................... 159
Chi-squared - Contingency tables ................................................................................................... 160
SOME CONCEPTS IN FREQUENTIST STATISTICS .................................................................................. 161
WHICH TEST SHOULD I USE? ............................................................................................................... 165
Comparing one sample to a known or hypothesized population mean. ........................................ 165
Testing relationships between two or more variables ................................................................... 165
Predicting outcomes ....................................................................................................................... 166
Testing for differences between two independent groups ............................................................ 166
Testing for differences between two related groups ..................................................................... 167
Testing for differences between three or more independent groups ............................................ 167
Testing for differences between three or more related groups ..................................................... 168
Test for interactions between 2 or more independent variables ................................................... 168
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PREFACE
JASP stands for Jeffrey’s Amazing Statistics Program in recognition of the pioneer of Bayesian
inference Sir Harold Jeffreys. This is a free multi-platform open-source statistics package, developed
and continually updated by a group of researchers at the University of Amsterdam. They aimed to
develop a free, open-source programme that includes both standard and more advanced statistical
techniques with a major emphasis on providing a simple intuitive user interface.
In contrast to many statistical packages, JASP provides a simple drag and drop interface, easy access
menus, intuitive analysis with real-time computation and display of all results. All tables and graphs
are presented in APA format and can be copied directly and/or saved independently. Tables can also
be exported from JASP in LaTeX format
JASP can be downloaded free from the website https://jasp-stats.org/ and is available for Windows,
Mac OS X and Linux. You can also download a pre-installed Windows version that will run directly from
a USB or external hard drive without the need to install it locally. The WIX installer for Windows
enables you to choose a path for the installation of JASP – however, this may be blocked in some
institutions by local Administrative rights.
The programme also includes a data library with an initial collection of over 50 datasets from Andy
Fields book, Discovering Statistics using IBM SPSS statistics1 and The Introduction to the Practice of
Statistics2 by Moore, McCabe and Craig.
Since May 2018 JASP can also be run directly in your browser via rollApp™ without having to install it
on your computer (https://www.rollapp.com/app/jasp). However, this may not be the latest version
of JASP.
Keep an eye on the JASP site since there are regular updates as well as helpful videos and blog posts!!
This book is a collection of standalone handouts covering the most common standard (frequentist)
statistical analyses used by students studying Biological Sciences. Datasets used in this document are
available for download from https://osf.io/bx6uv/
Dr Mark Goss-Sampson
Centre for Science and Medicine in Sport & Exercise
University of Greenwich
2020
1 A Field. (2017) Discovering Statistics Using IBM SPSS Statistics (5th Ed.) SAGE Publications. 2 D Moore, G McCabe, B Craig. (2011) Introduction to the Practice of Statistics (7th Ed.) W H Freeman.
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USING THE JASP ENVIRONMENT Open JASP.
The main menu can be accessed by clicking on the top-left icon.
Open:
JASP has its own .jasp format but can open a variety of
different dataset formats such as:
• .csv (comma separated values) can be saved in Excel
• .txt (plain text) also can be saved in Excel
• .tsv (tab-separated values) also can be saved in Excel
• .sav (IBM SPSS data file)
• .ods (Open Document spreadsheet)
You can open recent files, browse your computer files,
access the Open Science Framework (OSF) or open the
wide range of examples that are packaged with the Data
Library in JASP.
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Save/Save as:
Using these options the data file, any annotations and the analysis
can be saved in the .jasp format
Export:
Results can be exported to either an HTML file or as a PDF
Data can be exported to either a .csv, .tsv or .txt file
Sync data:
Used to synchronize with any updates in the current data file (also
can use Ctrl-Y)
Close:
As it states - it closes the current file but not JASP
Preferences:
There are three sections that users can use to tweak JASP to suit their needs
In the Data Preferences section users can:
• Synchronize/update the data automatically when the data file is saved (default)
• Set the default spreadsheet editor (i.e. Excel, SPSS etc)
• Change the threshold so that JASP more readily distinguishes between nominal and scale data
• Add a custom missing value code
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In the Results Preferences section users can:
• Set JASP to return exact p values i.e. P=0.00087 rather than P<.001
• Fix the number of decimals for data in tables – makes tables easier to read/publish
• Change the pixel resolution of the graph plots
• Select when copying graphs whether they have a white or transparent background.
In the Interface Preferences section users can now define a user font and pick between two different
themes; a light theme (default) and a dark theme. The preferred language currently supports English,
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German and Dutch only. In this section, there is also the ability to change the system size (zoom) for
accessibility and the scroll speeds.
In the Advanced Preferences section, most users will probably never have to change any of the default
settings.
Comparison of the dark and light themes in JASP
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JASP has a streamlined interface to switch between the spreadsheet, analysis and results views.
The vertical bars highlighted above allows for the windows to be dragged right or left by clicking and
dragging the three vertical dots
The individual windows can also be completely collapsed using the right or left arrow icons
If you click the Results icon a range of options is provided including:
• Edit title
• Copy
• Export results
• Add notes
• Remove all
• Refresh all
The ‘add notes’ option allows the results output to be easily annotated and then exported to an HTML
or PDF file by going to File > Export Results.
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The Add notes menu provides many options to change text font, colour size etc.
You can change the size of all the tables and graphs using ctrl+ (increase) ctrl- (decrease) ctrl= (back
to default size). Graphs can also be resized by dragging the bottom right corner of the graph.
As previously mentioned, all tables and figures are APA standard and can just be copied into any other
document. Since all images can be copied/saved with either a white or transparent background. This
can be selected in Preferences > Advanced as described earlier.
There are many further resources on using JASP on the website https://jasp-stats.org/
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DATA HANDLING IN JASP For this section open England injuries.csv
All files must have a header label in the first row. Once loaded, the dataset appears in the window:
For large datasets, there is a hand icon which allows easy scrolling through the data.
On import JASP makes a best guess at assigning data to the different variable types:
Nominal Ordinal Continuous
If JASP has incorrectly identified the data type just click on the appropriate variable data icon in the
column title to change it to the correct format.
If you have coded the data you can click on the variable name to open up the following window in
which you can label each code. These labels now replace the codes in the spreadsheet view. If you
save this as a .jasp file these codes, as well as all analyses and notes, will be saved automatically. This
makes the data analysis fully reproducible.
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In this window, you can also carry out simple filtering of data, for example, if you untick the Wales
label it will not be used in subsequent analyses.
Clicking this icon in the spreadsheet window opens up a much more comprehensive set of data
filtering options:
Using this option will not be covered in this document. For detailed information on using more
complex filters refer to the following link: https://jasp-stats.org/2018/06/27/how-to-filter-your-data-
in-jasp/
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By default, JASP plots data in the Value order (i.e. 1-4). The order can be changed by highlighting the
label and moving it up or down using the appropriate arrows:
Move up
Move down
Reverse order
Close
If you need to edit the data in the spreadsheet just double click on a cell and the data should open up
in the original spreadsheet i.e. Excel. Once you have edited your data and saved the original
spreadsheet JASP will automatically update to reflect the changes that were made, provided that you
have not changed the file name.
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JASP ANALYSIS MENU
The main analysis options can be accessed from the main toolbar. Currently, JASP offers the following
frequentist (parametric and non-parametric standard statistics) and alternative Bayesian tests:
Descriptives
• Descriptive stats
Regression
• Correlation
• Linear regression Logistic regression
T-Tests
• Independent
• Paired
• One sample
Frequencies
• Binomial test
• Multinomial test
• Contingency tables
• Log-linear regression*
ANOVA
• Independent
• Repeated measures
• ANCOVA
• MANOVA *
Factor
• Principal Component Analysis (PCA)*
• Exploratory Factor Analysis (EFA)*
• Confirmatory Factor Analysis (CFA)*
Mixed Models*
• Linear Mixed Models Generalised linear mixed models
* Not covered in this document
BY clicking on the + icon on the top-right menu bar you can also access advanced options that allow the addition of optional modules. Once ticked they will be added to the main analysis ribbon. These
include;
See the JASP website for more information on these advanced modules
Audit Network analysis
BAIN Reliability analysis
Distributions SEM
Equivalence tests Summary statistics
JAGS Visual modelling
Machine learning Learning Bayes
Meta-analysis (included in this guide) R (beta)
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Once you have selected your required analysis all the possible statistical options appear in the left
window and output in the right window.
JASP provides the ability to rename and ‘stack’ the results output thereby organising multiple
analyses.
The individual analyses can be renamed using the pen icon or deleted using the red cross.
By clicking on the analysis in this list will then take you to the appropriate part of the results output
window. They can also be rearranged by dragging and dropping each of the analyses.
The green + icon produces a copy of the chosen analysis
The blue information icon provides detailed information on each of the statistical procedures used
and includes a search option.
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DESCRIPTIVE STATISTICS Presentation of all the raw data is very difficult for a reader to visualise or to draw any inference on.
Descriptive statistics and related plots are a succinct way of describing and summarising data but do
not test any hypotheses. There are various types of statistics that are used to describe data:
• Measures of central tendency
• Measures of dispersion
• Percentile values
• Measures of distribution
• Descriptive plots
To explore these measures, load Descriptive data.csv into JASP. Go to Descriptives > Descriptive
statistics and move the Variable data to the Variables box on the right.
The Statistics menu can now be opened to see the various options available.
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CENTRAL TENDENCY.
This can be defined as the tendency for variable values to cluster around a central value. The three
ways of describing this central value are mean, median or mode. If the whole population is considered
we the term population mean / median/mode is used. If a sample/subset of the population is being
analysed the term sample mean/ median/mode is used. The measures of central tendency move
toward a constant value when the sample size is sufficient to be representative of the population.
In the Statistics options make sure that everything is unticked apart from mean, median and mode.
The mean, M or x̅ (17.71) is equal to the sum of all the values divided by the number of values in the
dataset i.e. the average of the values. It is used for describing continuous data. It provides a simple
statistical model of the centre of distribution of the values and is a theoretical estimate of the ‘typical
value’. However, it can be influenced heavily by ‘extreme’ scores.
The median, Mdn (17.9) is the middle value in a dataset that has been ordered from the smallest to
largest value and is the normal measure used for ordinal or non-parametric continuous data. Less
sensitive to outliers and skewed data
The mode (20.0) is the most frequent value in the dataset and is usually the highest bar in a distribution
histogram
DISPERSION
In the Statistics options make sure that the following options are ticked
Standard deviation, S or SD (6.94) is used to quantify the amount of dispersion of data values around
the mean. A low standard deviation indicates that the values are close to the mean, while a high
standard deviation indicates that the values are dispersed over a wider range.
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Variance (S2 = 48.1) is another estimate of how far the data is spread from the mean. It is also the
square of the standard deviation.
The standard error of the mean, SE (0.24) is a measure of how far the sample mean of the data is
expected to be from the true population mean. As the size of the sample data grows larger the SE
decreases compared to S and the true mean of the population is known with greater specificity.
MAD, median absolute deviation, a robust measure of the spread of data. It is relatively unaffected
by data that is not normally distributed. Reporting median +/- MAD for data that is not normally
distributed is equivalent to mean +/- SD for normally distributed data.
MAD Robust: Median absolute deviation of the data points, adjusted by a factor for asymptotically
normal consistency.
IQR - Interquartile Range is similar to the MAD but is less robust (see Boxplots).
Confidence intervals (CI), although not shown in the general Descriptive statistics output, these are
used in many other statistical tests. When sampling from a population to get an estimate of the mean,
confidence intervals are a range of values within which you are n% confident the true mean is
included. A 95% CI is, therefore, a range of values that one can be 95% certain contains the true mean
of the population. This is not the same as a range that contains 95% of ALL the values.
For example, in a normal distribution, 95% of the data are expected to be within ± 1.96 SD of the mean
and 99% within ± 2.576 SD.
95% CI = M ± 1.96 * the standard error of the mean.
Based on the data so far, M = 17.71, SE = 0.24, this will be 17.71 ± (1.96 * 0.24) or 17.71 ± 0.47.
Therefore the 95% CI for this dataset is 17.24 - 18.18 and suggests that the true mean is likely to be
within this range 95% of the time
QUARTILES
In the Statistics options make sure that everything is unticked apart from Quartiles.
Quartiles are where datasets are split into 4 equal quarters, normally based on rank ordering of
median values. For example, in this dataset
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1 1 2 2 3 3 4 4 4 4 5 5 5 6 7 8 8 9 10 10 10
25% 50% 75%
The median value that splits data by 50% = 50th percentile = 5
The median value of left side = 25th percentile = 3
The median value of right side = 75th percentile = 8
From this the Interquartile range (IQR) range can be calculated, this is the difference between the 75th
and 25th percentiles i.e. 5. These values are used to construct the descriptive boxplots later. The IQR
can also be shown by ticking this option in the Dispersion menu.
DISTRIBUTION
Skewness describes the shift of the distribution away from a normal distribution. Negative skewness
shows that the mode moves to the right resulting in a dominant left tail. Positive skewness shows
that the mode moves to the left resulting in a dominant right tail.
Kurtosis describes how heavy or light the tails are. Positive kurtosis results in an increase in the
“pointiness” of the distribution with heavy (longer) tails while negative kurtosis exhibit a much more
uniform or flatter distribution with light (shorter) tails.
Negative skewness Positive skewness
+ kurtosis
Normal
- kurtosis
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In the Statistics options make sure that everything is unticked apart from skewness, kurtosis and
Shapiro-Wilk test.
We can use the Descriptives output to calculate skewness and kurtosis. For a normal data distribution,
both values should be close to zero. The Shapiro-Wilk test is used to assess whether or not the data is
significantly different from a normal distribution. (see - Exploring data integrity in JASP for more
details).
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DESCRIPTIVE PLOTS IN JASP Currently, JASP produces a range of descriptive plots:
Again, using Descriptive data.csv with the variable data in the Variables box, go to the statistics
options and under Plots tick Distribution plots, Boxplots – Boxplot Element and Q-Q plots.
The Distribution plot is based on splitting the data into frequency bins, this is then overlaid with the
distribution curve. As mentioned before, the highest bar is the mode (most frequent value of the
dataset. In this case, the curve looks approximately symmetrical suggesting that the data is
approximately normally distributed. The second distribution plot is from another dataset which shows
that the data is positively skewed.
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The boxplots visualise several statistics described above in one plot:
• Median value
• 25 and 75% quartiles
• Interquartile range (IQR) i.e. 75% - 25% quartile values
• Maximum and minimum values plotted with outliers excluded
• Outliers are shown if requested
Maximum value
Median value
Minimum value
75% quartile
25% quartile
IQR
Top 25%
Bottom 25%
Outlier
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Go back to the statistics options, in Descriptive plots tick both Boxplot and Violin Element, look at how
the plot has changed. Next tick Boxplot, Violin and Jitter Elements. The Violin plot has taken the
smoothed distribution curve from the Distribution plot, rotated it 90o and superimposed it on the
boxplot. The jitter plot has further added all the data points.
A Q-Q plot (quantile-quantile plot) can be used to visually assess if a set of data comes from a normal
distribution. Q-Q plots take the sample data, sort it in ascending order, and then plot them against
quantiles (percentiles) calculated from a theoretical distribution. If the data is normally distributed,
the points will fall on or close to the 45-degree reference line. If the data is not normally distributed,
the points will deviate from the reference line.
Boxplot + Violin plot Boxplot + Violin + Jitter plot
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Scatter Plots
JASP can produce scatterplots of various types and to be able to include smooth or linear regression
lines. There are also options to add distributions to these either in the form of density plots or
histograms.
Pie charts
Also, users can plot piecharts when working with categorical or other frequency data.
Plot colour palettes
Users can choose from between 5 different colour palettes using the drop-down menu.
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SPLITTING DATA FILES If there is a grouping variable (categorical or ordinal) descriptive statistics and plots can be produced
for each group. Using Descriptive data.csv with the variable data in the Variables box now add Group
to the Split box.
The output will be as follows:
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EXPLORING DATA INTEGRITY Sample data is used to estimate parameters of the population whereby a parameter is a measurable
characteristic of a population, such as a mean, standard deviation, standard error or confidence
intervals etc.
What is the difference between a statistic and a parameter? If you randomly polled a selection of
students about the quality of their student bar and you find that 75% of them were happy with it. That
is a sample statistic since only a sample of the population were asked. You calculated what the
population was likely to do based on the sample. If you asked all the students in the university and
90% were happy you have a parameter since you asked the whole university population.
Bias can be defined as the tendency of a measurement to over or under-estimate the value of a
population parameter. There are many types of bias that can appear in research design and data
collection including:
• Participant selection bias – some being more likely to be selected for study than others
• Participant exclusion bias - due to the systematic exclusion of certain individuals from the
study
• Analytical bias - due to the way that the results are evaluated
However statistical bias can affect a) parameter estimates, b) standard errors and confidence intervals
or c) test statistics and p values. So how can we check for bias?
IS YOUR DATA CORRECT?
Outliers are data points that are abnormally outside all other data points. Outliers can be due to a
variety of things such as errors in data input or analytical errors at the point of data collection Boxplots
are an easy way to visualise such data points where outliers are outside the upper (75% + 1.5 * IQR)
or lower (25% - 1.5 * IQR) quartiles
Boxplots show:
• Median value
• 25 & 75% quartiles
• IQR – Inter quartile range
• Max & min values plotted
with outliers excluded
• Outliers shown if requested
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Load Exploring Data.csv into JASP. Under Descriptives > Descriptive Statistics, add Variable 1 to the
Variables box. In Plots tick the following Boxplots, Label Outliers, and BoxPlot Element.
The resulting Boxplot on the left looks very compressed and an obvious outlier is labelled as being in
row 38 of the dataset. This can be traced back to a data input error in which 91.7 was input instead of
917. The graph on the right shows the BoxPlot for the ‘clean’ data.
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How you deal with an outlier depends on the cause. Most parametric tests are highly sensitive to
outliers while non-parametric tests are generally not.
Correct it? – Check the original data to make sure that it isn’t an input error, if it is, correct it, and
rerun the analysis.
Keep it? - Even in datasets of normally distributed, data outliers may be expected for large sample
sizes and should not automatically be discarded if that is the case.
Delete it? – This is a controversial practice in small datasets where a normal distribution cannot be
assumed. Outliers resulting from an instrument reading error may be excluded but it should be verified
first.
Replace it? – Also known as winsorizing. This technique replaces the outlier values with the relevant
maximum and/or minimum values found after excluding the outlier.
Whatever method you use must be justified in your statistical methodology and subsequent analysis.
WE MAKE MANY ASSUMPTIONS ABOUT OUR DATA.
When using parametric tests, we make a series of assumptions about our data and bias will occur if
these assumptions are violated, in particular:
• Normality
• Homogeneity of variance or homoscedasticity
Many statistical tests are an omnibus of tests of which some will check these assumptions.
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TESTING THE ASSUMPTION OF NORMALITY
Normality does not mean necessarily that the data is normally distributed per se but it is whether or
not the dataset can be well modelled by a normal distribution. Normality can be explored in a variety
of ways:
• Numerically
• Visually / graphically
• Statistically
Numerically we can use the Descriptives output to calculate skewness and kurtosis. For a normal data
distribution, both values should be close to zero. To determine the significance of skewness or kurtosis
we calculate their z-scores by dividing them by their associated standard errors:
Skewness Z = skewness
Skewness standard error Kurtosis Z =
kurtosis
kurtosis standard error
Z score significance: p<0.05 if z >1.96 p<0.01 if z >2.58 p<0.001 if z >3.29
Using Exploring data.csv, go to Descriptives>Descriptive Statistics move Variable 3 to the Variables
box, in the Statistics drop-down menu select Mean, Std Deviation, Skewness and as shown below with
the corresponding output table.
It can be seen that both skewness and kurtosis are not close to 0. The positive skewness suggests that
data is distributed more on the left (see graphs later) while the negative kurtosis suggests a flat
distribution. When calculating their z scores it can be seen that the data is significantly skewed p<0.05.
Skewness Z = 0.839
0.337 = 2.49 Kurtosis Z =
-0.407
0.662 = 0.614
[As a note of caution skewness and kurtosis many appear significant in large datasets even though the
distribution is normal.]
Now add Variable 2 to the Variables box and in Plots tick Distribution plot. This will show the following
two graphs:
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It is quite easy to visualise that Variable 2 has a symmetrical distribution. Variable 3 is skewed to the
left as confirmed by the skewness Z score.
Another graphical check for normality is a Q-Q plot. Q-Q plots are available in Descriptives and are
also produced as part of the Assumption Checks used in linear regression and ANOVA. Q-Q plots show
the quantiles of the actual data against those expected for a normal distribution.
If data are normally distributed all the points will be close to the diagonal reference line. If the points
‘sag’ above or below the line there is a problem with kurtosis. If the points snake around the line then
the problem is skewness. Below are Q-Q plots for Variables 2 and 3. Compare these to the previous
distribution plots and the skewness/kurtosis z scores above.
Variable 2 Variable 3
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The following Q-Q plot scenarios are possible:
The Shapiro-Wilk test is a statistical way used by JASP to check the assumption of normality. It is also
used in the Independent (distribution of the two groups) and Paired (distribution of differences
between pairs) t-tests. The test results in a W value; where small values indicate your sample is not
normally distributed (the null hypothesis that your population is normally distributed if your values
are under a certain threshold can, therefore, be rejected).
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In Descriptives, the Shapiro-Wilk test can be selected in the Distribution tests. The Shapiro-Wilk
output table shows no significant deviation in normality for Variable 2 but a significant deviation
(p<.001) for Variable 3.
The most important limitation is that the test has can be biased by sample size. The larger the sample,
the more likely you’ll get a statistically significant result.
Testing the assumption of normality – A cautionary note!
For most parametric tests to be reliable, one of the assumptions is that the data is approximately
normally distributed. A normal distribution peaks in the middle and is symmetrical about the mean.
However, data does not need to be perfectly normally distributed for the tests to be reliable.
So, having gone on about testing for normality – is it necessary?
The Central Limit Theorem states that as the sample size gets larger i.e. >30 data points the
distribution of the sampling means approaches a normal distribution. So the more data points you
have the more normal the distribution will look and the closer your sample mean approximates the
population mean.
Large datasets may result in significant tests of normality i.e. Shapiro-Wilk or significant skewness and
kurtosis z-scores when the distribution graphs look fairly normal. Conversely, small datasets will
reduce the statistical power to detect non-normality.
However, data that does not meet the assumption of normality is going to result in poor results for
certain types of test (i.e. ones that state that the assumption must be met!). How closely does your
data need to be normally distributed? This is a judgment call best made by eyeballing the data.
WHAT DO I DO IF MY DATA IS NOT NORMALLY DISTRIBUTED?
Transform the data and redo the normality checks on the transformed data. Common transformations
include taking the log or square root of the data.
Use non-parametric tests since these are distribution-free tests and can be used instead of their
parametric equivalent.
TESTING HOMOGENEITY OF VARIANCE
Levene’s test is commonly used to test the null hypothesis that variances in different groups are equal.
The result from the test (F) is reported as a p-value, if not significant then you can say that the null
hypothesis stands — that the variances are equal; if the p-value is significant then the implication is
that the variances are unequal. Levene’s test is included in the Independent t-test and ANOVA in
JASP as part of the Assumption Checks.
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Using Exploring data.csv, go to T-Tests>Independent Samples t-test move Variable 1 to the Variables
box and Group to the Grouping variable and tick Assumption Checks > Equality of variances.
In this case, there is no significant difference in variance between the two groups F (1) = 0.218, p=.643.
The assumption of homoscedasticity (equal variance) is important in linear regression models as is
linearity. It assumes that the variance of the data around the regression line is the same for all
predictor data points. Heteroscedasticity (the violation of homoscedasticity) is present when the
variance differs across the values of an independent variable. This can be visually assessed in linear
regression by plotting actual residuals against predicted residuals
33 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
If homoscedasticity and linearity are not violated there should be no relationship between what the
model predicts and its errors as shown in the graph on the left. Any sort of funnelling (middle graph)
suggests that homoscedasticity has been violated and any curve (right graph) suggests that linearity
assumptions have not been met.
34 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
DATA TRANSFORMATION The ability to compute new variables or to transform data was introduced in version 0.9.1. In
some cases, it may be useful to compute the differences between repeated measures or, to
make a dataset more normally distributed, you can apply a log transform for example. When
a dataset is opened there will be a plus sign (+) at the end of the columns.
Clicking on the + opens up a small dialogue window where you can;
• Enter the name of a new variable or the transformed variable
• Select whether you enter the R code directly or use the commands built into JASP
• Select what data type is required
Once you have named the new variable and chose the other options – click create.
35 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
If you choose the manual option rather than the R code, this opens all the built-in create and
transform options. Although not obvious, you can scroll the left and right-hand options to see
more variables or more operators respectively.
For example, we want to create a column of data showing the difference between variable 2
and variable 3. Once you have entered the column name in the Create Computed Column
dialogue window, its name will appear in the spreadsheet window. The mathematical
operation now needs to be defined. In this case drag variable 2 into the equation box, drag
the ‘minus’ sign down and then drag in variable 3.
If you have made a mistake, i.e. used the wrong variable or operator, remove it by dragging
the item into the dustbin in the bottom right corner.
36 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
When you are happy with the equation/operation, click compute column and the data will be
entered.
If you decide that you do not want to keep the derived data, you can remove the column by
clicking the other dustbin icon next to the R.
Another example is to do a log transformation of the data. In the following case variable 1 has
been transformed by scrolling the operators on the left and selecting the log10(y) option.
Replace the “y” with the variable that you want to transform and then click Compute column.
When finished, click the X to close the dialogue.
37 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
The two graphs below show the untransformed and the log10 transformed data. The skewed
data has been transformed into a profile with a more normal distribution
The Export function will also export any new data variables that have been created.
Untransformed
Log10 transformed
38 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
EFFECT SIZE When performing a hypothesis test on data we determine the relevant statistic (r, t, F etc) and p-value
to decide whether to accept or reject the null hypothesis. A small p-value, <0.05 in most analyses
provides evidence against the null hypothesis whereas a large p-value >0.05 only means that there is
not enough evidence to reject the null hypothesis. A lower p-value is sometimes incorrectly
interpreted as meaning there is a stronger relationship of difference between variables. So what is
needed is not just null hypothesis testing but also a method of determining precisely how large the
effects seen in the data are.
An effect size is a statistical measure used to determine the strength of the relationship or difference
between variables. Unlike a p-value, effect sizes can be used to quantitatively compare the results
of different studies.
For example, comparing heights between 11 and 12-year-old children may show that the 12-year-
olds are significantly taller but it is difficult to visually see a difference i.e. small effect size. However,
a significant difference in height between 11 and 16-year-old children is obvious to see (large effect
size).
The effect size is usually measured in three ways:
• the standardized mean difference
• correlation coefficient
• odds ratio
When looking at differences between groups most techniques are primarily based on the differences
between the means divided by the average standard deviations. The values derived can then be used
to describe the magnitude of the differences. The effect sizes calculated in JASP for t-tests and ANOVA
are shown below:
39 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
When analysing bivariate or multivariate relationships the effect sizes are the correlation
coefficients:
When analysing categorical relationships via contingency tables i.e. chi-square test Phi is only used for
2x2 tables while Cramer’s V and be used for any table size.
For a 2 × 2 contingency table, we can also define the odds ratio measure of effect size.
40 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
ONE SAMPLE T-TEST Research is normally carried out in sample populations, but how close does the sample reflect the
whole population? The parametric one-sample t-test determines whether the sample mean is
statistically different from a known or hypothesized population mean.
The null hypothesis (Ho) tested is that the sample mean is equal to the population mean.
ASSUMPTIONS
Three assumptions are required for a one-sample t-test to provide a valid result:
• The test variable should be measured on a continuous scale.
• The test variable data should be independent i.e. no relationship between any of the data
points.
• The data should be approximately normally distributed
• There should be no significant outliers.
RUNNING THE ONE SAMPLE T-TEST
Open one sample t-test.csv, this contains two columns of data representing the height (cm) and body
masses (kg) of a sample population of males used in a study. In 2017 the average adult male in the UK
population was 178 cm tall and has a body mass of 83.6 kg.
Go to T-Tests > One-Sample t-test and in the first instance add height to the analysis box on the right.
Then tick the following options and add 178 as the test value:
41 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
UNDERSTANDING THE OUTPUT
The output should contain three tables.
The assumption check of normality (Shapiro-Wilk) is not significant suggesting that the heights are
normally distributed, therefore this assumption is not violated. If this showed a significant difference
the analysis should be repeated using the non-parametric equivalent, Wilcoxon’s signed-rank test
tested against the population median height.
This table shows that there are no significant differences between the means p =.706
The descriptive data shows that the mean height of the sample population was 177.6 cm compared
to the average 178 cm UK male.
Repeat the procedure by replacing height with mass and change the test value to 83.6.
42 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
The assumption check of normality (Shapiro-Wilk) is not significant suggesting that the masses are
normally distributed.
This table shows that there is a significant difference between the mean sample (72.9 kg) and
population body mass (83.6 kg) p <.001
REPORTING THE RESULTS
A one-sample t-test showed no significant difference in height compared to the population mean (t
(22) = -0.382, p= .706), however, the participants were significantly lighter than the UK male
population average (t (22) =-7.159, p<.001).
43 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
BINOMIAL TEST The binomial test is effectively a non-parametric version of the one-sample t-test for use with
dichotomous (i.e. yes/no) categorical datasets. This tests whether or not the sample frequency is
statistically different from a known or hypothesized population frequency.
The null hypothesis (Ho) tested is that the sample data frequency is equal to the expected population
frequency.
ASSUMPTIONS Three assumptions are required for a binomial test to provide a valid result:
• The test variable should be a dichotomous scale (such as yes/no, male/female etc.).
• The sample responses should be independent
• The sample size is less, but representative of the population
RUNNING THE BINOMIAL TEST
Open binomial.csv, this contains one column of data showing the number of students using either a
Windows laptop or a MacBook at University. In January 2018, when comparing just the two operating
systems, the UK market share of Windows was 86% and Mac IOS 14%.3
Go to Frequencies >Binomial test. Move the Laptop variable to the data window and set the Test value
to 0.86 (86%). Also, tick Descriptive plots.
3 https://www.statista.com/statistics/268237/global-market-share-held-by-operating-systems-since-
2009/
44 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
The following table and graph show that the frequencies of both laptops are significantly less than
86%. In particular, these students are using significantly fewer Windows laptops than was expected
compared to the UK market share.
Is this the same for MacBook users? Go back to the Options window and change the test value to
0.14 (14%). This time both frequencies are significantly higher than 14%. This shows that students
are using significantly more MacBooks than was expected compared to the UK market share.
45 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
REPORTING THE RESULTS
The UK proportion of Windows and MacBook users was reported to be 86% and 14% respectively. In
a cohort of University students (N=90), a Binomial test revealed that the proportion of students using
Windows laptops was significantly less (59.6%, p<.001) and those using MacBooks significantly more
(40.4%, p<.001) than expected.
46 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
MULTINOMIAL TEST The multinomial test is effectively an extended version of the Binomial test for use with categorical
datasets containing three or more factors. This tests whether or not the sample frequency is
statistically different from a hypothesized population frequency (multinomial test) or a known (Chi-
square ‘goodness-of-fit’ test).
The null hypothesis (Ho) tested is that the sample frequency is equal to the expected population
frequency.
ASSUMPTIONS Three assumptions are required for a multinomial test to provide a valid result:
• The test variable should be a categorical scale containing 3 or more factors
• The sample responses should be independent
• The sample size is less, but representative of the population
RUNNING THE MULTINOMIAL TEST
Open multinomial.csv. This contains three columns of data showing the number of different coloured
M&Ms counted in five bags. Without any prior knowledge, it could be assumed that the different
coloured M&Ms are equally distributed.
Go to Frequencies > Multinomial test. Move colour of the M&Ms to Factor and the observed number
of M&Ms to counts. Tick Descriptives and Descriptives Plots.
47 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
As can be seen in the Descriptive table, the test assumes an equal expectation for the proportions of
coloured M&Ms (36 of each colour). The Multinomial test results show that the observed distribution
is significantly different (p<.001) to an equal distribution.
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CHI-SQUARE ‘GOODNESS-OF-FIT’ TEST.
However, further research shows that the manufacturer produces coloured M&Ms in different ratios:
Colour Blue Brown Green Orange Red Yellow
Proportion 24 13 16 20 13 14
These values can now be used as the expected counts, so move the Expected variable to the Expected
Counts box. This automatically runs the χ2 ‘goodness-of-fit’ test leaving the Hypothesis options greyed
out.
As can be seen in the Descriptives table, JASP has calculated the expected numbers of the different
coloured M&Ms based on the manufacturers reported production ratio. The results of the test show
that the observed proportions of the different coloured M&Ms are significantly different (χ2 =74.5,
p<.001) to those proportions stated by the manufacturer.
49 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
MULTINOMIAL AND Χ2 ‘GOODNESS-OF-FIT’ TEST. JASP also provides another option whereby both tests can be run at the same time. Go back to the
Options window and only add Colour to the Factor and Observed to the Counts boxes, remove the
expected counts if the variable is still there. In Hypotheses now tick the χ2 test. This will open up a
small spreadsheet window showing the colour and Ho (a) with each cell have 1 in it. This is assuming
that the proportions of each colour are equal (multinomial test).
In this window, add another column which will automatically be labelled Ho (b). The expected
proportions of each colour can now be typed in.
Now when the analysis is run, the results of the tests for the two hypotheses are shown. Ho (a) is
testing the null hypothesis that the proportions of each colour are equally distributed, while Ho (b) is
testing the null hypothesis that the proportions are the same as those expected. As can be seen, both
hypotheses are rejected. In particular, evidence indicates that the colours of plain M&M's do not
match the manufacturers published proportions.
50 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
COMPARING TWO INDEPENDENT GROUPS
INDEPENDENT T-TEST The parametric independent t-test, also known as Student’s t-test, is used to determine if there is a
statistical difference between the means of two independent groups. The test requires a continuous
dependent variable (i.e. body mass) and an independent variable comprising 2 groups (i.e. males and
females).
This test produces a t-score which is a ration of the differences between the two groups and the
differences within the two groups:
t = 𝒎𝒆𝒂𝒏 𝒈𝒓𝒐𝒖𝒑 𝟏 − 𝒎𝒆𝒂𝒏 𝒈𝒓𝒐𝒖𝒑 𝟐
𝒔𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒆𝒓𝒓𝒐𝒓 𝒐𝒇 𝒕𝒉𝒆 𝒎𝒆𝒂𝒏 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆𝒔
A large t-score indicates that there is a greater difference between groups. The smaller the t-score,
the more similarity there is between groups. A t-score of 5 means that the groups are five times as
different from each other as they are within each other.
The null hypothesis (Ho) tested is that the population means from the two unrelated groups are equal
ASSUMPTIONS OF THE PARAMETRIC INDEPENDENT T-TEST
Group independence:
Both groups must be independent of each other. Each participant will only provide one data point for
one group. For example participant 1 can only be in either a male or female group – not both.
Repeated measures are assessed using the Paired t-test.
Normality of the dependent variable:
The dependent variable should also be measured on a continuous scale and be approximately
normally distributed with no significant outliers. This can be checked using the Shapiro-Wilk test. The
t-test is fairly robust and small deviations from normality are normally acceptable. However, this is
not the case if the group sizes are very different. A rule of thumb is that the ratio between the group
sizes should be <1.5 (i.e. group A = 12 participants and group B = >8 participants).
If normality is violated you can try transforming your data (for example log values, square root values)
or, and if the group sizes are very different, use the Mann-Whitney U test which is a non-parametric
equivalent that does not require the assumption of normality (see later).
X = mean
S = standard deviation
n = number of data points
51 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
Homogeneity of variance:
The variances of the dependent variable should be equal in each group. This can be tested using
Levene's Test of Equality of Variances.
If the Levene's Test is statistically significant, indicating that the group variances are unequal we can
correct for this violation by using an adjusted t-statistic based on the Welch method.
RUNNING THE INDEPENDENT T-TEST
Open Independent t-test.csv, this contains weight loss on a self-controlled 10-week diet between men
and women. Its good practice to check the Distribution and boxplots in Descriptives to visually check
for distribution and outliers.
Go to T-Tests > Independent Samples t-test and put weight loss in the Dependent variable box and
gender (independent variable) in the Grouping Variable box.
Unequal variance Equal variance
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In the analysis window tick the following options:
UNDERSTANDING THE OUTPUT
The output should consist of four tables and one graph. Firstly we need to check that the parametric
assumptions required are not violated.
Shapiro-Wilk test shows that both groups have normally distributed data, therefore, the assumption
of normality is not violated. If one or both were significant you should consider using the non-
parametric equivalent Mann-Whitney test.
53 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
Levene’s test shows that there is no difference in the variance, therefore, the assumption of
homogeneity of variance is not violated. If Levene’s test was significant Welch’s adjusted t-statistic,
degrees of freedom and p values should be reported.
This table shows the two computed t-statistics (Student and Welch). Remember the t-statistic is
derived from the mean difference divided by the standard error of the difference. Both show that
there is a significant statistical difference between the two groups (p<.001) and Cohen’s d suggests
that this is a large effect.
From the descriptive data, it can be seen that females had a higher weight loss than males.
REPORTING THE RESULTS
An independent t-test showed that females lost significantly more weight over 10 weeks dieting than
males t(85)=6.16, p<.001. Cohen’s d (1.322) suggests that this is a large effect.
54 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
MANN-WITNEY U TEST If you find that your data is not normally distributed (significant Shapiro-Wilk test result) or is ordinal
by nature, the equivalent non-parametric independent test is the Mann-Whitney U test.
Open Mann-Whitney pain.csv which contains subjective pain scores (0-10) with and without ice
therapy. NOTE: make sure that Treatment is categorical and pain score is ordinal. Go to T-Tests >
Independent t-tests and put pain score in the Dependent variable box and use Treatment as the
grouping variable.
In the analysis options only tick:
✓ Mann-Whitney
✓ Location parameter
✓ Effect size
There is no reason to repeat the assumption checks since Mann-Whitney does not require the
assumption of normality or homogeneity of variance required by parametric tests.
UNDERSTANDING THE OUTPUT
This time you will only get one table:
The Mann-Whitney U-statistic (JASP reports this as W since it is an adaptation of Wilcoxon’s signed-
rank test) is highly significant. U=207, p<.001.
The location parameter, the Hodges–Lehmann estimate, is the median difference between the two
groups. The rank-biserial correlation (rB) can be considered as an effect size and is interpreted the
same as Pearson’s r, so 0.84 is a large effect size.
For non-parametric data, you should report median and MAD (or IQR) values as your descriptive
statistics and use boxplots instead of line graphs and confidence intervals, SD/SE bars. Go to
Descriptive statistics, put Pain score into the variable box and Split the file by Treatment.
55 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
REPORTING THE RESULTS
A Mann-Whitney test showed that Ice therapy significantly reduces pain scores (Mdn = 3) compared
to the control group (Mdn = 7), U=207, p<.001.
56 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
COMPARING TWO RELATED GROUPS
PAIRED SAMPLES T-TEST As with the Independent t-test, there are both parametric and non-parametric options available in
JASP. The parametric paired-samples t-test (also known as the dependent sample t-test or repeated
measures t-test) compares the means between two related groups on the same continuous,
dependent variable. For example, looking at weight loss pre and post 10 weeks dieting.
The paired t statistic = mean of the differences between group pairs
the standard error of the mean differences
With the paired t-test, the null hypothesis (Ho) is that the pairwise difference between the two
groups is zero.
ASSUMPTIONS OF THE PARAMETRIC PAIRED SAMPLES T-TEST
Four assumptions are required for a paired t-test to provide a valid result:
• The dependent variable should be measured on a continuous scale.
• The independent variable should consist of 2 categorical related/matched groups, i.e. each
participant is matched in both groups
• The differences between the matched pairs should be approximately normally distributed
• There should be no significant outliers in the differences between the 2 groups.
RUNNING THE PAIRED SAMPLES T-TEST
Open Paired t-test.csv in JASP. This contains two columns of paired data, pre-diet body mass and post
4 weeks of dieting. Go to T-Tests > Paired Samples t-test. Ctrl-click both variables and add them to the
analysis box on the right.
In the analysis options tick the following:
57 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
UNDERSTANDING THE OUTPUT
The output should consist of three tables and one graph.
The assumption check of normality (Shapiro-Wilk) is not significant suggesting that the pairwise
differences are normally distributed, therefore the assumption is not violated. If this showed a
significant difference the analysis should be repeated using the non-parametric equivalent,
Wilcoxon’s signed-rank test.
This shows that there is a significant difference in body mass between the pre and post dieting
conditions, with a mean difference (location parameter) of 3.783kg. Cohen’s d states that this is a
large effect.
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The descriptive statistics and plot show that there was a reduction in body mass following 4 weeks of
dieting.
REPORTING THE RESULTS
On average participants lost 3.78 kg (SE: 0.29 kg) body mass following a 4-week diet plan. A paired
samples t-test showed this decrease to be significant (t (77) =13.04, p<.001). Cohen’s d suggests that
this is a large effect
59 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
RUNNING THE NON-PARAMETRIC PAIRED SAMPLES TEST
WILCOXON’S SIGNED RANK TEST
If you find that your data is not normally distributed (significant Shapiro-Wilk test result) or is ordinal
by nature, the equivalent non-parametric independent test is the Wilcoxon’s signed-rank test. Open
Wilcoxon’s rank.csv. This has two columns one with pre-anxiety and post hypnotherapy anxiety scores
(from 0 - 50). In the dataset view make sure that both variables are assigned to the ordinal data type.
Go to T-Tests > Paired Samples t-test and follow the same instructions as above but now only tick the
following options:
There will be only one table in the output:
The Wilcoxon W-statistic is highly significant, p<0.001.
The location parameter, the Hodges–Lehmann estimate, is the median difference between the two
groups. The rank-biserial correlation (rB) can be considered as an effect size and is interpreted the
same as Pearson’s r, so 0.48 is a medium to large effect size.
Effect size Trivial Small Medium Large
Rank -biserial (rB) <0.1
0.1
0.3
0.5
60 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
For non-parametric data, you should report median values as your descriptive statistics and use
boxplots instead of line graphs and confidence intervals, SD/SE bars.
REPORTING THE RESULTS
A Wilcoxon’s signed-rank test showed that hypnotherapy significantly reduces anxiety scores (Mdn =
15) compared to pre-therapy (Mdn =22) scores, W=322, p<.001.
61 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
CORRELATION ANALYSIS Correlation is a statistical technique that can be used to determine if, and how strongly, pairs of
variables are associated. Correlation is only appropriate for quantifiable data in which numbers are
meaningful, such as continuous or ordinal data. It cannot be used for purely categorical data for which
we have to use contingency table analysis (see Chi-square analysis in JASP).
Essentially do different variables co-vary? i.e. are changes in one variable reflected in similar changes
to another variable? If one variable deviates from its mean does the other variable deviate from its
mean in either the same or opposite direction? This can be assessed by measuring covariance,
however, this is not standardised. For example, we can measure the covariance of two variables which
are measured in meters, however, if we convert the same values to centimetres, we get the same
relationship but with a completely different covariance value.
To overcome this, standardised covariance is used which is known as Pearson’s correlation coefficient (or "r"). It ranges from -1.0 to +1.0. The closer r is to +1 or -1, the more closely the two
variables are related. If r is close to 0, there is no relationship. If r is (+) then as one variable increases
the other also increases. If r is (-) then as one increases, the other decreases (sometimes referred to
as an "inverse" correlation).
The correlation coefficient (r) should not be confused with R2 (coefficient of determination) or R
(multiple correlation coefficient as used in the regression analysis).
The main assumption in this analysis is that the data have a normal distribution and are linear. This
analysis will not work well with curvilinear relationships.
Covariance = 4.7 Covariance = 470
62 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
RUNNING CORRELATION
The analysis tests the null hypothesis (H0) that there is no association between the two variables
From the example, data open Jump height correlation.csv which contains 2 columns of data, jump
height (m) and explosive leg power (W). Firstly run the Descriptive statistics and check the boxplots
for any outliers.
To run the correlation analysis go to Regression > Correlation. Move the 2 variables to the analysis
box on the right. Tick
✓ Pearson,
✓ Report significance,
✓ Flag significant correlations
Under Plots
✓ Scatter plots
✓ Heatmap
63 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
UNDERSTANDING THE OUTPUT
The first table shows the correlation matrix with Pearson’s r value and its p-value. This shows a highly
significant correlation (p<.001) with a large r value close to 1 (r= 0.984) and that we can reject the null
hypothesis.
For simple correlations like this it is easier to look at the pairwise table (go back to analysis and tick
the Display pairwise table option. This replaces the correlation matrix in the results which may be
easier to read.
The Pearson’s r value is an effect size where <0.1 is trivial, 0.1 -0.3 is a small effect, 0.3 – 0.5 a moderate
effect and >0.5 a large effect.
The plot provides a simple visualisation of this strong positive correlation (r = 0.984, p<.001) which is
also highlighted by the heatmap (more relevant when looking at multiple correlations).
64 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
GOING ONE STEP FURTHER.
If you take the correlation coefficient r and square it you get the coefficient of determination (R2). This
is a statistical measure of the proportion of variance in one variable that is explained by the other
variable. Or:
R2= Explained variation / Total variation
R2 is always between 0 and 100% where:
• 0% indicates that the model explains none of the variability of the response data around its
mean and
• 100% indicates that the model explains all the variability of the response data around its
mean.
In the example above r = 0.984, so R2 = 0.968. This suggests that jump height accounts for 96.8% of
the variance in explosive leg power.
REPORTING THE RESULTS
Pearson’s correlation showed a significant correlation between jump height and leg power (r = 0.984,
p<.001) jump height accounting for 96.8% of the variance in leg power.
RUNNING NON-PARAMETRIC CORRELATION – Spearman’s and Kendall’s tau
If your data is ordinal or is continuous data that has violated the assumptions required for parametric
testing (normality and/or variance) you need to use the non-parametric alternatives to Pearson’s
correlation coefficient.
The alternatives are Spearman’s (rho) or Kendall’s (tau) correlation coefficients. Both are based on
ranking data and are not affected by outliers or normality/variance violations.
Spearman's rho is usually used for ordinal scale data and Kendall's tau is used in small samples or when
many values with the same score (ties). In most cases, Kendall’s tau and Spearman’s rank correlation
coefficients are very similar and thus invariably lead to the same inferences.
The effect sizes are the same as Pearson’s r. The main difference is that rho2 can be used as an
approximate non-parametric coefficient of determination but the same is not true for Kendall’s tau.
From the example data, open Non-parametric correlation.csv which contains 2 columns of data, a
creativity score and position in the ‘World’s biggest liar’ competition (thanks to Andy Field).
Run the analysis as before but now using Spearman and Kendall’s tau-b coefficients instead of
Pearson’s.
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As can be seen there is a significant correlation between creativity scores and final position in the
‘World’s biggest liar’ competition, the higher the score the better the final competition position.
However, the effect size is only moderate.
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NOTE OF CAUTION.
Correlation only gives information on the strength of association. It gives no information on the
direction i.e. which variable causes the other to change. So it cannot be used to state the one thing
causes the other. Often a significant correlation means absolutely nothing and is purely by chance
especially if you correlate thousands of variables. This can be seen in the following strange
correlations:
Pedestrians killed in a collision with a railway train correlates with rainfall in Missouri:
Number of honey-producing bee colonies (1000’s) correlates strongly with the marriage rate in
South Carolina (per 1000 marriages)
67 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
REGRESSION Whereas correlation tests for associations between variables, regression is the next step commonly
used for predictive analysis, i.e. to predict a dependent outcome variable from one (simple regression)
or more (multiple regression) independent predictor variables.
Regression results in a hypothetical model of the relationship between the outcome and predictor
variable(s). The model used is a linear one defined by the formula;
y = c + b*x + ε
• y = estimated dependent outcome variable score,
• c = constant,
• b = regression coefficient and
• x = score on the independent predictor variable
• ε = random error component (based on residuals)
Linear regression provides both the constant and regression coefficient(s).
Linear regression makes the following assumptions:
1. Linear relationship: important to check for outliers since linear regression is sensitive to their
effects.
2. Independence of variables
3. Multivariate normality: requires all variables to be normally distributed
4. Homoscedasticity: homogeneity of variance of the residuals
5. Minimal multicollinearity /autocorrelation: when the independent variables/residuals are
too highly correlated with each other.
With regard to sample sizes, there are many different ‘rules of thumb’ in the literature ranging from
10-15 data points per predictor in the model i.e. 4 predictor variables will each require between 40
and 60 data points each to 50 +(8 * number of predictors) for each variable. So for 4 variables that
would require 82 data point for each variable. Effectively the bigger your sample size the better your
model.
SUMS OF SQUARES (Boring, but the basis of evaluating the regression model.)
Most regression analysis will produce the best model available, but how good is it actually and how
much error is in the model?
This can be determined by looking at ‘the goodness of fit’ using the sums of squares. This is a measure
of how close the actual data points are close to the modelled regression line.
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The vertical difference between the data points and the predicted regression line is known as the
residuals. These values are squared to remove the negative numbers and then summed to give SSR.
This is effectively the error of the model or the ‘goodness of fit’, obviously the smaller the value the
less error in the model.
The vertical difference between the data points and the mean of the outcome variable can be
calculated. These values are squared to remove the negative numbers and then summed to give the
total sum of the squares SST. This shows how good the mean value is as a model of the outcome
scores.
Values above the
line are positive
Values below the
line are negative
69 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
The vertical difference between the mean of the outcome variable and the predicted regression line
is now determined. Again these values are squared to remove the negative numbers and then
summed to give the model sum of squares (SSM). This indicates how better the model is compared to
just using the mean of the outcome variable. SST is the total sum of the squares.
So, the larger the SSM the better the model is at predicting the outcome compared to the mean value
alone. If this is accompanied by a small SSR the model also has a small error.
R2 is similar to the coefficient of determination in correlation in that it shows how much of the
variation in the outcome variable can be predicted by the predictor variable(s).
R2 = SSM
SST
In regression, the model is assessed by the F statistic based on the improvement in the prediction of
the model SSM and the residual error SSR. The larger the F value the better the model.
F = Mean SSM
Mean SSR
70 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
SIMPLE REGRESSION Regression tests the null hypothesis (Ho) that there will be no significant prediction of the dependent
(outcome) variable by the predictor variable(s).
Open Rugby kick regression.csv. This dataset contains rugby kick data including distance kicked,
right/left leg strength and flexibility and bilateral leg strength.
Firstly go to Descriptives > Descriptive statistics and check the boxplots for any outliers. In this case,
there should be none, though it is good practice to check.
For this simple regression go to Regression > Linear regression and put distance into the Dependent
Variable (outcome) and R_Strength into the Covariates (Predictor) box. Tick the following options in
the Statistics options:
UNDERSTANDING THE OUTPUT
You will now get the following outputs:
Here it can be seen that the correlation (R) between the two variables is high (0.784). The R2 value of
0.614 tells us that right leg strength accounts for 61.4% of the variance in kick distance. Durbin-
Watson checks for correlations between residuals, which can invalidate the test. This should be above
1 and below 3 and ideally around 2.
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The ANOVA table shows all the sums of squares mentioned earlier. With regression being the model
and Residual being the error. The F-statistic is significant p=0.002. This tells us that the model is a
significantly better predictor of kicking distance that the mean distance.
Report as F (1, 11) = 17.53, p<.001.
This table gives the coefficients (unstandardized) that can be put into the linear equation.
y = c + b*x
y = estimated dependent outcome variable score,
c = constant (intercept)
b = regression coefficient (R_strength)
x = score on the independent predictor variable
For example for a leg strength of 60 kg the distance kicked can be predicted by the following:
Distance = 57.105 + (6.452 * 60) = 454.6 m
FURTHER ASSUMPTION CHECKS In Plots checks, tick the following two options:
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This will result in two graphs:
This graph shows a balanced random distribution of the residuals around the baseline suggesting that
the assumption of homoscedasticity has not been violated. (See Exploring data integrity in JASP for
further details.
The Q-Q plot shows that the standardized residuals fit nicely along the diagonal suggesting that both
assumptions or normality and linearity have also not been violated.
REPORTING THE RESULTS
Linear regression shows that right leg strength can significantly predict kicking distance F (1, 11) =
17.53, p<.001 using the following regression equation:
Distance = 57.105 + (6.452 * Right leg strength)
73 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
MULTIPLE REGRESSION The model used is still a linear one defined by the formula;
y = c + b*x + ε
▪ y = estimated dependent outcome variable score,
▪ c = constant,
▪ b = regression coefficient and
▪ x = score on the independent predictor variable
▪ ε = random error component (based on residuals)
However, we now have more than 1 regression coefficient and predictor score i.e.
y = c + b1*x1 + b2*x2 + b3*x3 …….. bn*xn
Data entry methods. If predictors are uncorrelated their order of entry has little effect on the model. In most cases,
predictor variables are correlated to some extent and thus, the order in which the predictors are
entered can make a difference. The different methods are subject to much debate in the area.
Forced entry (Enter): This is the default method in which all the predictors are forced into the model
in the order they appear in the Covariates box. This is considered to be the best method.
Blockwise entry (Hierarchical entry): The researcher, normally based on prior knowledge and previous
studies, decides the order in which the known predictors are entered first depending on their
importance in predicting the outcome. Additional predictors are added in further steps.
Stepwise (Backward entry): All predictors are initially entered in the model and then the contribution
of each is calculated. Predictors with less than a given level of contribution (p<0.1) are removed. This
process repeats until all the predictors are statistically significant.
Stepwise (Forward entry): The predictor with the highest simple correlation with the outcome variable
is entered first. Subsequent predictors selected based on the size of their semi-partial correlation with
the outcome variable. This is repeated until all predictors that contribute significant unique variance
to the model have been included in the model.
Stepwise entry: Same as the Forward method, except that every time a predictor is added to the
model, a removal test is made of the least useful predictor. The model is constantly reassessed to see
whether any redundant predictors can be removed.
There are many reported disadvantages of using stepwise data entry methods, however, Backward
entry methods can be useful for exploring previously unused predictors or for fine-tuning the model
to select the best predictors from the available options.
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RUNNING MULTIPLE REGRESSION
Open Rugby kick regression.csv that we used for simple regression. Go to Regression > Linear
regression and put distance into the Dependent Variable (outcome) and now add all the other
variables into the Covariates (Predictor) box.
In the Variable section leave the Method as Enter. Tick the following options in the Statistics options,
Estimates, Model fit, Collinearity diagnostics and Durbin-Watson.
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UNDERSTANDING THE OUTPUT
You will now get the following outputs:
This provides information on a model based on the H0 (no predictors) and the alternative H1.
The adjusted R2 (used for multiple predictors) shows that they can predict 68.1% of the outcome
variance. Durbin-Watson checks for correlations between residuals is between 1 and 3 as required.
The ANOVA table shows the F-statistic to be significant p=0.017 suggesting that the model is a
significantly better predictor of kicking distance that the mean distance.
This table shows both the H0 and H1 models and the constant (intercept) and regression coefficients
(unstandardized) for all the predictors forced into the model. Even though the ANOVA shows the
model to be significant none of the predictor regression coefficients is significant!
The collinearity statistics, Tolerance and VIF (Variance Inflation Factor) check the assumption of
multicollinearity. As a rule of thumb if VIF >10 and tolerance <0.1 the assumptions have been greatly
violated. If the average VIF >1 and tolerance <0.2 the model may be biased. In this case, the average
VIF is quite large (around 5).
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The casewise diagnostics table is empty! This is good news. This will highlight any cases (rows) that
have residuals which are 3 or more standard deviations away from the mean. These cases with the
largest errors may well be outliers. Too many outliers will have an impact on the model and should be
dealt with in the usual way (see Exploring Data Integrity).
As a comparison re-run the analyses but now choose Backward as the method of data entry.
The outputs are as follows:
JASP has now calculated 4 potential regression models. It can be seen that each consecutive model
increases the adjusted R2, with model 4 accounting for 73.5% of the outcome variance.
The Durbin-Watson score is also higher than with the forced entry method.
The ANOVA table indicates that each successive model is better as shown by the increasing F-value
and improving p-value.
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Model 1 is the same as the forced entry method first used. The table shows that as the least
significantly contributing predictors are sequentially removed, we end up with a model with two
significant predictor regression coefficients, right leg strength and bilateral leg strength.
Both tolerance and VIF are acceptable.
We now can report the Backward predictor entry results in a highly significant model F (2, 10) = 17.92,
p<.001 and a regression equation of
Distance = 46.251 + (3.914 * R_Strength) + (2.009 * Bilateral Strength)
TESTING FURTHER ASSUMPTIONS. As for the simple linear regression example, tick the following options.
78 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
The balanced distribution of the residuals around the baseline suggests that the assumption of
homoscedasticity has not been violated.
The Q-Q plot shows that the standardized residuals fit along the diagonal suggesting that both
assumptions or normality and linearity have also not been violated.
REPORTING THE RESULTS
Multiple linear regression using backward data entry shows that right leg and bilateral strength can
significantly predict kicking distance F(2,10) = 17.92, p<.001 using a regression equation of
Distance = 57.105 + (3.914 * R_Strength) + (2.009 * Bilateral Strength)
79 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
IN SUMMARY
R2 provides information on how much variance is explained by the model using the predictors
provided.
F-statistic provides information as to how good the model is.
The unstandardized (b)-value provides a constant which reflects the strength of the relationship
between the predictor(s) and the outcome variable.
Violation of assumptions can be checked using Durbin-Watson value, tolerance/VIF values, Residual
vs predicted and Q-Q plots.
80 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
LOGISTIC REGRESSION In simple and multiple linear regression outcome and predictor variable(s) were continuous data.
What if the outcome was a binary/categorical measure? Can, for example, a yes or no outcome be
predicted by other categorical or continuous variables? The answer is yes if binary logistic regression
is used. This method is used to predict the probability of a binary yes or no outcome.
The null hypothesis tested is that there is no relationship between the outcome and the predictor
variable(s).
As can be seen in the graph below, a linear regression line between the yes and no responses would
be meaningless as a prediction model. Instead, a sigmoidal logistic regression curve is fitted with a
minimum of 0 and a maximum of 1. It can be seen that some predictor values overlap between yes
and no. For example, a prediction value of 5 would give an equal 50% probability of being a yes or no
outcome. Thresholds are therefore calculated to determine if a predictor data value will be classified
as a yes or no outcome.
ASSUMPTIONS FOR BINARY LOGISTIC REGRESSION
• The dependent variable must be binary i.e. yes or no, male or female, good or bad.
• One or more independent (predictor variables) which can be continuous or categorical
variables.
• A linear relationship between any continuous independent variables and the logit
transformation (natural log of the odds that the outcome equals one of the categories) of the
dependent variable.
LOGISTIC REGRESSION METRICS
AIC (Akaike Information Criteria) and BIC (Bayesian Information Criteria) are measures of fit for the
model, the best model will have the lowest AIC and BIC values.
Outcome = No
Outcome = Yes
P ro
b a
b il
it y
o f
o u
tc o
m e
= Y
e s
81 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
Four pseudo R2 values are calculated in JASP, McFadden, Nagelkerke, Tjur and Cox & Snell. These are
analogous to R2 in linear regression and all give different values. What constitutes a good R2 value
varies, however, they are useful when comparing different models for the same data. The model with
the largest R2 statistic is considered to be the best.
The Wald test is used to determine statistical significance for each of the independent variables.
The confusion matrix is a table showing actual vs predicted outcomes and can be used to determine
the accuracy of the model. From this sensitivity and specificity can be derived.
Sensitivity is the percentage of cases that had the observed outcome was correctly predicted by the
model (i.e., true positives).
Specificity is the percentage of observations that were also correctly predicted as not having the
observed outcome (i.e., true negatives).
RUNNING LOGISTIC REGRESSION
Open Heart attack.csv in JASP. This contains 4 columns of data, Patient ID, did they have a second
heart attack (yes/no), whether they were prescribed exercise (yes/no) and their stress levels (high
value = high stress).
Put the outcome variable (2nd heart attack) into the Dependent variable, add the stress levels to
Covariates and Exercise prescription to Factors. Leave the data entry method as Enter.
In the Statistics options tick Estimates, Odds ratios, Confusion matrix, Sensitivity and Specificity.
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UNDERSTANDING THE OUTPUT
The initial output should comprise of 4 tables.
The model summary shows that H1 (with the lowest AIC and BIC scores) suggests a significant
relationship (X2(37) =21.257, p<.001) between the outcome (2nd heart attack) and the predictor
variables (exercise prescription and stress levels).
McFadden's R2 = 0.383. It is suggested that a range from 0.2 to 0.4 indicates a good model fit.
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Both stress level and exercise prescription are significant predictor variables (p=.031 and .022
respectively). The most important values in the coefficients table are the odds ratios. For the
continuous predictor, an odds ratio of greater than 1 suggests a positive relationship while < 1 implies
a negative relationship. This suggests that high-stress levels are significantly related to an increased
probability of having a second heart attack. Having an exercise intervention is related to a significantly
reduced probability of a second heart attack. The odds ratio of 0.13 can be interpreted as only having
a 13% probability of a 2nd heart attack if undergoing an exercise intervention.
The confusion matrix shows that the 15 true negative and positive cases were predicted by the model
while the error, false negatives and positives, were found in 5 cases. This is confirmed in the
Performance metrics where both sensitivity (% of cases that had the outcome correctly predicted) and
specificity (% of cases correctly predicted as not having the outcome (i.e., true negatives) are both
75%.
PLOTS
These findings can be easily visualised through the inferential plots.
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As stress levels increase the probability of having a second heart attack increases.
No exercise intervention increases the probability of a 2nd heart attack while it is reduced when it had
been put in place.
REPORTING THE RESULTS
Logistic regression was performed to ascertain the effects of stress and exercise intervention on the
likelihood that participants have a 2nd heart attack. The logistic regression model was statistically
significant, χ2 (37) = 21.257, p < .001. The model correctly classified 75.0% of cases. Increasing stress
was associated with an increased likelihood of a 2nd heart attack, but decreasing stress was associated
with a reduction in the likelihood. The presence of an exercise intervention programme reduced the
probability of a 2nd heart attack to 13%.
85 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
COMPARING MORE THAN TWO INDEPENDENT GROUPS
ANOVA
Whereas t-tests compare the means of two groups/conditions, one-way analysis of variance (ANOVA) compares the means of 3 or more groups/conditions. There are both independent and repeated
measures ANOVAs available in JASP. ANOVA has been described as an ‘omnibus test’ which results in
an F-statistic that compares whether the datasets overall explained variance is significantly greater
than the unexplained variance. The null hypothesis tested is that there is no significant difference
between the means of all the groups. If the null hypothesis is rejected, ANOVA just states that there
is a significant difference between the groups but not where those differences occur. To determine
where the group differences are, post hoc (From the Latin post hoc, "after this") tests are subsequently
used.
Why not just multiple pairwise comparisons? If there are 4 groups (A, B, C, D) for example and the
differences were compared using multiple t-tests:
• A vs. B P<0.05 95% no type I error
• A vs. C P<0.05 95% no type I error
• A vs. D P<0.05 95% no type I error
• B vs. C P<0.05 95% no type I error
• B vs. D P<0.05 95% no type I error
• C vs. D P<0.05 95% no type I error
Assuming that each test was independent, the overall probability would be:
0.95 * 0.95 * 0.95 * 0.95 * 0.95 * 0.95 = 0.735
This is known as familywise error or, cumulative Type I error, and in this case results in only a 73.5%
probability of no Type I error whereby the null hypothesis could be rejected when it is true. This is
overcome by using post hoc tests that make multiple pairwise comparisons with stricter acceptance
criteria to prevent familywise error.
ASSUMPTIONS
The independent ANOVA makes the same assumptions as most other parametric tests.
• The independent variable must be categorical and the dependent variable must be
continuous.
• The groups should be independent of each other.
• The dependent variable should be approximately normally distributed.
• There should be no significant outliers.
• There should be homogeneity of variance between the groups otherwise the p-value for the
F-statistic may not be reliable.
The first 2 assumptions are usually controlled through the use of appropriate research method design.
If the last three assumptions are violated then the non-parametric equivalent, Kruskal-Wallis should
be considered instead.
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CONTRASTS
Contrasts are ‘a priori’ tests (i.e. planned comparisons before any data were collected). As an
example, researchers may want to compare the effects of some new drugs to the currently
prescribed one. These should only be a small set of comparisons in an attempt to reduce
familywise error. The choice must be based on the scientific questions being asked, and
chosen during the experimental design. Hence the term planned comparisons. Therefore
they are looking at specified mean differences and therefore can be used if the ANOVA F test
is insignificant.
JASP provides 6 planned contrast enabling different types of comparisons:
Deviation: the mean of each level of the independent variable is compared to the overall
mean (the mean when all the levels are taken together).
Simple: the mean of each level is compared to the mean of a specified level, for example with
the mean of the control group.
Difference: the mean of each level is compared to the mean of the previous levels.
Helmert: the mean of each level is compared to the mean of the subsequent levels.
Repeated: By selecting this contrast, the mean of each level is compared to the mean of the
following level.2
Polynomial: tests polynomial trends in the data.
POST HOC TESTING
Post hoc tests are tests that were decided upon after the data have been collected. They can
only be carried out if the ANOVA F test is significant.
JASP provides 5 alternatives for use with the independent group ANOVA tests:
Bonferroni – can be very conservative but gives guaranteed control over Type I error at the risk of
reducing statistical power. Does not assume independence of the comparisons.
Holm – the Holm-Bonferroni test which is a sequential Bonferroni method that is less conservative
than the original Bonferroni test.
Tukey – one of the most commonly used tests and provides controlled Type I error for groups with
the same sample size and equal group variance.
Scheffe – controls for the overall confidence level when the group sample sizes are different.
Sidak – similar to Bonferroni but assumes that each comparison is independent of the others. Slightly
more powerful than Bonferroni.
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JASP also provides 4 Types of post hoc
Standard – as above
Games-Howell – used when you are unsure about the equality of group variances
Dunnett’s – used to compare all the groups to one group i.e. the control group
Dunn – a non-parametric post hoc test used for testing small sub-sets of pairs.
EFFECT SIZE
JASP provides 3 alternative effect size calculations for use with the independent group ANOVA tests:
Eta squared (η2) - accurate for the sample variance explained but overestimates the population
variance. This can make it difficult to compare the effect of a single variable in different studies.
Partial Eta squared (ηp2) – this solves the problem relating to population variance overestimation
allowing for comparison of the effect of the same variable in different studies.
Omega squared (ω2) – Normally, statistical bias gets very small as sample size increases, but for small
samples (n<30) ω2 provides an unbiased effect size measure.
Test Measure Trivial Small Medium Large
ANOVA Eta
Partial Eta
Omega squared
<0.1
<0.01
<0.01
0.1
0.01
0.01
0.25
0.06
0.06
0.37
0.14
0.14
RUNNING THE INDEPENDENT ANOVA
Load Independent ANOVA diets.csv. This contains A column containing the 3 diets used (A, B and C)
and another column containing the absolute amount of weight loss after 8 weeks on one of 3 different
diets For good practice check the descriptive statistics and the boxplots for any extreme outliers.
Go to ANOVA > ANOVA, put weight loss into the Dependent Variable and the Diet groupings into the
Fixed Factors box. In the first instance tick Descriptive statistics and ω2 as the effect size;
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In Assumptions checks, tick all options:
This should result in 3 tables and one Q-Q plot.
UNDERSTANDING THE OUTPUT
The main ANOVA table shows that the F-statistic is significant (p<.001) and that there is a large effect
size. Therefore, there is a significant difference between the means of the 3 diet groups.
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TESTING ASSUMPTIONS
Before accepting this any violations in the assumptions required for an ANOVA should be checked.
Levene’s test shows that homogeneity of variance is not significant. However, if Levene’s test shows
a significant difference in variance, the Brown-Forsythe or Welch correction should be reported.
The Q-Q plot shows that the data appear to be normally distributed and linear.
The descriptive statistics suggest that Diet 3 results in the highest weight loss after 8 weeks.
90 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
CONTRAST EXAMPLE
If for example, one planned to compare the effects of diets B and C to diet A. Click on the drop-
down menu and select ‘simple’ next to diet. This will test the significance between the first category
in the list with the remaining categories.
As can be seen, only diet C is significantly different from diet A (t(69) = 4.326, p<.001.
If the ANOVA reports no significant difference you can go no further in the analysis.
POST HOC TESTING
If the ANOVA is significant post hoc testing can now be carried out. In Post Hoc Tests add Diet to the
analysis box on the right, tick Standard type, use Tukey for the post hoc correction and tick flag
significant comparisons.
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Also in Descriptive Plots add the Factor Diet to the horizontal axis and tick display error bars.
Post hoc testing shows that there is no significant difference between weight loss on diets A and B.
However, It is significantly higher in diet C compared to diet A (p<.001) and diet B (p=.001). Cohen’s d
shows that these differences have a large effect size.
REPORTING THE RESULTS
Independent one way ANOVA showed a significant effect of the type of diet on weight loss after 10
weeks (F (2, 69) =46.184, p<.001, ω2 = 0.214.
Post hoc testing using Tukey’s correction revealed that diet C resulted in significantly greater weight
loss than diet A (p<.001) or diet B (p=.001). There were no significant differences in weight loss
between diets A and B (p=.777).
92 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
KRUSKAL-WALLIS – NON-PARAMETRIC ANOVA
If your data fails parametric assumption tests or is nominal, the Kruskal-Wallis H test is a non-
parametric equivalent to the independent samples ANOVA. It can be used for comparing two or more
independent samples of equal or different sample sizes. Like the Mann-Whitney and Wilcoxon’s tests,
it is a rank-based test.
As with the ANOVA, Kruskal-Wallis H test (also known as the "one-way ANOVA on ranks") is an
omnibus test which does not specify which specific groups of the independent variable are statistically
significantly different from each other. To do this, JASP provides the option for running Dunn’s post
hoc test. This multiple comparisons test can be very conservative in particular for large numbers of
comparisons.
Load Kruskal-Wallis ANOVA.csv dataset into JASP. This dataset contains subjective pain scores for
participants undergoing no treatment (control), cryotherapy or combined cryotherapy-compression
for delayed onset muscle soreness after exercise.
RUNNING THE KRUSKAL-WALLIS TEST
Go to ANOVA >ANOVA. In the analysis window add Pain score to the dependent variable and
treatment to the fixed factors. Check that the pain score is set to ordinal. This will automatically run
the normal independent ANOVA. Under Assumption Checks tick both Homogeneity tests and Q-Q
plots.
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Although the ANOVA indicates a significant result, the data has not met the assumptions of
homogeneity of variance as seen by the significant Levene’s test and only shows linearity in the middle
of the Q-Q plot and curves off at the extremities indicating more extreme values. Added to the fact
that the dependent variable is based on subjective pain scores suggest the use of a non-parametric
alternative.
Return to the statistics options and open the Nonparametrics option at the bottom. For the Kruskal-
Wallis test Move the Treatment variable to the box on the right. In Post Hoc tests move treatment to
the right box and tick Dunn’s post hoc test and flag significant comparisons.
UNDERSTANDING THE OUTPUT
Two tables are shown in the output. The Kruskal-Wallis test shows that there is a significant difference
between the three treatment modalities.
94 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
The Dunn’s post hoc test provides its own p-value as well as those for Bonferroni and Holm’s
Bonferroni correction. As can be seen, both treatment conditions are significantly different from the
controls but not from each other.
REPORTING THE RESULTS
Pain scores were significantly affected by treatment modality H (2) = 19.693, p<.001. Pairwise
comparisons showed that both cryotherapy and cryotherapy with compression significantly reduces
pain scores (p=.002 and p<.001 respectively) compared to the control group. There were no significant
differences between cryotherapy and cryotherapy with compression (p=.102).
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COMPARING MORE THAN TWO RELATED GROUPS
RMANOVA
The one-way repeated measures ANOVA (RMANOVA) is used to assess if there is a difference in means between 3 or more groups (where the participants are the same in each group) that have been
tested multiple times or under different conditions. Such a research design, for example, could be that
the same participants were tested for an outcome measure at 1, 2 and 3 weeks or that the outcome
was tested under conditions 1, 2 and 3.
The null hypothesis tested is that there is no significant difference between the means of the
differences between all the groups.
The independent variable should be categorical and the dependent variable needs to be a continuous
measure. In this analysis the independent categories are termed levels i.e. these are the related
groups. So in the case where an outcome was measured at weeks 1, 2 and 3, the 3 levels would be
week 1, week 2 and week 3.
The F-statistic is calculated by dividing the mean squares for the variable (variance explained by the
model) by its error mean squares (unexplained variance). The larger the F-statistic, the more likely it
is that the independent variable will have had a significant effect on the dependent variable.
ASSUMPTIONS
The RMANOVA makes the same assumptions as most other parametric tests.
• The dependent variable should be approximately normally distributed.
• There should be no significant outliers.
• Sphericity, which relates to the equality of the variances of the differences between levels of
the repeated measures factor.
If the assumptions are violated then the non-parametric equivalent, Friedman’s test should be
considered instead and is described later in this section.
SPHERICITY
If a study has 3 levels (A, B and C) sphericity assumes the following:
Variance (A-B) ≈ Variance (A-C) ≈ Variance (B-C)
RMANOVA checks the assumption of sphericity using Mauchly’s (pronounced Mockley’s) test of
sphericity. This tests the null hypothesis that the variances of the differences are equal. In many
cases, repeated measures violate the assumption of sphericity which can lead to Type I error. If this is
the case corrections to the F-statistic can be applied.
JASP offers two methods of correcting the F-statistic, the Greenhouse-Geisser and the Huynh-Feldt
epsilon (ε) corrections. A general rule of thumb is that if the ε values are <0.75 then use the
Greenhouse-Geisser correction and if they are >0.75 then use the Huynh-Feldt correction.
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POST HOC TESTING
Post hoc testing is limited in RMANOVA, JASP provides two alternatives:
Bonferroni – can be very conservative but gives guaranteed control over Type I error at the risk of
reducing statistical power.
Holm – the Holm-Bonferroni test which is a sequential Bonferroni method that is less conservative
than the original Bonferroni test.
If you ask for either Tukey or Scheffe post hoc corrections JASP will return a NaN (not a number) error.
EFFECT SIZE
JASP provides the same alternative effect size calculations that are used with the independent group
ANOVA tests:
Eta squared (η2) - accurate for the sample variance explained but overestimates the population
variance. This can make it difficult to compare the effect of a single variable in different studies.
Partial Eta squared (ηp2) – this solves the problem relating to population variance overestimation
allowing for comparison of the effect of the same variable in different studies. This appears to be the
most commonly reported effect size in repeated measures ANOVA
Omega squared (ω2) – Normally, statistical bias gets very small as sample size increases, but for small
samples (n<30) ω2 provides an unbiased effect size measure.
Levels of effect size:
Test Measure Trivial Small Medium Large
ANOVA Eta
Partial Eta
Omega squared
<0.1
<0.01
<0.01
0.1
0.01
0.01
0.25
0.06
0.06
0.37
0.14
0.14
RUNNING THE REPEATED MEASURES ANOVA
Load Repeated ANOVA cholesterol.csv. This contains one column with the participant IDs and 3
columns one for each repeated measurement of blood cholesterol following an intervention. For good
practice check the descriptive statistics and the boxplots for any extreme outliers.
Go to ANOVA > Repeated measures ANOVA. As stated above, the independent variable (repeated
measures factor) has levels, in this case, there are 3 levels. Rename RM Factor 1 to Time post-
intervention and then rename 3 levels to Week 0, week 3 and week 6 accordingly.
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Once these have been done they will appear in the Repeated Measures Cells. Now add the appropriate
data to the appropriate level.
Tick Descriptive Statistics, Estimates of effect size and ω2.
Under Assumption Checks tick Sphericity tests and all Sphericity correction options.
The output should consist of 4 tables. The third table, between-subject effects, can be ignored for this
analysis.
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UNDERSTANDING THE OUTPUT
The within-subjects effects table reports a large F-statistic which is highly significant (p<.001) and has
a small to medium effect size (0.058). This table shows the statistics for sphericity assumed (none) and
the two correction methods. The main differences are in the degrees of freedom (df) and the value of
the mean square. Under the table, it is noted that the assumption of sphericity has been violated.
The following table gives the results of Mauchly’s test of sphericity. It can be seen that there is a
significant difference (p<.001) in the variances of the differences between the groups. Greenhouse-
Geisser and the Huynh-Feldt epsilon (ε) values are below 0.75. Therefore the ANOVA result should be
reported based on the Greenhouse-Geisser correction:
To provide a cleaner table, go back to Assumption Checks and only tick Greenhouse-Geisser for
sphericity correction.
There is a significant difference between the means of the differences between all the groups F (1.235,
21.0) =212.3, p<.001, ω2 = 0.058.
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The descriptive data suggest that blood cholesterol levels were higher at week 0 compared to weeks
3 and 6.
However, if the ANOVA reports no significant difference you can go no further in the
analysis.
POST HOC TESTING
If the ANOVA is significant, post hoc testing can now be carried out. In Post Hoc Tests add Time post-
intervention to the analysis box on the right, tick Effect size and, in this case, use Holm for the post
hoc correction.
Also in Descriptive Plots add the Factor – Time post-intervention to the horizontal axis and tick display
error bars.
Post hoc testing shows that there are significant differences in blood cholesterol levels between all of
the time point combinations and are associated with large effect sizes.
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REPORTING THE RESULTS
Since Mauchly’s test of sphericity was significant, the Greenhouse-Geisser correction was used. This
showed that cholesterol levels differed significantly between F (1.235, 21.0) =212.3, p<.001, ω2 =
0.058.
Post hoc testing using the Bonferroni correction revealed that cholesterol levels decreased
significantly as time increased, weeks 0 – 3 (mean difference=0.566 units, p<.001) and weeks 3 – 6
(mean difference = 0.063 units, p=.004).
FRIEDMAN’S REPEATED MEASURES ANOVA If parametric assumptions are violated or the data is ordinal you should consider using the non-
parametric alternative, Friedman’s test. Similar to the Kruskal-Wallis test, the Friedman’s test is used
for one-way repeated measures analysis of variance by ranks and doesn’t assume the data comes from
a particular distribution. This test is another omnibus test which does not specify which specific groups
of the independent variable are statistically significantly different from each other. To do this, JASP
provides the option for running Conover’s post hoc test if Friedman’s test is significant.
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Load Friedman RMANOVA.csv into JASP. This has 3 columns of subjective pain ratings measured at
18, 36 and 48 hours post-exercise. Check that the pain scores are set to ordinal data.
RUNNING THE FRIEDMAN’S TEST
Go to ANOVA > Repeated measures ANOVA. The independent variable (repeated measures factor)
has 3 levels. Rename RM Factor 1 to Time and then rename 3 levels to 18 hours, 36 hours and w48
hours accordingly.
Once these have been done they will appear in the Repeated Measures Cells. Now add the appropriate
dataset to the appropriate level.
This will automatically produce the standard repeated measures within-subjects ANOVA table. To run
the Friedman’s test, expand the Nonparametrics tab, move Time to the RM factor box and tick
Conover’s post hoc tests.
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UNDERSTANDING THE OUTPUT
Two tables should be produced.
Friedman’s test shows that time has a significant effect on pain perception. Connor’s post hoc pairwise
comparisons show that all pain perception is significantly different between each time point.
REPORTING THE RESULTS
Time has a significant effect on subjective pain scores χ2 (2) = 26.77, p<.001. Pairwise comparisons
showed that pain perception is significantly different between each time point (all p<0.001).
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COMPARING INDEPENDENT GROUPS AND THE EFFECTS OF COVARIATES
ANCOVA
ANOVA can be used to compare the means of one variable (dependent) in two or more groups,
whereas analysis of covariance (ANCOVA) sits between ANOVA and regression and compares the means of one (dependent) variable in two or more groups while taking into account the variability of
other continuous variables (COVARIATES). ANCOVA checks for differences in ‘adjusted’ means (i.e.
adjusted for the effects of the covariate). A covariate may not usually be part of the main research
question but could influence the dependent variable and therefore needs to be adjusted or controlled
for. As long as a good covariate is used ANCOVA will have improved statistical power and control over
error.
Control for – to subtract statistically the effects of a variable (a control variable) to see what a
relationship would be without it (Vogt 1977).
Hold constant – to “subtract” the effects of a variable from a complex relationship to study what the
relationship would be if the variable were, in fact, a constant. Holding a variable constant essentially
means assigning it an average value (Vogt 1977).
Statistical control – using statistical techniques to isolate or “subtract” variance in the dependent
variable attributable to variables that are not the subject of the study (Vogt, 1999).
For example, when looking for a difference in weight loss between three diets it would be appropriate
to take into account the individuals pre-trial bodyweight since heavier people may lose
proportionately more weight.
Type of diet
(Factor)
Weight loss
Starting body weight
(Covariate)
Independent
variables
Dependent
variable
ANOVA
ANCOVA
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The null hypothesis tested is that there is no significant difference between the ‘adjusted’ means of
all the groups.
ASSUMPTIONS
ANCOVA makes the same assumptions as the independent ANOVA makes. However, there are two
further assumptions:
• The relationship between the dependent and covariate variables are linear.
• Homogeneity of regression i.e. the regression lines for each of the independent groups are
parallel to each other.
POST HOC TESTING
JASP provides 4 alternatives for use with the independent group ANOVA tests:
Bonferroni – can be very conservative but gives guaranteed control over Type I error at the risk of
reducing statistical power.
Holm – the Holm-Bonferroni test which is a sequential Bonferroni method that is less conservative
than the original Bonferroni test.
Tukey – one of the most commonly used tests and provides controlled Type I error for groups with
the same sample size and equal group variance.
Scheffe – controls for the overall confidence level when the group sample sizes are different.
JASP also provides 4 Types
Standard – as above
Games-Howell – used when you are unsure about the equality of group variances
Dunnett’s – used to compare all the groups to one group i.e. the control group
Dunn – a non-parametric post hoc test used for testing small sub-sets of pairs.
Covariate
D e
p e
n d
e n
t v
a ri
a b
le
Diet 1
Diet 2
Diet 3
Covariate
D e
p e
n d
e n
t v
a ri
a b
le
Diet 1
Diet 2
Diet 3
Homogeneity of regression Assumption violated
Homogeneity of regression
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EFFECT SIZE
JASP provides 3 alternative effect size calculations for use with the independent group ANOVA tests:
Eta squared (η2) - accurate for the sample variance explained but overestimates the population
variance. This can make it difficult to compare the effect of a single variable in different studies.
Partial Eta squared (ηp2) – this solves the problem relating to population variance overestimation
allowing for comparison of the effect of the same variable in different studies.
Omega squared (ω2) – Normally, statistical bias gets very small as sample size increases, but for small
samples (n<30) ω2 provides an unbiased effect size measure.
Test Measure Trivial Small Medium Large
ANOVA Eta Partial Eta Omega squared
<0.1 <0.01 <0.01
0.1 0.01 0.01
0.25 0.06 0.06
0.37 0.14 0.14
RUNNING THE INDEPENDENT ANCOVA
Load ANCOVA hangover.csv. This dataset has been adapted from the one provided by Andy Field
(2017). The morning after a Fresher’s ball students were given either water, coffee or a Barocca to
drink. Two hours later they reported how well they felt (from 0 – awful to 10 –very well). At the same
time, data were collected on how drunk they were the night before (0-10).
Initially, run an ANOVA with wellness as the dependent variable and the type of drink as the fixed
factor.
As can be seen from the results, homogeneity of variance has not been violated while the ANOVA
shows that there is no significant difference in the wellness scores between any of the morning drinks.
F(2,27)=1.714, p=.199. However, this may be related to how drunk the students were the night before!
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Go to ANOVA > ANCOVA, put wellness as the dependent variable and the type of drink as the fixed
factor. Now add drunkenness to the Covariate(s) box. In the first instance tick Descriptive statistics
and ω2 as the effect size;
In Assumption Checks tick both options
In Marginal Means move drink to the right
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This should result in 4 tables and one Q-Q plot.
UNDERSTANDING THE OUTPUT
It can be seen that the covariate (drunkenness) significantly predicts wellness (p<.001). The effects of
the type of drink on wellness, when adjusted for the effects of drunkenness are now significant
(p=.003).
It can be seen that Levene’s test is significant, unlike in ANOVA, no homogeneity of variance
corrections (i.e. Welch) are provided. For ANCOVA this can be ignored. The Q-Q plot appears to be
normal.
The descriptive statistics show the unadjusted means for wellness in the three drink groups.
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The marginal means are now the wellness means having been adjusted for the effects of the covariate
(drunkenness).
TESTING FURTHER ASSUMPTIONS
As previously mentioned the assumption of homogeneity of regression is important in ANCOVA. This
can be tested by looking at the interaction between the type of drink and the drunkenness scores. Go
to Model, drink and drunkenness will have been automatically added as individual Model terms. Now
highlight both drink and drunkenness and add them both to Model terms.
The ANOVA table now has an extra row showing the interaction between the type of drink and
drunkenness. This is not significant (p=.885), i.e. the relationships between drunkenness and wellness
are the same in each drink group. If this is significant there will be concerns over the validity of the
main ANCOVA analysis.
Having checked this, go back and remove the interaction term from the Model terms.
If the ANCOVA reports no significant difference you can go no further in the analysis.
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POST HOC TESTING
If the ANCOVA is significant post hoc testing can now be carried out. In Post Hoc Tests add Drink to
the analysis box on the right, tick Effect size and, in this case, use Tukey for the post hoc correction.
Also, tick flag significant comparisons.
Post hoc testing shows that there is no significant difference between coffee and water on wellness.
However, wellness scores were significantly higher after drinking a Barocca.
This can be seen from the Descriptive plots.
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REPORTING THE RESULTS
The covariate, drunkenness, was significantly related to the morning after wellness, F (1,26) = 33.03,
p<.001, ω2 = 0.427. There was also a significant effect of drink on wellness after controlling for
drunkenness, F (2, 26) = 7.47, p=.003, ω2 = 0.173.
Post hoc testing using Tukey’s correction revealed that drinking a Barocca resulted in significantly
greater wellness compared to water (p<.004) or coffee (p=.01). There were no significant differences
in wellness between water and coffee (p=.973).
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TWO-WAY INDEPENDENT ANOVA One-way ANOVA tests situations when only one independent variable is manipulated, two-way
ANOVA is used when more than 1 independent variable has been manipulated. In this case,
independent variables are known as factors.
FACTOR 1 FACTOR 2
CONDITION 1 Group 1 Dependent variable Group 2 Dependent variable
CONDITION 2 Group 1 Dependent variable Group 2 Dependent variable
CONDITION 3 Group 1 Dependent variable Group 2 Dependent variable
The factors are split into levels, therefore, in this case, Factor 1 has 3 levels and Factor 2 has 2 levels.
A “main effect” is the effect of one of the independent variables on the dependent variable, ignoring
the effects of any other independent variables. There are 2 main effects tested both of which are
“between-subjects”: in this case comparing differences between factor 1 (i.e. condition) and
differences between factor 2 (i.e. groups). Interaction is where one factor influences the other factor.
The two-way independent ANOVA is another omnibus test that is used to test 2 null hypotheses:
1. There is no significant between-subject effect i.e. no significant difference between the
means of the groups in either of the factors.
2. There is no significant interaction effect i.e. no significant group differences across
conditions.
ASSUMPTIONS
Like all other parametric tests, mixed factor ANOVA makes a series of assumptions which should either
be addressed in the research design or can the tested for.
• The independent variables (factors) should have at least two categorical independent groups
(levels).
• The dependent variable should be continuous and approximately normally distributed for all
combinations of factors.
• There should be homogeneity of variance for each of the combination of factors.
• There should be no significant outliers.
RUNNING TWO-WAY INDEPENDENT ANOVA
Open 2-way independent ANOVA.csv in JASP. This comprises on 3 columns of data, Factor 1 – gender
with 2 levels (male and female), Factor 2 - supplement with 3 levels (control, carbohydrate CHO and
protein) and the dependent variable (explosive jump power. In Descriptive statistics check the data
for significant outliers. Go to ANOVA >ANOVA, add Jump power to the Dependent variable, Gender
and Supplement to the Fixed factors.
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Tick Descriptive statistics and Estimates of effect size (ω2).
In Descriptive plots, add supplement to the horizontal axis and Gender to separate lines. In Additional
Options,
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UNDERSTANDING THE OUTPUT
The output should comprise 2 tables and one plot.
The ANOVA table shows that there are significant main effects for both Gender and Supplement
(p=0.003 and p<.001 respectively) with medium and large effect sizes respectively. This suggests that
there is a significant difference in jump power between genders, irrespective of Supplement, and
significant differences between supplements, irrespective of Gender.
There is also a significant interaction between Gender and Supplement (p<.001) which also has a
medium to large effect size (0.138). This suggests that the differences in jump power between genders
are affected somehow by the type of supplement used.
The Descriptive statistics and plot suggest that the main differences are between genders when using
a protein supplement.
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TESTING ASSUMPTIONS
In Assumption Checks, tick Homogeneity tests and Q-Q plot of residuals.
Levene’s test shows no significant difference in variance within the dependent variable groups, thus
homogeneity of variance has not been violated.
The Q-Q plot shows that the data appear to be normally distributed and linear. We can now accept
the ANOVA result since none of these assumptions has been violated.
However, if the ANOVA reports no significant difference you can go no further with the
analysis.
SIMPLE MAIN EFFECTS
Go to the analysis options and Simple Main Effects. Here add Gender to the Simple effect
factor and Supplement to the Moderator Factor 1. Simple main effects are effectively limited
pairwise comparisons.
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This table shows that there are no gender differences in jump power between the control or
CHO groups (p=.116 and p=0.058 respectively). However, there is a significant difference
(p<.001) in jump power between genders in the protein supplement group.
CONTRAST TESTING
There are two ways of testing for a difference between (combinations of) cells: post hoc tests
and contrast analysis. JASP has a range of different contrast tests available, including custom
contrasts. For example, we can contrast the three different supplements. Open up the
Contrasts menu and next to Supplement click on the drop-down menu and select custom.
This will add another series of options to this window.
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In this window contrasts can be added, in this case, there are three contrasts which can be
defined:
This will result in the following table:
Comparing this table to the post hoc analysis below the estimates of the differences in
marginal means are the same, as well as their standard errors and t-statistics. However, both
the p-values and confidence intervals vary: the corrected p-values are typically higher, and
the confidence intervals wider, for the post hoc analysis.
CHO vs Protein
CHO vs Control
Control vs Protein
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POST HOC TESTING
If the ANOVA is significant post hoc testing can now be carried out. In Post Hoc Tests add Supplement
and the Gender*Supplement to the analysis box on the right, tick Effect size and, in this case, use
Tukey for the post hoc correction. Also for ease of viewing tick Flag significant comparisons.
Post hoc testing is not done for Gender since there are only 2 levels.
Post hoc testing shows no significant difference between the control and CHO, supplement group,
irrespective of Gender, but significant differences between Control and Protein (p<.001) and between
CHO and Protein (p<.001).
The post hoc comparisons for the interactions decomposes the results further.
REPORTING THE RESULTS
A two-way ANOVA was used to examine the effect of gender and supplement type on explosive jump
power. There were significant main effects for both gender (F (1, 42) = 9.59, p=.003, ω2 = 0.058) and
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Supplement (F (2, 42) = 30.07, p<.001, ω2 = 0.477). There was a statistically significant interaction
between the effects of gender and supplement on explosive jump power (F (2, 42) = 11.1, p<.001, ω2
= 0.138).
Tukey’s post hoc correction showed that explosive leg power was significantly higher in the protein
group compared to the control or CHO groups (t=-1.919, p<.001 and t=-1.782, p<.001 respectively).
Simple main effects showed that jump power was significantly higher in males on a protein
supplement compared to females (F (1) =28.06, p<.001).
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TWO-WAY REPEATED MEASURES ANOVA Two-Way Repeated Measures ANOVA means that there are two factors in the experiment, for
example, different treatments and different conditions. "Repeated-measures" means that the same
subject received more than one treatment and/or more than one condition.
Independent variable (Factor 2)
Independent variable (Factor 1) = time
Participant Time 1 Time 2 Time 3
Condition 1 1 Dependent variable
Dependent variable
Dependent variable
2 Dependent variable
Dependent variable
Dependent variable
3 Dependent variable
Dependent variable
Dependent variable
Condition 2 1 Dependent variable
Dependent variable
Dependent variable
2 Dependent variable
Dependent variable
Dependent variable
3 Dependent variable
Dependent variable
Dependent variable
The factors are split into levels, therefore, in this case, Factor 1 has 3 repeated levels and Factor 2 has
2 repeated levels.
A “main effect” is the effect of one of the independent variables on the dependent variable, ignoring
the effects of any other independent variables. There are 2 main effects tested both of which are
“between-subjects”: in this case comparing differences between factor 1 (i.e. condition) and
differences between factor 2 (i.e. groups). Interaction is where one factor influences the other factor.
The two-way repeated ANOVA is another omnibus test that is used to test the following main effect
null hypotheses:
H01: the dependent variable scores are the same for each level in factor 1 (ignoring factor 2).
H02: the dependent variable scores are the same for each level in factor 2 (ignoring factor 1).
The null hypothesis for the interaction between the two factors is:
H03: the two factors are independent or that interaction effect is not present.
ASSUMPTIONS
Like all other parametric tests, two-way repeated ANOVA makes a series of assumptions which should
either be addressed in the research design or can the tested for.
• The independent variables (factors) should have at least two categorical related groups
(levels).
• The dependent variable should be continuous and approximately normally distributed for all
combinations of factors.
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• Sphericity i.e. the variances of the differences between all combinations of related groups
must be equal.
• There should be no significant outliers.
RUNNING TWO-WAY REPEATED MEASURES ANOVA
Open 2-way repeated ANOVA.csv in JASP. This comprises of 4 columns of data (“sit and reach”
flexibility scores for two factors, Factor 1 with 2 levels (stretch and no stretch) and Factor 2 with 2
levels (warm-up and no warm-up). In Descriptive statistics check the data for significant outliers. Go
to ANOVA > Repeated Measures ANOVA. Firstly each Factor and its levels should be defined, for RM
Factor 1 – define this as Stretching and its levels as stretch and no stretch. Then define RM Factor 2 as
Warm-up and its levels as warm-up and no warm-up. Then add the appropriate column of data to the
assigned repeated measures cells.
Also, tick Descriptive statistics and estimates of effect size - ω2.
In Descriptive plots add the Stretching factor to the horizontal axis and Warm-up factor to
separate lines. Tick the display error bars option.
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UNDERSTANDING THE OUTPUT
The output should comprise 3 tables and one plot. The Between-Subjects Effects table can be ignored
in this analysis.
The ANOVA within-subjects effects table shows that there are significant main effects for both stretch
(p<.001) and warm-up (p<.001) on sit and reach distance. Both of these are associated with large effect
sizes. There is also a significant interaction between stretch and warm-up (p<.001), this suggests that
that the effects of performing a stretch on sit and reach distance are different depending on whether
or not a warm-up had been performed. These findings can be seen in both the descriptive data and
plot.
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TESTING ASSUMPTIONS
In this case, there are no assumption checks. Sphericity can only be tested when there are at least three levels and homogeneity requires at least two unrelated data sets. If a factor has more than 2 levels Mauchly’s test of Sphericity should also be run and the appropriate corrected F value used if necessary (See Repeated Measures ANOVA for details).
However, if the ANOVA reports no significant difference you can go no further with the
analysis.
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SIMPLE MAIN EFFECTS
Now go to the analysis options and Simple Main Effects. Here add Warm up to the Simple effect factor
and Stretch to the Moderator Factor 1. Simple main effects are effectively pairwise comparisons.
This table shows that when moderating for warm-up there is a significant difference (p<.001) in sit
and reach performance when a stretch was also carried out but not without a stretch (p=.072).
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We can now moderate for stretch by changing the Simple Main Effects to use Stretch as the simple
effect factor and warm up as the moderator factor. We can also replot the descriptives with a warm-
up on the horizontal axis and stretch as separate lines.
In this case, when controlling for Stretch there were significant differences between both warm-up
and no warm-up.
Both of these simple main effects can be visualised in their descriptive plots.
POST HOC TESTING
If the ANOVA is significant post hoc testing can now be carried out. In Post Hoc Tests add stretch,
warm-up and the Stretching*warm-up interaction to the analysis box on the right, tick Effect size and,
in this case, use Holm for the post hoc correction. Tick Flag significant comparisons.
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Post hoc testing for the main effects confirms that there are significant differences in sit and reach
distance when comparing the two levels of each factor. This is further decomposed in the Post hoc
comparisons for the interaction.
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REPORTING THE RESULTS
A two-way ANOVA was used to examine the effect of stretch and warm-up type on sit and teach
performance. There were significant main effects for stretch (F (1, 11) = 123.4, p<.001, ω2 = 0.647) and
warm-up (F (1, 11) = 68.69, p<.001, ω2 = 0.404). There was a statistically significant interaction
between the effects of stretch and warm up on sit and reach performance (F (1, 11) = 29.64, p<.001,
ω2 = 0.215).
Simple main effects showed that sit and reach performance was significantly higher when both a
stretch and warm-up had been done (F (1) =112.6, p<.001).
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MIXED FACTOR ANOVA Mixed factor ANOVA (another two-way ANOVA) is a combination of both independent and
repeated measures ANOVA involving more than 1 independent variable (known as factors).
The factors are split into levels, therefore, in this case, Factor 1 has 3 levels and Factor 2 has
2 levels. This results in 6 possible combinations.
A “main effect” is the effect of one of the independent variables on the dependent variable,
ignoring the effects of any other independent variables. There are 2 main effects tested: in
this case comparing data across factor 1 (i.e. time) is known as the “within-subjects” factor
while comparing differences between factor 2 (i.e. groups) is known as the “between-
subjects” factor. Interaction is where one factor influences the other factor.
The main effect of time or condition tests the following i.e. irrespective of which group the
data is in:
The main effect of group tests the following i.e. irrespective of which condition the data is in:
Simple main effects are effectively pairwise comparisons:
Independent variable (Factor 2)
Independent variable (Factor 1) = time or condition
Time/condition 1 Time/condition 2 Time/condition 3 Group 1 Dependent variable Dependent variable Dependent variable
Group 2 Dependent variable Dependent variable Dependent variable
Independent variable (Factor 2)
Independent variable (Factor 1) = time or condition
Time/condition 1 Time/condition 2 Time/condition 3 Group 1
All data All data All data Group 2
Independent variable (Factor 2)
Independent variable (Factor 1) = time or condition
Time/condition 1 Time/condition 2 Time/condition 3 Group 1 All data Group 2 All data
Independent variable (Factor 2)
Independent variable (Factor 1) = time or condition
Time/condition 1 Time/condition 2 Time/condition 3 Group 1 Data Data Data
Group 2 Data Data Data
* *
*
*
* * *
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A mixed factor ANOVA is another omnibus test that is used to test 3 null hypotheses:
3. There is no significant within-subject effect i.e. no significant difference between the
means of the differences between all the conditions/times.
4. There is no significant between-subject effect i.e. no significant difference between
the means of the groups.
5. There is no significant interaction effect i.e. no significant group differences across
conditions/time
ASSUMPTIONS
Like all other parametric tests, mixed factor ANOVA makes a series of assumptions which
should either be addressed in the research design or can the tested for.
• The “within-subjects” factor should contain at least two related (repeated measures)
categorical groups (levels)
• The “between-subjects” factor should have at least two categorical independent
groups (levels).
• The dependent variable should be continuous and approximately normally distributed
for all combinations of factors.
• There should be homogeneity of variance for each of the groups and, if more than 2
levels) sphericity between the related groups.
• There should be no significant outliers.
RUNNING THE MIXED FACTOR ANOVA
Open 2-way Mixed ANOVA.csv in JASP. This contains 4 columns of data relating to the type
of weightlifting grip and speed of the lift at 3 different loads (%1RM). Column 1 contains the
grip type, columns 2-4 contain the 3 repeated measures (30, 50 and70%). Check for significant
outliers using boxplots then go to ANOVA > Repeated measures ANOVA.
Define the Repeated Measures Factor, %1RM, and add 3 levels (30, 50 and 70%). Add the
appropriate variable to the Repeated measures Cells and add Grip to the Between-Subjects
Factors:
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Additionally, tick Descriptive statistics and Estimates of effect size (ω2).
In Descriptive plots, move %1RM to the horizontal axis and Grip to separate lines. It is now
possible to add a title for the vertical axis.
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UNDERSTANDING THE OUTPUT
The output should initially comprise of 3 tables and 1 graph.
For the main effect for %1RM, the within-subjects effects table reports a large F-statistic
which is highly significant (p<.001) and has a large effect size (0.744). Therefore, irrespective
of grip type, there is a significant difference between the three %1RM loads.
However, JASP has reported under the table that the assumption of sphericity has been
violated. This will be addressed in the next section.
Finally, there is a significant interaction between %1RM and grip (p<.001) which also has a
large effect size (0.499). This suggests that the differences between the %1RM loads are
affected somehow by the type of grip used.
For the main effect for grip, the between-subjects table shows a significant difference
between grips (p< .001), irrespective of %1RM.
From the descriptive data and the plot, it appears that there is a larger difference between
the two grips at the high 70% RM load.
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TESTING ASSUMPTIONS
In Assumptions Checks, tick Sphericity tests, Sphericity corrections and Homogeneity tests.
Mauchly’s test of sphericity is significant so that assumption has been violated, therefore, the
Greenhouse-Geisser correction should be used since epsilon is <0.75. Go back to Assumption
Checks and in Sphericity corrections leave Greenhouse-Geisser only ticked. This will result in
an updated Within-Subjects Effects table:
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Levene’s test shows that there is no difference in variance in the dependent variable between
the two grip types.
However, if the ANOVA reports no significant difference you can go no
further in the analysis.
POST HOC TESTING
If the ANOVA is significant post hoc testing can now be carried out. In Post Hoc Tests add
%1RM to the analysis box on the right, tick Effect size and, in this case, use Holm for the post
hoc correction. Only Bonferroni or Holm’s corrections are available for repeated measures.
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The post hoc tests show that irrespective of grip type each load is significantly different from each of
the other loads, and as seen from the plot, lift velocity significantly decreases as load increases.
Finally, In Simple main effects add Grip to the Simple effect factor and %1RM to Moderator
factor 1
These results show that there is a significant difference in lift speed between the two grips at
30% 1RM and also at the higher 70% 1RM loads (p=0.035 and p<0.001 respectively).
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REPORTING THE RESULTS
Using the Greenhouse-Geisser correction, there was a significant main effect of load (F= (1.48,
26.64) = 115.45, p<.001). Bonferroni corrected post hoc testing showed that there was a
significant sequential decline in lift speed from 30-50% 1RM (p=.035) and 50-70% 1RM
(p<.001).
There was a significant main effect for grip type (F (1, 18) = 20.925, p<.001) showing an overall
higher lift speed using the traditional rather than the reverse grip.
Using the Greenhouse-Geisser correction, there was a significant %1RM x Grip interaction (F
(1.48, 26.64) = 12.00, p<.001) showing that the type of grip affected lift velocity over the
%1RM loads.
135 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
CHI-SQUARE TEST FOR ASSOCIATION The chi-square (χ2) test for independence (also known as Pearson's χ2 test or the χ2 test of association)
can be used to determine if a relationship exists between two or more categorical variables. The test
produces a contingency table, or cross-tabulation, which displays the cross-grouping of the categorical
variables.
The χ2 test checks the null hypothesis that there is no association between two categorical variables.
It compares the observed frequencies of the data with frequencies which would be expected if there
was no association between the two variables.
The analysis requires two assumptions to be met:
1. The two variables must be categorical data (nominal or ordinal)
2. Each variable should comprise two or more independent categorical groups
Most statistical tests fit a model to the observed data with a null hypothesis that there is no difference
between the observed and modelled (expected) data. The error or deviation of the model is calculated
as:
Deviation = ∑ (𝒐𝒃𝒔𝒆𝒓𝒗𝒆𝒅 −𝒎𝒐𝒅𝒆𝒍) 𝟐
Most parametric models are based around population means and standard deviations. The χ2 model,
however, is based on expected frequencies.
How are the expected frequencies calculated? For example, we categorised 100 people into male,
female, short and tall. If there was an equal distribution between the 4 categories expected frequency
= 100/4 or 25% but the actual observed data does not have an equal frequency distribution.
Equal
Distribution
Male Female Row
Total
Tall 25 25 50
Short 25 25 50
Column Total 50 50
The model based on expected values can be calculated by:
Model (expected) = (𝑟𝑜𝑤 𝑡𝑜𝑡𝑎𝑙 𝑥 𝑐𝑜𝑙𝑢𝑚𝑛 𝑡𝑜𝑡𝑎𝑙)/100
Model – tall male = (81 x 71) /100 = 57.5
Model – tall female = (81 x 29) /100 = 23.5
Model – small male = (19 x 71) /100 = 13.5
Model – small female = (19 x 29) /100 = 5.5
Observed
Distribution
Male Female Row
Total
Tall 57 24 81
Short 14 5 19
Column Total 71 29
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These values can then be added to the contingency table:
Male (M) Female (F) Row Total
Tall (T) 57 24 81
Expected 57.5 23.5
Short (S) 14 5 19
Expected 13.5 5.5
Column Total 71 29
the χ2 statistic is derived from ∑ (𝐨𝐛𝐬𝐞𝐫𝐯𝐞𝐝 −𝐞𝐱𝐩𝐞𝐜𝐭𝐞𝐝)
𝐞𝐱𝐩𝐞𝐜𝐭𝐞𝐝
𝟐
Validity
χ2 tests are only valid when you have a reasonable sample size, that is, less than 20% of cells have an
expected count of less than 5 and none have an expected count of less than 1.
RUNNING THE ANALYSIS
The dataset Titanic survival is a classic dataset used for machine learning and contains data on 1309
passengers and crew who were on board the Titanic when it sank in 1912. We can use this to look at
associations between survival and other factors. The dependent variable is ‘Survival’ and possible
independent values are all the other variables.
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By convention, the independent variable is usually placed in the contingency table columns and the
dependent variable is placed in the rows.
Open Titanic survival chi square.csv in JASP, add survived to rows as the dependent variable and sex
into columns as the independent variable.
In statistics, tick all the following options:
In Cells, tick the following:
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UNDERSTANDING THE OUTPUT
First, look at the Contingency table output.
Remember that χ2 tests are only valid when you have a reasonable sample size, i.e. less than 20% of
cells have an expected count of less than 5 and none have an expected count of less than 1.
From this table, looking at % within rows, it can be seen that more males died on the Titanic compared
to females and more females survived compared to males. But is there a significant association
between gender and survival?
The statistical results are shown below:
χ2 statistic (χ2 (1) = 365.9, p <.001) suggest that there is a significant association between gender and
survival.
χ2 continuity correction can be used to prevent overestimation of statistical significance for small
datasets. This is mainly used when at least one cell of the table has an expected count smaller than 5.
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As a note of caution, this correction may overcorrect and result in an overly conservative result that
fails to reject the null hypothesis when it should (a type II error).
The likelihood ratio is an alternative to the Pearson chi-square. It is based on maximum-likelihood
theory. For large samples, it is identical to Pearson χ2. It is recommended in particular for small
samples sizes i.e. <30.
Nominal measures, Phi (2 x 2 contingency tables only) and Cramer's V (most popular) are both tests
of the strength of association (i.e. effect sizes). Both values are in the range of 0 (no association) to 1
(complete association). It can be seen that the strength of association between the variables has a
large effect size.
The Contingency coefficient is an adjusted Phi value and is only suggested for large contingency tables
such as 5 x 5 tables or larger.
Effect size 4 df Small Moderate Large
Phi and Cramer’s V (2x2 only) 1 0.1 0.3 0.5
Cramer’s V 2 0.07 0.21 0.35
Cramer’s V 3 0.06 0.17 0.29
Cramer’s V 4 0.05 0.15 0.25
Cramer’s V 5 0.04 0.13 0.22
JASP also provides the Odds ratio (OR) which is used to compare the relative odds of the occurrence
of the outcome of interest (survival), given exposure to the variable of interest (in this case gender).
For some reason, JASP calculates OR as a natural log. To convert this from a log value calculate the
natural antilog value (using Microsoft calculator, input number then click on Inv followed by ex), in this
case, it is 11.3. This suggests that male passengers had 11.3 times more chance of dying than females.
4 Kim HY. Statistical notes for clinical researchers: Chi-squared test and Fisher's exact test. Restor. Dent. Endod. 2017; 42(2):152-155.
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How is this calculated? Use the counts from the contingency table in the following:
Odds[males] = Died/Survived = 682/162 = 4.209
Odds[females] = Died/Survived = 127/339 = 0.374
OR = Odds[males] / Odds [females] = 11.3
GOING ONE STEP FURTHER.
We can also further decompose the contingency table as a form of post hoc testing by converting the
counts and expected counts in each cell to a standardised residual. This can tell us if the observed
counts and expected counts are significantly different in each cell.
The standardized residual for a cell in a table is a version of the standard z-score, calculated as
z = observed — expected
√expected
In the special case where df = 1, the calculation of the standardized residual incorporates a correction
factor:
z = |observed — expected| — 0.5
√expected
The resulting value of z is then given a positive sign if observed>expected and a negative sign if
observed<expected. Z-score significances are shown below.
z-score P-value
<-1.96 or > 1.96 <0.05
<-2.58 or > 2.58 <0.01
<-3.29 or > 3.29 <0.001
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When the z-scores are calculated for each cell in the contingency table we can see that significantly
fewer women died than expected and significantly more males died than expected p<.001.
Female No z= - 9.5
Male No z = 7.0
Female Yes z = 12.0
Male Yes z = -8.9
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META-ANALYSIS IN JASP Background
A meta-analysis is a statistical analysis that integrates results from multiple studies, providing a single
numerical value of the overall treatment effect for that group of studies. The difference between
statistical data analysis and a meta-analysis is shown below. Effectively each study becomes a
“participant” in the meta-analysis.
Statistical analysis Meta-analysis
Participant 1 Individual data Study 1 Study data
Participant 2 Individual data Study 2 Study data
Participant 3 Individual data Study 3 Study data
Participant 4 Individual data Study 4 Study data
Participant 5 Individual data Study 5 Study data
Overall group data & statistics Overall study group data & statistics
Effect size and calculations
To perform a Meta-analysis in JASP the overall effect size (ES) and standard error (SE) of the study
needs to be provided. An ES is a dimensionless estimate (no units) that indicates both direction and
magnitude of the effect/outcome. The standard error measures the dispersion of different sample
means taken from the same population and can be estimated from a single sample standard deviation.
Some studies will provide this information, although many do not. However, they will provide results
such as:
• Central tendencies and dispersion
• T or F statistics
• P-values
• Correlation coefficients
• Chi-square
All of these can be converted to an ES and the SE also determined using David Wilsons “Meta-analysis
effect size calculator”
https://campbellcollaboration.org/escalc/html/EffectSizeCalculator-Home.php
For example, a study comparing a treatment to a control group may provide the variable mean, SD
and n of each group. From this the ES, a standardised mean difference (d) and estimated error variance
(v) can be calculated:
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The 95% confidence intervals (CI) can be used to calculate the SE using the following equation:
SE = (upper CI limit – lower CI limit) / 3.92. NB: 3.92 = 2 x SD =95%
SE = (1.7487- (-0.0791) / 3.92
SE = 0.466
A quicker way is to find the square root of the estimated error variance (v)
SE = √0.2174 = 0.466
Therefore, for this study the overall ES = 0.835 and SE = 0.466.
These data can put into Excel and then used to perform a meta-analysis in JASP. The overall ES derived
from the meta-analysis is calculated by combining the effect sizes of the included studies.
To interpret the meta-analysis output, one needs to understand the following concepts,
heterogeneity, the model, effect size, and the forest plot.
Heterogeneity:
Heterogeneity describes any variability that may exist between the different studies. It is the opposite
of homogeneity, which means that the population/data/results are the same. There are 3 types:
• Clinical: Differences in participants, interventions, or outcomes
• Methodological: Differences in study design, risk of bias
• Statistical: Variation in intervention effects or results
If there is no variability, then data can be described as homogenous. Meta-analysis is concerned with
statistical heterogeneity. Statistical heterogeneity used to describe variability among data/studies and
occurs when the treatment effect estimates of a set of data/studies vary among one another. Studies
with methodological flaws and small studies may overestimate treatment effects and can contribute
to statistical heterogeneity.
The diagram above5 shows examples of forest plots exhibiting low and high heterogeneity. In the low
condition, all the studies are generally lined up to the right of the vertical axis and all confidence
intervals are overlapping. In the high condition, the studies are spread over either side of the vertical
decision line and there is little overlap of the confidence intervals. Apart from visual observation of
5 https://s4be.cochrane.org/blog/2018/11/29/what-is-heterogeneity/
Low
heterogeneity
High
heterogeneity
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the level of heterogeneity meta-analysis provides quantitative statistical methods to measure
heterogeneity.
Tests for heterogeneity
Q Cochran Q test - values are given for 2 tests outlined below, (these tests have low statistical
power when only a few studies are included in the meta-analysis):
‘Omnibus test of the Model Coefficients’ tests the null hypothesis that all the ES are zero.
‘Test for residual heterogeneity’ tests the null hypothesis that all the effect sizes in the studies
are all equal (homogeneity of effect sizes).
τ2 Tau square is an estimate of the total amount of heterogeneity. This is interpreted as
systematic unexplained differences between the observed effects of the separate studies. It
is not affected by the number of studies but it is often hard to interpret how relevant the value
is from a practical standpoint.
I2 This measures the extent of heterogeneity not caused by sampling error. If there is high
heterogeneity possible sub-group analysis could be done. If the value is very low, there is no
point doing further subgroup analyses. I2 is not sensitive to changes in the number of studies
and is therefore used extensively in medical and psychological research, especially since there
is a “rule of thumb” to interpret it. A rough guide for interpretations has been suggested as
follows6:
• 0% to 40%: might not be important.
• 30% to 60%: may represent moderate heterogeneity.
• 50% to 90%: may represent substantial heterogeneity.
• 75% to 100%: considerable heterogeneity.
H2 The between-study variance determined by equating the Q statistic to its expected value. H2
has a value of 1 in the case of homogeneity and heterogeneity is assumed to be present when
H2>1.
Meta-analysis models
There are two models commonly used in meta-analysis and each makes different assumptions relating
to the observed differences among the studies
Fixed effects model: this assumes that all studies share a common true ES i.e. the data is
homogeneous. All factors that could influence the ES are the same in all the study samples and
therefore very little heterogeneity. Between-study differences are assumed to be due to chance and
thus not incorporated into the model. Therefore, each study included in the meta-analysis is
estimating the same population treatment effect, which, in theory, represents the true population
treatment effect. Each study is weighted where more weight is given to studies with large samples
sizes i.e. more information.
6 Cochrane Handbook for Systematic Reviews of Interventions
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This model answers the following question “What is the best estimate of the population effect size?”
Random effects model: this assumes a distribution of the treatment effect for some populations. i.e.
that the different studies are estimating different, yet related, intervention effects. Therefore,
heterogeneity cannot be explained because it is due to chance. This model assigns a more balanced
weighting between studies.
This model answers the question “What is the average treatment effect”.
It is therefore important to check for significant heterogeneity to
select the correct model for the meta-analysis.
Random effects model selection.
JASP provides 8 random effects estimators to estimate the 4 indices of heterogeneity (see tests for
heterogeneity above). All use a slightly different approach resulting in different pooled ES
estimates and CIs. To date, the most often used estimator in medical research is the
DerSimonian-Laird estimator however, recent studies have shown better estimates of
between-study variance using Maximum-Likelihood, Empirical Bayes and Sidak-Jonkman
methods.
The Forest Plot.
This plot provides a visual summary of the meta-analysis findings. Graphically it represents the ES and
95% CI for each study and an estimate of the overall ES and 95% CI for all the studies that were
included. As previously mentioned, it can be used to visually assess the level of heterogeneity.
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As a rule of thumb, if all the individual study CIs cover the final combined ES and its 95% CI (diamond)
then there is low heterogeneity (or high homogeneity). In this case, 6 studies do not intersect with the
diamond.
Funnel plot
Looking like an inverted funnel, this is a scatter plot of the intervention effect estimates from individual
studies against each study’s size or precision. The intervention effect should increase as the size of the
study increases. Studies with low power will cluster at the bottom while higher power studies will be
near the top. Ideally, the data points should have a symmetrical distribution. Funnel plot asymmetry
indicates that the meta-analysis results may be biased. Potential reasons for bias include:
• Publication bias (the tendency of authors/publishers to only publish studies with significant
results)
• True heterogeneity
• Artefact (wrong choice of effect measure)
• Data irregularities (methodological design, data analysis etc)
Funnel plot asymmetry can be analysed statistically using either a Rank-test or Eggers test.
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Running a meta-analysis in JASP
If you have not added the Meta-analysis module, click the blue cross at the top right corner
of JASP and tick ‘Meta-Analysis’, this will add it to the main menu ribbon. Go to ‘Meta-
Analysis’ and click ‘Classical Meta-Analysis’.
Open meta analysis.csv. This contains the effect sizes and SEM from 14 studies for changes
in DOMS pain scores in groups receiving no therapy vs cryotherapy. To start, add ES to ‘Effect
Size’, SEM to ‘Effect Size Standard Error’ and Author to ‘Study Labels’. Change the ‘Method’
to ‘Fixed Effects’.
Initial output
This should produce two tables:
The significance of the ES is shown in the ‘Fixed and Random Effects’ table:
Omnibus test of model coefficients tests the H0 that all the coefficients in the second table
are all zero i.e. the interventions have no significant effect. As can be seen, this is significant
and H0 can be rejected, thus the intervention has a significant effect.
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The test of residual heterogeneity tests the H0 that all the effect sizes in all the studies are
equal. If this is not significant a fixed-effects model should be used. However, in this case, it
is significant and therefore a random-effects model would be more appropriate.
The coefficients table provides an estimate (Wald test) of the overall effect size of the
combined study and its significance. This confirms the result of the Omnibus test in the first
table.
Now we know that a random-effects model is more appropriate to this dataset, return to the
options and change the ‘Method’ to ‘Restricted ML (Maximum Likelihood)’. This will then add
another table showing the different estimates of residual homogeneity:
Both τ2 and I2 show excess variance (heterogeneity) between the studies thus supporting the
use of a random-effects model.
Further options.
Return to the Statistics options and in model fit, tick ‘Forest Plot’, ‘Funnel Plot’ and ‘Rank Test
for Funnel Plot Asymmetry’:
The Forest plot shows the weighted effect sizes (size of the squares reflects the weight of each
study) and CIs used to determine the combined ES (diamond).
It can be easily seen that the overall effects of cryotherapy have a significant positive effect
on DOMS scores (ES = 0.66).
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The funnel plot shows that the observed effects sizes appear to be symmetrically distributed
around the vertical axis (based on the overall effect size estimate, in this case, 0.66) and lie
within the 95% confidence triangle. Asymmetry is often reported as being indicative of
publication bias. This plot is accompanied by the ‘Rank Correlation Test’ for funnel plot
asymmetry which in this case is non-significant (p=.912).
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EXPERIMENTAL DESIGN AND DATA LAYOUT IN EXCEL FOR JASP IMPORT.
Independent t-test
Design example:
Independent variable Group 1 Group 2
Dependent variable Data Data
Independent variable Dependent variable
Categorical Continuous
More dependent variables can be added if required
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Paired samples t-test
Design example:
Independent variable Pre-test Post-test
Participant Dependent variable
1 Data Data
2 Data Data
3 Data Data
..n Data Data
Pre-test Post-test
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Correlation
Design example:
Simple correlation
Participant Variable 1 Variable 2 Variable 3 Variable 4 Variable ..n
1 Data Data Data Data Data
2 Data Data Data Data Data
3 Data Data Data Data Data
…n Data Data Data Data Data
Multiple correlations
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Regression
Design example:
Simple Regression
Participant Outcome Predictor 1 Predictor 2 Predictor 3 Predictor ..n
1 Data Data Data Data Data
2 Data Data Data Data Data
3 Data Data Data Data Data
…n Data Data Data Data Data
Multiple regression
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Logistic Regression
Design example:
Dependent Variable (categorical)
Factor (categorical)
Covariate (continuous)
Participant Outcome Predictor 1 Predictor 2
1 Data Data Data
2 Data Data Data
3 Data Data Data
…n Data Data Data
More factors and covariates can be added if required
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One-way Independent ANOVA
Design example:
Independent variable Group 1 Group 2 Group 3 Group…n
Dependent variable Data Data Data Data
Independent variable Dependent variable
(Categorical) (Continuous)
More dependent variables can be added if required
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One-way repeated measures ANOVA
Design example:
Independent variable (Factor)
Participant Level 1 Level 2 Level 3 Level..n
1 Data Data Data Data
2 Data Data Data Data
3 Data Data Data Data
4 Data Data Data Data
..n Data Data Data Data
Factor (time)
Levels
(Related groups)
More levels can be added if required
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Two-way Independent ANOVA
Design example:
Factor 1 Supplement 1 Supplement 2
Factor 2 Dose 1 Dose 2 Dose 3 Dose 1 Dose 2 Dose 3
Dependent variable
Data Data Data Data Data Data
Factor 1 Factor 2 Dependent variable
More factors and dependent variables can be added if required
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Two-way Repeated measures ANOVA
Design example:
Factor 1 Interventions
Level 1 i.e. intervention 1
Level 2 i.e. intervention 2
Factor 2 Time
Level 1 i.e time 1
Level 2 i.e time 2
Level 3 i.e time 3
Level 1 i.e time 1
Level 2 i.e time 2
Level 3 i.e time 3
1 Data Data Data Data Data Data
2 Data Data Data Data Data Data
3 Data Data Data Data Data Data
..n Data Data Data Data Data Data
Factor 1 levels 1-n Factor 2 levels 1-n
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Two-way Mixed Factor ANOVA
Design example:
Factor 1 (Between subjects)
Group 1 Group 2
Factor 2 levels (Repeated measures)
Trial 1 Trial 2 Trial 3 Trial 1 Trial 2 Trial 3
1 Data Data Data Data Data Data
2 Data Data Data Data Data Data
3 Data Data Data Data Data Data
..n Data Data Data Data Data Data
Factor 1 Factor 2 levels
(Categorical) (Continuous)
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Chi-squared - Contingency tables
Design example:
Participant Response 1 Response 2 Response 3 Response…n
1 Data Data Data Data
2 Data Data Data Data
3 Data Data Data Data
..n Data Data Data Data
All data should be categorical
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SOME CONCEPTS IN FREQUENTIST STATISTICS The frequentist approach is the most commonly taught and used statistical methodology. It
describes sample data based on the frequency or proportion of the data from repeated
studies through which the probability of events is defined.
Frequentist statistics uses rigid frameworks including hypothesis testing, p values and
confidence intervals etc.
Hypothesis testing
A hypothesis can be defined as “a supposition or proposed explanation made based on
limited evidence as a starting point for further investigation”.
There are two simple types of hypotheses, a null hypothesis (H0) and an alternative or
experimental hypothesis (H1). The null hypothesis is the default position for most statistical
analyses in which it is stated that there is no relationship or difference between groups. The
alternative hypothesis states that there is a relationship or difference between groups has n
a direction of difference/relationship. For example, if a study was carried out to look at the
effects of a supplement on sprint time in one group of participants compared to the placebo
group:
H0 = there is no difference in sprint times between the two groups
H1 = there is a difference in sprint times between the two groups
H2 = group 1 is greater than group 2
H3 = group 1 is less than group 2
Hypothesis testing refers to the strictly predefined procedures used to accept or reject the
hypotheses and the probability that this could be purely by chance. The confidence at which
a null hypothesis is accepted or rejected is called the level of significance. The level of
significance is denoted by α, usually 0.05 (5%). This is the level of probability of accepting an
effect as true (95%) and that there is only 5% of the result being purely by chance.
Different types of hypothesis can easily be selected in JASP, however, the null hypothesis is
always the default.
162 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
Type I and II errors
The probability of rejecting the null hypothesis, when it is, in fact, true, is called Type I error
whereas the probability of accepting the null hypothesis when it is not true is called Type II
error.
The truth Not guilty (H0) Guilty (H1)
The verdict
Guilty (H1)
Type I error An innocent person goes to prison
Correct decision
Not guilty (H0) Correct decision
Type II error A guilty person goes free
Type I error is deemed the worst error to make in statistical analyses.
Statistical power is defined as the probability that the test will reject the null hypothesis when
the alternative hypothesis is true. For a set level of significance, if the sample size increases,
the probability of Type II error decreases, which therefore increases the statistical power.
Testing the hypothesis
The essence of hypothesis testing is to first define the null (or alternative) hypothesis, set
the criterion level α, usually 0.05 (5%), collect and analyse sample data. Use a test statistic to
determine how far (or the number of standard deviations) the sample mean is from the
population mean stated in the null hypothesis. The test statistic is then compared to a critical
value. This is a cut-off value defining the boundary where less than 5% of the sample means
can be obtained if the null hypothesis is true.
If the probability of obtaining a difference between the means by chance is less than 5% when
the null hypothesis has been proposed, the null hypothesis is rejected and the alternative
hypothesis can be accepted.
The p-value is the probability of obtaining a sample outcome, given that the value stated in
the null hypothesis is true. If the p-value is less than 5% (p < .05) the null hypothesis is
rejected. When the p-value is greater than 5% (p > .05), we accept the null hypothesis.
Effect size
An effect size is a standard measure that can be calculated from any number of statistical
analyses. If the null hypothesis is rejected the result is significant. This significance only
evaluates the probability of obtaining the sample outcome by chance but does not indicate
how big a difference (practical significance), nor can it be used to compare across different
studies.
163 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
The effect size indicates the magnitude of the difference between the groups. So for example,
if there was a significant decrease in 100m sprint times in a supplement compared to a
placebo group, the effect size would indicate how much more effective the intervention was.
Some common effect sizes are shown below.
Test Measure Trivial Small Medium Large
Between means Cohen’s d <0.2 0.2 0.5 0.8
Correlation Correlation coefficient (r) Rank -biserial (rB) Spearman’s rho
<0.1 <0.1 <0.1
0.1 0.1 0.1
0.3 0.3 0.3
0.5 0.5 0.5
Multiple Regression Multiple correlation coefficient (R)
<0.10 0.1 0.3 0.5
ANOVA Eta Partial Eta Omega squared
<0.1 <0.01 <0.01
0.1 0.01 0.01
0.25 0.06 0.06
0.37 0.14 0.14
Chi-squared Phi (2x2 tables only) Cramer’s V Odds ratio (2x2 tables only)
<0.1 <0.1 <1.5
0.1 0.1 1.5
0.3 0.3 3.5
0.5 0.5 9.0
In small datasets, there may be a moderate to large effect size but no significant differences.
This could suggest that the analysis lacked statistical power and that increasing the number
of data points may show a significant outcome. Conversely, when using large datasets,
significant testing can be misleading since small or trivial effects may produce statistically
significant results.
PARAMETRIC vs NON-PARAMETRIC TESTING
Most research collects information from a sample of the population of interest, it is normally
impossible to collect data from the whole population. We do, however, want to see how well
the collected data reflects the population in terms of the population mean, standard
deviations, proportions etc. based on parametric distribution functions. These measures are
the population parameters. Parameter estimates of these in the sample population are
statistics. Parametric statistics require assumptions to be made of the data including the
normality of distribution and homogeneity of variance.
In some cases these assumptions may be violated in that the data may be noticeably skewed:
164 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
Sometimes transforming the data can rectify this but not always. It is also common to collect
ordinal data (i.e. Likert scale ratings) for which terms such as mean and standard deviation
are meaningless. As such there are no parameters associated with ordinal (non-parametric)
data. The non-parametric counterparts include median values and quartiles.
In both of the cases described non-parametric statistical tests are available. There are
equivalents of most common classical parametric tests. These tests don’t assume normally
distributed data or population parameters and are based on sorting the data into ranks from
lowest to highest values. All subsequent calculations are done with these ranks rather than
with the actual data values.
165 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
WHICH TEST SHOULD I USE?
Comparing one sample to a known or hypothesized population mean.
Testing relationships between two or more variables
Data type
Continuous Ordinal Nominal
Are parametric
assumptions met?
Yes No
Pearson’s
correlation
Spearman’s or
Kendall’s tau
Chi-square
contingency
tables
Continuous Ordinal Nominal
2 categories >2 categories
One-sample
t-test One-sample
median test Binomial test Multinomial test
or Chi-square
‘goodness of fit’
Data type
Currently not available in
JASP
166 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
Predicting outcomes
Testing for differences between two independent groups
Data type
Continuous Ordinal Nominal
More than one
predictor variable?
No Yes
Simple
regression
Ordinal
regression
Logistic
regression
tables
Multiple
regression
Currently not available in JASP
Data type
Continuous Ordinal Nominal
Are parametric
assumptions met?
Yes No
Independent
t-test
Mann-Whitney U
test
Chi-square or
Fischer’s Exact
test
167 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
Testing for differences between two related groups
Testing for differences between three or more independent groups
Data type
Continuous Ordinal Nominal
Are parametric
assumptions met?
Yes No
Paired samples t-test Wilcoxon’s test McNemar’s test
Currently not available in
JASP
Data type
Continuous Ordinal Nominal
Are parametric
assumptions met?
Yes No
ANOVA Kruskall-Wallis Chi-square
Contingency tables
168 | P a g e JASP 0.14 - Dr Mark Goss-Sampson
Testing for differences between three or more related groups
Test for interactions between 2 or more independent variables
Data type
Continuous Ordinal Nominal
Are parametric
assumptions met?
Yes No
RMANOVA Friedman test Repeated measures
logistic regression
Data type
Continuous Ordinal Nominal
Are parametric
assumptions met?
Yes No
Two-way
ANOVA
Ordered logistic
regression
Factorial logistic
regression
Applied Economics and Finance
Vol. 3, No. 2; May 2016
ISSN 2332-7294 E-ISSN 2332-7308
Published by Redfame Publishing
URL: http://aef.redfame.com
15
Capital Budgeting Theory and Practice:
A Review and Agenda for Future Research
Lingesiya Kengatharan 1
1 Department of Financial Management, University of Jaffna, Sri Lanka.
Correspondence: Lingesiya Kengatharan, Department of Financial Management, University of Jaffna, Sri Lanka.
Received: December 21, 2015 Accepted: January 14, 2016 Available online: February 4, 2016
doi:10.11114/aef.v3i2.1261 URL: http://dx.doi.org/10.11114/aef.v3i2.1261
Abstract
The main purpose of this research was to delineate unearth lacunae in the extant capital budgeting theory and practice
during the last two decades and ipso facto become springboard for future scholarships. Web of science search and iCat
search were used to locate research papers published during the last twenty years. Four criteria have been applied in
selection of research papers: be an empirical study, published in English language, appeared in peer reviewed journal
and full text research papers. These papers were collected from multiple databases including OneFile (GALE), SciVerse
ScienceDirect (Elsevier), Informa - Taylor & Francis (CrossRef), Wiley (CrossRef), Business (JSTOR), Arts & Sciences
(JSTOR), Proquest ,MEDLINE (NLM), and Wiley Online Library. Search parameters covered capital budgeting,
capital budgeting decision, capital budgeting theory, capital budgeting practices, capital budgeting methods, capital
budgeting models, capital budgeting tools, capital budgeting techniques, capital budgeting process and investment
decision. Thematic text analyses have been explored to analyses them. Recent studies lent credence on the use of more
sophisticated capital budgeting techniques along with many capital budgeting tools for incorporating risk.
Notwithstanding, it drew a distinction between developed and developing countries. Moreover, factors impinging on
choice of capital budgeting practice were identified, and bereft of behavioral finance and event study methodological
approach were highlighted. More extensive studies are imperative to build robust knowledge of capital budgeting theory
and practice in the chaotic environment. This research was well thought out in its design and contributed by stating the
known and unknown arena of capital budgeting during the last two decades. This scholarship facilitates to academics,
practitioners, policy makers, and stakeholders of the company.
Keywords: Capital budgeting theory and practices, capital budgeting tools for incorporating risk, discount rate
1. Introduction
Predominantly, area of capital and capital budgeting of financial management have been attracted many researchers
during the last five decades and the seminal studies culminated with presenting many theories (e.g., Markowitz,1952;
Modigliani & Miller,1958; Markowitz,1959; Miller & Modigliani,1961; Fama,1970; Black & Scholes,1973; Ross, 1976;
Roll,1977; Myers,1977; Myers,1984; Jensen,1986; Ritter,1991;Graham & Harvey, 2001; Myers,2003; Halov &
Heider,2004; Atkeson & Cole,2005;) and models (e.g.,Markowitz,1952; Sharpe,1963; Sharpe,1964; Linter,1965;
Roll,1977) time to time. Notwithstanding, due to the globalization, environmental changes and cutting edge advanced
technological developments, theories and models developed in the past do not applicable today and many of them are
criticized and their applicability in practice is intriguing (e.g., Malkiel, 2003; Bornholt, 2013). A curious instance
illustrated by Brounen, de Jong and Koedijk (2004) is that ‗Nobel Prize winning concepts like the capital asset pricing
model and capital structure theorems have been praised and taught in class rooms, but to what the extent to these
celebrated notions have also found their way into corporate board rooms remains somewhat opaque‘ (p.72). ‗Traditional
capital budgeting methods have been heavily criticized of discouraging the adoption of advanced manufacturing
technology and thus undermining the competitiveness of Western firms‘ (Slagmulder, Bruggeman & Wassenhove, 1995,
p.121). In a similar vein, many research scholars on their seminal scholarships argued that there are gaps in theory of
capital budgeting and its applicability (e.g., Mukheijee & Henderson, 1987; Arnold & Hatzopoulos, 2000; Graham &
Harvey, 2001; Cooper, Morgan, Redman & Smith, 2002; Brounen et al., 2004; Kersyte, 2011).
Firms operating in a dynamic environment must respond to changes to beat competitors and to sustain, survive and
grow in markets (Ghahremani, Aghaie & Abedzadeh, 2012). Most changes impinge on capital investment decisions,
Applied Economics and Finance Vol. 3, No. 2; 2016
16
which can invariably involve large sums of money over the long period (e.g.,Peterson & Fabozzi, 2002, Cooper et al.,
2002; Dayananda, Irons, Harrison, Herbohn & Rowland, 2002) and these decisions are critical in managing strategic
change and sustaining long term corporate performance (Emmanuel, Harris & Komakech, 2010). Capital investment
decision can be acquisitions, investing new facilities, new product development, employing new technology and
adoption of new business processes or some combination of these (Emmanuel et al., 2010). Capital budgeting
investment decisions are critical to survival and long term success for firms due to many factors and those factors are
commonly named as uncertainty. The global financial crisis is epitomized this truth. One of the most intractable issues
confronted by researchers is how to identify, capture, and evaluate uncertainties associated with long term projects
(Haka, 2006). Sources of uncertainty range from the mundane (cash flow estimation, number and sources of estimation
error, etc.) to the more esoteric (complementarities among investments, options presented by investment opportunities,
opportunity cost of investments, etc.) (Haka, 2006). Since capital investment decision deals with large sum of fund,
scrupulous attention has been given in making decision. ‗Capital budgeting is as the procedures, routines, methods and
techniques used to identify investment opportunities, to develop initial ideas into specific investment proposals, to
evaluate and select a project and to control the investment project to assess forecast accuracy‘(Segelod,1997). Albeit
there are number of capital budgeting methods assist in making decision, number of other uncertainty factors have
deleterious penetration into making capital budgeting decision.
Nowadays, complex methods are used for making capital budgeting decision rather purely depends on theories of
capital budgeting because of uncertainty and other contingency factors (Singh, Jain &Yadav, 2012; Zhang, Huang
&Tang, 2011; Kersyte, 2011; Bock & Truck, 2011; Byrne & Davis, 2005;Cooper et al, 2002; Arnold & Hatzopoulos,
2000; Mao, 1970; and Dickerson, 1963). After the advent of full-fledged globalization and in the era of cutthroat
competition (Verma, Gupta & Batra, 2009), advanced developments in technologies, other macro environmental factors
and demographic factors are intruding into capital budgeting practices (Verbeeten, 2006). In a world of geo -political,
social as well as economic uncertainty, strategic financial management is of process of change, in turn requiring a re-
examination of the fundamental assumption (e.g, efficient market hypothesis, Fama,1970) that cut across traditional
boundaries of the financial management (Hill, 2008). With limited credit and other sources of financing in today‘s
uncertain and challenging economic environment, also required to be scrupulously evaluated the profitability and
successfulness of proposed capital investments and allocate limited capital is more vital than ever (Kester & Robbins,
2011).
Over the last 20 years, there have been many changes and challenges in making financial decision due to the global
financial crisis, fluctuations in value of money, advanced technology, interest rate, exchange rate and inflation rates‘
risks and dramatic changes in economic and business environment both in national as well as in global markets. Thus,
there is need to re- examine and re- study for re-building capital budgeting practices since it has considerable impact on
investment decision making. The investment decision making is not a simple or straightforward approach, the risk is an
important element in making investment decision. There are number of risk techniques employed by companies for
evaluating investment projects. However, there is problem in setting up theoretical model and applying that model into
practice (e.g: Arnold & Hatzopoulos, 2000; Digkerson, 1963). Thus, the theory is not purely able to apply at all times.
Sometimes theories developed in the past do not applicable today. There is no doubt, over the last two decades
corporate practices regarding capital budgeting practices have not been static, diverged from the theories.
This study presents systematic review on capital budgeting practices literature published in the last two decades. The
systematic review of literature is referred to as 'principally justified by the manner in which the reviewer proceeds, stage
by stage, with full transparency and explicitness about what is (and what is not) done, typically using a protocol to
guide the process' (Young, Ashby, Boaz & Grayson, 2002, p.220). Through this review, updating information about the
capital budgeting techniques which being used by firms and to compare the current usage of various techniques,
methods with those found in previous studies. This study is thus accumulatively builds a robust knowledge in the area
of capital budgeting practices and identifying unearth gaps will become springboard for future research. Therefore, this
research guides the researchers to reflect on and assess where they are in an area of capital budgeting practices and
guide future research directions.
1.1 Objective of the study
Examining empirical research on capital budgeting practices to date has been very useful in explaining importance of
capital budgeting practices for the long time success of the business organization. Nowadays, complex methods are used
for making capital budgeting decision rather purely depends on theories of capital budgeting. Advanced developments
in technologies, other macro environmental factors and demographic factors are intruding into capital budgeting
practices and thus some of the theories become out of use in well developed countries (e.g: payback period). Thus, the
main aim of this research is to demonstrate unearth gaps in the existing capital budgeting practices literature and to
suggest the directions for the future research .It will further attempt to
Applied Economics and Finance Vol. 3, No. 2; 2016
17
- Explain the capital budgeting theories and practices in different countries and demonstrate the disparities between theories and practices of capital budgeting
- Identify the factors that determine use of capital budgeting practices of a country or firm
1.2 Problem Statement
During the past twenty years (1993-2013), the theory of capital budgeting has been characterized by the many increased
applications on the basis of risk and uncertainty resulting from global economic, technological and advanced educational
changes e.g: inflation risk, interest rate and exchange rate risk. Capital budgeting is the backbone of the financial
management. Modern financial management theory generally assumes that the primary objective of a firm is to
maximize the wealth of its owners (Atrill, 2009). Uncertainty and risk are the major influence in making investment
decision and thus Mao (1970) says ‗a central aspect of any theory of capital budgeting is the concept of risk‘ (p.352). In
order to implement the objective of modern financial management theory, ‗financial executives need criteria for
choosing between alternative time patterns of project evaluations within his planning horizon' (Mao, 1970). There are
complexities in making investment decision and the theory could not always applicable in all situations. Problem
statement of this study is how far capital budgeting theory differentiates with practice and to demonstrate the nature
of the gaps in existing capital budgeting literature.
1.3 Research Questions
On the basis of background of research, the following research questions have been developed as the way to attain
research objectives.
What are the capital budgeting theories and practices used by firms? Are there any disparities between
the capital budgeting theories and practices? If so how?
What are the factors determines the use of capital budgeting practices? Are there different across
countries? If so how?
What are the gaps in the existing capital budgeting literature?
2. Methodology
The main objective of this study is to find out gaps in extant capital budgeting literature during the past 20 years of
study. The methodology covers research philosophy, research approach, research strategy, methods of data collection
and data analysis. These entire methodological spheres used throughout the research have been below discussed in
details.
2.1 Research Philosophy
One of the dominant philosophical concepts is the ‗ontological assumption‘ that enquires about nature of reality, and
any study absence of this assumption would be treated as 'blinded' (Easterby-Smith, Thorpe & Lowe, 2002, p. 27). This
research assumes that capital budgeting practices are different across firms/ nations and the ways of looking at capital
budgeting practices are not same at all the time. It can be further articulated that even when there are number of capital
budgeting theories, we cannot expect similar application at all situations and thus it is subject to changes. Thus, the
ontological assumption is of constructionism. Constructionist ontology‘s view that world is being internally constructed
and both individually and collectively generate meaning where we are not sure about what is real! Consequently, people
guess reality of the world with the experience of external indicators.
Another important philosophical assumption is the epistemological assumption. It enquires about what should be taken
as acceptable knowledge in a particular field (Easterby-Smith et al., 2002). The traditional practices do not applicable in
the contemporary borderless global businesses and thus try to understand the factors determine the use of capital
budgeting practices. It guides how can we understand and determine capital budgeting practices in different context and
in different geographical location. The knowledge can be attainable by text analysis with subject methods. Thus, it
offers what is already known about capital budgeting practices and captures the gaps in extant literature by
systematically reviewing literature.
This research takes interpretive approach on epistemology for answering research questions. The reality is not
independent of individual thought and thus all the research findings are not similar with one another (Blaikie, 2007).
Thus, this multiple reality is called ‗subjectivism‘. Findings could vary in different context such as nature of
measurement tools, geographical location, company‘s size, organizational practices, types of sectors and form of
methodology used. Thus, this research is organized by collecting relevant literature review and interpreting concepts of
relationship between researchers and research. Inductive approach is thus suited by exploring thematic text analysis.
2.2 Research approach
The research strategy leads to design qualitative research approach. This research covered sufficient researches carried
Applied Economics and Finance Vol. 3, No. 2; 2016
18
out during the past two decades in the area of capital budgeting. This research analyzed past literature by identifying
relevant themes and then thematic text analysis was employed. Thus, this research is ‗subjective‘ and adopts inductive
approach in order to answering research questions.
2.3 Research strategy
Research strategy tells about how research should be designed for answering a set of developed research questions and
consequently research aims are attained. As this research covers last twenty years of research papers carried out in the
area of capital budgeting from 1993 to 2013, this study adapts research strategy of longitudinal research design.
However, the collection of literature covers broad areas including different sectors, different locations/countries and
different size of firms. Thus, the systematic literature review sometimes takes comparative research design as well.
2.4 Data collection methods
Web of science search and iCat search were used to locate research papers published during the last twenty years. Web
of science is a mass search engine linking with mass database covering more than 10000 journals and 110 000
conference proceedings. However, all most all the databases (online the full text of electronic resources) have been
covered by iCat search which is subscribed and launched by Kingston University, London. Kingston University
library‘s access service was exploited for collecting all the research papers. Search parameters includes capital
budgeting, capital budgeting decision, capital budgeting theory, capital budgeting practices, capital budgeting methods,
capital budgeting models, capital budgeting tools, capital budgeting techniques, capital budgeting process and
investment decision.
Initially, there are 363 research papers identified during the last 20 years. Of them, 201 research papers were screened
and considered for this research to be reviewed based on the following criteria.
- An empirical study (i.e., sampling process, measurement , analysis): 363 papers were identified - Published in English language: Of 363, 264 were published in English. - Should be published in peer reviewed journal : Of 264, 239 were published in a peer reviewed journals - Full text research papers: Of 239, 201 papers were full text journal
These papers were collected from following databases: OneFile (GALE), SciVerse ScienceDirect (Elsevier), Informa -
Taylor & Francis (CrossRef), Wiley (CrossRef), Business (JSTOR), Arts & Sciences (JSTOR), MEDLINE (NLM),
SpringerLink, Wiley Online Library , Inderscience Journals , ERIC (U.S. Dept. of Education), Sage Publications
(CrossRef), INFORMS Journals, Health Reference Center Academic (Gale), University of Chicago Press Journals,
Emerald Management eJournals, Directory of Open Access Journals (DOAJ),IngentaConnect, IEEE (CrossRef). All
these papers were spread over across many journals including Journal of Banking and Finance, The Journal of Finance,
Journal of Accounting and Economics, Management Decision, Journal of Cleaner Production, Journal of Financial
Economics, Management Science, European Journal of Operational Research, Accounting Review, Journal of
Economic Behavior and Organization, Long Range Planning, Energy Policy, Accounting, Organizations and Society,
Computers and Mathematics with Applications.
2.5 Data analysis
As discussed, at the outset, Miles and Huberman‘s (1984) proposed strategy was carried out that involves collection,
reduction, displays and conclusions. Based on the set criteria, 363 research papers were reduced to 201 and they
analyzed using a coding procedure. Initially, collected research papers were grouped into themes or topics. Theme
represents the focused area of research and it is selective coding on grounded theory (Corbin & Strauss, 1990). Themes
were in terms of current theory and practices of capital budgeting, factors influencing on capital budgeting practices/
determinants of capital budgeting practices, capital budgeting methods/ models, supplementary tools for the capital
budgeting methods, influences of capital budgeting practices on investment decisions, component of capital budgeting
process, capital budgeting stages, and global capital budgeting practices.
A thematic analysis was employed to capture key themes and concepts in chosen research papers. In doing so, open
coding, as suggested by Strauss and Corbin (1998), was adopted. The analysis was focused on the concepts related to
capital budgeting practices and theories, research design, research sampling techniques, research approach, year of
publication, nature of industry and so on. The results of this analysis were presented below.
3. Results
3.1 Multi-disciplinary concepts of capital budgeting
During the past twenty years, a total of 202 research papers appeared in peer reviewed indexed journals were identified
across many academic journals. Majority of the papers appeared in Engineering Economist (N= 32) yielding 15.92%
followed by Managerial Finance (27), Public Budgeting & Finance (16), Financial Management(9), Journal of Banking
and Finance (8), Journal of Business Finance & Accounting (6), Accounting Education(5), Management Accounting
Applied Economics and Finance Vol. 3, No. 2; 2016
19
Research(5), The Journal of Finance(5), Journal of Corporate Accounting & Finance (4), Management Decision (4) and
The Review of Financial Studies. All of these journals represented 62.20 % of research papers in capital budgeting in
the last two decades. The reminder of the research papers appeared in many journals. Capital budgeting is thus
multi-disciplinary aspects and applied across many discipline. The table 1 below summarizes entire list of journals
contained capital budgeting research papers.
Table 1. Name of the journals: Capital budgeting research papers appeared during the past twenty years
Name of the Journal Number of
paper
published
Percentage
Engineering Economist 32 15.92%
Managerial Finance 27 13.43%
Public Budgeting & Finance 16 7.96%
Financial Management 9 4.48%
Journal of Banking and Finance 8 3.98%
Journal of Business Finance & Accounting 6 2.99%
Accounting Education 5 2.49%
Management Accounting Research 5 2.49%
The Journal of Finance 5 2.49%
Journal of Corporate Accounting & Finance 4 1.99%
Management Decision 4 1.99%
The Review of Financial Studies 4 1.99%
Three papers in each journal: Healthcare Financial
Management, Information Sciences, International Journal of
Energy Research, International Journal of Production Economics,
Journal of Financial Economics, Management Science,
Operations Research, The Journal of Business, Theoretical and
Applied Economics.
3 1.49%
Two papers in each journal: Accounting & Finance, Accounting
and Business Research, Accounting, Organizations and Society,
Computers & Industrial Engineering, Computers and
Mathematics with Applications, Contemporary Accounting
Research, European Financial Management, European Journal of
Operational Research, Health care strategic management,
Industrial Management & Data Systems, International Journal of
Business and Management, Journal of Accounting and
Economics, Journal of Computational and Applied Mathematics,
Journal of Information Technology, Journal of Marketing
Management, Journal of Small Business Management, Journal of
the International Academy for Case Studies, Journal of the
Operational Research Society, Long Range Planning, Managerial
and Decision Economics, The Bond Buyer, The Financial
Review.
2 1.00%
One paper in each journal: Academy of Marketing Studies
Journal, Accounting Review, Agricultural Finance Review.
1 0.50%
Applied Economics and Finance Vol. 3, No. 2; 2016
20
Applied Financial Economics, Australasian Radiology, Australian
Journal of Management, BuR : Business Research, Business
Forum, Wntr-Spring, Business Process Management Journal,
Canadian Journal of Anesthesia/Journal canadien d'anesthésie,
Computational Management Science, Computers and Chemical
Engineering, Cornell Hotel and Restaurant Administration
Quarterly, Energy Policy, European Management Journal, Expert
Systems With Applications, Forest Products Journal, Fuzzy Sets
and Systems, Healthcare financial management. IEEE
Transactions on Engineering Management, Industrial
Management, International Journal of Commerce and
Management, International Journal of Information Technology &
Decision Making, International Journal of Project Management,
International Journal of Quality & Reliability Management,
International Transactions in Operational Research, Journal of
Accounting Research , Journal of Cleaner Production, Journal of
Economic Behavior and Organization, Journal of Empirical
Finance, Journal of Hospitality & Tourism Research, Journal of
International Financial Management & Accounting, Journal of
International Money and Finance, Journal of Management
Accounting Research, Journal of Managerial Issues, Journal of
Property Investment & Finance, Journal of Public Health
Dentistry, Journal of Retail Banking, Journal of Risk and
Insurance, Journal of Teaching in International Business, journal
of the Healthcare Financial Management, Knowledge-Based
Systems, Management Accounting Quarterly, Mid-Atlantic
Journal of Business, Naval Research Logistics (NRL), New
Directions for Higher Education, Operations-Research-Spektrum,
Quarterly Review of Economics and Finance, Real Estate
Economics, Review of Agricultural Economics, Review of
Business, Review of Finance and Banking, Review of
Quantitative Finance and Accounting, Scandinavian Journal of
Management, South East European Journal of Economics and
Business, Strategic Finance, The Accounting Review, The
European Journal of Finance, The Financier, Spring-Winter, The
McKinsey Quarterly, Tsinghua Science & Technology, UTMS
Journal of Economics, Vision: The Journal of Business
Perspective, Journal of advances in management research.
Percentages calculated in terms of number of papers appeared in each journal (N = 202).
3.2 Major themes identified in Capital budgeting research
A total of 201 research papers in capital budgeting have been meticulously reviewed and consequently following major
themes have been identified: capital budgeting theory and practices, capital budgeting theory and practices in developed
countries, capital budgeting theory and practices in developing countries and factor affecting capital budgeting decision.
Findings discusses under identified themes.
3.3 Capital budgeting theory and practices
Capital budgeting decisions are crucial and complex and have attracted many research scholars in this field. According
to Dayananda et al. (2002), capital budgeting is the process of deciding investment projects which create in
maximization of shareholder value. Capital budgeting is mostly dealt with sizable investments in long term assets.
Assets can be either tangible such as building, plant, or equipment or intangible assets such as patents, new technology
or trade mark (Brealey & Myers, 2003). Capital budgeting is not a short term aspects, generally prepared a year in
advance and extendable to five, ten or even fifteen years in future (Brickley, 2006). And thus, Peterson and Fabozzi
(2002) define capital budgeting is the process of analyzing and selecting investment opportunities in long term assets
where its benefits last for more than one year.
Capital budgeting is a fundamental and used everywhere as a tool for planning, control, and allocation of scare
resources among competing demands. Capital budgeting is a vital part in financial planning and decision making since
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21
capital budgeting tools leads better decision making and be able to justify selection of specific capital investments
among competing alternatives (Sekwat,1999).Decision to choose the best investment project among competing projects
is of critical and being taken by top management (Bowman & Hurry, 1993; McGrath, Ferrier & Mendelow, 2004) and
considerable attention is thus to be given to investigating the methods used in evaluating and selecting investment
projects (Sangster, 1993; Segelod, 1998).
The most prevalent capital budgeting techniques in the public finance literature include payback period (PB),
accounting rate of return (ARR), net present value (NPV), internal rate of return (IRR), benefit-cost ratio (BCR), and
profitability index (PI) (e.g., Sekwat,1999;Cooper et al.,2002). Among these methods, four methods .viz., NPV, IRR,
PB and ARR, have been identified as a predominant method and used in many studies (e.g., Pike,1996; Kester, Chang,
Echanis, Haikal, Isa, Skully,Tsui & Wang, 1999; Hermes, Smid, & Yao , 2007).
The PB model determines the length of time required to recover exactly the invested cash outlay. On the other hand, the
ARR is calculated as the ratio of the investment‘s average after tax income to its average book value (Cooper et al.,
2002). The PB period has been criticized for failing to make accurate assessments of project value as it does not
consider use of cash flows, time value of money, risk in a systematic manner and further it does not identify investment
projects that will maximize profits, therefore PB does not have theoretical justification (Pike, 1988; Lefley,1996).
Research scholars and practitioners criticized the ARR due to the ignorance of the time value of money (e.g., Cooper et
al., 2002; Ross, Waterfield, Jordan & Roberts, 2005). And PB methods failed to consider return from the capital
investment after the initial outlay recovered, yet it is also oft- used methods (e.g., Graham & Harvey, 2001; Brounen et
al., 2004; Bennouna, Meredith and Marchant, 2010). Researchers argued that the reasons behind widespread use of PB
method are of its easiness and of providing information about recovery of initial investment.
Thus, in the next generation, the NPV model came into practice where it measures the difference between present value
of the money in and present value of the money out (Cooper et al., 2002). If the NPV is positive, the capital investment
is accepted and vice versa. Alternatively, the IRR determines the rate at which capital investment can be acceptable and
thus equates the cost of the capital investment to the present value of that project (Cooper et al., 2002). In finance, the
methods of assessing capital budgeting using the concepts of the time value of money is called discounted cash
flow (DCF) analysis. The NPV and IRR methods are called discounted cash flow (DCF) methods. The PB and ARR
methods are considered to be non-DCF methods. ‗Capital budgeting theory assumes that projects are evaluated based on
economic merit. Building upon certain economic assumptions, including the time value of money, risk aversion, and an
assumed goal of value maximization, sophisticated investment appraisal techniques such as NPV and IRR, have been
advocated in the literature‘ (Slagmulder et al., 1995,p.123).Notwithstanding, several researchers criticized that requisite
necessary information for NPV and IRR is commonly not known with certainty owing to longer periods, uncertainty
in future, higher degree of risk, ignore the size of the investment and absence of logical comparison on time value of
money (e.g., Sekwat,1999;Cooper et al.,2002; Hermes et al., 2007).Thus, in order to overcome both the time value of
money and the size of the investment, the PI model has been emerged. It is the ratio of the capital investment to its
outlay and the decision being made in terms of the highest PI (Cooper et al., 2002). If this method used carelessly with
constrained investment resources, it generates bad results (Brealey & Myers, 2003).
However, Graham and Harvey (2001) reported that twelve capital budgeting methods were in practice: NPV, IRR,
Annuity, Earning multiple (P/E), Adjusted present value (APV), PB, Discounted Payback, PI, ARR, Sensitivity analysis,
Value at risk and real options. However, all of them are not in usable at all situations in capital budgeting practices. For
example, IRR should not be the best method if investments are mutually exclusive or have multiple rates of return,
however, IRR is oft-exploited methods in practice (Graham & Harvey, 2001; Brounen et al., 2004; Bennouna et al.,
2010).
Of these methods, discounted payback considers time value of money but it still ignores cash flows after initial outlay
recovered. Value-at-risk (VAR) is to measure 'the worst expected loss over a given horizon under normal market
conditions at a given confidence level' (Jorion, 2006; p.12), is a relatively new method. The APV additionally covers the
value of financial side-effects of an investment to NPV, and treated as having no drawbacks principally (Ross et al.,
2005).
The greatest problems of the traditional present value models are that its complete reliance on quantifiable cash flows.
However, in a contemporary high tech world, many new projects entail complete redesign of the manufacturing
environment and computerized design is of paramount important to be innovative, higher qualities and speedier
response (Cooper et al., 2002). And thus, the theory of capital budgeting is diverged from its practices.
The complex nature of the capital investment in today‘s world incubates many new models into practices including
multi-attribute decision model, and analytical hierarchy process that are more subjective (Cooper et al., 2002). Modern
theoretical developments in finance views that DCF methods are not the best methods to select capital investment
projects: they have severe drawbacks in the analysis of investment projects if the information about future investment
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decision is not available (Brennan & Schwartz, 1992; Trigeorgis, 1993; Dixit & Pindyck, 1994). In such a situation,
Real Options Reasoning (ROR) and Game Theory (GT) serves as better analytical tools to evaluate such investment
projects (Smit & Ankum, 1993). GT stresses that firm is having an incentive to invest early in the case of fear of
pre-emption (Smit, 2003)
Real option theory: Real option is closely related to corporate capital investment decision-making and has been
introduced as an alternative approach for investment appraisal under uncertainty. The starting point for real options
research was the criticism of traditional strategic investment decision-making and capital budgeting methods. In general,
a real option represents or reflects the option or options that a company has when it comes to deciding whether to invest
in a project, delay, put it on hold, expand or reduce an investment, or any other flexibility that it may have (Rigopoulos,
2014). ROT involves the use of investment evaluation tools and processes that properly account for both uncertainty and
the company‘s ability to react to new information (Verbeeten, 2006). ROT has operating flexibility (which enables the
management to make or revise decisions at a future time, such as expansion or abandonment of the project) and the
strategic option value (resulting from interdependence with future and follow-up investments, such as implementation in
phases and the postponement of investments) (Verbeeten, 2006). Many researchers have argued that the use of real
options analysis has an advantage over NPV, since NPV is not able to capture the value of managerial flexibility (e.g.,
Ingersoll & Ross, 1992; Trigeorgis, 1993; Dixit & Pindyck, 1994). For example, the management could delay, expand,
abandon, temporarily close or alter the operation during the project‘ life. Ross et al. (2005) argued that most capital
investment projects have options (i.e., the option to expand, the option to modify, the option to abandon), which have
value per se. Although this method has not been applied on a large scale in practice (Hermes et al., 2007), it is mostly
applicable in specific industries or situations. DCF techniques are used concurrently with real options in order to
determine the true NPV (Amram & Howe, 2002). Many research scholars have found that only a few firms have
employed real options (Graham & Harvey, 2001; Ryan & Ryan, 2002; Brounen et al., 2004; Block, 2007; Truong,
Partington & Peat,2008; Verma et al., 2009; Bennouna et al., 2010; Shinoda,2010, Singh et al.,2012; Andres, Fuente &
Martin,2015).
It is obvious that widespread use of sophisticated capital budgeting during the last two decades. Many earliest studies
investigated about capital budgeting decision rule, in contrast, recent researches attempted to focus on the use of
sophisticated capital budgeting practices (e.g., Miller & Waller, 2003). Application of sophisticated capital budgeting is
more complex, and required the firms to be able to expend cost, time and effort (Busby & Pitts, 1997; Miller & Waller,
2003). Thus, it is important to think about the appropriate level use of sophisticated capital budgeting practices to the net
benefits against costs. Anyhow, theory, in contrast, suggests that if uncertainty exists, use of sophisticated capital
budgeting practices is valuable and the costs would be offset by the gains from successful investments (Verbeeten, 2006).
If uncertainty exists, additional information needed to solve the problem of investment dilemma (Miller & Waller, 2003).
It was identified that Canadian firms seem to be increasingly using sophisticated methods when dealing with risk (for
example, sensitivity analysis, decision-tree analysis, Monte Carlo simulation, ROR, GT) (Bennouna et al., 2010).
Nowadays, there are number of other methods including the project-dependent (risk-adjusted) cost of capital (PDCC), the
weighted average cost of capital (WACC), the cost of debt (CD) used in capital budgeting practices. Among them PDCC
and WACC are said to be sophisticated method and CD is the least sophisticated method (Hermes et al., 2007).
3.4 Capital budgeting tools for incorporating risk
Overall, uncertainty affects future cash flows and causes estimation difficulties. Therefore, various risk analysis and
management science techniques have been developed to supplement the traditional present value based decision models.
Scholarship on the practice of capital budgeting in many countries has found that firms are increasingly employing
more sophisticated capital budgeting techniques in order to make investment decisions over several years (Klammer,
1973; Klammer & Walker,1984; Pike,1988; Jog & Srivastava,1995; Gilbert & Reichart,1995; Farragher, Kleiman &
Sahu,1999; Arnold & Hatzopoulos, 2000; Brounen et al.,2004; Truong et al., 2008; Baker, Dutta & Saadi,2011). In the
contemporary world, there are a number of sophisticated capital budgeting methods including the oft-cited: Monte
Carlo Simulations, Game theory decision rules , Real option pricing, Using certainty equivalents, Decision trees, CAPM
analysis / ß analysis, Adjusting expected values, Sensitivity analysis/break-even analysis, Scenario analysis, Adaptation
of required return/discount rate, IRR, NPV, uncertainty absorption in cash flows, and PB (e.g., Arnold & Hatzopoulos,
2000; Hall, 2000; Graham & Harvey, 2001; Ryan & Ryan, 2002; Murto & Keppo, 2002; Cooper et al., 2002; Smit, 2003;
Sandahl & Sjogren, 2003; Brounen et al., 2004; Lazaridis, 2004; Lord, Shanahan & Bogd, 2004; du Toit & Pie naar,
2005;Verbeeten, 2006; Elumilade, Asaolu & Ologunde, 2006; Hermes et al., 2007; Leon et al., 2008; Correia &
Cramer, 2008; Verma et al., 2009; Bennouna et al., 2010; Shinoda, 2010; Hall & Millard, 2010; Dragota et al, 2010;
Poudel et al., 2009; Kester & Robbins, 2011; Maroyi & Poll, 2012; Singh et al., 2012; Andres et al., 2015). Thus, the
complex models of capital budgeting practices are dependent on not only the use of DCF techniques, but also proper
cash flows, discount rates and the risk analysis (Brigham & Ehrhardt, 2002).
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3.5Classification of Capital budgeting Practices
Capital budgeting practices help managers to select n out of N investment projects with the highest profits and an
acceptable ‗risk of ruin‘ (Verbeeten, 2006, p.108). By and large, all capital budgeting practices can be subsumed into the
categories of sophisticated, advanced and naive (e.g., Haka, 1987; Haka, Gordon & Pinches, 1985; Verbeeten, 2006;
Wolffsen, 2012). Naive practices includes PB, the adaptation of required payback and ARR, and the advanced /NPV
based, including Sensitivity analysis/break-even analysis, scenario analysis, the adaptation of required return/discount
rate, IRR, NPV, uncertainty absorption in cash flows, MIRR and PI. Farragher et al. (2001) suggested that a degree of
sophistication is represented by the use of DCF techniques and incorporating risk into the analysis. Sophisticated capital
budgeting methods generally include Monte Carlo simulations, GT, RO, using certainty equivalents, decision trees,
CAPM analysis / ß analysis, and adjusting expected values (Verbeeten, 2006; Wolffsen, 2012).
3.6 Capital budgeting theory and practices in developed countries
This section clearly discusses the capital budgeting theory and practices especially in developed countries. As
aforementioned, the capital budgeting practices are the investment decision taken for increasing shareholders value
(Dayananda et al., 2002).
Many studies have been conducted about capital budgeting practices in U.S. and Europe (e.g., Pike, 1996; Sangster,
1993; Block, 2007; Herme et al., 2007). Chadwell-Hatfield et al.(1997) conducted a survey among 118 manufacturing
firms in the U.S. Results showed that NPV (84%) and IRR (70%) were preferred primary methods. However, it was
clearly observed that two thirds of firms relied on shorter PB periods rather IRR or NPV. A seminal study carried out by
Graham and Harvey (2001) about ‗the theory and practice of corporate finance: evidence from the field‘ and the sample
consisted of 392 CFOs in the USA. In larger firms with high debt ratio, CFOs with MBA were more likely to use DCF
(75% NPV and IRR) than their counterparts. Larger firms applied risk-adjusted discount rate whereas small firms opted
for Monte Carlo simulation for adjusting risk. In addition, their findings further argued that PB method has not used
as a primary tool, however, it kept as a vital secondary tool. Very similar results were reported in Ryan and Ryan‘s
(2002) study where sample consisted of Fortune 1000 companies. Results were found that NPV was most popular
technique, followed by IRR. Most of the firms used sensitivity analysis, scenario analysis, inflation adjusted cash flows,
economic value added, and incremental IRR along with NPV and IRR. Block (1997) studied about capital budgeting
techniques across small business firms operating in the United States. The most popular method was the PB (42.7%),
followed by ARR (22.4%). Notwithstanding, researchers connotes that small business owners seemed to be increasingly
using DCF as the primary method for evaluating.
Cooper et al. (2002) studied capital budgeting practices in fortune 500 companies in America. Sample consisted of 102
chief financial officers reported that commonly used primary capital budgeting model is the IRR and the second is the
payback. Ken and Cherukuri (1991) found that IRR was mostly preferred method in larger companies operating in the
U.S. NPV was the next preferred method. The widely used discount rate was the WACC (78%) and the risk was
commonly measured by sensitivity analysis (80%).Almost similar results were reported in the survey of Fortune 100
firms by Bierman in 1993.
Arnold and Hatzopoulos (2000) conducted a study on "The gap between theory and practice in Capital Budgeting:
Evidence from the UK for 300 UK companies (comprising 100 large, 100 medium and small 100). Results of study
indicate that UK companies have increasingly adopted the analysis of financial textbooks prescribed. Stage has been
reached in which only a small minority do not make use of discounted cash flows, formal risk analysis, adjustment
corresponding inflation and post-audit in their study. Study reported however, managers still using simple rules of
thumb techniques in UK
Jog and Srivastava (1995) conducted a survey of capital budgeting practices in Corporate Canada and the results
showed that the most preferred method was the PB. Similar results were found in the UK in Pike‘s (1996) study. Further
results indicated that decreased use of ARR in Canada and the United Kingdom, respectively. It was identified that
Canadian firms seem to be increasingly using sophisticated methods when dealing with risk (for example, sensitivity
analysis, decision-tree analysis, Monte Carlo simulation, ROR, GT) (Bennouna et al., 2010).
Drury, Braund and Tayles (1993) surveyed 300 manufacturing companies in the UK about their capital budgeting
practices. Results showed that PB (86%) and IRR (80%) were mostly preferred methods across the sample. The widely
used risk analysis was the sensitivity analysis. In a seminal study of Brounen et al. (2004), four European countries
viz., U.K., France, Germany and the Netherlands consisting of 313 companies during 2002 and 2003 were examined.
Their result showed that 47% and 67% of the UK companies were used NPV and PB respectively as a primary tool for
evaluating capital budgeting decision whereas companies in Netherlands were used 70% of NPV and 65% of PB
methods. However, companies in France and Germany reported lower usages of both methods (42% for NPV, 50 % for
PB and 44% for NPV, 51 % for PB respectively). Previous studies have mainly conducted in the U.S. and the UK and
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limited number of studies are also available for the Netherlands (e.g., Herst, Poirters & Spekreijse, 1997; Brounen et al.,
2004).
Many researches recognized that DCF is the dominant in capital budgeting evaluation methods in the UK (e.g., Arnold
& Hatzopoulos, 2000), the USA (e.g., Ryan & Ryan, 2002) and in Canada (e.g., Payne et al., 1999). However, most of
the US firms use DCF techniques in comparison with firms in European countries (e.g., Brounen et al., 2004). There is
still some reluctance in this field due to the technical aspects of DCF (e.g., Cary, 2008; Magni, 2009). In 1993, Bierman
and Smidt opined that the DCF methods are the pre-eminent investment decision tool and thus, it is imperative to
manager to learn about its uses. Anyhow, NPV, IRR and PB are the most popular methods among North American and
Western European companies (Graham & Harvey, 2001; Brounen et al., 2004).
Sekwat (1999) studied capital budgeting practices among 321 Tennessee municipal governments. His results showed
that most of the municipal government‘s organizations are using benefit cost ratio (62.5 %) and payback methods
(61.5%), and financial officers were in reluctant using IRR, ARR and even NPV methods. Holmen (2005) conducted a
survey of capital budgeting techniques, used for FDI‘s by Swedish firms and found that larger firms were preferred to
use NPV and IRR methods. However, the most preferred method was the PB (79%). In a survey of capital budgeting
practices of Australian listed companies, Truong et al., 2008 found that NPV, IRR and PB were the most popular capital
budgeting evaluation methods. Researchers were also identified the use of real option across the sample but not yet part
of the mainstream.
In 2009, Kester and Robbins surveyed about capital budgeting techniques used by Irish listed companies. Results
revealed that they use DCF methods and reported that most prevalent method was NPV, followed by PB, and IRR.
Scenario analysis and sensitivity analyses were found to be most important tools for incorporating risk. WACC was the
most important widespread method employed for calculating discount rate. On the other hand, Lazaridis (2004) studied
capital budgeting practices in Cyprus. The PB was found as the most preferred method and not NPV.
Shinoda (2010) carried out a survey of capital budgeting in Japan. Questionnaire has been administered to collect data
from a sample of 225 companies listed on Tokyo Stock Exchange. Results showed that firms were using combination of
PB and NPV for evaluating capital investment projects.
In summary, many studies have found that increasing use of sophisticated capital budgeting techniques among many
developed countries: US, UK, European and Australian companies (Freeman & Hobbes, 1991;Shao & Shao, 1996;
Pike, 1996; Herst, Poirters & Spekreijse, 1997; Brounen et al., 2004 ; Truong et al., 2008). However, US companies
seem to be using more DCF methods as compared to European countries.
3.7 Capital budgeting theory and practices in developing countries
There is dearth of studies carried out on capital budgeting practices in developing countries during the last two decades.
In comparison with developed countries, the results of the most studies show a different picture. In most of the
developing countries, PB method was the dominant methods in evaluating capital investment. Kester et al.(1999)
surveyed a total of 226 companies across six countries: Australia, Hong Kong, Indonesia, Malaysia, Philippines and
Singapore. Results showed that PB is still important method and the DCF methods have become increasingly important.
In five Asian countries, 95% of firms used PB method and 88% of them use NPV in evaluating projects. However, both
methods were treated as equally important. Kester et al. (1999) noted that sophistication of capital budgeting techniques
within the developing countries in Asia has been increased very rapidly during the last decade.
Babu and Sharma (1996) studied Indian industries‘ capital budgeting practices and the findings showed that 90% of the
companies were using capital budgeting methods. Of them 75% of companies reported that they were adopting DCF
methods in evaluating capital budgeting, among them IRR was most popular. Sensitivity analysis was found to be
popular in assessing risk. In 1998, Jain and Kumar studied about comparative capital budgeting practices: the Indian
context and sampled 96 nongovernment companies where listed in Bombay Stock Exchange and five companies of
South East Asia. They observed that most preferred capital budgeting techniques was the PB (80% companies),
followed by NPV and IRR. Sensitivity analysis was the preferred risk assessment method.
Cherukuri (1996) surveyed about capital budgeting practices: a comparative study of India and select South East Asian
countries,‖ with those of Hong Kong, Malaysia and Singapore and a sample consisted of top 300 non-government
companies. This study found that of DCF methods, 51% of companies used IRR, followed by NPV (30%). Of non DCF
methods,PB (38%) is the dominant method and the next widely used method was ARR (19%). The non DCF methods
were used as supplement to DCF methods. WACC is the widely used discount rate and Sensitivity analysis was mainly
used for risk assessment. A recent survey of capital budgeting Practices in corporate India, conducted by Verma et
al.(2009), took a sample of 30 manufacturing companies in India. The results confirmed findings of Cherukuri
(1996).This study showed that most preferred method is IRR (56.7%), followed by NPV (50%) and PB (36.7%).WACC
(43.3%) is the widely used discount rate and Sensitivity analysis (36.7%) was mainly used for risk assessment.
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Researchers further observed that increasing adoption of DCF rather traditional use of non-discounted techniques. In
2012, Singh et al. studied on capital budgeting decision sampling from 31 listed companies in India. Albeit capital
budgeting decision continued in India, all sampled firms reported that they are using DCF techniques in combining with
non-DCF techniques. Of discounted cash flow techniques, more than three quarters of the sampled companies use IRR
which more preferred than NPV that used by half of the sampled companies. Further it has been reported that half of
the companies use real option techniques in selecting their capital investment projects. Long term capital is of financing
source to finance fixed assets (net) and working capital (net) in India. Most of the variables are country specific;
researchers call for further detailed research considering sectorial analysis of the constituent sectors of the sample
companies would be shed new light on this area.
Hermes et al. (2007) carried out a comparative study of the Dutch and Chinese firms about capital budgeting practices.
66.7% of the Dutch CFOs stated that they used WACC and only 9.5 % of them used PDCC. Small firms use CD most
often (22.7%) in comparison with larger firms (5.0%). In the Dutch firms, 89% of CFOs reported that they used NPV
methods however, 2% of CFOs stated that they used the ARR which is the least popular method. In contrast, 53.3% of
Chinese firms indicated that they use WACC, and just 15.7% of CFOs of Chinese firms use PDCC. However, 28.9% of
CFOs reported that they use CD which is higher than that of the Dutch counterparts. Chinese CFOs stated that they
more likely to use NPV and PB methods (89% and 84% respectively) in evaluating capital budgeting projects. Thus, on
average, Dutch CFOs use more sophisticated capital budgeting techniques than Chinese CFOs do.
In 2008, Leon et al. conducted a survey of capital budgeting practices of listed companies in Indonesia. DCF was
mainly adopted methods in those companies as primary evaluation tool for capital investment projects. The most
prevalent risk assessment tools were scenario and sensitivity analysis. Results supported that CAPM was not so popular
Recently, a survey of capital budgeting practices have been conducted by Khamees, Al-Fayoumi, and Al-Thuneibat
(2010) in Jordan. Results reported that both DCF and non DCF method were still popular in evaluating capital
budgeting investment. Surprisingly, the most popular method was PI, followed by PB.
Most recently, Maroyi and Poll (2012) conducted a survey of capital budgeting practices in listed mining companies in
South Africa. Results showed that NPV, IRR and PB were the most prevalent methods in evaluating larger investment
projects. Results further indicated that PB was found to be continual use of method. Following table summarizes the
key findings on capital budgeting literature
Table 1. Key findings on capital budgeting studies during last two decades (from 1993 to 2013)
Author/s Population Most popular capital
budgeting method
Methods for evaluating
risk in Capital Budgeting
Drury, Braund & Tayles
(1993)
300 UK
Manufacturing
companies
PBP and IRR Sensitivity analysis.
Babu & Sharma (1995) 73 Indian
companies
DCF Methods Sensitivity analysis and
adjustment of discount
rate methods
Jog & Srivastava
(1995)
582 Canadian
companies
IRR and PBP Sensitivity analysis
Pike (1996) Large UK
companies
PBP
Kester & Chang (1996) 54 companies IRR and PBP Scenario and sensitivity
analysis
Farragher, Kleiman &
Sahu (1999)
379 US
companies in the
Standard & Poor‘s
industrial index
DCF Methods : NPV Capital Assets Pricing
Model
Sekwat (1999) 166 Finance
Officers of
Municipal
Governments
Cost-Benefit Ratio and
PBP
Applied Economics and Finance Vol. 3, No. 2; 2016
26
(Tennessee)
Kester , Chang,
Echanis, Haikal, . Isa,
Skully, Tsui, & , Wang
(1999)
226 companies in
Australia, Hong
Kong, Indonesia,
Malaysia, The
Philippines and
Singapore in 1996-
1997
Equal importance to
discounted and
non-discounted cash flow
techniques in evaluating
projects
Scenario analysis and
sensitivity analysis
Arnold & Hatzopoulos
(2000)
300 UK Companies DCF is widely using by
the selected UK firms.
Hall (2000) 65 Respondents
(South Africa)
IRR
Graham & Harvey
(2001)
392 Chief
CFOs of
companies in the
U.S.
NPV and IRR Large firms- risk adjusted
discount rate Small firms-
Monte Carlo Simulation
Ryan & Ryan (2002) 205US Companies NPV and IRR Sensitivity analysis,
Scenario analysis,
inflation adjusted cash
flows, economic value
added, and incremental
IRR
Sandahl & Sjogren
(2003)
129 Swedish
Corporations
PBP Annuity
Lord, Shanahan & Boyd
(2004)
29 Local authorities
of New Zealand
Local Government
Cost Benefit Ratio
Brounen, deJong &
Koedijk (2004)
Four European
countries viz.,
U.K., France,
Germany and the
Netherlands
consisting of 313
companies during
2002 and 2003
Primary tools were in UK
– NPV and PBP, in
Netherland – NPV and
PBP , France and
Germany reported lower
usages of both methods
(42% for NPV, 50 % for
PB and 44% for NPV,
51 % for PB
respectively).
Lazaridis (2004) Small Medium
Sized Companies
(Cyprus)
PBP Statistical Risk Analysis,
Scenario Analysis
Elumilade, Asaolu &
Ologunde (2006)
94 firms from
Nigerian stock
exchange (Nigeria)
PBP, ARR , and NPV Linear programming
Lam, Wang & Lam
(2007)
157 Hong Kong
Building
Contractors
PBP and Average
Accounting Rate of
Return
Shortening Payback
Period, Raising Required
Rate of Return
Dedi & Orsag (2007) 200 firms
selected from 400
of the best Croatian
firms & 34 banks
IRR, PBP (cost of capital
is calculated by WACC)
Risk-adjusted discount
rate, Certainty equivalents
for cash flows
Applied Economics and Finance Vol. 3, No. 2; 2016
27
from a ranking of
Croatian
banks
Truong, Partington &
Peat (2008)
87 Australian
companies
NPV, IRR and PBP Real options techniques
have gained a foothold in
capital budgeting but are
not yet part of the
mainstream. Capital
Assets Pricing Model is
found to be the most
popular method used in
the estimation of the cost
of equity capital
Leon, Isa & Kester
(2008)
229 Listed
Companies
(Indonesia)
DCF Techniques Scenario and Sensitivity
Analysis
Zubairi (2008) 35 firms listed on
KSE (Pakistan)
Bigger size companies
give greater preference to
IRR, while smaller firms
rely more on NPV.
Also smaller firms are
keener in estimating the
PBP as compared to larger
companies.
Verma, Gupta & Batra
(2009)
100 manufacturing
companies (India)
NPV and IRR Weighted Average Cost of
Capital (WACC) was to
calculate the Cost of
Capital. Sensitivity‘
analysis
Hall & Millard (2010) South African
industrial
companies listed
on the JSE
Securities
Exchange for at
Least ten years.
IRR
Dragota. Tatu, ,Pele,
Vintila, & Semenescu
(2010)
Professors in the
economic field,
having
competences in
Corporate Finance
and teaching in
Romania.
NPV, IRR or PI, Discount
Rate used for the
investment projects
analysis is the weighted
average cost of capital.
Sensitivity Analysis ,
Monte Carlo Method and
the Scenarios Technique
Shinoda (2010) 225 firms listed on
the Tokyo Stock
Exchange
PBP and NPV
Poudel, Sugimoto,
Yamamoto, Nishiwaki ,
& Kano (2010)
50 Farms (Nepal) Benefit-Cost ratio (B/C),
NPV, IRR, and PBP
Sensitivity Analysis
Bennouna,Meredith &
Marchant (2010)
88 Large Firms Trends towards
sophisticated techniques
(DCF) have continued. Of
those which did, the
The majority of Canadian
firms use risk analysis
tools mainly sensitivity
analysis followed by
Applied Economics and Finance Vol. 3, No. 2; 2016
28
Source: Survey data
3.7 Factor affecting capital budgeting
In practice, there are numerous factors that heavily influence on capital budgeting decision. Behavioral finance become
increasingly important and intrudes into capital budgeting theory, and the knowledge on behavioral finance derived
from sociology and psychology. The behavioral finance states that capital investment decision is not solely dependent
on quantitative data, but the decision is also strongly influenced by qualitative data including institution and personal
values, tolerance to risk, situational context and so on. More recently, Ben-David, Graham, and Harvey‘s (2008) study
of CFOs found that overconfidence was a key driver of investment, however optimism found to be more marginal effect
on investment. Larrick, Burson, and Soll (2007) found that the degree of individuals overconfident is strongly
associated with their thinking that make them to feel that they are better than average. Overconfident managers
generally prefer to overinvest and the overconfident tends to attract more mergers, starting new firms and initiate more
investment. Similarly, Brown and Sarma (2007) stated that CEO overconfidence affect the frequency of corporate
acquisitions of a firm. If past returns on investment are high, CFOs would become more confident on their estimate o f
future returns. A group of 55 managers working in small firms of computer industry have been studied by Simon and
Houghton (2003). Findings showed that managers with greater overconfidence would prefer to introduce more risky
products and seem to fail many times. In early 1990s, some studies found that managerial overconfidence tends to
innovation (Staw, 1991) and to plant expansion (Nutt, 1993). Glaser, Schafers, and Weber (2008) surveyed senior
managers behavior and they observed that when managers are optimistic, they increase their exposure to firm specific
risk when transaction on invest more and in turn increase investment cash flow sensitivity.
Size of the firm is one of the major determinants in capital budgeting practices (e.g., Ho & Pike, 1992; Graham &
Harvey, 2001; Farragher et al., 2001; Brounen et al., 2004; Verbeeten, 2006). Researches supported that large firms
adopts more innovative capital budgeting methods, say, sophisticated capital budgeting practices, to a large extent than
smaller firms do (e.g., Rogers, 1995; Williams & Seaman, 2001) since the larger firms have the capacity and resources
to use sophisticated capital budgeting practices (Ho & Pike, 1992). Payne et al.(1999) and Ryan and Ryan (2002)
documented that large firms were more inclined to use more sophisticated capital budgeting practices. This is due to the
larger firms involves larger projects and the use of sophisticated capital budgeting practices become less costly (Payne
et al., 1999; Hermes et al., 2007).There was a positive relationship between firm size and the use of DCF methods.
Findings have also been confirmed in Hermes et al.‘s (2007) studies. Trahan and Gitman (1995) connotes that large
companies exploited DCF methods (88 % for NPV and 91 % IRR) than small companies (65% for NPV and 54% for
IRR). It was further confirmed in Segelod‘s (1998) study and he found that major firms uses PB model for evaluating
small investments, however, for the large investment decision at least of the DCF methods is in practice. In 2001,
Graham and Harvey studied about capital budgeting methods and firm size in the U.S. and results showed that there is a
significant negative relationship between size and PB. Brounen et al.(2004) found that company size was positively
correlated with the use of capital budgeting methods, large companies use NPV, IRR, and sensitivity analysis more than
small companies.
majority favored NPV and
IRR.
scenario analysis and
risk-adjusted discount
rate. Use of real options is
limited (8%).
Kester & Robbins
(2011)
18 Chief Financial
Officers of
companies listed
on the Irish Stock
Exchange
More Sophisticated
Discounted Cash Flow
Techniques.
(Weighted Average Cost
of Capital is to evaluate
all proposed capital
investments).
Scenario Analysis and
Sensitivity Analysis
Singh, Jain & Yadav
(2012)
31 listed
Companies (India)
More sophisticated DCF
techniques.
Sensitivity analysis
Maroyi & Pol l (2012) 13 Companies
Listed in the
Mining Sector of
the Johannesburg
Securities
Exchange (JSE).
NPV Real option
Applied Economics and Finance Vol. 3, No. 2; 2016
29
Generally, ownership structure has greater influence on any managerial decision making and resultant effect on firm‘s
performance (Warfield,Wild & Wild, 1995; Klassen, 1997). Greater managerial ownership has been identified to be
increased use of recommended capital budgeting methods and thus less likely to experience financial distress (Donker,
Santen & Zahir, 2009). It is oft-reported that what managers actually do they ignore profitability investment (even if it
offers positive NPV), if accounting rate of return is too low, and thus top management willed to sacrifice long term
value to meet accounting targets (Graham, Harvey & Rajgopal, 2005).The ownership sometime classify as listed at the
stock exchange or non listed (Hermes et al., 2007). Listed firms were used accurate estimation of cost of equity ,and
cost of capital and more likely to NPV or IRR than non listed (Hermes et al., 2007).
Nature of the industries were also identified as the determinant of capital budgeting practices, for example financial
services industry and the building, construction and utilities industries, have been interest of using more sophisticated
capital budgeting practices than other industries (Verbeeten, 2006). Further, many empirical researches in the past
showed that capital budgeting practices are different across industries (e.g., Ho & Pike, 1998). For example,
widespread use of real option or game theory are more prevalent in the pharmaceutical industry (e.g., Bowman &
Moskowitz, 2001; McGrath & Nerkar, 2004), the extraction industry (e.g., Trigeorgis, 1993), and the financial services
industry and the high-tech industry (e.g., Billington, Johnson & Triantis, 2003).
Education of CFOs was recognized as the determinant of capital budgeting. There was a general argument that CFO
with higher education has fewer problems in understanding more sophisticated capital budgeting techniques and they
thus have the capacity to use them. For example, in Chinese firms, CFOs with higher level of education use cost of debt
less often in comparison with less educated CFOs. Thus, a positive relationship identified between educational
background of CFOs and the use of sophisticated methods (Hermes et al., 2007). Among the U.S. sample, there was a
positive association has been found between CEO education and use of IRR (Graham & Harvey, 2001) and the findings
has been confirmed in the Netherlands, Germany and France, but not in the UK (Brounen et al., 2004). The reasons for
more widespread use of DCF are the availability of computer software that used in computation (e.g., Pike, 1996) and
increased level of formal education of managers (e.g., Pike, 1996; Sangster, 1993). A few studies found that age of the
CFOs was also a determinant of capital budgeting methods. For example, older CFOs could be reluctant to adopt new
techniques, and instead prefer to relaying on older methods (e.g., Hermes et al., 2007).
Since capital investment involves in long term, uncertainty /risk would play a vital role in capital investment decision
making. Generally, uncertainty refers to as the gap between information available and information required to make any
decision. Complete information is unavailable in long run and thus, uncertainty is the dominant factor in capital
investment (Simerly & Li, 2000; Zhu & Weyant, 2003). Nature and type of uncertainty could be, including raw material
uncertainties, input market uncertainties, labor uncertainties, political uncertainties, production uncertainties, output
market uncertainties, liability uncertainties, interest uncertainties, inflation uncertainties, policy uncertainties, exchange
rate uncertainties, competitive uncertainties and society uncertainties. Uncertainties have been treated with adopting
sophisticated capital budgeting practices, for example, use of ROR and/or GT tools (e.g., Bowman & Hurry, 1993; Zhu
& Weyant, 2003). The main concepts of the ROR demonstrates that specific uncertainties (rather than in general) that
would affect capital budgeting practices (Dixit & Pindyck, 1994). Game theory specifies that the optimal investment
criterion can also be changed by specific uncertainties (Smit, 2003). Thus, specific uncertainties need to be tackled with
using different capital budgeting methods. The research findings supported that sophisticated capital budgeting practices
are crucial and useful if financial uncertainties i.e., exchange rate, interest exist. However, social uncertainties, market
uncertainties, and input uncertainties have not sufficiently supported to influence on use of sophisticated capital
budgeting practices. Rather, theoretical background, many experts in capital budgeting area is expected to offer the
capacity and willingness to adopt contemporary capital budgeting practices (e.g., Libby & Waterhouse, 1996, Williams
& Seaman, 2001). Theory and a few empirical research states that specific uncertainties affect capital budgeting
practices, for example, Ho and Pike (1998) found that there is a positive relationship between socioeconomic
uncertainty (i.e., governmental regulations, trade unions actions) and the application of risk analysis techniques,
however, the empirical evidence on these relationship with sophisticated capital budgeting practices are scarce
(Verbeeten, 2006).
Recognition, assessment and reflection of the risk/uncertainty are intriguing. Nowadays, there are number of risk
analysis method available such as sensitivity analysis, scenario analysis, decision trees, computer simulation and Monte
Carlo analysis. In Graham and Harvey‘s (2001) study, participants recognized market risk and they also reported other
risk factors including interest rate, inflation, size, foreign exchange rate. Surprisingly, they found that at least half of the
firm did nothing to adjust WACC (firm‘s average risk) to incorporate project risk. However, in 1996, Shao and Shao
reported that firms employed more on risk adjusted cash flows than risk-adjusted discount rates. Across their sample,
they found that sensitivity analysis was the principal assessment technique. In contrast, Gitman and Vandenberg (2000)
found in their study that 39 % of firms were adjusting their rates against adjusting risk for cash flows. Through there are
number of sophisticated risk analysis models available, the applicability of those models were prone to barriers. The
Applied Economics and Finance Vol. 3, No. 2; 2016
30
reasons for their reluctant have been reported as; it is not practical, depending on unrealistic assumption, difficulties in
explaining to the top management and the difficulties in applying (Trahan & Gitman, 1995). Notwithstanding progress
in risk identification, assessment and adjustment has been reported, none of the studies have not been looked at actual
risk analysis, its process and management inputs to improve or usage of existing risk assessment and adjustment models.
Sophisticated capital budgeting practices would help to identify many different types of investment projects in terms of
uncertainty.A range of risk across the many investment projects would create diversification. Diversification generally
helps to maximize the income from investments at minimum risk. A positive relationship has been found between
diversification and use of sophisticated capital budgeting practices (Verbeeten, 2006). Recently, Holmen and Pramborg
(2009) reported that the use of payback method has been positively combined with political risk.
Klammer (1993), and Shank and Govindarajan (1992) suggested that nonfinancial consideration have been integrated
into capital budgeting practices. For example, corporate management integrated into capital budgeting and thus the
decision depends on some of the strategic management tools such as value chain analysis, cost drivers analysis, and
completive advantage analysis. According to Carr and Tomkins (1996), the most successful companies were found to
be using nonfinancial strategic information in making investment decision among their sample of 51 case studies in the
UK, the U.S., and the German companies. However, it is argued that nonfinancial methods were prevalence when the
firms did not adequately implement DCF methods (Carr &Tomkins 1996). However, any studies have not been carried
out the use of non financial methods linking to DCF analysis. It has been argued that increasing acceptance of DCF
analysis ignores the use of nonfinancial methods (e.g., Graham & Harvey 2001; Ryan &d Ryan, 2002).
Capital budgeting practices are different and may have ―country effect‖ influence. This can be attributed to the some
level of economic factors that determine choice of capital budgeting practices. It is recommend furthering research in
indentifying country effect on capital budgeting practices with respect to the level of economic, human, financial and
technological improvement. Shahrokh (2002) argued that capital budgeting is very complex, determined by many
factors including: terminal values, foreign currency fluctuations, long-term inflation rates, subsidized financing, and
Political risk. In Sekwat‘s (1999) study of capital budgeting practices in Tennessee municipal governments, the decision
in using capital budgeting techniques are based on simple, versatile and flexibility of those techniques. Notwithstanding,
he further argued that the usage of techniques in practices is in conjunction with qualitative factors such as ethical, legal,
or political considerations. He concluded that since government funds the capital projects, political factors plays a
critical role in making capital investment decisions.
3.8 Disparities between capital budgeting theory and practices
Capital budgeting theory recommends in using DCF methods (NPV, IRR, MIRR, PI and DPB) and non DFC methods
(PB and ARR) for making capital budgeting decision. However, all most all the firms in developed and developing
countries inclined to use sophisticated capital budgeting methods along with many capital budgeting tools for
incorporating risk (i.e., sensitivity analysis, real options) and sophisticated discounted rate (i.e., Weighted Average Cost
of Capital, Cost of Debt, CAPM) (e.g., Arnold & Hatzopoulos, 2000; Graham & Harvey, 2001; Ryan & Ryan, 2002;
Cooper et al., 2002; Brounen et al., 2004; Hermes et al.,2007; Bennouna et al., 2010; Maquieira , Preve and Allende,
2012).
Nemours factors have been identified as the determinant of capital budgeting during the last two decades including size
of the firm, ownership structure, nature of industries, educational qualification of CFOs, experience of CFOs, age of
CFOs, uncertainty(for example, interest rate, inflation, foreign exchange rate), nonfinancial consideration and other
factors (i.e, economic, human, technology, finance, ethical and political). Among them, some factors (for example, size
of the firm, educational qualification of CFOs, experience of CFOs, age of CFOs) were positively associated with the
use of sophisticated capital budgeting practices. However, in some cases, economic, political and technological factors
directly and indirectly affect choice of the capital budgeting practices. (e.g., Bowman & Moskowitz, 2001; Zhu &
Weyant, 2003; McGrath & Nerkar, 2004; Verbeeten, 2006; Donker et al., 2009). Moreover, the factors determining
capital budgeting practice connotes that to certain extent capital budgeting practice prone to ‗country effect infl uence‘,
for example economic factor, cutting edge technology (i.e., decision support system), political factors, accounting
policies, accounting standards and other infrastructure facilities. Although capital budgeting theory was applicable
regardless of countries, to certain extent the actual practices of capital budgeting (for example selection of capital
investment) vary (e.g., Graham & Harvey, 2001; Shahrokh ,2002).‗In practice uncertainty, information asymmetry,
multiple (conflicting) objectives, real options and multi -period multi project considerations greatly complicate capital
budgeting beyond the focus of the theory‘ (Arnold & Hatzopoulos, 2000, p.609). A consideration of the impact of
information asymmetry, real options and other complications on the capital budgeting exercise gives one the view that
there is no unique correct technique and that there is a need for multiple methods (Arnold & Hatzopoulos, 2000). Thus,
all these factors impinge on choice of the capital budgeting practices, and consequently, there are disparities between
theory and practices.
Applied Economics and Finance Vol. 3, No. 2; 2016
31
Studies on the practice of capital budgeting in many countries have found that firms increasingly employ more
sophisticated capital budgeting techniques to make investment decisions over several years (Klammer, 1973; Klammer
& Walker, 1984; Pike, 1988; Klammer, Koch & Wilner, 1991; Jog & Srivastava, 1995; Gilbert & Reichart, 1995;
Farragher et al., 1999;Arnold & Hatzopoulos, 2000; Graham & Harvey, 2001; Mustapha & Mooi, 2001; Ryan & Ryan,
2002; Brounen et al., 2004; Hermes et al., 2007; Truong et al., 2008; Baker et al., 2011; Singh et al., 2012). When
comparing a developed economy with an emerging economy, the developed economy has highly developed capital
markets with high levels of liquidity, meaningful regulatory bodies, large market capitalization, and high levels of per
capita income (Geary, 2012). An emerging market is in the process of rapid growth and development with lower per
capita income, less mature capital markets and very small capital projects, compared with developed countries. Therefore,
obviously, emerging market economies pose challenges in applying capital budgeting techniques, owing to less
developed capital markets and the difficulty of setting key parameters.
3.9 Answering to the research questions: Summary of the findings
It is crucial to answer the research questions in order to attain research aims. The first question enquired about ―what are
the capital budgeting theories and practices used by firms? Are there any disparities between the capital budgeting
theories and practices? If so how?‖ The answers for these questions have been well documented during the last twenty
years of studies. Capital budgeting theory recommends in using DCF methods (NPV, IRR, MIRR, and DPB) and non
DFC methods (PB and ARR) for making capital budgeting decision. However, all most all the firms in developed and
developing countries inclined to use sophisticated capital budgeting methods along with many capital budgeting tools
for incorporating risk (i.e., sensitivity analysis, real options) and sophisticated discounted rate (i.e., WACC, CD, CAPM)
(e.g., Arnold & Hatzopoulos, 2000; Graham & Harvey, 2001; Ryan & Ryan, 2002; Cooper et al., 2002; Brounen et al.,
2004; Hermes et al., 2007; Bennouna et al., 2010; Maquieira et al., 2012). Thus it can be concluded that there are
some disparities between capital budgeting theory and practice. The next research‘s question further backs up to this
question.
The second question asked about ―what are the factors determines the use of capital budgeting practices? Are there
different across countries? If so how?‖ Nemours factors have been identified as the determinant of capital budgeting
during the last two decades including size of the firm, ownership structure, nature of industries, educational
qualification of CFOs, experience of CFOs, age of CFOs, uncertainty(for example, interest rate, inflation, foreign
exchange rate), nonfinancial consideration and other factors (i.e, economic, human, technology, finance, ethical and
political). Among them, some factors (for example, size of the firm, educational qualification of CFOs, experience of
CFOs, age of CFOs) were positively associated with the use of sophisticated capital budgeting practices. However, in
some cases, economic, political and technological factors directly and indirectly affect choice of the capital budgeting
practices. (e.g., Bowman & Moskowitz, 2001; Zhu & Weyant, 2003; McGrath & Nerkar, 2004; Verbeeten, 2006;
Donker et al.,2009). Moreover, the factors determining capital budgeting practice connotes that to certain extent capital
budgeting practice prone to ―country effect influence‖, for example economic factor, cutting edge technology (i.e.,
decision support system), political factors, accounting policies, accounting standards and other infrastructure facilities.
Although capital budgeting theory was applicable regardless of countries, to certain extent the actual practices of capital
budgeting (for example selection of capital investment) vary (e.g., Graham & Harvey, 2001; Shahrokh , 2002). Thus, all
these factors impinge on choice of the capital budgeting practices, and consequently, there are disparities between
theory and practices.
The last question asked about ―what are the gaps in the existing capital budgeting literature?‖ Traditional financial
theory suggests that the decision makers are rational, however, modern theory suggests that decision have influenced by
many cognitive illusions (Leon et al., 2008; Tayib & Hussin, 2011). Thus behavioral finance came into play in capital
budgeting decision making. Capital budgeting research connected with behavioral finance have not been studied any
developing countries during the last twenty years. Literature says behavioral finance is a dominant theory determining
capital budgeting decision, confirmed in many studies carried out in developed countries. Thus, there is a complete
dearth of research in Asian studies in case of behavioral finance penetration on capital budgeting practices.
No studies have been attempted to identify relationship between supportive capital information system (software
products to make the required analysis easier in comparison with manual system) and capital budgeting decision
making. Thus it has been identified as a gap between information system and choice and practice of capital budgeting
(Bennouna et al., 2010). Similarly, the environment in which organization are working impact on quality decision. Thus,
researcher should concentrate on scanning organizational environment to make good investment decision rather purely
depends on financial theory. Thus it is paramount important in the current context.
Almost all the research carried out during the last two decades adopted limited methodological aspects. For example,
cross sectional research design, case study and some form of qualitative study were more popular (e.g., Butler et al.,
1993; Verbeeten, 2006; Hermes et al.,2007; Maquieira et al., 2012). However, in modern world, some form of event
Applied Economics and Finance Vol. 3, No. 2; 2016
32
study methodology would be seminal for providing greater insights into capital budgeting practices. Thus, a gap has
been identified in use of methodological concepts.
Renowned researchers found that nowadays most of the large companies are inclined to use sophisticated capital
budgeting practices. However, it is intriguing question whether SCBP are important to all types of investment (e.g.
expansion, replacements, mergers and takeovers) and all type of industries, and those techniques outperform than non
SCBP. Thus, these conundrums need to be well investigated.
Many research scholars have argued that capital budgeting influenced by ―country effect influence‖ (e.g., Graham &
Harvey, 2001; Shahrokh, 2002; Hermes et al., 2007), for example, economic policies, taxation system, accounting
policies, conductive social climate, culture of people, technological factor (i.e., decision support system), government
control, political factors, infrastructure facilities. Therefore, more extensive studies are imperative from unsearched
countries to build robust knowledge.
Many studies conducted in developed counties have found that firms use more sophisticated capital budgeting practices
(Graham & Harvey, 2001; Brounen et al.,2004). Nonetheless, when comparing with developed countries, more
sophisticated capital budgeting practices are not prevalent in developing countries. Thus, future research scholars need
to consider the challenges faced by CFOs with regard to the use of sophisticated capital budgeting practices (i.e.
organizational barriers/knowledge gap of CFOs, technological challenges) as they lead to increased performance.
Another opportunity for future research is the investigation of other organizational characteristics (e.g. business unit
strategies, reward and incentive structures, distribution of decision rights and financial structure) that have been shown
to affect capital budgeting practices. Renowned researchers have found that nowadays, most large companies are
inclined to use sophisticated capital budgeting practices (SCBP).
3.10 Policy recommendation
Many research scholars criticized that many researches on capital budgeting were opt-testing the methods of capital
budgeting and its practices. They were purely finding that actual what methods were in practice. However, in practice,
there are enormous factors affecting the capital budgeting practice and it has ―country effect‖ too. In line up with this
argument, this research was well thought out in its design and become springboard for future research. This study
contributed by stating the known and unknown arena of capital budgeting during the last two decades.
In the cutting edge technology world, the way of doing things have been changed and challenging. For example,
decision support system become more prevent in making decision and more advanced technological sphere penetrates
into assessing capital budgeting practices than ever before. Thus, this research would make awareness to top
management, policy makers, practitioners and stakeholders of the company.
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This work is licensed under a Creative Commons Attribution 3.0 License.
Use the following data set to answer the following questions. To earn full credit show all of your calculations and other work. Explain your answers. Don’t just write a number.
The 26 students who signed up for General Psychology reported their GPA. Each person was matched with another person on the basis of the GPAs, and two groups were formed. One group was taught with the traditional lecture method by Professor Nouveau. The other class could access the Web for the same lectures whenever they wished. At the end of the term, both classes took the same comprehensive final exam, and they also filled out a "Satisfaction Questionnaire." Scores on both measures are shown below.
Analyze the data with t tests and effect size indexes. Write a conclusion.
You can use the JASP Software to perform your analysis. Make sure you include the analysis output in your submission. Also, explain your results in detail.
|
Comprehensive Final Exam Scores |
Satisfaction Scores |
||
|
Traditional Section |
Online Section |
Traditional Section |
Online Section |
|
50 |
56 |
25 |
31 |
|
72 |
75 |
18 |
19 |
|
64 |
62 |
40 |
38 |
|
82 |
90 |
31 |
35 |
|
89 |
91 |
17 |
24 |
|
65 |
65 |
22 |
20 |
|
74 |
72 |
14 |
18 |
|
85 |
87 |
36 |
35 |
|
80 |
76 |
27 |
31 |
|
65 |
79 |
22 |
27 |
|
82 |
77 |
23 |
27 |
|
75 |
78 |
28 |
28 |
|
64 |
70 |
20 |
23 |
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