gardnerAltmanPlot
Gardner-Altman plot for two-sample effect size
Syntax
Description
gardnerAltmanPlot(___,
generates a Gardner-Altman plot for options specified using one or more of the
Name=Value
)Name=Value
arguments in combination with any of the previous
syntaxes.
returns
a graphics array H
= gardnerAltmanPlot(___)H
for the plot using any of the previous syntaxes.
Examples
Compute Mean Effect Size for Two Independent Samples
Load Fisher's iris data and define the variables for which to compare.
load fisheriris species2 = categorical(species); x = meas(species2=='setosa'); y = meas(species2=='virginica');
Compute the median difference effect size of the observations from two independent samples.
effect = meanEffectSize(x,y,Effect="mediandiff")
effect=1×2 table
Effect ConfidenceIntervals
______ ___________________
MedianDifference -1.5 -1.8259 -1.3
meanEffectSize
by default assumes independent samples (that is, Paired=false
). The function uses bootstrapping to estimate the confidence intervals when effect type is median difference.
Visualize the median difference effect size using Gardner-Altman plot.
gardnerAltmanPlot(x,y,Effect="mediandiff");
Gardner-Altman plot displays the two sample data on the left. The median of the sample Y
corresponds to the zero effect size on the effect size axis, which is the yellow axis line on the right. The median of the sample X
corresponds to the value of the effect size on the effect size axis. The plot displays the actual median difference effect size value and the confidence intervals with the vertical error bar.
Specify Bootstrap Options
Load Fisher's iris data and define the variables for which to compare.
load fisheriris species2 = categorical(species); x = meas(species2=='setosa'); y = meas(species2=='virginica');
Compute the Cohen's d effect size for the observations from two independent samples and compute the 95% confidence intervals for the effect size. meanEffectSize
by default uses the exact formula based on the noncentral t-distribution to estimate the confidence intervals when the effect type is Cohen's d. Specify the bootstrapping options:
Tell
meanEffectSize
to use bootstrapping for confidence interval computation.Use parallel computing for bootstrapping computations. You need Parallel Computing Toolbox™ for this option.
Use 3000 bootstrap replicas.
rng(123) % For reproducibility effect = meanEffectSize(x,y,Effect="cohen",ConfidenceIntervalType="bootstrap", ... BootstrapOptions=statset(UseParallel=true),NumBootstraps=3000)
Starting parallel pool (parpool) using the 'local' profile ... Connected to the parallel pool (number of workers: 6).
effect=1×2 table
Effect ConfidenceIntervals
_______ ___________________
CohensD -3.0536 -3.5611 -2.3219
Use the same options in Gardner-Altman plot to visualize the effect size.
gardnerAltmanPlot(x,y,Effect="cohen",ConfidenceIntervalType="bootstrap", ... BootstrapOptions=statset(UseParallel=true),NumBootstraps=3000);
Gardner-Altman plot displays the two sample data on the left. The mean of the sample Y
corresponds to the zero effect size on the effect size axis, which is the yellow axis line on the right. The mean of the sample X
corresponds to the value of the effect size on the effect size axis. The plot displays the Cohen's d effect size value and the confidence intervals with the vertical error bar.
Compare Paired Sample Means
Load exam grades data and define the variables for which to compare.
load examgrades
x = grades(:,1);
y = grades(:,2);
Compute the mean difference effect size of the grades from the paired samples and 95% confidence intervals for the effect size.
effect = meanEffectSize(x,y,Paired=true)
effect=1×2 table
Effect ConfidenceIntervals
________ ___________________
MeanDifference 0.016667 -1.3311 1.3644
meanEffectSize
uses the exact method to estimate the confidence intervals when you use mean difference effect size.
You can use a different effect size type (note that you can't use Glass's delta for paired samples). Use robust Cohen's d to compare the paired-sample means. Compute the 97% confidence intervals for the effect size.
effect = meanEffectSize(x,y,Paired=true,Effect="robustcohen",Alpha=0.03)
effect=1×2 table
Effect ConfidenceIntervals
________ ___________________
RobustCohensD 0.059128 -0.1405 0.26573
meanEffectSize
uses bootstrapping to estimate the confidence intervals when you use robust Cohen's d as the effect size.
Visualize the effect size using Gardner-Altman plot. Again use robust Cohen's d as the effect size and compute the 97% confidence intervals.
gardnerAltmanPlot(x,y,Paired=true,Effect="robustcohen",Alpha=0.03);
Gardner-Altman plot displays the paired data on the left. Blue lines show the values that are increasing and the red lines show the values that are decreasing from the first sample to the corresponding values in the paired sample, respectively. On the right side of the plot, you see the robust Cohen's d effect size with the 97% confidence intervals.
Input Arguments
X
— Input data
numeric vector
Input data, specified as a numeric vector.
Data Types: single
| double
Y
— Input data
numeric vector
Input data, specified as a numeric vector.
Data Types: single
| double
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Example: Effect="cliff",Alpha=0.03,ConfidenceIntervalType="bootstrap",VarianceType=unequal
specifies to use the Cliff's Delta effect size and compute the 97% confidence intervals
using bootstrapping and assume the samples come from populations with unequal
variances..
Alpha
— Confidence level
0.05 (default) | value from 0 to 1
Confidence level, specified as a numeric value from 0 to 1. Default value of Alpha
, 0.05, corresponds to 95% confidence level.
Example: Alpha=0.025
Data Types: single
| double
BootstrapOptions
— Options for bootstrap confidence interval computation in parallel
structure
Options for bootstrap confidence interval computation in parallel, specified as a structure generated by using statset('bootci')
. gardnerAltmanPlot
uses the following fields:
Field | Description |
---|---|
'Streams' | A
In this case, use a cell array the same size as the parallel pool. If a parallel pool is not open, then |
'UseParallel' | The default is false , indicating serial computation. |
'UseSubstreams' | Set to true to compute in parallel in a reproducible fashion. The default is false . To compute reproducibly, set Streams to a type allowing substreams: 'mlfg6331_64' or 'mrg32k3a' . |
Computing bootstrap confidence intervals in parallel requires Parallel Computing Toolbox™.
Example: BootstrapOptions=options
Data Types: struct
ConfidenceIntervalType
— Type of confidence interval
"exact"
| "bootstrap"
| "none"
Type of confidence interval to compute, specified as "exact"
,
"bootstrap"
, or "none"
. The default is
"exact"
when there is an exact formula for the effect size or
"bootstrap"
otherwise. "none"
is for not
computing any confidence intervals.
Default is "exact"
for Cliff's Delta, Glass's delta, mean
difference, and Cohen's d and "bootstrap"
for Kolmogorov-Smirnov
statistic, median difference, and Robust Cohen's d. If you specify confidence interval
type as "exact"
for Kolmogorov-Smirnov statistic, median difference,
and Robust Cohen's d, gardnerAltmanPlot
returns an error.
Example: ConfidenceIntervalType="none"
Data Types: string
| char
Effect
— Effect size to compute
"meandiff"
(default) | "cohen"
| "cliff"
| "glass"
| "kstest"
| "mediandiff"
| "robustcohen"
Effect size to compute, specified as one of the following:
Effect size option | Definition |
---|---|
"cohen" | Cohen's d for two-sample input. |
"cliff" | Cliff's Delta. |
"glass" | Glass's delta. |
"kstest" | Kolmogorov-Smirnov statistic. |
"mediandiff" | Median difference. |
"meandiff" | Mean difference. |
"robustcohen" | Robust Cohen's d for two-sample input. |
Example: Effect="glass"
Data Types: string
| char
NumBootstraps
— Number of bootstrap replicas
1000 (default) | positive integer
Number of bootstrap replicas to use when computing the bootstrap confidence intervals, specified as a positive integer.
Example: NumBootstraps=1500
Data Types: single
| double
Paired
— Indicator for paired samples
false
(default) | true
Indicator for paired samples, specified as a logical value.
If
Paired
is"true"
, thenVarianceType
must be"equal"
.If
Paired
is"true"
, thenEffect
cannot be"glass"
.
Example: Paired="true"
Data Types: logical
VarianceType
— Population variance assumption for two samples
"equal"
(default) | "unequal"
Population variance assumption for two samples, specified as either "equal"
or "unequal"
. If Paired
is "true"
, then VarianceType
must be "equal"
.
Example: VarianceType="unequal"
Data Types: string
| char
Output Arguments
H
— Graphics array for the plot
1-by-5 graphics array (default) | 1-by-4 graphics array
Graphics array for the plot, specified as a 1-by-5 or a 1-by-4 graphics array depending on the effect size you choose and whether the data is unpaired or paired as follows.
Unpaired data (
Paired="false"
):"meandiff"
,"glass"
,"cohen"
or"robustcohen"
effect size:H
is a 1-by-5 graphics array that holds twoScatter
objects, oneErrorBar
object, and twoLine
objects.The
Scatter
objects represent the scatter plots of the input data inX
andY
, theErrorBar
object represents the effect size and its confidence intervals, and theLine
objects represent the mean line from the control sample to the effect size and the other sample to 0."mediandiff"
effect size:H
is a 1-by-5 graphics array that holds twoScatter
objects, oneErrorBar
object, and twoLine
objects that represent the median lines from the two sample data sets."cliff"
or"kstest"
effect size:H
is a 1-by-4 graphics array that holds twoScatter
objects, oneErrorBar
object, and oneLine
object that represents the line at 0 to indicate no effect between the two groups.
Paired data (
Paired="true"
):H
holds 1-3Line
objects, representing the lines for the pairs that increased, decreased, or stayed the same and oneErrorBar
object, representing the effect size and its confidence intervals.
For all cases, if you choose not to generate any confidence intervals for the effect
(ConfidenceIntervalType="none"
), the ErrorBar
object will instead be a Line
object.
Use H
to inspect and adjust the properties of the objects. To
learn more about the object properties, see Scatter Properties, ErrorBar Properties, and Line Properties, respectively.
Algorithms
Effect Sizes
Cliff's Delta
Unpaired data
where n1 is the size of the first sample and n2 is the size of the second sample.
Paired data
gardnerAltmanPlot
uses the between-group delta, which is comparing the differences fromx
andy
, but excluding comparisons of paired data. For n paired samples, this results in n(n-1) comparisons [3].
Cohen's d
gardnerAltmanPlot
computes the unbiased estimate of Cohen's d, which is also known as Hedge's g.One-sample
Two-sample
where df is the degrees of freedom, μ is the known population mean to compare against, s is the pooled standard deviation, and J(df) is the bias correction term. The pooled standard deviation and the bias correction term are defined as follows, respectively.
where n1 is the size of the first sample and n2 is the size of the second sample.
where is the gamma function.
Cohen's d follows a noncentral t-distribution, and uses that to derive the confidence intervals. Hence,
gardnerAltmanPlot
by default uses"exact"
to compute the confidence intervals for the effect size. See [1] and [4] to see the derivation of the confidence intervals for paired versus unpaired input data.Glass's Delta
where sx is the standard deviation of the control group.
gardnerAltmanPlot
uses the data inx
as the control group. If you wish to use the other sample as the control group, you can swap the data inx
andy
and swap the sign of the test result.Similar to Cohen's d, Glass's delta also follows a noncentral t-distribution, and uses that to derive the confidence intervals. Hence,
gardnerAltmanPlot
by default uses"exact"
to compute the confidence intervals for the effect size [4]. You can't use this effect size for paired samples.Kolmogorov-Smirnov Test Statistic
This two-sample test statistic is the same as given in Two-Sample Kolmogorov-Smirnov Test.
gardnerAltmanPlot
uses bootstrapping to compute the confidence intervals.Mean Difference
One-sample
Two-sample
gardnerAltmanPlot
computes the confidence intervals using the t-distribution (using pooled standard deviation in the two-sample case. In case of unequal variance assumption for two samples, the confidence intervals are called Welch-Satterthwaite confidence intervals). The function by default uses the"exact"
method to compute the confidence intervals.Median Difference
gardnerAltmanPlot
computes the confidence intervals using bootstrapping for this effect size.Robust Cohen's d
where and are the 20% trimmed mean of data in
x
andy
, respectively. is the pooled 20% Winsorized variance [2].gardnerAltmanPlot
computes the confidence intervals using bootstrapping for this effect size.
References
[1] Cousineau, Denis and Jean-Christophe Goulet-Pelletier. "A study of confidence intervals for Cohen's d in within-subject designs with new proposals". The Quantitative Methods for Psychology, 2021.
[2] Algina, James, H. J. Keselman, and R. D. Penfield. "An Alternative to Cohen's Standardized Mean Difference Effect Size: A Robust Parameter and Confidence Interval in the Two Independent Groups Case". Psychological Methods, Vol. 10, No. 3, pp. 317–328. 2005.
[3] Hess, Melinda and Jeffrey Kromrey. "Robust Confidence Intervals for Effect Sizes: A Comparative Study of Cohen's d and Cliff's Delta Under Non-normality and Heterogeneous Variances". Annual Meeting of the American Educational Research Association. 2004.
[4] Delacre, Marie, Daniel Lakens, Christophe Ley, Limin Liu, and Christophe Leys. "Why Hedges G's Based on the Non-pooled Standard Deviation Should Be Reported with Welch's T-test". 2021.
[5] Gardner, M. J. and D. G. Altman. "Confidence intervals rather than P values: estimation rather than hypothesis testing". BMJ, Vol. 292 No. 6522, pp. 746–750. 1986.
Extended Capabilities
Automatic Parallel Support
Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.
To run in parallel, specify the 'Options'
name-value argument in the call
to this function and set the 'UseParallel'
field of the options
structure to true
using statset
.
For example: 'Options',statset('UseParallel',true)
For more information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).
Version History
Introduced in R2022a
See Also
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