# gardnerAltmanPlot

## Syntax

## Description

`gardnerAltmanPlot(___,`

specifies options using one or more name-value arguments in addition to any of the input
argument combinations in previous syntaxes. For example, you can specify the type of the
effect size to compute or the number of bootstrap replicas to use when computing the
bootstrap confidence intervals.`Name=Value`

)

returns
a graphics array `H`

= gardnerAltmanPlot(___)`H`

.

## Examples

### Compute Median-Difference Effect Size for Two Independent Samples

Load Fisher's iris data and define the variables for which to compute the median-difference effect size.

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 -1.8 -1.3

By default, the `meanEffectSize`

function assumes that the samples are independent (that is, `Paired=false`

). The function uses bootstrapping to estimate the confidence intervals when the effect type is median-difference.

Visualize the median-difference effect size using the Gardner-Altman plot.

`gardnerAltmanPlot(x,y,Effect="mediandiff");`

The Gardner-Altman plot displays the two data samples 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.

### Compute Cohen's d Effect Size and Specify Bootstrap Options

Load Fisher's iris data and define the variables for which to compare the Cohen's d effect size.

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. By default,the `meanEffectSize`

function uses the exact formula based on the noncentral *t-*distribution to estimate the confidence intervals when the effect size type is Cohen's d. Specify the bootstrapping options as follows:

Set

`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 'Processes' profile ... Connected to the parallel pool (number of workers: 6).

`effect=`*1×2 table*
Effect ConfidenceIntervals
_______ ___________________
CohensD -3.0536 -3.5621 -2.3468

Visualize the Cohen's d effect size using the Gardner-Altman plot with the same options set.

gardnerAltmanPlot(x,y,Effect="cohen",ConfidenceIntervalType="bootstrap", ... BootstrapOptions=statset(UseParallel=true),NumBootstraps=3000);

The Gardner-Altman plot displays the two data samples 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 to compare.

```
load examgrades
x = grades(:,1);
y = grades(:,2);
```

Compute the mean-difference effect size of the grades from the paired samples, and compute the 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

The `meanEffectSize`

function uses the exact method to estimate the confidence intervals when you use the mean-difference effect size.

You can specify a different effect size type. (Note that you cannot 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

The `meanEffectSize`

function uses bootstrapping to estimate the confidence intervals when the effect size type is robust Cohen's d.

Visualize the effect size using the Gardner-Altman plot. Specify robust Cohen's d as the effect size, and compute the 97% confidence intervals.

`gardnerAltmanPlot(x,y,Paired=true,Effect="robustcohen",Alpha=0.03);`

The Gardner-Altman plot displays the paired data on the left. The 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. Right side of the plot displays the robust Cohen's d effect size with the 97% confidence interval.

## 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, 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 of the confidence intervals for the effect size, specified as a numeric value from 0 to 1. Default value of 0.05 corresponds to the 95% confidence level.

**Example: **`Alpha=0.025`

**Data Types: **`single`

| `double`

`BootstrapOptions`

— Options for computing bootstrap confidence intervals in parallel

structure

Options for computing bootstrap confidence intervals in parallel, specified as a
structure generated by using `statset("bootci")`

.
`gardnerAltmanPlot`

uses the following fields.

Field | Description |
---|---|

`Streams` | A You have an open parallel pool. `UseParallel` is`true` .`UseSubstreams` is`false` .
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 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 value is
`"exact"`

when there is an exact formula for the effect size or
`"bootstrap"`

otherwise. Specify `"none"`

is when
you do not want to compute any confidence intervals.

The default value 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 options.

Option | Definition |
---|---|

`"cohen"` | Cohen's d for two-sample input |

`"cliff"` | Cliff's Delta |

`"glass"` | Glass's delta; not supported by |

`"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 `false`

or
`true`

.

If

`Paired`

is`true`

, then`VarianceType`

must be`"equal"`

.If

`Paired`

is`true`

, then`Effect`

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 `"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, returned as a 1-by-5 or 1-by-4 graphics array depending on the specified effect size and whether the data is unpaired or paired.

Unpaired data (

`Paired="false"`

)For the

`"meandiff"`

,`"glass"`

,`"cohen"`

or`"robustcohen"`

effect size:`H`

is a 1-by-5 graphics array that holds two`Scatter`

objects, one`ErrorBar`

object, and two`Line`

objects.The

`Scatter`

objects represent the scatter plots of the input data in`X`

and`Y`

. The`ErrorBar`

object represents the effect size and its confidence intervals. The`Line`

objects represent the mean line from the control sample to the effect size and from the other sample to 0.For the

`"mediandiff"`

effect size:`H`

is a 1-by-5 graphics array that holds two`Scatter`

objects, one`ErrorBar`

object, and two`Line`

objects that represent the median lines from the two sample data sets.For the

`"cliff"`

or`"kstest"`

effect size:`H`

is a 1-by-4 graphics array that holds two`Scatter`

objects, one`ErrorBar`

object, and one`Line`

object that represents the line at 0 to indicate no effect between the two groups.

Paired data (

`Paired="true"`

)`H`

holds one to three`Line`

objects representing the lines for the pairs that increased, decreased, or stayed the same, and one`ErrorBar`

object representing the effect size and its confidence intervals.

For all cases, if you choose not to generate confidence intervals for the effect
(`ConfidenceIntervalType="none"`

), the `ErrorBar`

object is, 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.

## Algorithms

### Effect Size

*Cliff's Delta*Unpaired data

$$\delta =\frac{{\displaystyle \sum _{i,j=1}^{{n}_{1},{n}_{2}}\left({x}_{i}>{y}_{j}\right)-\left({x}_{i}<{y}_{j}\right)}}{{n}_{1}*{n}_{2}},$$

where

*n*is the size of the first sample, and_{1}*n*is the size of the second sample._{2}Paired data

`gardnerAltmanPlot`

uses the between-group delta, which compares the differences between`x`

and`y`

, but excludes comparisons of paired data. For*n*paired samples, the result is*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

$$d=J\left(df\right)*\frac{\left(\overline{x}-\mu \right)}{s}$$

Two-sample

$$d=J\left(df\right)*\frac{\left(\overline{x}-\overline{y}\right)}{s}$$

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 is defined as$$s=\sqrt{\frac{\left({n}_{1}-1\right){s}_{1}^{2}+\left({n}_{2}-1\right){s}_{2}^{2}}{{n}_{1}+{n}_{2}-2}},$$

where

*n*is the size of the first sample and_{1}*n*is the size of the second sample._{2}The bias correction term are defined as

$$J\left(df\right)=\frac{\Gamma \left(df/2\right)}{\sqrt{df/2}\Gamma \left(\left(df-1\right)/2\right)},$$

where $$\Gamma \left(\right)$$ is the gamma function.

Cohen's d follows a noncentral

*t*-distribution and uses it to derive the confidence intervals. So, by default,`gardnerAltmanPlot`

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*$$D=\frac{\overline{x}-\overline{y}}{{s}_{x}},$$

where

*s*is the standard deviation of the control group._{x}`gardnerAltmanPlot`

uses the data in`x`

as the control group. If you wish to use the other sample as the control group, you can swap the data in`x`

and`y`

and swap the sign of the test result.Similar to Cohen's d, Glass's delta also follows a noncentral

*t*-distribution and uses it to derive the confidence intervals. So, by default,`gardnerAltmanPlot`

uses`"exact"`

to compute the confidence intervals for the effect size [4]. You cannot 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

$$m=\left(\overline{x}-\mu \right)$$

Two-sample

$$m=\left(\overline{x}-\overline{y}\right)$$

`gardnerAltmanPlot`

computes the confidence intervals using the*t*-distribution with pooled standard deviation in the two-sample case. (In the case of unequal variance assumption for two samples, the confidence intervals are called Welch-Satterthwaite confidence intervals). By default, the function uses`"exact"`

to compute the confidence intervals for the effect size.*Median Difference*$$M\text{=}median\left(x\right)-median\left(y\right)$$

`gardnerAltmanPlot`

computes the confidence intervals using bootstrapping for the effect size.*Robust Cohen's d*$$d=0.643*J\left(df\right)*\frac{\left({\overline{x}}_{t}-{\overline{y}}_{t}\right)}{{s}_{w}},$$

where $${\overline{x}}_{t}$$ and $${\overline{y}}_{t}$$ are the 20% trimmed means of data in

`x`

and`y`

, respectively. $${\text{s}}_{w}$$ is the pooled 20% Winsorized variance [2].`gardnerAltmanPlot`

computes the confidence intervals using bootstrapping for the 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* 17, no. 1 (March 2021): 51--75. https://doi.org/10.20982/tqmp.17.1.p051.

[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* 10, no. 3 (Sept 2005): 317–28. https://doi.org/10.1037/1082-989X.10.3.317.

[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*, 292 no. 6522 (March 1986): 746–50. https://doi.org/10.1136/bmj.292.6522.746.

## 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`

:

`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**

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