# meanEffectSize

One-sample or two-sample effect size computations

Since R2022a

## Syntax

``Effect = meanEffectSize(X)``
``Effect = meanEffectSize(X,Y)``
``Effect = meanEffectSize(X,Y,Name=Value)``

## Description

example

````Effect = meanEffectSize(X)` computes the mean-difference effect size for a single sample `X` against the default mean value of 0.```

example

````Effect = meanEffectSize(X,Y)` computes the mean-difference effect size for two samples `X` and `Y`.```

example

````Effect = meanEffectSize(X,Y,Name=Value)` specifies options using one or more of the name-value arguments. 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.```

## Examples

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Load the stock returns data and define the variable for which to compute the mean-difference effect size.

```load stockreturns x = stocks(:,1);```

Compute the mean-difference effect size of the stock returns compared to the default mean value of 0, and compute the 95% confidence intervals for the effect size.

`effect = meanEffectSize(x)`
```effect=1×2 table Effect ConfidenceIntervals ________ ______________________ MeanDifference -0.20597 -0.41283 0.00087954 ```

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

You can also specify the mean value you want to compare against.

`effect = meanEffectSize(x,Mean=-1)`
```effect=1×2 table Effect ConfidenceIntervals _______ ___________________ MeanDifference 0.79403 0.58717 1.0009 ```

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.

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.

```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

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Input data, specified as a numeric vector.

Data Types: `single` | `double`

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.

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`

Options for computing bootstrap confidence intervals in parallel, specified as a structure generated by using `statset("bootci")`. `meanEffectSize` uses the following fields.

FieldDescription
`Streams`

A `RandStream` object or cell array of such objects. If you do not specify `Streams`, `meanEffectSize` uses the default stream or streams. If you specify `Streams`, use a single object except when all of the following conditions exist:

• 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 `Streams` must supply a single random number stream.

`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`

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, `meanEffectSize` returns an error.

Example: `ConfidenceIntervalType="none"`

Data Types: `string` | `char`

Effect size type to compute, specified as one of or a cell array of the following built-in options.

Options for single-sample input

Effect size optionDefinition
`"cohen"`Cohen's d for single-sample input.
`"meandiff"`Mean difference.
`"robustcohen"`Robust Cohen's d for single-sample input.

Options for two-sample input

OptionDefinition
`"cohen"`Cohen's d for two-sample input
`"cliff"`Cliff's Delta
`"glass"`

Glass's delta; not supported by `meanEffectSize` for paired data

`"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` | `cell`

Known population mean value to compare against, specified as a scalar value. This option is only for single-sample data.

Example: `Mean=10`

Data Types: `single` | `double`

Number of bootstrap replicas to use when computing the bootstrap confidence intervals, specified as a positive integer.

Example: `NumBootstraps=1500`

Data Types: `single` | `double`

Indicator for paired samples, specified as `false` or `true`.

Example: `Paired=true`

Data Types: `logical`

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

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Effect size information, returned as a table. `Effect` has a row for each effect size computed and a column for the value of the effect size, and a column for the confidence intervals for that effect size, if they are computed.

Data Types: `table`

## Algorithms

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### Effect Size

• Cliff's Delta

• Unpaired data

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

where n1 is the size of the first sample, and n2 is the size of the second sample.

• Paired data

`meanEffectSize` 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

`meanEffectSize` 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 n1 is the size of the first sample and n2 is the size of the second sample.

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, `meanEffectSize` 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 sx is the standard deviation of the control group. `meanEffectSize` 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, `meanEffectSize` 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. `meanEffectSize` uses bootstrapping to compute the confidence intervals.

• Mean Difference

• One-sample

`$m=\left(\overline{x}-\mu \right)$`
• Two-sample

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

`meanEffectSize` 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

`meanEffectSize` 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].

`meanEffectSize` 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.

## Version History

Introduced in R2022a