library(knitr) options(knitr.kable.NA = "") knitr::opts_chunk$set(comment = ">") options(digits = 3) set.seed(7)

This vignette provides a review of effect sizes for comparisons of groups, which
are typically achieved with the `t.test()`

and `wilcox.test()`

functions.

library(effectsize) options(es.use_symbols = TRUE) # get nice symbols when printing! (On Windows, requires R >= 4.2.0)

For *t*-tests, it is common to report an effect size representing a standardized
difference between the two compared samples' means. These measures range from
$-\infty$ to $+\infty$, with negative values indicating the second group's mean
is larger (and vice versa).

For two independent samples, the difference between the means is standardized based on the pooled standard deviation of both samples (assumed to be equal in the population):

t.test(mpg ~ am, data = mtcars, var.equal = TRUE) cohens_d(mpg ~ am, data = mtcars)

Hedges' *g* provides a bias correction for small sample sizes ($N < 20$).

hedges_g(mpg ~ am, data = mtcars)

If variances cannot be assumed to be equal, it is possible to get estimates that are not based on the pooled standard deviation:

t.test(mpg ~ am, data = mtcars, var.equal = FALSE) cohens_d(mpg ~ am, data = mtcars, pooled_sd = FALSE) hedges_g(mpg ~ am, data = mtcars, pooled_sd = FALSE)

In cases where the differences between the variances are substantial, it is also
common to standardize the difference based only on the standard deviation of one
of the groups (usually the "control" group); this effect size is known as Glass'
$\Delta$ (delta) (Note that the standard deviation is taken from the *second* sample).

glass_delta(mpg ~ am, data = mtcars)

For a one-sided hypothesis, it is also possible to construct one-sided confidence intervals:

t.test(mpg ~ am, data = mtcars, var.equal = TRUE, alternative = "less") cohens_d(mpg ~ am, data = mtcars, pooled_sd = TRUE, alternative = "less")

In the case of a one-sample test, the effect size represents the standardized distance of the mean of the sample from the null value.

t.test(mtcars$wt, mu = 2.7) cohens_d(mtcars$wt, mu = 2.7) hedges_g(mtcars$wt, mu = 2.7)

For paired-samples, the difference is standardized by the variation in the differences. This effect size, known as Cohen's $d_z$, represents the difference in terms of its homogeneity (a small but stable difference will have a large $d_z$).

t.test(extra ~ group, data = sleep, paired = TRUE) cohens_d(extra ~ group, data = sleep, paired = TRUE) hedges_g(extra ~ group, data = sleep, paired = TRUE)

A Bayesian estimate of Cohen's *d* can also be provided based on `BayesFactor`

's
version of a *t*-test via the `effectsize()`

function:

library(BayesFactor) BFt <- ttestBF(formula = mpg ~ am, data = mtcars) effectsize(BFt, type = "d")

When examining multivariate differences (e.g., with Hotelling's $T^2$ test), Mahalanobis' *D* can be used as the multivariate equivalent for Cohen's *d*. Unlike Cohen's *d* which is a measure of standardized *differences*, Mahalanobis' *D* is a measure of standardized *distances*. As such, it cannot be negative, and ranges from 0 (no distance between the multivariate distributions) to $+\infty$.

mahalanobis_d(mpg + hp + cyl ~ am, data = mtcars)

The rank-biserial correlation ($r_{rb}$) is a measure of dominance: larger
values indicate that more of *X* is larger than more of *Y*, with a value of
$(-1)$ indicates that *all* observations in the second group are larger than the
first, and a value of $(+1)$ indicates that *all* observations in the first
group are larger than the second.

These effect sizes should be reported with the Wilcoxon (Mann-Whitney) test or
the signed-rank test (both available in `wilcox.test()`

).

A <- c(48, 48, 77, 86, 85, 85) B <- c(14, 34, 34, 77) wilcox.test(A, B) # aka Mann–Whitney U test rank_biserial(A, B)

For one sample, $r_{rb}$ measures the symmetry around $\mu$ (mu; the null value), with 0 indicating perfect symmetry, $(-1)$ indicates that all observations fall below $\mu$, and $(+1)$ indicates that all observations fall above $\mu$.

x <- c(1.15, 0.88, 0.90, 0.74, 1.21, 1.36, 0.89) wilcox.test(x, mu = 1) # aka Signed-Rank test rank_biserial(x, mu = 1)

For paired samples, $r_{rb}$ measures the symmetry of the (paired) *differences*
around $\mu$ as for the one sample case.

x <- c(1.83, 0.50, 1.62, 2.48, 1.68, 1.88, 1.55, 3.06, 1.30) y <- c(0.88, 0.65, 0.60, 2.05, 1.06, 1.29, 1.06, 3.14, 1.29) wilcox.test(x, y, paired = TRUE) # aka Signed-Rank test rank_biserial(x, y, paired = TRUE)

Related effect sizes are the *common language effect sizes* which present information about group differences in terms of probability.

These measures indicate the degree two independent distributions overlap: Cohen's $U_1$ is the proportion of the total of both distributions that does not overlap, while *Overlap (OVL)* is the proportional overlap between the distributions.

cohens_u1(mpg ~ am, data = mtcars) p_overlap(mpg ~ am, data = mtcars)

Note the by default, these functions return the parametric versions of these effect sizes: these assume equal normal variance in both populations. When these assumptions are not met, the values produced will be biased in unknown ways. In such cases, we should use the non-parametric versions ($U_1$ is not defined):

p_overlap(mpg ~ am, data = mtcars, parametric = FALSE)

*Probability of superiority* is the probability that, when sampling an observation from each of the groups at random, that the observation from the second group will be larger than the sample from the first group.

p_superiority(mpg ~ am, data = mtcars)

Here, this indicates that if we were to randomly draw a sample from `am==0`

and from `am==1`

, 15% of the time, the first will have a larger `mpg`

values than the second.

Cohen's $U_2$ is the proportion of one of the groups that exceeds the same proportion in the other group, and Cohen's $U_3$ is the proportion of the second group that is smaller than the median of the first group.

cohens_u2(mpg ~ am, data = mtcars) cohens_u3(mpg ~ am, data = mtcars)

Here too we have a non-parametric versions when the assumptions of equal variance of normal populations:

p_superiority(mpg ~ am, data = mtcars, parametric = FALSE) cohens_u2(mpg ~ am, data = mtcars, parametric = FALSE) cohens_u3(mpg ~ am, data = mtcars, parametric = FALSE)

For one sample, *probability of superiority* is the probability that, when sampling an observation at random, it will be larger than $\mu$.

p_superiority(mtcars$wt, mu = 2.75) p_superiority(mtcars$wt, mu = 2.75, parametric = FALSE)

For paired samples, *probability of superiority* is the probability that, when sampling an observation at random, its *difference* will be larger than $\mu$.

p_superiority(extra ~ group, data = sleep, paired = TRUE, mu = -1) p_superiority(extra ~ group, data = sleep, paired = TRUE, mu = -1, parametric = FALSE)

A Bayesian estimate of (the parametric version of) these effect sizes can also
be provided based on `BayesFactor`

's version of a *t*-test via the
`effectsize()`

function:

effectsize(BFt, type = "p_superiority") effectsize(BFt, type = "u1") effectsize(BFt, type = "u2") effectsize(BFt, type = "u3") effectsize(BFt, type = "overlap")

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