Standardized Differences

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)

Standardized Differences

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).

Two Independent Samples

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)
> 
>   Two Sample t-test
> 
> data:  mpg by am
> t = -4, df = 30, p-value = 3e-04
> alternative hypothesis: true difference in means between group 0 and group 1 is not equal to 0
> 95 percent confidence interval:
>  -10.85  -3.64
> sample estimates:
> mean in group 0 mean in group 1 
>            17.1            24.4
cohens_d(mpg ~ am, data = mtcars)
> Cohen's d |         95% CI
> --------------------------
> -1.48     | [-2.27, -0.67]
> 
> - Estimated using pooled SD.

Hedges’ g provides a small-sample bias correction (for small sample sizes, \(N < 20\)).

hedges_g(mpg ~ am, data = mtcars)
> Hedges' g |         95% CI
> --------------------------
> -1.44     | [-2.21, -0.65]
> 
> - Estimated using pooled SD.

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)
> 
>   Welch Two Sample t-test
> 
> data:  mpg by am
> t = -4, df = 18, p-value = 0.001
> alternative hypothesis: true difference in means between group 0 and group 1 is not equal to 0
> 95 percent confidence interval:
>  -11.28  -3.21
> sample estimates:
> mean in group 0 mean in group 1 
>            17.1            24.4
cohens_d(mpg ~ am, data = mtcars, pooled_sd = FALSE)
> Cohen's d |         95% CI
> --------------------------
> -1.41     | [-2.26, -0.53]
> 
> - Estimated using un-pooled SD.
hedges_g(mpg ~ am, data = mtcars, pooled_sd = FALSE)
> Hedges' g |         95% CI
> --------------------------
> -1.35     | [-2.17, -0.51]
> 
> - Estimated using un-pooled SD.

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)
> Glass' Δ (adj.) |         95% CI
> --------------------------------
> -1.10           | [-1.80, -0.37]

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")
> 
>   Two Sample t-test
> 
> data:  mpg by am
> t = -4, df = 30, p-value = 1e-04
> alternative hypothesis: true difference in means between group 0 and group 1 is less than 0
> 95 percent confidence interval:
>   -Inf -4.25
> sample estimates:
> mean in group 0 mean in group 1 
>            17.1            24.4
cohens_d(mpg ~ am, data = mtcars, pooled_sd = TRUE, alternative = "less")
> Cohen's d |        95% CI
> -------------------------
> -1.48     | [-Inf, -0.80]
> 
> - Estimated using pooled SD.
> - One-sided CIs: lower bound fixed at [-Inf].

One Sample

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$wt, mu = 2.7)
> 
>   One Sample t-test
> 
> data:  mtcars$wt
> t = 3, df = 31, p-value = 0.005
> alternative hypothesis: true mean is not equal to 2.7
> 95 percent confidence interval:
>  2.86 3.57
> sample estimates:
> mean of x 
>      3.22
cohens_d(mtcars$wt, mu = 2.7)
> Cohen's d |       95% CI
> ------------------------
> 0.53      | [0.15, 0.90]
> 
> - Deviation from a difference of 2.7.
hedges_g(mtcars$wt, mu = 2.7)
> Hedges' g |       95% CI
> ------------------------
> 0.52      | [0.15, 0.87]
> 
> - Deviation from a difference of 2.7.

Paired Samples

In a repeated-measures design, the same subjects are measured in multiple conditions or time points. Unlike the case of independent groups, there are multiple sources of variation that can be used to standardized the differences between the means of the conditions / times, and each provides a unique standardized mean difference.

The most basic option is compute from the paired samples difference scores and compute a one-sample effect size. 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 \(d_z\)).

sleep_wide <- datawizard::data_to_wide(sleep,
  id_cols = "ID",
  values_from = "extra",
  names_from = "group",
  names_prefix = "extra_"
)

t.test(sleep_wide[["extra_1"]], sleep_wide[["extra_2"]], paired = TRUE)
> 
>   Paired t-test
> 
> data:  sleep_wide[["extra_1"]] and sleep_wide[["extra_2"]]
> t = -4, df = 9, p-value = 0.003
> alternative hypothesis: true mean difference is not equal to 0
> 95 percent confidence interval:
>  -2.46 -0.70
> sample estimates:
> mean difference 
>           -1.58
repeated_measures_d(sleep_wide[["extra_1"]], sleep_wide[["extra_2"]], method = "z")
> d (z) |         95% CI
> ----------------------
> -1.17 | [-1.94, -0.41]
> 
> - Adjusted for small sample bias.
# same as:
hedges_g(sleep_wide[["extra_1"]] - sleep_wide[["extra_2"]])
> Hedges' g |         95% CI
> --------------------------
> -1.17     | [-1.94, -0.38]

Other options try to get close to the value that would have been reached if the samples were independant (see more info in the documentation of repeated_measures_d()):

repeated_measures_d(sleep_wide[["extra_1"]], sleep_wide[["extra_2"]])
> dᵣₘ   |         95% CI
> ----------------------
> -0.75 | [-1.17, -0.33]
> 
> - Adjusted for small sample bias.
repeated_measures_d(sleep_wide[["extra_1"]], sleep_wide[["extra_2"]], method = "av")
> dₐᵥ   |         95% CI
> ----------------------
> -0.76 | [-1.13, -0.39]
> 
> - Adjusted for small sample bias.
repeated_measures_d(sleep_wide[["extra_1"]], sleep_wide[["extra_2"]], method = "b")
> Becker's d |         95% CI
> ---------------------------
> -0.72      | [-1.20, -0.24]
> 
> - Adjusted for small sample bias.
repeated_measures_d(sleep_wide[["extra_1"]], sleep_wide[["extra_2"]], method = "d")
> Cohen's d |         95% CI
> --------------------------
> -0.80     | [-1.29, -0.30]
> 
> - Adjusted for small sample bias.
# all closer to:
cohens_d(sleep_wide[["extra_1"]], sleep_wide[["extra_2"]], ci = NULL)
> Cohen's d
> ---------
> -0.83    
> 
> - Estimated using pooled SD.

For data containing repetition in each condition/subject, another effect size (residual d) is also available:

data("rouder2016")

head(rouder2016)
>   id cond    rt
> 1  1    1 0.560
> 2  1    1 0.930
> 3  1    1 0.795
> 4  1    1 0.615
> 5  1    1 1.028
> 6  1    1 0.845
repeated_measures_d(rt ~ cond | id, data = rouder2016, method = "r")
> dᵣ    |         95% CI
> ----------------------
> -0.26 | [-0.33, -0.18]
> 
> - Adjusted for small sample bias.

For a Bayesian t-test

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")

(Multivariate) Standardized Distances

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)
> Mahalanobis' D |      95% CI
> ----------------------------
> 2.14           | [1.22, Inf]
> 
> - One-sided CIs: upper bound fixed at [Inf].

Means Ratio

Instead of the difference between means, we can also look at the ratio between means. This effect size is only applicable to ratio scale outcomes (variables with an absolute zero).

Lucky for us, miles-per-gallon is on a ratio scale!

means_ratio(mpg ~ am, data = mtcars)
> Means Ratio (adj.) |       95% CI
> ---------------------------------
> 0.70               | [0.59, 0.83]

Values range between \(0\) and \(\infty\), with values smaller than 1 indicating that the second mean is larger than the first, values larger than 1 indicating that the second mean is smaller than the first, and values of 1 indicating that the means are equal.

Dominance Effect Sizes

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()).

Two Independent Samples

A <- c(48, 48, 77, 86, 85, 85)
B <- c(14, 34, 34, 77)

wilcox.test(A, B, exact = FALSE) # aka Mann–Whitney U test
> 
>   Wilcoxon rank sum test with continuity correction
> 
> data:  A and B
> W = 22, p-value = 0.05
> alternative hypothesis: true location shift is not equal to 0
rank_biserial(A, B)
> r (rank biserial) |       95% CI
> --------------------------------
> 0.79              | [0.30, 0.95]

One Sample

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
> 
>   Wilcoxon signed rank exact test
> 
> data:  x
> V = 16, p-value = 0.8
> alternative hypothesis: true location is not equal to 1
rank_biserial(x, mu = 1)
> r (rank biserial) |        95% CI
> ---------------------------------
> 0.14              | [-0.59, 0.75]
> 
> - Deviation from a difference of 1.

Paired Samples

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
> 
>   Wilcoxon signed rank exact test
> 
> data:  x and y
> V = 40, p-value = 0.04
> alternative hypothesis: true location shift is not equal to 0
rank_biserial(x, y, paired = TRUE)
> r (rank biserial) |       95% CI
> --------------------------------
> 0.78              | [0.30, 0.94]

Common Language Effect Sizes

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

Two Independent Samples

Measures of (Non)Overlap

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)
> Cohen's U1 |       95% CI
> -------------------------
> 0.70       | [0.42, 0.85]
p_overlap(mpg ~ am, data = mtcars)
> Overlap |       95% CI
> ----------------------
> 0.46    | [0.26, 0.74]

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)
> Overlap |       95% CI
> ----------------------
> 0.69    | [0.27, 0.99]
> 
> - Non-parametric CLES

Probabilistic Measures

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)
> Pr(superiority) |       95% CI
> ------------------------------
> 0.15            | [0.05, 0.32]

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)
> Cohen's U2 |       95% CI
> -------------------------
> 0.77       | [0.63, 0.87]
cohens_u3(mpg ~ am, data = mtcars)
> Cohen's U3 |       95% CI
> -------------------------
> 0.07       | [0.01, 0.25]

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)
> Pr(superiority) |       95% CI
> ------------------------------
> 0.17            | [0.08, 0.32]
> 
> - Non-parametric CLES
cohens_u2(mpg ~ am, data = mtcars, parametric = FALSE)
> Cohen's U2 |       95% CI
> -------------------------
> 0.80       | [0.50, 1.00]
> 
> - Non-parametric CLES
cohens_u3(mpg ~ am, data = mtcars, parametric = FALSE)
> Cohen's U3 |       95% CI
> -------------------------
> 0.15       | [0.00, 0.42]
> 
> - Non-parametric CLES

One Sample and Paired Samples

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$wt, mu = 2.75)
> Pr(superiority) |       95% CI
> ------------------------------
> 0.63            | [0.53, 0.72]
p_superiority(mtcars$wt, mu = 2.75, parametric = FALSE)
> Pr(superiority) |       95% CI
> ------------------------------
> 0.74            | [0.57, 0.86]
> 
> - Non-parametric CLES

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(sleep_wide[["extra_1"]], sleep_wide[["extra_2"]],
  paired = TRUE, mu = -1
)
> For paired samples, 'repeated_measures_d()' provides more options.
> Pr(superiority) |       95% CI
> ------------------------------
> 0.37            | [0.22, 0.56]
p_superiority(sleep_wide[["extra_1"]], sleep_wide[["extra_2"]],
  paired = TRUE, mu = -1,
  parametric = FALSE
)
> Pr(superiority) |       95% CI
> ------------------------------
> 0.19            | [0.06, 0.49]
> 
> - Non-parametric CLES

For a Bayesian t-test

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")

References