Standardized Differences
Source:vignettes/standardized_differences.Rmd
standardized_differences.RmdThis 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:
> 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")> Cohen's d | 95% CI
> --------------------------
> -1.29 | [-2.11, -0.52]
>
> - Estimated using pooled SD.(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 | [0.30, 1.00]
>
> - Non-parametric CLESProbabilistic 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, 0.37]
>
> - Non-parametric CLESOne 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 CLESFor 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 CLESFor 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")> Pr(superiority) | 95% CI
> ------------------------------
> 0.18 | [0.07, 0.36]
effectsize(BFt, type = "u1")> Cohen's U1 | 95% CI
> -------------------------
> 0.65 | [0.32, 0.83]
effectsize(BFt, type = "u2")> Cohen's U2 | 95% CI
> -------------------------
> 0.74 | [0.60, 0.85]
effectsize(BFt, type = "u3")> Cohen's U3 | 95% CI
> -------------------------
> 0.10 | [0.02, 0.31]
effectsize(BFt, type = "overlap")> Overlap | 95% CI
> ----------------------
> 0.52 | [0.29, 0.80]