# Standardized Differences

Source:`vignettes/standardized_differences.Rmd`

`standardized_differences.Rmd`

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 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 | 95% CI
> -------------------------
> -1.17 | [-1.93, -0.39]
```

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

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)`

```
>
> Paired t-test
>
> data: extra by group
> 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
```

`cohens_d(extra ~ group, data = sleep, paired = TRUE)`

```
> Cohen's d | 95% CI
> --------------------------
> -1.28 | [-2.12, -0.41]
```

`hedges_g(extra ~ group, data = sleep, paired = TRUE)`

```
> Hedges' g | 95% CI
> --------------------------
> -1.17 | [-1.94, -0.38]
```

### 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 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, 0.37]
>
> - 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(extra ~ group,
data = sleep,
paired = TRUE, mu = -1
)
```

```
> Pr(superiority) | 95% CI
> ------------------------------
> 0.37 | [0.22, 0.56]
```

```
p_superiority(extra ~ group,
data = sleep,
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")`

```
> 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]
```