Compute effect size indices for standardized differences: Cohen's *d*,
Hedges' *g* and Glass’s *delta* (\(\Delta\)). (This function returns the
**population** estimate.)

Both Cohen's *d* and Hedges' *g* are the estimated the standardized
difference between the means of two populations. Hedges' *g* provides a bias
correction (using the exact method) to Cohen's *d* for small sample sizes.
For sample sizes > 20, the results for both statistics are roughly
equivalent. Glass’s *delta* is appropriate when the standard deviations are
significantly different between the populations, as it uses only the *second*
group's standard deviation.

## Usage

```
cohens_d(
x,
y = NULL,
data = NULL,
pooled_sd = TRUE,
mu = 0,
paired = FALSE,
ci = 0.95,
alternative = "two.sided",
verbose = TRUE,
...
)
hedges_g(
x,
y = NULL,
data = NULL,
pooled_sd = TRUE,
mu = 0,
paired = FALSE,
ci = 0.95,
alternative = "two.sided",
verbose = TRUE,
...
)
glass_delta(
x,
y = NULL,
data = NULL,
mu = 0,
ci = 0.95,
alternative = "two.sided",
verbose = TRUE,
...
)
```

## Arguments

- x
A formula, a numeric vector, or a character name of one in

`data`

.- y
A numeric vector, a grouping (character / factor) vector, a or a character name of one in

`data`

. Ignored if`x`

is a formula.- data
An optional data frame containing the variables.

- pooled_sd
If

`TRUE`

(default), a`sd_pooled()`

is used (assuming equal variance). Else the mean SD from both groups is used instead.- mu
a number indicating the true value of the mean (or difference in means if you are performing a two sample test).

- paired
If

`TRUE`

, the values of`x`

and`y`

are considered as paired. This produces an effect size that is equivalent to the one-sample effect size on`x - y`

.- ci
Confidence Interval (CI) level

- alternative
a character string specifying the alternative hypothesis; Controls the type of CI returned:

`"two.sided"`

(default, two-sided CI),`"greater"`

or`"less"`

(one-sided CI). Partial matching is allowed (e.g.,`"g"`

,`"l"`

,`"two"`

...). See*One-Sided CIs*in effectsize_CIs.- verbose
Toggle warnings and messages on or off.

- ...
Arguments passed to or from other methods. When

`x`

is a formula, these can be`subset`

and`na.action`

.

## Value

A data frame with the effect size ( `Cohens_d`

, `Hedges_g`

,
`Glass_delta`

) and their CIs (`CI_low`

and `CI_high`

).

## Details

Set `pooled_sd = FALSE`

for effect sizes that are to accompany a Welch's
*t*-test (Delacre et al, 2021).

## Note

The indices here give the population estimated standardized difference. Some statistical packages give the sample estimate instead (without applying Bessel's correction).

## Confidence (Compatibility) Intervals (CIs)

Unless stated otherwise, confidence (compatibility) intervals (CIs) are
estimated using the noncentrality parameter method (also called the "pivot
method"). This method finds the noncentrality parameter ("*ncp*") of a
noncentral *t*, *F*, or \(\chi^2\) distribution that places the observed
*t*, *F*, or \(\chi^2\) test statistic at the desired probability point of
the distribution. For example, if the observed *t* statistic is 2.0, with 50
degrees of freedom, for which cumulative noncentral *t* distribution is *t* =
2.0 the .025 quantile (answer: the noncentral *t* distribution with *ncp* =
.04)? After estimating these confidence bounds on the *ncp*, they are
converted into the effect size metric to obtain a confidence interval for the
effect size (Steiger, 2004).

For additional details on estimation and troubleshooting, see effectsize_CIs.

## CIs and Significance Tests

"Confidence intervals on measures of effect size convey all the information
in a hypothesis test, and more." (Steiger, 2004). Confidence (compatibility)
intervals and p values are complementary summaries of parameter uncertainty
given the observed data. A dichotomous hypothesis test could be performed
with either a CI or a p value. The 100 (1 - \(\alpha\))% confidence
interval contains all of the parameter values for which *p* > \(\alpha\)
for the current data and model. For example, a 95% confidence interval
contains all of the values for which p > .05.

Note that a confidence interval including 0 *does not* indicate that the null
(no effect) is true. Rather, it suggests that the observed data together with
the model and its assumptions combined do not provided clear evidence against
a parameter value of 0 (same as with any other value in the interval), with
the level of this evidence defined by the chosen \(\alpha\) level (Rafi &
Greenland, 2020; Schweder & Hjort, 2016; Xie & Singh, 2013). To infer no
effect, additional judgments about what parameter values are "close enough"
to 0 to be negligible are needed ("equivalence testing"; Bauer & Kiesser,
1996).

## References

Algina, J., Keselman, H. J., & Penfield, R. D. (2006). Confidence intervals for an effect size when variances are not equal. Journal of Modern Applied Statistical Methods, 5(1), 2.

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd Ed.). New York: Routledge.

Delacre, M., Lakens, D., Ley, C., Liu, L., & Leys, C. (2021, May 7). Why Hedges’ g*s based on the non-pooled standard deviation should be reported with Welch's t-test. https://doi.org/10.31234/osf.io/tu6mp

Hedges, L. V. & Olkin, I. (1985). Statistical methods for meta-analysis. Orlando, FL: Academic Press.

Hunter, J. E., & Schmidt, F. L. (2004). Methods of meta-analysis: Correcting error and bias in research findings. Sage.

## See also

Other effect size indices:
`cles()`

,
`effectsize.BFBayesFactor()`

,
`eta_squared()`

,
`phi()`

,
`rank_biserial()`

## Examples

```
# \donttest{
data(mtcars)
mtcars$am <- factor(mtcars$am)
# Two Independent Samples ----------
(d <- cohens_d(mpg ~ am, data = mtcars))
#> Cohen's d | 95% CI
#> --------------------------
#> -1.48 | [-2.27, -0.67]
#>
#> - Estimated using pooled SD.
# Same as:
# cohens_d("mpg", "am", data = mtcars)
# cohens_d(mtcars$mpg[mtcars$am=="0"], mtcars$mpg[mtcars$am=="1"])
# More options:
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.
cohens_d(mpg ~ am, data = mtcars, mu = -5)
#> Cohen's d | 95% CI
#> -------------------------
#> -0.46 | [-1.17, 0.26]
#>
#> - Deviation from a difference of -5.
#> - Estimated using pooled SD.
cohens_d(mpg ~ am, data = mtcars, alternative = "less")
#> Cohen's d | 95% CI
#> -------------------------
#> -1.48 | [-Inf, -0.80]
#>
#> - Estimated using pooled SD.
#> - One-sided CIs: lower bound fixed at [-Inf].
hedges_g(mpg ~ am, data = mtcars)
#> Hedges' g | 95% CI
#> --------------------------
#> -1.44 | [-2.21, -0.65]
#>
#> - Estimated using pooled SD.
glass_delta(mpg ~ am, data = mtcars)
#> Glass' delta | 95% CI
#> -----------------------------
#> -1.17 | [-1.93, -0.39]
# One Sample ----------
cohens_d(wt ~ 1, data = mtcars)
#> Cohen's d | 95% CI
#> ------------------------
#> 3.29 | [2.44, 4.24]
# same as:
# cohens_d("wt", data = mtcars)
# cohens_d(mtcars$wt)
# More options:
cohens_d(wt ~ 1, data = mtcars, mu = 3)
#> Cohen's d | 95% CI
#> -------------------------
#> 0.22 | [-0.13, 0.58]
#>
#> - Deviation from a difference of 3.
hedges_g(wt ~ 1, data = mtcars, mu = 3)
#> Hedges' g | 95% CI
#> -------------------------
#> 0.22 | [-0.13, 0.57]
#>
#> - Deviation from a difference of 3.
# Paired Samples ----------
data(sleep)
cohens_d(Pair(extra[group == 1], extra[group == 2]) ~ 1, data = sleep)
#> Cohen's d | 95% CI
#> --------------------------
#> -1.28 | [-2.23, -0.44]
# same as:
# cohens_d(sleep$extra[sleep$group == 1], sleep$extra[sleep$group == 2], paired = TRUE)
# More options:
cohens_d(Pair(extra[group == 1], extra[group == 2]) ~ 1, data = sleep, mu = -1)
#> Cohen's d | 95% CI
#> -------------------------
#> -0.47 | [-1.18, 0.21]
#>
#> - Deviation from a difference of -1.
hedges_g(Pair(extra[group == 1], extra[group == 2]) ~ 1, data = sleep)
#> Hedges' g | 95% CI
#> --------------------------
#> -1.17 | [-2.04, -0.40]
# Interpretation -----------------------
interpret_cohens_d(-1.48, rules = "cohen1988")
#> [1] "large"
#> (Rules: cohen1988)
#>
interpret_hedges_g(-1.48, rules = "sawilowsky2009")
#> [1] "very large"
#> (Rules: sawilowsky2009)
#>
interpret_glass_delta(-1.48, rules = "gignac2016")
#> [1] "large"
#> (Rules: gignac2016)
#>
# Or:
interpret(d, rules = "sawilowsky2009")
#> Cohen's d | 95% CI | Interpretation
#> -------------------------------------------
#> -1.48 | [-2.27, -0.67] | very large
#>
#> - Estimated using pooled SD.
#> - Interpretation rule: sawilowsky2009
# Common Language Effect Sizes
d_to_cles(1.48)
#> $`Pr(superiority)`
#> [1] 0.852339
#>
#> $`Cohen's U3`
#> [1] 0.9305634
#>
#> $Overlap
#> [1] 0.4593
#>
# Or:
print(d, append_CLES = TRUE)
#> Cohen's d | 95% CI
#> --------------------------
#> -1.48 | [-2.27, -0.67]
#>
#> - Estimated using pooled SD.
#>
#> ## Common Language Effect Sizes:
#> Parameter | Coefficient | 95% CI
#> --------------------------------------------
#> Pr(superiority) | 0.15 | [0.05, 0.32]
#> Cohen's U3 | 0.07 | [0.01, 0.25]
#> Overlap | 0.46 | [0.26, 0.74]
# }
```