misty (version 0.7.0)

cohens.d: Cohen's d

Description

This function computes Cohen's d for one-sample, two-sample (i.e., between-subject design), and paired-sample designs (i.e., within-subject design) for one or more variables, optionally by a grouping and/or split variable. In a two-sample design, the function computes the standardized mean difference by dividing the difference between means of the two groups of observations by the weighted pooled standard deviation (i.e., Cohen's \(d_s\) according to Lakens, 2013) by default. In a paired-sample design, the function computes the standardized mean difference by dividing the mean of the difference scores by the standard deviation of the difference scores (i.e., Cohen's \(d_z\) according to Lakens, 2013) by default. Note that by default Cohen's d is computed without applying the correction factor for removing the small sample bias (i.e., Hedges' g).

Usage

cohens.d(x, ...)
# S3 method for default
cohens.d(x, y = NULL, mu = 0, paired = FALSE, weighted = TRUE, cor = TRUE,
         ref = NULL, correct = FALSE, alternative = c("two.sided", "less", "greater"),
         conf.level = 0.95, group = NULL, split = NULL, sort.var = FALSE,
         digits = 2, as.na = NULL, write = NULL, append = TRUE,
         check = TRUE, output = TRUE, ...)
# S3 method for formula
cohens.d(formula, data, weighted = TRUE, cor = TRUE, ref = NULL,
         correct = FALSE, alternative = c("two.sided", "less", "greater"),
         conf.level = 0.95, group = NULL, split = NULL, sort.var = FALSE,
         na.omit = FALSE, digits = 2, as.na = NULL, write = NULL, append = TRUE,
         check = TRUE, output = TRUE, ...)

Value

Returns an object of class misty.object, which is a list with following entries:

call

function call

type

type of analysis

sample

type of sample, i.e., one-, two-, or, paired-sample

data

matrix or data frame specified in x

args

specification of function arguments

result

result table

Arguments

x

a numeric vector or data frame.

...

further arguments to be passed to or from methods.

y

a numeric vector.

mu

a numeric value indicating the reference mean.

paired

logical: if TRUE, Cohen's d for a paired-sample design is computed.

weighted

logical: if TRUE (default), the weighted pooled standard deviation is used to compute the standardized mean difference between two groups of a two-sample design (i.e., paired = FALSE), while standard deviation of the difference scores is used to compute the standardized mean difference in a paired-sample design (i.e., paired = TRUE).

cor

logical: if TRUE (default), paired = TRUE, and weighted = FALSE, Cohen's d for a paired-sample design while controlling for the correlation between the two sets of measurement is computed. Note that this argument is only used in a paired-sample design (i.e., paired = TRUE) when specifying weighted = FALSE.

ref

character string "x" or "y" for specifying the reference reference group when using the default cohens.d() function or a numeric value or character string indicating the reference group in a two-sample design when using the formula cohens.d() function. The standard deviation of the reference variable or reference group is used to standardized the mean difference. Note that this argument is only used in a two-sample design (i.e., paired = FALSE).

correct

logical: if TRUE, correction factor to remove positive bias in small samples is used.

alternative

a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less".

conf.level

a numeric value between 0 and 1 indicating the confidence level of the interval.

group

a numeric vector, character vector or factor as grouping variable.

split

a numeric vector, character vector or factor as split variable.

sort.var

logical: if TRUE, output table is sorted by variables when specifying group.

digits

an integer value indicating the number of decimal places to be used for displaying results.

as.na

a numeric vector indicating user-defined missing values, i.e. these values are converted to NA before conducting the analysis. Note that as.na() function is only applied to y but not to group in a two-sample design, while as.na() function is applied to pre and post in a paired-sample design.

write

a character string naming a text file with file extension ".txt" (e.g., "Output.txt") for writing the output into a text file.

append

logical: if TRUE (default), output will be appended to an existing text file with extension .txt specified in write, if FALSE existing text file will be overwritten.

check

logical: if TRUE (default), argument specification is checked.

output

logical: if TRUE (default), output is shown on the console.

formula

a formula of the form y ~ group for one outcome variable or cbind(y1, y2, y3) ~ group for more than one outcome variable where y is a numeric variable giving the data values and group a numeric variable, character variable or factor with two values or factor levels giving the corresponding groups.

data

a matrix or data frame containing the variables in the formula formula.

na.omit

logical: if TRUE, incomplete cases are removed before conducting the analysis (i.e., listwise deletion) when specifying more than one outcome variable.

Author

Takuya Yanagida

Details

Cohen (1988, p.67) proposed to compute the standardized mean difference in a two-sample design by dividing the mean difference by the unweighted pooled standard deviation (i.e., weighted = FALSE).

Glass et al. (1981, p. 29) suggested to use the standard deviation of the control group (e.g., ref = 0 if the control group is coded with 0) to compute the standardized mean difference in a two-sample design (i.e., Glass's \(\Delta\)) since the standard deviation of the control group is unaffected by the treatment and will therefore more closely reflect the population standard deviation.

Hedges (1981, p. 110) recommended to weight each group's standard deviation by its sample size resulting in a weighted and pooled standard deviation (i.e., weighted = TRUE, default). According to Hedges and Olkin (1985, p. 81), the standardized mean difference based on the weighted and pooled standard deviation has a positive small sample bias, i.e., standardized mean difference is overestimated in small samples (i.e., sample size less than 20 or less than 10 in each group). However, a correction factor can be applied to remove the small sample bias (i.e., correct = TRUE). Note that the function uses a gamma function for computing the correction factor, while a approximation method is used if computation based on the gamma function fails.

Note that the terminology is inconsistent because the standardized mean difference based on the weighted and pooled standard deviation is usually called Cohen's d, but sometimes called Hedges' g. Oftentimes, Cohen's d is called Hedges' d as soon as the small sample correction factor is applied. Cumming and Calin-Jageman (2017, p.171) recommended to avoid the term Hedges' g , but to report which standard deviation was used to standardized the mean difference (e.g., unweighted/weighted pooled standard deviation, or the standard deviation of the control group) and whether a small sample correction factor was applied.

As for the terminology according to Lakens (2013), in a two-sample design (i.e., paired = FALSE) Cohen's \(d_s\) is computed when using weighted = TRUE (default) and Hedges's \(g_s\) is computed when using correct = TRUE in addition. In a paired-sample design (i.e., paired = TRUE), Cohen's \(d_z\) is computed when using weighted = TRUE, default, while Cohen's \(d_{rm}\) is computed when using weighted = FALSE and cor = TRUE, default and Cohen's \(d_{av}\) is computed when using weighted = FALSE and cor = FALSE. Corresponding Hedges' \(g_z\), \(g_{rm}\), and \(g_{av}\) are computed when using correct = TRUE in addition.

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Academic Press.

Cumming, G., & Calin-Jageman, R. (2017). Introduction to the new statistics: Estimation, open science, & beyond. Routledge.

Glass. G. V., McGaw, B., & Smith, M. L. (1981). Meta-analysis in social research. Sage Publication.

Goulet-Pelletier, J.-C., & Cousineau, D. (2018) A review of effect sizes and their confidence intervals, Part I: The Cohen's d family. The Quantitative Methods for Psychology, 14, 242-265. https://doi.org/10.20982/tqmp.14.4.p242

Hedges, L. V. (1981). Distribution theory for Glass's estimator of effect size and related estimators. Journal of Educational Statistics, 6(3), 106-128.

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

Lakens, D. (2013). Calculating and reporting effect sizes to facilitate cumulative science: A practical primer for t-tests and ANOVAs. Frontiers in Psychology, 4, 1-12. https://doi.org/10.3389/fpsyg.2013.00863

See Also

test.t, test.z, effsize, cor.matrix, na.auxiliary

Examples

#----------------------------------------------------------------------------
# One-sample design

# Example 1a: Cohen's d.z with two-sided 95% CI
# population mean = 3
cohens.d(mtcars$mpg, mu = 20)

# Example 1b: Cohen's d.z (aka Hedges' g.z) with two-sided 95% CI
# population mean = 3, with small sample correction factor
cohens.d(mtcars$mpg, mu = 20, correct = TRUE)

# Example 1c: Cohen's d.z with two-sided 95% CI
# population mean = 3, by 'vs' separately
cohens.d(mtcars$mpg, mu = 20, group = mtcars$vs)

# Example 1d: Cohen's d.z with two-sided 95% CI
# population mean = 20, split analysis by 'vs'
cohens.d(mtcars$mpg, mu = 20, split = mtcars$vs)

# Example 1e: Cohen's d.z with two-sided 95% CI
# population mean = 3, by 'vs' separately, split by 'am'
cohens.d(mtcars$mpg, mu = 20, group = mtcars$vs, split = mtcars$am)

#----------------------------------------------------------------------------
# Two-sample design

# Example 2a: Cohen's d.s with two-sided 95% CI
# weighted pooled SD
cohens.d(mpg ~ vs, data = mtcars)

# Example 2b: Cohen's d.s with two-sided 99% CI
# weighted pooled SD
cohens.d(mpg ~ vs, data = mtcars, conf.level = 0.99)

# Example 2c: Cohen's d.s with one-sided 99% CI
# weighted pooled SD
cohens.d(mpg ~ vs, data = mtcars, alternative = "greater", conf.level = 0.99)

# Example 2d: Cohen's d.s for more than one variable with two-sided 95% CI
# weighted pooled SD
cohens.d(cbind(mpg, disp, hp) ~ vs, data = mtcars)

# Example 2e: Cohen's d with two-sided 95% CI
# unweighted SD
cohens.d(mpg ~ vs, data = mtcars, weighted = FALSE)

# Example 2f: Cohen's d.s (aka Hedges' g.s) with two-sided 95% CI
# weighted pooled SD, with small sample correction factor
cohens.d(mpg ~ vs, data = mtcars, correct = TRUE)

# Example 2g: Cohen's d (aka Hedges' g) with two-sided 95% CI
# Unweighted SD, with small sample correction factor
cohens.d(mpg ~ vs, data = mtcars, weighted = FALSE, correct = TRUE)

# Example 2h: Cohen's d (aka Glass's delta) with two-sided 95% CI
# SD of reference group 1
cohens.d(mpg ~ vs, data = mtcars, ref = 0)

# Example 2i: Cohen's d.s with two-sided 95% CI
# weighted pooled SD, by 'am' separately
cohens.d(mpg ~ vs, data = mtcars, group = mtcars$am)

# Example 2j: Cohen's d.s with two-sided 95% CI
# weighted pooled SD, split analysis by 'am'
cohens.d(mpg ~ vs, data = mtcars, split = mtcars$am)

#----------------------------------------------------------------------------
# Paired-sample design

# Example 3a: Cohen's d.z with two-sided 95% CI
# SD of the difference scores
cohens.d(mtcars$drat, mtcars$wt, paired = TRUE)

# Example 3b: Cohen's d.z with one-sided 99% CI
# SD of the difference scores
cohens.d(mtcars$drat, mtcars$wt, paired = TRUE, alternative = "greater",
         conf.level = 0.99)

# Example 3c: Cohen's d.rm with two-sided 95% CI
# controlling for the correlation between measures
cohens.d(mtcars$drat, mtcars$wt, paired = TRUE, weighted = FALSE)

# Example 3d: Cohen's d.av with two-sided 95% CI
# without controlling for the correlation between measures
cohens.d(mtcars$drat, mtcars$wt, paired = TRUE, weighted = FALSE, cor = FALSE)

# Example 3e: Cohen's d.z (aka Hedges' g.z) with two-sided 95% CI
# SD of the differnece scores
cohens.d(mtcars$drat, mtcars$wt, paired = TRUE, correct = TRUE)

# Example 3f: Cohen's d.rm (aka Hedges' g.rm) with two-sided 95% CI
# controlling for the correlation between measures
cohens.d(mtcars$drat, mtcars$wt, paired = TRUE, weighted = FALSE, correct = TRUE)

# Example 3g: Cohen's d.av (aka Hedges' g.av) with two-sided 95% CI
# without controlling for the correlation between measures
cohens.d(mtcars$drat, mtcars$wt, paired = TRUE, weighted = FALSE, cor = FALSE,
         correct = TRUE)

# Example 3h: Cohen's d.z with two-sided 95% CI
# SD of the difference scores, by 'vs' separately
cohens.d(mtcars$drat, mtcars$wt, paired = TRUE, group = mtcars$vs)

# Example 3i:  Cohen's d.z with two-sided 95% CI
# SD of the difference scores, split analysis by 'vs'
cohens.d(mtcars$drat, mtcars$wt, paired = TRUE, split = mtcars$vs)