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: list with the input specified in x, group, and split
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 takuya.yanagida@univie.ac.at

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

Examples

Run this code

dat1 <- data.frame(group1 = c(1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2,
                              1, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1),
                   group2 = c(1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2,
                              1, 2, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2),
                   group3 = c(1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1,
                              1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1),
                   x1 = c(3, 2, 5, 3, 6, 3, 2, 4, 6, 5, 3, 3, 5, 4,
                          4, 3, 5, 3, 2, 3, 3, 6, 6, 7, 5, 6, 6, 4),
                   x2 = c(4, 4, 3, 6, 4, 7, 3, 5, 3, 3, 4, 2, 3, 6,
                          3, 5, 2, 6, 8, 3, 2, 5, 4, 5, 3, 2, 2, 4),
                   x3 = c(7, 6, 5, 6, 4, 2, 8, 3, 6, 1, 2, 5, 8, 6,
                          2, 5, 3, 1, 6, 4, 5, 5, 3, 6, 3, 2, 2, 4))

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

# Example 1: Cohen's d.z with two-sided 95% CI
# population mean = 3
cohens.d(dat1$x1, mu = 3)

# Example 2: Cohen's d.z (aka Hedges' g.z) with two-sided 95% CI
# population mean = 3, with small sample correction factor
cohens.d(dat1$x1, mu = 3, correct = TRUE)

# Example 3: Cohen's d.z for more than one variable with two-sided 95% CI
# population mean = 3
cohens.d(dat1[, c("x1", "x2", "x3")], mu = 3)

# Example 4: Cohen's d.z with two-sided 95% CI
# population mean = 3, by group1 separately
cohens.d(dat1$x1, mu = 3, group = dat1$group1)

# Example 5: Cohen's d.z for more than one variable with two-sided 95% CI
# population mean = 3, by group1 separately
cohens.d(dat1[, c("x1", "x2", "x3")], mu = 3, group = dat1$group1)

# Example 6: Cohen's d.z with two-sided 95% CI
# population mean = 3, split analysis by group1
cohens.d(dat1$x1, mu = 3, split = dat1$group1)

# Example 7: Cohen's d.z for more than one variable with two-sided 95% CI
# population mean = 3, split analysis by group1
cohens.d(dat1[, c("x1", "x2", "x3")], mu = 3, split = dat1$group1)

# Example 8: Cohen's d.z with two-sided 95% CI
# population mean = 3, by group1 separately1, split by group2
cohens.d(dat1$x1, mu = 3, group = dat1$group1, split = dat1$group2)

# Example 9: Cohen's d.z for more than one variable with two-sided 95% CI
# population mean = 3, by group1 separately1, split by group2
cohens.d(dat1[, c("x1", "x2", "x3")], mu = 3, group = dat1$group1,
         split = dat1$group2)

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

# Example 10: Cohen's d.s with two-sided 95% CI
# weighted pooled SD
cohens.d(x1 ~ group1, data = dat1)

# Example 11: Cohen's d.s with two-sided 99% CI
# weighted pooled SD
cohens.d(x1 ~ group1, data = dat1, conf.level = 0.99)

# Example 12: Cohen's d.s with one-sided 99% CI
# weighted pooled SD
cohens.d(x1 ~ group1, data = dat1, alternative = "greater")

# Example 13: Cohen's d.s with two-sided 99% CI
# weighted pooled SD
cohens.d(x1 ~ group1, data = dat1, conf.level = 0.99)

# Example 14: Cohen's d.s with one-sided 95%% CI
# weighted pooled SD
cohens.d(x1 ~ group1, data = dat1, alternative = "greater")

# Example 15: Cohen's d.s for more than one variable with two-sided 95% CI
# weighted pooled SD
cohens.d(cbind(x1, x2, x3) ~ group1, data = dat1)

# Example 16: Cohen's d with two-sided 95% CI
# unweighted SD
cohens.d(x1 ~ group1, data = dat1, weighted = FALSE)

# Example 17: Cohen's d.s (aka Hedges' g.s) with two-sided 95% CI
# weighted pooled SD, with small sample correction factor
cohens.d(x1 ~ group1, data = dat1, correct = TRUE)

# Example 18: Cohen's d (aka Hedges' g) with two-sided 95% CI
# Unweighted SD, with small sample correction factor
cohens.d(x1 ~ group1, data = dat1, weighted = FALSE, correct = TRUE)

# Example 19: Cohen's d (aka Glass's delta) with two-sided 95% CI
# SD of reference group 1
cohens.d(x1 ~ group1, data = dat1, ref = 1)

# Example 20: Cohen's d.s with two-sided 95% CI
# weighted pooled SD, by group2 separately
cohens.d(x1 ~ group1, data = dat1, group = dat1$group2)

# Example 21: Cohen's d.s for more than one variable with two-sided 95% CI
# weighted pooled SD, by group2 separately
cohens.d(cbind(x1, x2, x3) ~ group1, data = dat1, group = dat1$group2)

# Example 22: Cohen's d.s with two-sided 95% CI
# weighted pooled SD, split analysis by group2
cohens.d(x1 ~ group1, data = dat1, split = dat1$group2)

# Example 23: Cohen's d.s for more than one variable with two-sided 95% CI
# weighted pooled SD, split analysis by group2
cohens.d(cbind(x1, x2, x3) ~ group1, data = dat1, split = dat1$group2)

# Example 24: Cohen's d.s with two-sided 95% CI
# weighted pooled SD, by group2 separately, split analysis by group3
cohens.d(x1 ~ group1, data = dat1,
         group = dat1$group2, split = dat1$group3)

# Example 25: Cohen's d.s for more than one variable with two-sided 95% CI
# weighted pooled SD, by group2 separately, split analysis by group3
cohens.d(cbind(x1, x2, x3) ~ group1, data = dat1,
         group = dat1$group2, split = dat1$group3)

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

# Example 26: Cohen's d.z with two-sided 95% CI
# SD of the difference scores
cohens.d(dat1$x1, dat1$x2, paired = TRUE)

# Example 27: Cohen's d.z with two-sided 99% CI
# SD of the difference scores
cohens.d(dat1$x1, dat1$x2, paired = TRUE, conf.level = 0.99)

# Example 28: Cohen's d.z with one-sided 95% CI
# SD of the difference scores
cohens.d(dat1$x1, dat1$x2, paired = TRUE, alternative = "greater")

# Example 29: Cohen's d.rm with two-sided 95% CI
# controlling for the correlation between measures
cohens.d(dat1$x1, dat1$x2, paired = TRUE, weighted = FALSE)

# Example 30: Cohen's d.av with two-sided 95% CI
# without controlling for the correlation between measures
cohens.d(dat1$x1, dat1$x2, paired = TRUE, weighted = FALSE, cor = FALSE)

# Example 31: Cohen's d.z (aka Hedges' g.z) with two-sided 95% CI
# SD of the differnece scores
cohens.d(dat1$x1, dat1$x2, paired = TRUE, correct = TRUE)

# Example 32: Cohen's d.rm (aka Hedges' g.rm) with two-sided 95% CI
# controlling for the correlation between measures
cohens.d(dat1$x1, dat1$x2, paired = TRUE, weighted = FALSE, correct = TRUE)

# Example 33: Cohen's d.av (aka Hedges' g.av) with two-sided 95% CI
# without controlling for the correlation between measures
cohens.d(dat1$x1, dat1$x2, paired = TRUE, weighted = FALSE, cor = FALSE,
         correct = TRUE)

# Example 34: Cohen's d.z with two-sided 95% CI
# SD of the difference scores, by group1 separately
cohens.d(dat1$x1, dat1$x2, paired = TRUE, group = dat1$group1)

# Example 35:  Cohen's d.z with two-sided 95% CI
# SD of the difference scores, split analysis by group1
cohens.d(dat1$x1, dat1$x2, paired = TRUE, split = dat1$group1)

# Example 36: Cohen's d.z with two-sided 95% CI
# SD of the difference scores, by group1 separately, split analysis by group2
cohens.d(dat1$x1, dat1$x2, paired = TRUE,
         group = dat1$group1, split = dat1$group2)

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