MKdescr (version 0.8)

SMD: Compute Standardized Mean Difference (SMD)

Description

The function computes the standardized mean difference, where a bias correction can be applied.

Usage

SMD(x, y, bias.cor = TRUE, var.equal = FALSE, na.rm = FALSE)

Value

SMD value.

Arguments

x

numeric vector, data of group 1.

y

numeric vector, data of group 2.

bias.cor

a logical variable indicating whether a bias correction should be performed.

var.equal

a logical variable indicating whether to treat the two variances as being equal. If TRUE then the pooled variance is used to estimate the variance otherwise the Welch-Satterthwaite approximation is used.

na.rm

logical. Should missing values be removed?

Author

Matthias Kohl

Details

The function compute the (bias-corrected) standardized mean difference.

If bias.cor = FALSE and var.equal = TRUE, the result corresponds to Cohen's d (Cohen (1988)).

If bias.cor = TRUE and var.equal = TRUE, the result corresponds to Hedges' g (Hedges (1981)).

If bias.cor = FALSE and var.equal = FALSE, the result is closely related to the test statistic of Welch's t test (Aoki (2020)).

If bias.cor = TRUE and var.equal = FALSE, the result corresponds to Aoki's e (Aoki (2020)) which incorporates a Welch-Satterthwaite approximation in combination with a bias correction.

References

Aoki, S. (2020). Effect sizes of the differences between means without assuming variance equality and between a mean and a constant. Heliyon, 6(1), e03306.

Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences. Routledge. ISBN 978-1-134-74270-7.

Hedges, L. V. (1981). Distribution theory for Glass's estimator of effectsize and related estimators. Journal of Educational Statistics 6, 107-128.

Examples

n1 <- 200
x <- rnorm(n1)
n2 <- 300
y <- rnorm(n2, mean = 3, sd = 2)
## true value
(0-3)/sqrt((1 + n1/n2*2^2)/(n1/n2+1))
## estimates
## Aoki's e
SMD(x, y)
## Hedges' g
SMD(x, y, var.equal = TRUE)
## standardized test statistic of Welch's t-test
SMD(x, y, bias.cor = FALSE)
## Cohen's d
SMD(x, y, bias.cor = FALSE, var.equal = TRUE)

## Example from Aoki (2020)
SMD(0:4, c(0, 0, 1, 2, 2))
SMD(0:4, c(0, 0, 1, 2, 2), var.equal = TRUE)
SMD(0:4, c(0, 0, 1, 2, 2), bias.cor = FALSE)
SMD(0:4, c(0, 0, 1, 2, 2), bias.cor = FALSE, var.equal = TRUE)