Solved – Cohen’s d for 2×2 interaction

cohens-deffect-sizeinteraction

I have a significant 2×2 interaction in a between-subjects design. I would normally report omega-squared as my effect-size measure, but I've been asked to provide Cohen's d instead? Can Cohen's d be computed for an interaction? If so, how?

Best Answer

You can translate pretty much any of the effect sizes to Cohen's d, so start with omega and work your way over. Equations are available for them all online... the wikipedia page for effect size is good for this.

In your particular case though you can get a numerator for cohen's d as (a1 - b1) - (a2 - b2). That's the simple effect of a 2x2 interaction with, a and b as the variables (e.g. sex and handedness) and the numbers are levels of the variables (e.g. male / female and left / right). Your denominator is just the square root of the MSE from the ANOVA*.

*assuming homogeneity of variance

Related Solutions

Effect Size Metrics – Difference Between Cohen’s d and Hedges’ g

Both Cohen's d and Hedges' g pool variances on the assumption of equal population variances, but g pools using n - 1 for each sample instead of n, which provides a better estimate, especially the smaller the sample sizes. Both d and g are somewhat positively biased, but only negligibly for moderate or larger sample sizes. The bias is reduced using g*. The d by Glass does not assume equal variances, so it uses the sd of a control group or baseline comparison group as the standardizer for the difference between the two means.

These effect sizes and Cliff's and other nonparametric effect sizes are discussed in detail in my book:

Grissom, R. J., & Kim, J, J. (2005). Effect sizes for research: A broad practical approach. Mahwah, NJ: Erlbaum.

R ANOVA Effect Size – How to Measure Omega Squared

A function to compute omega squared is straightforward to write. This function takes the object returned by the aov test, and calculates and returns and omega squared:

omega_sq <- function(aovm){
    sum_stats <- summary(aovm)[[1]]
    SSm <- sum_stats[["Sum Sq"]][1]
    SSr <- sum_stats[["Sum Sq"]][2]
    DFm <- sum_stats[["Df"]][1]
    MSr <- sum_stats[["Mean Sq"]][2]
    W2 <- (SSm-DFm*MSr)/(SSm+SSr+MSr)
    return(W2)
}

edit: updated function for n-way aov models:

omega_sq <- function(aov_in, neg2zero=T){
    aovtab <- summary(aov_in)[[1]]
    n_terms <- length(aovtab[["Sum Sq"]]) - 1
    output <- rep(-1, n_terms)
    SSr <- aovtab[["Sum Sq"]][n_terms + 1]
    MSr <- aovtab[["Mean Sq"]][n_terms + 1]
    SSt <- sum(aovtab[["Sum Sq"]])
    for(i in 1:n_terms){
        SSm <- aovtab[["Sum Sq"]][i]
        DFm <- aovtab[["Df"]][i]
        output[i] <- (SSm-DFm*MSr)/(SSt+MSr)
        if(neg2zero & output[i] < 0){output[i] <- 0}
    }
    names(output) <- rownames(aovtab)[1:n_terms]

    return(output)
}

Best Answer

Related Solutions

Effect Size Metrics – Difference Between Cohen’s d and Hedges’ g

R ANOVA Effect Size – How to Measure Omega Squared

Related Question