Kravitz, Duan, and Braslow (2004) defined heterogeneity of treatment effects (HTE) as the standard deviation for the individual treatment effects (ITEs) across a target population, and provided the following formula for the HTE:
where SD denotes the pooled standard deviation of the outcome across treatment arms, A and B, and ρ denotes the correlation between the outcome for individuals under treatment A compared with treatment B.
Formula (1) is actually missing a term, and should be modified as follows:
the geometric mean for the variances of the outcomes across arms, and
the arithmetic mean for the variances of the outcomes across arms
More specifically, let YAi (YBi) denote the outcome for the i-th patient under treatment A (B), and VA (VB) denote the variance for the outcome across patients in the target population under treatment A (B). Then,
When the variance is the same under the two treatment conditions, i.e., when VA= VB, the factor c is one, the correct formula (2) simplifies to the wrong formula (1). When the variance differs between the two conditions, the factor c is smaller than one. Unless the two variances differ enormously, the factor c is usually pretty close to one. For example, if the two variances differ by fourfold, say, VA= 1 and VB= 4, we have VGM= 2, VAM= 2.5, thus c= VGM/VAM= 2/2.5 = 0.8. Therefore, even when the two variances differ as substantial as fourfold, the factor c is still pretty close to one, i.e., the wrong formula (1) is a reasonable approximation to the correct formula (2).
We have provided below the derivation for formula (2). The individual treatment effect for the i-th individual, ITEi, is given by
By our definition, HTE is the standard deviation for ITE across individuals, therefore
Formula (2) follows by taking the square root on the two sides of the above equation.