Effect Modification and Interaction

Effect-Measure Modification (Heterogeneity)

To make the concepts and distinctions precise, suppose we are studying the effects that changes in a variable X will have on a subsequent variable Y, in the presence of a background variable Z that precedes X and Y. For example, X might be treatment level such as dose or treatment arm, Y might be a health outcome variable such as life expectancy following treatment, and Z might be sex (1 = female,0 = male). To measure effects, write Yx for the outcome one would have if administered treatment level x of X; for example, if X = 1 for active treatment, X = 0 for placebo, then Y1 is the outcome a subject will have if X = 1is administered, and Y0 is the outcome a subject will have if X = 0 is administered. The Yx are often called potential outcomes.

One measure of the effect of changing X from 0 to 1 on the outcome is the difference Y1 − Y0; for example, if Y were life expectancy, Y1 − Y0 would be the change in life expectancy. If this difference varied with sex in a systematic fashion, one could say that the difference was modified by sex, or that there was heterogeneity of the difference across sex. Another common measure of effect is the ratio Y1=Y0; if this ratio varied with sex in a systematic fashion, one could say that the ratio was modified by sex.

For purely algebraic reasons, two measures may be modified in very different ways by the same variable. Furthermore, if both X and Z affect Y, absence of modification of the difference implies modification of the ratio, and vice versa. As a simple example, suppose for the subjects under study Y1 = 20 and Y0 = 10 for all the males, but Y1 = 30 and Y0 = 15 for all the females. Then Y1 − Y0 = 10 for males but Y1 − Y0 = 15 for females, so there is a 5-year modification of the [Page 304]difference measure by sex. But suppose we measured the effects by expectancy ratios Y1/Y0, instead of differences. Then Y1/Y0 = 20= 10 = 2 for males and Y1/Y0 = 30= 15 = 2 for females as well, so there is no modification of the ratio measure by sex.

Consider next an example in which Y1 = 20 and Y0 = 10 for all the males, and Y1 = 30 and Y0 = 20 for all the females. Then Y1 − Y0 = 10 for both males and females, so there is no modification of the difference by sex. But Y1/Y0 = 20/10 = 2 for males and Y1/Y0 = 30/20 = 1.5 for females, so there is modification of the ratio by sex.

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