Treatment heterogeneity and potential outcomes in linear mixed effects models

Studies commonly focus on estimating a mean treatment effect in a population. However, in some applications the variability of treatment effects across individual units may help to characterize the overall effect of a treatment across the population. Consider a set of treatments, {T,C}, where T denotes some treatment that might be applied to an experimental unit and C denotes a control. For each of N experimental units, the duplet {rTᵢ,rCᵢ}, i = 1,2, … , N, represents the potential response of the i th experimental unit if treatment were applied and the response of the experimental unit if control were applied, respectively. The causal effect of T compared to C is the difference between the two potential responses, rTᵢ - rCᵢ. Much work has been done to elucidate the statistical properties of a causal effect, given a set of particular assumptions. Gadbury and others have reported on this for some simple designs and primarily focused on finite population randomization based inference. When designs become more complicated, the randomization based approach becomes increasingly difficult.