Doctor of Philosophy / Department of Statistics / Gary L. Gadbury / The potential for high variability in treatment effects across individuals has been recognized as an important consideration in clinical studies. Surprisingly, little attention has been given to evaluating this variability in design of clinical trials or analyses of resulting data. High variation in a treatment’s efficacy or safety across individuals (referred to herein as treatment heterogeneity) may have important consequences because the optimal treatment choice for an individual may be different from that suggested by a study of average effects. We call this an individual qualitative interaction (IQI), borrowing terminology from earlier work - referring to a qualitative interaction (QI) being present when the optimal treatment varies across ‘groups’ of individuals. At least three techniques have been proposed to investigate treatment heterogeneity: techniques to detect a QI, use of measures such as the density overlap of two outcome variables under different treatments, and use of cross-over designs to observe ‘individual effects.’ Connections, limitations, and the required assumptions are compared among these techniques through a quantity frequently referred to as subject-treatment (S-T) interaction, but shown here to be the probability of an IQI (PIQI). Their association is studied utilizing a potential outcomes framework that can add insights to results from usual data analyses and to study design features to more directly assess treatment heterogeneity. Particular attention is given to the density overlap of two outcome variables, each representing an individual’s ‘potential’ response under a different treatment. Connections are made between the overlap quantified as the proportion of similar responses (PSR) and the PIQI. Given a bivariate normal model, the maximum PIQI is shown to be an upper bound for ½ the PSR. Additionally, the characterization of a conditional PSR allows for the PIQI boundaries to be developed within subgroups defined over observable covariates so that the subset contribution to treatment heterogeneity may be identified. The possibility of similar boundaries is explored outside the normal model using the skew normal distribution. Furthermore, a bivariate PIQI is developed along with its PSR counterpart to help characterize treatment heterogeneity resulting from a bivariate response such as the efficacy and safety of a treatment.

Doctor of Philosophy / Department of Statistics / Gary L. Gadbury / 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 {r[subscript]i, r[subscript]Ci}, i=1,2,…,N, represents the potential response of the i[superscript]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, r[subscript]Ti- r[subscript]Ci. 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. Since linear mixed effects models are particularly useful for modeling data from complex designs, their role in modeling treatment heterogeneity is investigated. It is shown that an individual treatment effect can be conceptualized as a linear combination of fixed treatment effects and random effects. The random effects are assumed to have variance components specified in a mixed effects “potential outcomes” model when both potential outcomes, r[subscript]T,r[subscript]C, are variables in the model. The variance of the individual causal effect is used to quantify treatment heterogeneity. Post treatment assignment, however, only one of the two potential outcomes is observable for a unit. It is then shown that the variance component for treatment heterogeneity becomes non-estimable in an analysis of observed data. Furthermore, estimable variance components in the observed data model are demonstrated to arise from linear combinations of the non-estimable variance components in the potential outcomes model. Mixed effects models are considered in context of a particular design in an effort to illuminate the loss of information incurred when moving from a potential outcomes framework to an observed data analysis.