Propensity score matching is commonly used to estimate causal effects of treatments. However, when using data with a hierarchical structure, we need to take the multilevel nature of the data into account. In this thesis the estimation of propensity scores with multilevel models is presented to extend propensity score matching for use with multilevel data. A Monte Carlo simulation study is performed to evaluate several different estimators. It is shown that propensity score estimators ignoring the multilevel structure of the data are biased, while fixed effects models produce unbiased results. An empirical study of the causal effect of truancy on mathematical ability for Swedish 9th graders is also performed, where it is shown that truancy has a negative effect on mathematical ability.

Observational studies are an indispensable complement to randomized clinical trials (RCT) for comparison of treatment effectiveness. Often RCTs cannot be carried out due to the costs of the trial, ethical questions and rarity of the outcome. When noncompliance and missing data are prevalent, RCTs become more like observational studies. The main problem is to adjust for the selection bias in the observational study. One increasingly used method is propensity-score matching. Compared to traditional multi-covariate matching methods, matching on the propensity score alleviates the curse of dimensionality. It allows investigators to balance multiple covariate distributions between treatment groups by matching on a single score. This thesis focuses on the large sample properties of the matching estimators of the treatment effect. The first part of this thesis deals with problems of the analytic supports of the logit propensity score and various matching methods. The second part of this thesis focuses on the matching estimators of additive and multiplicative treatment effects. We derive the asymptotic order of the biases and asymptotic distributions of the matching estimators. We also derive the large sample variance estimators for the treatment effect estimators. The methods and theoretical results are applied and checked in a series of simulation studies. The third part of this thesis is devoted to a comparison between propensity-score matching and multiple linear regression using simulation.

Factorial designs have been widely used in many scientific and industrial settings, where it is important to distinguish "active'' or real factorial effects from "inactive" or noise factorial effects used to estimate residual or "error" terms. We propose a new approach to screen for active factorial effects from such experiments that utilizes the potential outcomes framework and is based on sequential posterior predictive model checks. One advantage of the proposed method lies in its ability to broaden the standard definition of active effects and to link their definition to the population of interest. Another important aspect of this approach is its conceptual connection to Fisherian randomization tests. As in the literature in design of experiments, the unreplicated case receives special attention and extensive simulation studies demonstrate the superiority of the proposed Bayesian approach over existing methods. The unreplicated case is also thoroughly explored. Extensions to three level and fractional factorial designs are discussed and illustrated using a classical seat belt example for the former and part of a stem-cell research collaborative project for the latter. / Statistics

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.

We examine three topics related to experimental design in this dissertation. Two are related to the analysis of experimental data and the other focuses on the design of paired comparison experiments, in this case knockout tournaments. The two analysis topics are motivated by how to estimate and test causal effects when the assignment mechanism fails to create balanced treatment groups. In Chapter 2, we apply conditional randomization tests to experiments where, through random chance, the treatment groups differ in their covariate distributions. In Chapter 4, we apply principal stratification to factorial experiments where the subjects fail to comply with their assigned treatment. The sources of imbalance differ, but, in both cases, ignoring the imbalance can lead to incorrect conclusions. In Chapter 3, we consider designing knockout tournaments to maximize different objectives given a prior distribution on the strengths of the players. These objectives include maximizing the probability the best player wins the tournament. Our emphasis on balance in the other two chapters comes from a desire to create a fair comparison between treatments. However, in this case, the design uses the prior information to intentionally bias the tournament in favor of the better players. / Statistics

Doctor of Philosophy / Department of Statistics / Gary Gadbury / In biomedical studies, the treatment main effect is often expressed in terms of an “average difference.” A treatment that appears superior based on the average effect may not be superior for all subjects in a population if there is substantial “subject-treatment interaction.” A parameter quantifying subject-treatment interaction is inestimable in two sample completely randomized designs. Crossover designs have been suggested as a way to estimate the variability in individual treatment effects since an “individual treatment effect” can be measured. However, variability in these observed individual effects may include variability due to the treatment plus inherent variability of a response over time. We use the “Neyman - Rubin Model of Causal Inference” (Neyman, 1923; Rubin, 1974) for analyses. This dissertation consists of two parts: The quantitative and qualitative response analyses. The quantitative part focuses on disentangling the variability due to treatment effects from variability due to time effects using suitable crossover designs. Next, we propose a variable that defines the variance of a true individual treatment effect in a two crossover designs and show that they are not directly estimable but the mean effect is estimable. Furthermore, we show the variance of individual treatment effects is biased under both designs. The bias depends on time effects. Under certain design considerations, linear combinations of time effects can be estimated, making it possible to separate the variability due to time from that due to treatment. The qualitative section involves a binary response and is centered on estimating the average treatment effect and bounding a probability of a negative effect, a parameter which relates to the individual treatment effect variability. Using a stated joint probability distribution of potential outcomes, we express the probability of the observed outcomes under a two treatment, two periods crossover design. Maximum likelihood estimates of these probabilities are found using an iterative numerical method. From these, we propose bounds for an inestimable probability of negative effect. Tighter bounds are obtained with information from subjects that receive the same treatments over the two periods. Finally, we simulate an example of observed count data to illustrate estimation of the bounds.

The aim of this thesis is to develop and explore models in, and related to, the sufficient cause framework, and additive interaction. Additive interaction is closely connected with public health interventions and can be used to make inferences about the sufficient causes in order to find the mechanisms behind an outcome, for instance a disease. In paper A we extend the additive interaction, and interventions, to include continuous exposures. We show that there does not exist a model that does not lead to inconsistent conclusions about the interaction. The sufficient cause framework can also be expressed using Boolean functions, which is expanded upon in paper B. In this paper we define a new model based on the multifactor potential outcome model (MFPO) and independence of causal influence models (ICI). In paper C we discuss the modeling and estimation of additive interaction in relation to if the exposures are harmful or protective conditioned on some other exposure. If there is uncertainty about the effects direction there can be errors in the testing of the interaction effect. / Målet med denna avhandling är att utveckla, och utforska modeller i det så kallade sufficent cause ramverket, och additiv interaktion. Additiv interaktion är nära kopplat till interventioner inom epidemiology och sociologi, men kan också användas för statistiska tester för sufficient causes för att förstå mekanimser bakom ett utfall, tex en sjukdom. I artikel A så expanderar vi modellen för additiv interaktion och interventioner till att också inkludera kontinuerliga variabler. Vi visar att det inte finns någon modell som inte leder till motsägelser i slutsatsen om interaktionen. Sufficient cause ramverket kan också utryckas via Boolska funktioner, vilket byggs vidare på i artikel B. I den artikeln definerar vi en modell baserad på mutltifactor potential outcome modellen (MFPO) och independence of causal influence modellen (ICI). I artikel C diskuterar vi modelleringen och estimering av additiv interaktion i relation till om variablerna har skadlig eller skyddande effekt betingat på någon annan variabel. Om det finns osäkerhet kring en effekts riktning så kan det leda till fel i testerna för den additiva interaktionen. / <p>Examinator: Professor Henrik Hult, Matematik, KTH</p>