Mediation is used to separate direct and indirect effects of an exposure variable on anoutcome variable. In this thesis, a mediation model is extended to account for censoredmediator and outcome variable. The two-part framework is used to account for thecensoring. The counterfactual based causal effects of this model are derived. A MonteCarlo study is performed to evaluate the behaviour of the causal effects accounting forcensoring, together with a comparison with methods for estimating the causal effectswithout accounting for censoring. The results of the Monte Carlo study show that theeffects accounting for censoring have substantially smaller bias when censoring is present.The proposed effects also seem to have a low cost with unbiased estimates for samplesizes as small as 100 for the two-part mediator model. In the case of limited mediatorand outcome, sample sizes larger than 300 is required for reliable improvements. A smallsensitivity analysis stresses the need of further development of the two-part models.

Estimating causal effects in clinical trials is often complicated by treatment noncompliance and missing outcomes. In time-to-event studies, estimation is further complicated by censoring. Censoring is a type of missing outcome, the mechanism of which may be non-ignorable. While new estimates have recently been proposed to account for noncompliance and missing outcomes, few studies have specifically considered time-to-event outcomes, where even the intention-to-treat (ITT) estimator is potentially biased for estimating causal effects of assigned treatment. In this thesis, we develop a series of parametric potential-outcome (PPO) survival models, for the analysis of randomised controlled trials (RCT) with time-to-event outcomes and noncompliance. Both ignorable and non-ignorable censoring mechanisms are considered. We approach model-fitting from a likelihood-based perspective, using the EM algorithm to locate maximum likelihood estimators. We are not aware of any previous work that addresses these complications jointly. In addition, we give new formulations for the average causal effect (ACE) and the complier average causal effect (CACE) to suit survival analysis. To illustrate the likelihood-based method proposed in this thesis, the HIP breast cancer trial data \citep{Baker98, Shapiro88} were re-analysed using specific PPO-survival models, the Weibull and log-normal based PPO-survival models, which assume that the failure time and censored time distributions both follow Weibull or log-normal distributions. Furthermore, an extended PPO-survival model is also derived in this thesis, which permits investigation into the impact of causal effect after accommodating certain pre-treatment covariates. This is an important contribution to the potential outcomes, survival and RCT literature. For comparison, the Frangakis-Rubin (F-R) model \citep{Frangakis99} is also applied to the HIP breast cancer trial data. To date, the F-R model has not yet been applied to any time-to-event data in the literature.

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>