Analysis of non-linear covariates effects and temporal treatment effect in Cox-type models

Abstract

This thesis focuses on the statistical analysis of time to event data that the effects of one or more continuous explanatory variables are not linear or the treatment effect varies over time, say the waning efficacy of some chemoprevention interventions. Standard Cox proportional hazards model cannot be applied to those situations. Modifications of the standard Cox model to accommodate the above situations are proposed.

Motivated by a breast cancer data set, we first consider the estimation of the potentially non-linear age effect based on the generalized partly linear survival models. Appropriate adjustment of the non-linear age effects is warranted to ensure a valid statistical inference on other fixed effects. A simple and efficient sieve maximum likelihood estimation method that can be implemented easily using any standard statistical software is proposed. A data-driven algorithm to determine the optimal number and location of the knots in the estimation of the non-linear age effect is adopted. This algorithm is able to identify some possible change points where the investigated covariate effect is very different before and after these points. The performance of the proposed method is evaluated by simulation studies. For illustration purpose, the method is applied to the breast cancer data set from the public domain to study the non-linear effects of age-at-onset on the disease free survival of the patients.

The next problem considered is the estimation of a time-varying treatment effect probably due to waning of the treatment efficacy. Two special features are attached to this special problem. The first one is the possibility of multiple episodes of the disease from the same subject over time, leading to recurrent events nature of the data. The second one is that the treatment is administered intermittently to the subjects to offer protection for the control of infectious disease. The continual administration of the treatment is mainly because of the waning efficacy of the treatment. The primary goal of this study is to estimate the time-varying treatment effect, which generally declines over time. One main objective of this study is to provide a method to choose the optimal interval between two consecutive supplementary treatments (boosters) to maintain a high level of protection to the subjects at all time. Another important question is to determine whether the intervention will have a harmful effect to the subjects in the long run. Both the fully parametric time-varying treatment effect and the fully nonparametric treatment effect are considered based on the Andersen-Gill type Cox model for recurrent data. The partial likelihood approach is applicable to estimate the parameters. Furthermore, intra-class or within-subject correlation may not be ignorable in clinical studies with recurrent event or clustered data. The marginal approach is considered. To ensure a valid statistical inference on the fixed effects, robust variance estimate is proposed to adjust for the dependent nature of the recurrent event data. The method is applied to data from a phase III clinical trial for malaria control.