Analysis of change-point in covariate effects in survival models

Abstract

This thesis focuses on the statistical analysis of survival data where the effect of one continuous explanatory variable on the response function is not linear as in standard generalized linear models. It has wide applications in the field of clinical medicine and epidemiology. The thesis comprises three parts, each devoted to resolving a statistical problem on nonlinear effects encountered in medical applications.

The first part of this thesis focuses on addressing the nuisance nonlinear effect of a continuous explanatory variable. For example, age is known to be an important prognostic factor in many cancer studies. The age effect on the risk is more or less constant and stays at a low level for young patients aged 60 or below, but it increases with age for patients aged above 60. While the primary interest of most cancer studies is the treatment effect, a valid statistical inference procedure, that appropriately adjusts for the nonlinear effect of the nuisance factor age, is warranted. A class of partly linear models is considered to address this problem.

The nuisance nonlinear effect can be estimated nonparametrically using polynomial splines as in most partly linear models. In medicine, the severity of some diseases are often characterized by some proxy measures, like obesity characterized by BMI, cardiovascular diseases characterized by diastolic blood pressure, diabetes mellitus characterized by blood sugar level. The disease status classifications are for preliminary diagnostic purposes based on previous experience that subjects with a measurement exceeding the threshold value is of strong evidence of having the disease. These threshold values are often called change-points in statistical models in a sense that the trends of the effects of measurements are very different before and after the change-points. Unlike the first part, the change-point associated factor is not nuisance while statistical inference on the change-points is the main objective of the study. The second part of the thesis probes into testing for the existence of a change-point in the effect of a continuous surrogate measurement. Three tests are proposed and their performance are studied both theoretically and empirically.

For multi-level disease status classifications, the work above is extended to study the multiple change-points problem. For example, normal fasting blood glucose level for non-diabetics is between the two change-points 70 and 130 mg/dL. Confidence interval estimation method for the change-point locations is not available in the literature. The third part of the thesis aims to fill the research gap by proposing a sequential test for determining the optimal number of change-points in the effect of the surrogate measurement on the risk function. Given the optimal number of change-points, a bootstrap method is proposed for confidence interval estimation of the change-point locations.