View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by D-Scholarship@Pitt REGRESSION ON MEDIAN RESIDUAL LIFE FUNCTION FOR CENSORED SURVIVAL DATA by Hanna Bandos M.S., V.N. Karazin Kharkiv National University, 2000 Submitted to the Graduate Faculty of The Department of Biostatistics Graduate School of Public Health in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 2007 Bp, UNIVERSITY OF PITTSBURGH Graduate School of Public Health This dissertation was presented by Hanna Bandos It was defended on July 26, 2007 and approved by Dissertation Advisor: Jong-Hyeon Jeong, PhD Associate Professor Biostatistics Graduate School of Public Health University of Pittsburgh Dissertation Co-Advisor: Joseph P. Costantino, DrPH Professor Biostatistics Graduate School of Public Health University of Pittsburgh Janice S. Dorman, MS, PhD Professor Health Promotion & Development School of Nursing University of Pittsburgh Howard E. Rockette, PhD Professor Biostatistics Graduate School of Public Health University of Pittsburgh ii Copyright © by Hanna Bandos 2007 iii REGRESSION ON MEDIAN RESIDUAL LIFE FUNCTION FOR CENSORED SURVIVAL DATA Hanna Bandos, PhD University of Pittsburgh, 2007 In the analysis of time-to-event data, the median residual life (MERL) function has been promoted by many researchers as a practically relevant summary of the residual life distribution. Formally the MERL function at a time point is defined as the median of the remaining lifetimes among survivors beyond that particular time point. Despite its widely recognized usefulness, there is no commonly accepted approach to model the median residual life function. In this dissertation we introduce two novel regression techniques that model the relationship between the MERL function and covariates of interest at multiple time points simultaneously; proportional median residual life model and accelerated median residual life model. These models have a conceptual similarity to the well-known proportional hazards and accelerated failure time (AFT) models. Inference procedures that we propose for these models permit the data to be right censored. For the semiparametric analysis under the proportional MERL model, we propose an estimating equation for the regression coefficients. The bootstrap resampling technique is utilized to evaluate the standard errors of the regression coefficient estimates. A simulation study is performed to investigate the proposed inferential approach. The developed method is applied to a real data example from a breast cancer study conducted by the National Surgical Adjuvant Breast and Bowel Project (NSABP). We also propose parametric and semiparametric (under the AFT assumption) inference procedures under the accelerated MERL model. The maximum likelihood inference is iv considered for the parametric inference and the Buckley and James method is used to estimate the median residual lifetimes semiparametrically under the AFT assumption. A simulation study is performed to validate the proposed maximum likelihood inference procedure. A generated dataset is used to illustrate statistical analysis via both estimation approaches. It is very important from a public health perspective to be able to identify the risk factors for a specific disease or condition. The regression techniques presented in this work enable researchers to identify the patients’ characteristics that affect their survival experience and describe advantages of a preventive or therapeutic intervention by means of median residual life function in a clinically relevant and intuitively appealing way. v TABLE OF CONTENTS ACKNOWLEDGEMENT........................................................................................................... X 1.0 INTRODUCTION........................................................................................................ 1 1.1 PROPORTIONAL MEDIAN RESIDUAL LIFE MODEL............................. 2 1.2 ACCELERATED MEDIAN RESIDUAL LIFE MODEL............................... 3 2.0 BACKGROUND .......................................................................................................... 5 2.1 MEDIAN RESIDUAL LIFE FUNCTION ........................................................ 5 2.1.1 Overview ........................................................................................................ 5 2.1.2 Definition and properties ............................................................................. 6 2.1.3 Estimation of the MERL function............................................................... 8 2.2 REGRESSION MODELS ON SURVIVAL DATA........................................ 10 2.2.1 Cox proportional hazards model............................................................... 11 2.2.2 Accelerated failure time model.................................................................. 11 2.2.3 Regression model for a simple median and other related techniques.... 12 3.0 PROPORTIONAL MEDIAN RESIDUAL LIFE MODEL ................................... 15 3.1 PROPORTIONAL MEDIAN RESIDUAL MODEL ..................................... 16 3.1.1 Model description and estimating equations............................................ 16 3.1.2 Estimating procedure ................................................................................. 19 3.1.3 Estimation interval...................................................................................... 22 vi 3.1.4 Variance of the parameter estimates......................................................... 22 3.1.5 Checking the proportionality assumption ................................................ 23 3.1.6 Parametric distributions and proportional MERL model...................... 24 3.2 SIMULATION STUDY..................................................................................... 25 3.2.1 Simulation scenarios ................................................................................... 25 3.2.2 Simulation results ....................................................................................... 26 3.3 EXAMPLE ......................................................................................................... 29 3.4 DISCUSSION..................................................................................................... 34 4.0 ACCELERATED MEDIAN RESIDUAL LIFE MODEL ..................................... 37 4.1 ACCELERATED MERL MODEL.................................................................. 37 4.2 PARAMETRIC APPROACH .......................................................................... 39 4.2.1 Parametric distributions and accelerated MERL model ........................ 39 4.2.2 Maximum likelihood estimation................................................................ 43 4.2.3 MLE for the Weibull distribution ............................................................. 44 4.2.4 Simulation study ......................................................................................... 45 4.3 ACCELERATED MERL MODEL UNDER THE AFT ASSUMPTION .... 48 4.4 EXAMPLE ......................................................................................................... 50 4.5 SOME RELATIONSHIPS FOR THE MERL FUNCTIONS ....................... 54 4.5.1 Relationships under the accelerated MERL model ................................. 54 4.5.2 Relationship under the Cox proportional hazards model....................... 56 4.6 DISCUSSION..................................................................................................... 57 5.0 DISCUSSION AND FUTURE RESEARCH........................................................... 59 BIBLIOGRAPHY....................................................................................................................... 62 vii LIST OF TABLES Table 3-1 Proportional MERL model (empirical bias and standard deviation).................... 27 Table 3-2 Proportional MERL model (bootstrap standard error and average length of the estimation interval, n = 200)................................................................................. 28 Table 3-3 Proportional MERL model (the rejection rate when the true regression parameter is β, n = 200) ......................................................................................................... 28 Table 3-4 B-04 results for the proportional MERL model.................................................... 31 Table 4-1 Accelerated MERL model (parameter estimation bias and standard errors)........ 46 Table 4-2 Accelerated MERL model (probability of type I error)........................................ 47 Table 4-3 Accelerated MERL model (power, n = 200) ........................................................ 47 viii LIST OF FIGURES Figure 3-1 Survival functions with MERL functions proportional with parameter 2........... 17 Figure 3-2 Median residual life functions proportional with parameter 2 ............................ 18 Figure 3-3 B-04 node status as a covariate............................................................................ 32 Figure 3-4 B-04 pathological tumor size as a covariate........................................................ 33 Figure 3-5 B-04 node status and pathological tumor size as covariates................................ 33 Figure 4-1 Survival functions with MERL functions accelerated by the factor 2................. 38 Figure 4-2 Median residual life functions accelerated by the factor 2.................................. 39 Figure 4-3 Accelerated MERL model (ML vs.