Time-Dependent Hazard Ratio: Modeling and Hypothesis Testing with Application in Lupus Nephritis
Journal article by Michal Abrahamowicz, Todd Mackenzie, John M. Esdaile; Journal of the American Statistical Association, Vol. 91, 1996
Journal Article Excerpt
Time-Dependent Hazard Ratio: Modeling and Hypothesis Testing with Application in Lupus Nephritis. by Michal Abrahamowicz , Todd MacKenzie , John M. Esdaile 1. INTRODUCTION The proportional hazards model is the most popular regression method for analyzing censored survival data. It requires the assumption (the proportional hazards, or PH, assumption) that the hazard ratio, which describes the effect of a predictor on survival, is constant over the entire follow-up period (Cox 1972). However, the PH assumption may not hold in some survival studies. Predictors of interest are often measured only at time zero, even if their values may later change, so that a time-dependent covariate must be treated as a fixed one. Further, the impact of a predictor, which is by definition fixed in time, may change during follow-up. For example, in studies of chronic diseases, the hazard ratio of disease over control may increase if damage to the affected organs increases with disease duration. Numerous tests of the PH hypothesis have been proposed (Lin and Wei 1991). However, after having rejected the PH hypothesis, it is not obvious how to summarize the effect of a predictor. Such a difficulty arose in our investigation of the role of prognostic factors for mortality in lupus nephritis, a rare rheumatologic disease that affects mostly young women. Eighty-seven lupus patients undergoing renal biopsy between 1967 and 1983 at the Yale-New Haven Hospital were followed since biopsy until death or the end of 1990 (Esdaile, Abrahamowicz, MacKenzie, Hayslett, and Kashgarian 1994). There were 35 deaths in the cohort. Comparison of the results of published short- and long-term studies suggested that the PH assumption may be violated in the case of variable "disease duration" (the length of period prior to biopsy when the disease was active but not treated). Indeed, in our preliminary analyses the Cox model estimates of the hazard ratio were 1.16 (p = .22) and 1.64 (p = .005) for the entire 16-year follow-up and for the initial 5-year period. Thus to describe the effect of disease duration accurately, we need to model the changes in the hazard ratio over time. Different nonparametric regression methods have been proposed to estimate the hazard ratio as a function of time (HR function). Gray (1992) and Kooperberg, Stone, and Truong (1995) used low-order regression splines (step functions and breaking lines). Verweij and Houwelingen (1995) smoothed local hazard ratios by introducing a penalty for the first difference. Hess (1994) worked with natural cubic splines. Zucker and Karr (1990) and Hastie and Tibshirani (1993) relied on smoothing splines. However, several issues of practical interest remain to be addressed. As adaptive modeling techniques are often used, the impact of model selection on the accuracy of estimation and inference must be studied. Little is known about the behavior of the proposed methods in small samples, and simulation results are scarce. It is not clear how to test the null hypothesis of no association against the compound alternative of either constant or time-dependent HR. In this article we propose a method that uses regression splines to model a time-dependent hazard ratio. Our objective is to provide a flexible tool that is simple to use yet able to yield smooth, clinically plausible estimates and reliable inference. In Section 2 we describe maximum partial likelihood estimation, model selection, and inference. We discuss in detail testing of the hypothesis of no association against a possibly time-dependent predictor effect. In Section 3 we present the results of simulations under different assumptions about the shape of the HR function. We investigate the accuracy of the estimates and of pointwise confidence intervals. We compare the model-based test of the PH assumption with two conventional tests, and evaluate the power of the proposed procedure for testing the hypothesis of no association. In Section 4 we reexamine the association between duration of untreated disease and survival in lupus nephritis and analyze a well-known primary biliary cirrhosis (PBC) data set. We summarize our findings and suggest some directions for future work in the final section. 2. ESTIMATION AND INFERENCE 2.1 Estimation We consider the following model, [Lambda](t; x) = [[Lambda].sub.0](t) exp ([summation of] [[Beta].sub.i](t)[x.sub.i] where i = 1 to p), (1) where x = ([x.sub.1], . . ., [x.sub.p]) is a vector of p covariates, [[Lambda].sub.0](t) is an unspecified baseline hazard function corresponding to x = 0, and [[Beta].sub.i](t) is the logarithm of the hazard ratio at time t corresponding to a unit increase in covariate [x.sub.i].Model (1) generalizes the PH model as the constant log hazard ratios [[Beta].sub.i] ... |
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