Hazard Function Estimation Using B-Splines

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. . International Biometric Society is collaborating with JSTOR to digitize, preserve and extend access to Biometrics. SUMMARY A flexible parametric procedure is given to model the hazard function as a linear combination of cubic B-splines and to obtain maximum likelihood estimates from censored survival data. The approach yields smooth estimates of the hazard and suivivorship functions that are intermediate in structure between strongly parametric and non-parametric models. A simple method is described for selecting the number and location of knots. Simulation results show favorable root mean square error compared to non-parametric estimates for both the hazard and survivorship functions. Three methods are given to calculate confidence intervals based on the delta method, profile likelihood, and bootstrap, respectively. The procedure is applied to estimate hazard rates for acquired immu-nodeficiency syndrome (AIDS) following infection with human immunodeficiency virus (HIV). Spline methods can accommodate complex censoring mechanisms such as those that arise in the AIDS setting. To illustrate, HIV infection incidence is estimated for a cohort of hemophiliacs in which the dates of HIV infection are interval-censored and some subjects were born after the onset of the HIV epidemic. 1. Introduction This report develops flexible parametric methods based on splines to estimate the hazard function hi(t) that describes the rate of disease onset among persons still susceptible at time t. The observed data derive from a random sample Yi, i = 1, . n. , n of disease onset times with survivorship function S(t) = PY > t}. The Yi are right-censored by independent random variables Ci drawn from H(C) = P{C > c}. One observes Xi = min(Yi, Ci) and 3i = I(Y. > Ci), the indicator for the event that Yi is censored. The hazard function