Testing under the extended Koziol-Green model

Abstract

Suppose that Y1,Y2, ...,Yn are the response variables in a survival study that are independent copies of a positive random variable Y with an unknown continuous distribution function F. In the random censorship model, these variables are subject to random right censoring in that the observable variables are Zi = min(Yi;Ci) and di = 1 if Yi <= Ci (i = 1,2,...,n), where C1,C2,...,Cn are independent copies of another positive random variable C. To draw inference about the survival distribution of the response variable, it is imperative to make a non-verifiable assumption about the relationship between the survival time Y and censoring time C (Tsiatis, 1975). In some settings, the censoring time is additionally informative to the survival time through its distribution. When the survival and censoring times are independent, and the distribution function of the censoring time is a power to that of the survival time, we have the random censorship Koziol-Green model, which is characterized by the independence of the observable variables Zi and di (i =1,2,...,n). However, in some applications, Zi and di are not independent. By introducing a general copula function, we extend the Koziol-Green model to accommodate the possible dependence between Zi and di. As in Braekers and Veraverbeke (2005), we also allow for possible dependence between the survival time and censoring time by means of an Archimedean copula function. In this presentation, we shall give a semiparametric goodness-of-fit testing procedure for the copula function under the extended model. The goodness-of-fit statistics are based on an underlying empirical quantity for which some asymptotic results will be shown. Also, we will describe a bootstrap procedure to approximate the null distribution of the test statistics. Finally, some finite sample properties as well as a real life application of the testing procedure will be presented.