Case-Control Current Status Data

Nicholas P. Jewell, Division of Biostatistics, School of Public Health, University of California, Berkeley
Mark J. van der Laan, Division of Biostatistics, School of Public Health, University of California, Berkeley

Abstract

Current status observation on survival times has recently been widely studied. An extreme form of interval censoring, this data structure refers to situations where the only available information on a survival random variable, T, is whether or not T exceeds a random independent monitoring time C, a binary random variable, Y. To date, nonparametric analyses of current status data have assumed the availability of i.i.d. random samples of the random variable (Y, C), or a similar random sample at each of a set of fixed monitoring times. In many situations, it is useful to consider a case-control sampling scheme. Here, cases refer to a random sample of observations on C from the sub-population where T is less than or equal to C. On the other hand, controls provide a random sample of observations from the sub-population where T is greater than C. In this paper, we examine the identifiability of the distribution function F of T from such case-control current status data, showing that F is identified up to a one parameter family of distribution functions. With supplementary information on the relative population frequency of cases/controls, a simple weighted version of the nonparametric maximum likelihood estimator for prospective current status data provides a natural estimate for case-control samples. Following the parametric results of Scott and Wild (1997), we show that this estimator is, in fact, nonparametric maximum likelihood.