Cross-Sectional Sampling, Bias, Dependence, and Composite Likelihood

A population that can be joined at a known sequence of discrete times is sampled cross-sectionally, and the sojourn times of individuals in the sample are observed. It is well known that cross-sectioning leads to length-bias, but less well known that it may result also in dependence among the observations, which is often ignored. It is therefore important to understand and to account for this dependence when estimating the distribution of sojourn times in the population. In this paper, we study conditions under which observed sojourn times are independent and conditions under which treating observations as independent, using the product of marginals in spite of dependence, results in proper inference. The latter is known as the Composite Likelihood approach. We study parametric and nonparametric inference based on Composite Likelihood, and provide conditions for consistency, and further asymptotic properties, including normal and non-normal distributional limits of estimators. We show that Composite Likelihood leads to good estimators under certain conditions, and illustrate that it may fail without them. The theoretical study is supported by simulations. We apply the proposed methods to two data sets collected by cross-sectional designs: data on hospitalization time after bowel and hernia surgeries, and data on service times at our university.