Piecewise Constant Cross-Ratio Estimation for Association in Bivariate Survival Data with Application to Studying Markers of Menopausal Transition

A question of significant interest in female reproductive aging is to identify bleeding criteria for the menopausal transition. Although various bleeding criteria, or markers, have been proposed for the menopausal transition, their validity has not been adequately examined. The Tremin Trust data are collected from a long-term cohort study that followed a group of women throughout their whole reproductive life, and provide a unique opportunity for assessing the association between age at onset of a bleeding marker and age onset of menopause. Formal statistical analysis of this dependence is challenging give the fact that both the marker event and menopause are subject to right censoring and their association depends on age at the marker event. We propose using cross-ratio to measure their dependence, which is assumed to be a piecewise constant function of age at onset of the marker event. Two estimation procedures using direct two-stage method and sequential two-stage method are proposed, while the latter is extended to allow for covariates in marginal survival functions. The proposed methods are applied to the analysis of the Tremin Trust data, and their performance is evaluated using simulations.