Piecewise Constant Cross-Ratio Estimation for Association of Age at a Marker Event and Age at Menopause

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