Modeling menstrual cycle length using a mixture distribution

In reproductive health studies, epidemiologists are often interested in examining the effects of covariates on menstrual cycle length which is a convenient, noninvasive measure of women's ovarian and reproductive function. Previous literature (Harlow and Zeger, 1991) suggests that the distribution of cycle length is a mixture of a major symmetric distribution and a component featuring a long right tail. Motivated by the shape of this marginal distribution, we propose a mixture distribution for cycle length, representing standard cycles from a Normal distribution and nonstandard cycles from a shifted Weibull distribution. The parameters are estimated using an estimating equation derived from the score function of an independence working model. The fitted mixture distribution agrees well with the distribution estimated using nonparametric approaches. We propose two measures to help determine whether a cycle is standard or nonstandard, developing tools necessary to identify characteristics of the menstrual cycles that are biologically indicative of ovarian dysfunction. We model the effect of a woman's age on the mean and variation of both standard and nonstandard cycle lengths using multiple measurements of women.