Joint modeling of cross‐sectional health outcomes and longitudinal predictors via mixtures of means and variances

Joint models naturally link longitudinal covariates to disease outcomes. Many joint models have been developed in the context of cancer research and HIV/AIDS clinical trials, where a mixed-effect model is outlined for the longitudinal trajectories and a primary outcome model is defined for the disease outcome. The primary outcome models are often postulated as: (1) shared random effects (SRE) models, where covariates include a functional form of the random effects in the mixed-effect submodel, and (2) latent class (LC) models, where there exists heterogeneity (latent classes) in the longitudinal mean profiles, and subjects in a particular latent class share the same risk of event, conditional on other covariates. For SRE models, the random effects are used to capture the main features in the longitudinal trajectories that predict the outcomes. The concept of “shared parameters” was first used in Wu and Carroll (1988) to model non-ignorable missing data, and later by Henderson et al. (2000) to jointly analyze longitudinal and time-to-event data; also see Tsiatis and Davidian (2004), Ibrahim et al. (2001, 2010) for excellent general reviews of these models. In the LC model literature, growth mixture models (Verbeke and Lesaffre (1996), Muthen and Shedden 1999) are extensions of random growth curve models, creating distinct subgroups where individual trajectories vary around group-specific mean trajectories. Considering time-to-event outcomes, Proust-Lima et al. (2012) studied the joint LC modeling in detail and contrasted its use in terms of goodness of fit, prediction accuracy and model performances with that of joint SRE models. Using a prostate cancer study data consisting of four well-separated classes of longitudinal mean trajectories, they illustrated that, in comparison to a LC model, the use of SRE model alone was not sufficient to fully capture the relationship between class-specific outcomes and the heterogeneity among different classes. They also reported that only a mild advantage of LC remained for prediction of outcomes from an external data set of similar nature. In this paper, we study the associations between longitudinal hormone levels and menopausal symptoms for a group of middle-aged women. The Penn Ovarian Aging Study (Freeman et al., 2011) is a longitudinal study consisting of a population-based sample of 436 women aged 35-47 years selected via random digit dialing in Philadelphia County, PA during 1996-97. At each annual assessment, measurements and a blood sample were collected two times approximately a month apart. One goal of the study is to explore associations between reproductive hormone levels and symptoms in the transition to menopause. Changes in hormone levels alter menstrual bleeding patterns prior to menopause marking the end of a woman’s reproductive years. This course of events coincides for a majority of women with the development of hot flashes, sleep disorders, and bone loss, among other symptoms. While researchers have focused on the associations between these symptoms and hormone levels, the impact of within woman rate of change and variability in hormones, such as Follicle Stimulating Hormone (FSH), is not well understood. To evaluate that hypothesis that subject-level hormone fluctuation may accentuate menopausal symptoms (Freeman et al., 2006), we investigate methods that model both longitudinal profiles and residual variability of the FSH and simultaneously link them with the risk of experiencing severe hot flashes (SHF). While most joint models have treated within-subject variability as a nuisance parameter, recently a small literature has developed to evaluate the associations between longitudinal within-subject variability and the primary outcomes (Sammel et al., 2001; Elliott 2007, Elliott et al. 2012). Thus, in this dataset we have longitudinal measures with heterogeneity both in trajectory and variability that may be predictive of a binary outcome. There is evidence that these trajectories and variabilities may cluster into possibly clinically relevant groupings, so we consider a mixture model for FSH hormone that also includes latent classes for the subject-level trajectories and variability. This leads to two potential candidate models for the outcome: a “multiple shared random effects” (MSRE) model whose predictors are subject-specific random coefficients, and a latent class (LC) model whose predictors are the LC memberships. Since it is not clear which approach is best, we examine the robustness and predictive accuracy of each approach via simulation study. Our key focus is not on one primary-outcome model or the other, but their contrasts and the information they jointly provide.