Correlated probit analysis of repeatedly measured ordinal and continuous outcomes with application to the Health and Retirement Study

The Health and Retirement Study was designed to evaluate changes in health and labor force participation during and after the transition from working to retirement. Every 2 years, participants provided information about their self-rated health (SRH), body mass index (BMI), smoking status, and other characteristics. Our goal was to assess the effects of smoking and gender on trajectories of change in BMI and SRH over time. Joint longitudinal analysis of outcome measures is preferable to separate analyses because it allows to account for the correlation between the measures, to test the effects of predictors while controlling type I error, and potentially to improve efficiency. However, because SRH is an ordinal measure while BMI is continuous, formulating a joint model and parameter estimation is challenging. A joint correlated probit model allowed us to seamlessly account for the correlations between the measures over time. Established estimating procedures for such models are based on quasi-likelihood or numerical approximations that may be biased or fail to converge. Therefore, we proposed a novel expectation-maximization algorithm for parameter estimation and a Monte Carlo bootstrap approach for standard errors approximation. Expectation-maximization algorithms have been previously considered for combinations of binary and/or continuous repeated measures; however, modifications were needed to handle combinations of ordinal and continuous responses. A simulation study demonstrated that the algorithm converged and provided approximately unbiased estimates with sufficiently large sample sizes. In the Health and Retirement Study, male gender and smoking were independently associated with steeper deterioration in self-rated health and with lower average BMI. Copyright © 2016 John Wiley & Sons, Ltd.