Joint Correlated Responses and Feedback Effect with Time-dependent Covariates

This chapter is presented in two parts. In part 1, we discuss the simultaneous response generalized method of moments (GMM) model; this model addresses cases of multiple outcomes with coefficients that account for time-dependent covariates through partitioning. This method uses valid moment conditions to ensure efficient parameter estimation. The partitioned coefficients allow us to measure the effect of each covariate on the outcome at different periods. The use of a marginal model negates the need for any additional distributional assumptions, as is often present in joint modeling. A numerical example demonstrating the fit of risk factors in smoking and social alcohol drinking on obesity among adolescents is provided. In part 2, correlated observations due to the association between responses at a particular time and the predictors at later times are analyzed. The feedback effects include the relation between responses at the present and the covariates at a later time. This is not always relevant and is often ignored. In any event, feedback must be accounted for as they can have different effects on the regression coefficients. In particular, there are models that reflect the direct impact and the delayed impact of covariates on the response, utilizing valid moment conditions to estimate the relevant regression coefficients. This added feedback effect gives rise to a two-stage model. It accounts for the feedback, thus modeling the direct impact, as well as the delayed impact of the covariates on future responses and vice versa. The fit of this model is shown by modeling child morbidity and its impact on future values of BMI in the Philippines. In addition, we revisit the obesity status example and its feedback effects on physical activity and depression levels from the Add Health data.