  In cohort studies, non-random medication use can pose barriers to estimation
of the natural history trend in a mean biomarker value (namely, the association
between a predictor of interest and a biomarker outcome that would be observed
in the absence of biomarker-specific treatment). Common causes of treatment and
outcomes are often unmeasured, obscuring our ability to easily account for
medication use with commonly invoked assumptions such as ignorability. Further,
absent some variable satisfying the exclusion restriction, use of instrumental
variable approaches may be difficult to justify. Heckman's hybrid model with
structural shift (sometimes referred to less specifically as the treatment
effects model) can be used to correct endogeneity bias via a homogeneity
assumption (i.e., that average treatment effects do not vary across covariates)
and parametric specification of a joint model for the outcome and treatment. In
recent work, we relaxed the homogeneity assumption by allowing observed
covariates to serve as treatment effect modifiers. While this method has been
shown to be reasonably robust in settings of cross-sectional data, application
of this methodology to settings of longitudinal data remains unexplored. We
demonstrate how the assumptions of the treatment effects model can be extended
to accommodate clustered data arising from longitudinal studies. Our proposed
approach is semi-parametric in nature in that valid inference can be obtained
without the need to specify the longitudinal correlation structure. As an
illustrative example, we use data from the Multi-Ethnic Study of
Atherosclerosis to evaluate trends in low-density lipoprotein by age and
gender. We confirm that our generalization of the treatment effects model can
serve as a useful tool to uncover natural history trends in longitudinal data
that are obscured by endogenous treatment.
