Using structural-nested models to estimate the effect of cluster-level adherence on individual-level outcomes with a three-armed cluster-randomized trial

Much attention has been paid to estimating the causal effect of adherence to a randomized protocol using instrumental variables to adjust for unmeasured confounding. Researchers tend to use the instrumental variable within one of the three main frameworks: regression with an endogenous variable, principal stratification, or structural-nested modeling. We found in our literature review that even in simple settings, causal interpretations of analyses with endogenous regressors can be ambiguous or rely on a strong assumption that can be difficult to interpret. Principal stratification and structural-nested modeling are alternative frameworks that render unambiguous causal interpretations based on assumptions that are, arguably, easier to interpret. Our interest stems from a wish to estimate the effect of cluster-level adherence on individual-level binary outcomes with a three-armed cluster-randomized trial and polytomous adherence. Principal stratification approaches to this problem are quite challenging because of the sheer number of principal strata involved. Therefore, we developed a structural-nested modeling approach and, in the process, extended the methodology to accommodate cluster-randomized trials with unequal probability of selecting individuals. Furthermore, we developed a method to implement the approach with relatively simple programming. The approach works quite well, but when the structural-nested model does not fit the data, there is no solution to the estimating equation. We investigate the performance of the approach using simulated data, and we also use the approach to estimate the effect on pupil absence of school-level adherence to a randomized water, sanitation, and hygiene intervention in western Kenya.