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Coarse Structural Nested Mean Models (SNMMs) provide useful tools to estimatetreatment effects from longitudinal observational data with time-dependentconfounders. Coarse SNMMs lead to a large class of estimators,within which anoptimal estimator can be derived under the conditions of well-specified modelsfor the treatment effect, for treatment initiation, and for nuisance regressionoutcomes (Lok & Griner, 2015). The key assumption lies in a well-specifiedmodel for the treatment effect; however, there is no existing guidance tospecify the treatment effect model, and model misspecification leads to biasedestimators, preventing valid inference. To test whether the treatment effectmodel matches the data well, we derive a goodness-of-fit (GOF) test procedurebased on overidentification restrictions tests (Sargan, 1958; Hansen, 1982). Weshow that our GOF statistic is doubly-robust in the sense that with a correcttreatment effect model, if either the treatment initiation model or thenuisance regression outcome model is correctly specified, the GOF statistic hasa Chi-squared limiting distribution with degrees of freedom equal to the numberof overidentification restrictions. We demonstrate the empirical relevance ofour methods using simulation designs based on an actual dataset. In addition,we apply the GOF test procedure to study how the initiation time of highlyactive antiretroviral treatment (HAART) after infection predicts the one-yeartreatment effect in HIV-positive patients with acute and early infection.

biometrika

Doubly Robust Goodness-of-Fit Test of Coarse Structural Nested Mean  Models with Application to Initiating combination antiretroviral treatment in  HIV-Positive Patients