Coarse Structural Nested Mean Models (SNMMs) provide useful tools to estimate treatment effects from longitudinal observational data in the presence of time-dependent confounders. Coarse SNMMs lead to a large class of estimators. An optimal estimator was derived within the class of coarse SNMMs(Lok et al., 2014). The key assumption lies in a well-specified model for the treatment effect; however there is no existing guidance to specify the treatment effect model. Misspecification of the treatment effect model leads to biased estimators, preventing valid inference.
To test whether the treatment effect model matches the data well, we derive a goodness-of-fit (GOF) test procedure based on overidentification restrictions tests (Hansen, 1982) and show that our GOF statistic is double-robust and consistent. Our simulation shows that the asymptotic distribution of the GOF statistic provides an accurate approximation to the finite sample behavior of the GOF statistic. In addition, we apply the GOF test procedure in the study of the role of initiation timing of highly active antiretroviral treatment (HAART) after infection on one-year treatment effect in HIV-positive patients.
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