  Coarse Structural Nested Mean Models (SNMMs) provide useful tools to estimate
treatment effects from longitudinal observational data with time-dependent
confounders. Coarse SNMMs lead to a large class of estimators,within which an
optimal estimator can be derived under the conditions of well-specified models
for the treatment effect, for treatment initiation, and for nuisance regression
outcomes (Lok & Griner, 2015). 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, and model misspecification 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 (Sargan, 1958; Hansen, 1982). We
show that our GOF statistic is doubly-robust in the sense that with a correct
treatment effect model, if either the treatment initiation model or the
nuisance regression outcome model is correctly specified, the GOF statistic has
a Chi-squared limiting distribution with degrees of freedom equal to the number
of overidentification restrictions. We demonstrate the empirical relevance of
our methods using simulation designs based on an actual dataset. In addition,
we apply the GOF test procedure to study how the initiation time of highly
active antiretroviral treatment (HAART) after infection predicts the one-year
treatment effect in HIV-positive patients with acute and early infection.
