A goodness-of-fit test for structural nested mean models

Coarse structural nested mean models are tools for estimating treatment effects from longitudinal observational data with time-dependent confounding. There is, however, no guidance on how to specify the treatment effect model, and model misspecification can lead to bias. We derive a goodness-of-fit test based on modified over-identification restrictions tests for evaluating a treatment effect model, and show that our test is doubly robust in the sense that, with a correct treatment effect model, the test has the correct Type I error if either the treatment initiation model or a nuisance regression outcome model is correctly specified. In a simulation study, we show that the test has correct Type I error and can detect model misspecification. We use the test to study how the timing of antiretroviral treatment initiation after HIV infection predicts the effect of one year of treatment in HIV-positive patients with acute and early infection.