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

TL;DR

This paper develops a doubly robust goodness-of-fit test for coarse structural nested mean models to assess model adequacy in estimating treatment effects from longitudinal observational data, with applications to HIV treatment timing.

Contribution

It introduces a novel GOF test based on overidentification restrictions that is doubly robust, ensuring valid inference even if some models are misspecified.

Findings

The GOF test maintains correct size under model misspecification if either the treatment or nuisance model is correct.

Simulation studies demonstrate the test's effectiveness in finite samples.

Application to HIV data reveals insights into treatment initiation timing and its effects.

Abstract

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.