Testing model fit in longitudinal data analysis against alternatives with omitted covariates

Several types of common model misspecifications can be re-formulated as problems of omitted covariates. These include situations with unmeasured confounders, measurement errors in observed covariates and informative censoring. Longitudinal data present special opportunities for detecting omitted covariates that are related to the observed ones differently across time than across individuals. This situation arises with period and cohort effects, as well as with usual formulations of classical measurement error in observed covariates. In this article we focus on testing for the existence of omitted covariates in longitudinal data analysis when models are fit by generalized estimation equations. When omitted covariates are present, specification of the correct link function conditionally on only observed covariates under the alternative usually involves complicated numerical integration. We propose a quasi-score test statistic that avoids the need to fit such alternative models. The statistic is asymptotically chi-square distributed under the null hypothesis of no omitted covariates with degrees of freedom determined by the assumed alternative structure. We study the significance level and the power of the quasi-score test in linear and logistic regression models. The test is then applied to an analysis of excessive daytime sleepiness.