Testing hypothesis and estimation in the presence of omitted confounders or latent variables

Several types of common model misspecifications can be reformulated as problems of missing covariates. These include situations with unmeasured confounders, measurement errors in observed covariates and informative censoring. The missing covariates can take either continuous or discrete forms. This thesis is divided into two parts. One part of the thesis is testing model fit in the longitudinal data analysis against alternatives with omitted continuous covariates, the other part of the thesis is estimating model parameters and drawing statistical inferences in the presence of a missing binary covariate.; 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 the first part of the thesis, we focus on testing for the existence of omitted continuous 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 applied to an analysis of excessive daytime sleepiness.; When the missing covariate takes a binary form, and it relates to the observed covariate via certain common link functions, we have the opportunities to fit a nonlinear model conditioning on only the observed covariate. In the second part of the thesis, we test the existence of a hidden indicator using Wald-type chi-square test statistics, estimate the nonlinear model parameters using an iterative algorithm, calibrate risks and apply the bootstrap resampling approach to construct confidence intervals for the primary parameters. The proposed procedure is applied to an analysis of a clinical study of schizophrenia.