Cluster-specific nonignorably missing, endogenous, and continuous regressors in multilevel model for binary outcome

In multilevel regression models for observational clustered data, regressors can be correlated with cluster-level error components, namely endogenous, due to omitted cluster-level covariates, measurement error, and simultaneity. When endogeneity is ignored, regression coefficient estimators can be severely biased. To deal with endogeneity, instrument variable methods have been widely used. However, the instrument variable method often requires external instrument variables with certain conditions that cannot be verified empirically. Methods that use the within-cluster variations of the endogenous variable work under the restriction that either the outcome or the endogenous variable has a linear relationship with the cluster-level random effect. We propose a new method for binary outcome when it follows a logistic mixed-effects model and the endogenous variable is normally distributed but not linear in the random effect. The proposed estimator capitalizes on the nested data structure without requiring external instrument variables. We show that the proposed estimator is consistent and asymptotically normal. Furthermore, our method can be applied when the endogenous variable is missing in a cluster-specific nonignorable mechanism, without requiring that the missing mechanism be correctly specified. We evaluate the finite sample performance of the proposed approach via simulation and apply the method to a health care study using a San Diego inpatient dataset. Our study demonstrates that the clustered structure can be exploited to draw valid analysis of multilevel data with correlated effects.