Constrained Maximum Likelihood Estimation for Model Calibration Using Summary-level Information from External Big Data Sources

Abstract

Information from various public and private data sources of extremely large sample sizes are now increasingly available for research purposes. Statistical methods are needed for utilizing information from such big data sources while analyzing data from individual studies that may collect more detailed information required for addressing specific hypotheses of interest. In this article, we consider the problem of building regression models based on individual-level data from an "internal" study while utilizing summary-level information, such as information on parameters for reduced models, from an "external" big data source. We identify a set of very general constraints that link internal and external models. These constraints are used to develop a framework for semiparametric maximum likelihood inference that allows the distribution of covariates to be estimated using either the internal sample or an external reference sample. We develop extensions for handling complex stratified sampling designs, such as case-control sampling, for the internal study. Asymptotic theory and variance estimators are developed for each case. We use simulation studies and a real data application to assess the performance of the proposed methods in contrast to the generalized regression (GR) calibration methodology that is popular in the sample survey literature.

Keywords: Case-control study; Empirical likelihood; Generalized regression estimator; Misspecified model; Profile-likelihood.

Figure 1

Geometric interpretation of the problem. The true distribution function P₀(Y|X) for Y given (X, Z) across all values of β is depicted in the left panel, with the true value of β, i.e., β₀ shown. In the right top panel, we see the true distribution function P₀(Y|X) for Y given X across all values of β with β₀ shown. The assumed external data model is given in the bottom right panel across all values of θ, with the solution θ* to (1) being shown as the minimizer of the Kullback-Leibler distance between the fixed P₀(Y|X) and the model space of g_θ(Y|X).