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.

Empirical Likelihood Estimation Using Auxiliary Summary Information with Different Covariate Distributions

“…This estimator has been shown to be more efficient thanβ MLE , the maximum likelihood estimator based on current study data alone (Qin 2000;Chatterjee et al 2016). Asymptotically equivalent estimators can be…”

“…This moment equality provides a constraint on f (X , Z ), and an application of semi-empirical likelihood (Qin 2000;Chatterjee et al 2016) leads to an estimator of β 0 defined through…”

“…However, there is much recent interest in using big data bases that are external to the current study (e.g. Qin et al 2015;Chatterjee et al 2016;Huang et al 2016). In many such contexts assumption (a) may be plausible but assumption (b) is more likely to be violated; see, for example, Keiding and Louis (2016) who note that conditional features or distributions are more likely to be "transportable" from one population to another than are marginal distributions.…”

“…The only model we specify is for the conditional distribution of the response given covariates of interest. The approach was proposed by Han and Lawless (2016) in a discussion of Chatterjee et al (2016), but not developed or studied there. When the variability associated with the auxiliary summary information is negligible compared to that in the study data, we show the proposed estimators are more efficient than the maximum likelihood estimator based on the study data alone.…”

“…More recently, studies in medicine and public health have received attention (e.g., Qin et al 2015;Chatterjee et al 2016;Huang et al 2016). The auxiliary data typically have much less detailed covariate information than the study data, and are often in summary or aggregate form.…”