Abstract: Abstract: Statistical models and inferences are typically based on measurements made on individual participants in a study (individual-level data). However, there is growing interest in improving statistical inference by taking advantage of aggregated summary-level data from other studies, such as statistics used in meta-analyses. Although the generalized method of moments (GMM) provides a flexible way of doing so, integrating external summary information does not always improve efficiency. Here, we provide a necessary and sufficient condition under which external summary information can be beneficial. We further extend the GMM to incorporate summary data generated from a population with a covariate distribution that is different from that of the individual-level data. Lastly, we compare the GMM with other integration procedures.

Key words and phrases: Empirical likelihood, generalized linear model, generalized method of moments, meta-analysis, summary statistics.