Consistency of the Estimator for the Common Mean in Fixed-Effect Meta-Analyses

Fixed-effect meta-analyses aim to estimate the common mean parameter by the best linear unbiased estimator. Besides unbiasedness, consistency is one of the most fundamental requirements for the common mean estimator to be valid. However, conditions for the consistency of the common mean estimator have not been discussed in the literature. This article fills this gap by clarifying conditions for making the common mean estimator consistent in fixed-effect meta-analyses. In this article, five theorems are devised, which state regularity conditions for the common mean estimator to be consistent. These theorems are novel applications of the classical large sample theory to meta-analyses. Numerical illustrations are also given to help understand the needs of the regularity conditions. Three real datasets illustrate the practical consequences of the devised theorems. This article concludes that the inconsistency of the common mean estimator occurs under some conditions in real meta-analyses.

Automated summary

This article fills the research gap by clarifying the conditions for consistency of the common mean estimator in fixed-effect meta-analyses. Five theorems are devised to state the regularity conditions for the estimator to be consistent. These theorems are novel applications of large sample theory to meta-analyses. Numerical illustrations and real datasets demonstrate the practical consequences of the theorems. The article concludes that the common mean estimator can be inconsistent under certain conditions in real meta-analyses.