Clinical heterogeneity in random-effect meta-analysis: Between-study boundary estimate problem

Random-effect meta-analysis is commonly applied to estimate overall effects with unexplained heterogeneity across studies. However, standard methods, including (restricted) maximum likelihood (ML or REML), frequently produce (near) zero estimates for between-study variance parameters. Consequently, these methods are reduced to simple and unrealistic fixed-effect models, resulting in an ignorance of the substantial clinical heterogeneity and sometimes leading to incorrect conclusions. To solve the boundary estimate problem, we propose (1) an adjusted maximum likelihood method for the between-study variance that maximizes a likelihood defined as a product of a standard likelihood and a Gaussian class of adjustment factor and (2) a framework using sensitivity analysis by developing a new criterion to check for the occurrence of the boundary estimate. Although the adjustment introduces bias to the overall effects to ensure strictly positive estimates of the between-study variance when the number of studies K is small, the bias asymptotically approaches zero, resulting in the same estimates derived from the REML method. Moreover, the adjusted maximum likelihood estimator of the between-study variance is consistent for large K, and interestingly, the REML method and our method are equivalent in terms of mean squared error criterion, up to O(K^-1 ). We illustrate our approach with a motivating example to examine the controversial result of a meta-analysis for 24 randomized controlled trials of human albumin. Numerical evaluations show that our approach produces no boundary estimates but similar synthesized results with the standard maximum likelihood methods as those produced by conventional methods, especially with a small number of studies.