In random-effects meta-analysis, the between-study variance quantifies the heterogeneity among studies. Without selective publication, the restricted maximum likelihood (REML) method is recommended for this variance, since the maximum likelihood (ML) method underestimates it when the number of studies is small. However, under selective publication, referred to as publication bias (PB), the published studies no longer follow the normal-normal model, and the REML cannot be directly applied. Thus, the existing sensitivity analysis methods for PB based on selection functions estimate the between-study variance by the ML conditional on published studies and inherit its underestimation. This underestimation also affects the estimate of the common effect and its confidence interval. To improve the estimation of the common effect and the heterogeneity, we propose a REML-type estimator of the between-study variance for the selection-function-based sensitivity analysis. It centers the profile score by its approximate mean and reduces exactly to the REML without PB. The proposed sensitivity analysis method is a more accurate alternative to the ML-based sensitivity analysis in meta-analyses with a small number of studies, and it is the first REML-type estimator in the presence of PB. Simulation studies substantiate that the proposed method reduces the median bias of the between-study variance and improves the estimation error and coverage probability for the common effect compared with the conditional ML. We illustrate the proposed method with two meta-analyses on high maintenance-dose clopidogrel and nicotine gum for smoking cessation. The proposed method would give useful guidance on the between-study heterogeneity in sensitivity analysis for PB.

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