  Publication bias is a major concern in conducting systematic reviews and
meta-analyses. Various sensitivity analysis or bias-correction methods have
been developed based on selection models and they have some advantages over the
widely used bias-correction method of the trim-and-fill method. However,
likelihood methods based on selection models may have difficulty in obtaining
precise estimates and reasonable confidence intervals or require a complicated
sensitivity analysis process. In this paper, we develop a simple publication
bias adjustment method utilizing information on conducted but still unpublished
trials from clinical trial registries. We introduce an estimating equation for
parameter estimation in the selection function by regarding the publication
bias issue as a missing data problem under missing not at random. With the
estimated selection function, we introduce the inverse probability weighting
(IPW) method to estimate the overall mean across studies. Furthermore, the IPW
versions of heterogeneity measures such as the between-study variance and the
I2 measure are proposed. We propose methods to construct asymptotic confidence
intervals and suggest intervals based on parametric bootstrapping as an
alternative. Through numerical experiments, we observed that the estimators
successfully eliminate biases and the confidence intervals had empirical
coverage probabilities close to the nominal level. On the other hand, the
asymptotic confidence interval is much wider in some scenarios than the
bootstrap confidence interval. Therefore, the latter is recommended for
practical use.
