Confidence intervals and P-values for meta-analysis with publication bias.

We study publication bias in meta-analysis by supposing there is a population (y, sigma) of studies which give treatment effect estimates y approximately N(theta, sigma(2)). A selection function describes the probability that each study is selected for review. The overall estimate of theta depends on the studies selected, and hence on the (unknown) selection function. Our previous paper, Copas and Jackson (2004, Biometrics 60, 146-153), studied the maximum bias over all possible selection functions which satisfy the weak condition that large studies (small sigma) are as likely, or more likely, to be selected than small studies (large sigma). This led to a worst-case sensitivity analysis, controlling for the overall fraction of studies selected. However, no account was taken of the effect of selection on the uncertainty in estimation. This article extends the previous work by finding corresponding confidence intervals and P-values, and hence a new sensitivity analysis for publication bias. Two examples are discussed.