SummaryFixed effects meta analysis can be thought of as least squares analysis of the radial plot, the plot of standardized treatment effect against precision (reciprocal of the standard deviation) for the studies in a systematic review. For example, the least squares slope through the origin estimates the treatment effect, and a widely used test for publication bias is equivalent to testing the significance of the regression intercept. However, the usual theory assumes that the within-study variances are known, whereas in practice they are estimated. This leads to extra variability in the points of the radial plot which can lead to a marked distortion in inferences derived from these regression calculations. This is illustrated by a clinical trials example from the Cochrane Database. We derive approximations to the sampling properties of the radial plot and suggest bias corrections to some of the commonly used methods of meta analysis. A simulation study suggests that these bias corrections are effective in controlling significance levels of tests and coverage of confidence intervals.

The modern theory of likelihood inference provides improved inferences in many parametric models, with little more effort than is required for application of standard first-order theory. We outline the relevant computations, and illustrate the calculations using a dilution assay, a zero-inflated Poisson regression model, and a short time series. In each case the effect of the higher order correction can be appreciable.