# Paper No. 09-02

**JB Copas and C Lozada-Can**

**The Radial Plot in Meta Analysis: Approximations and Applications**

**Abstract: **Fixed 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.

**Keywords****:** Meta analysis; Radial plot; Bias corrections; Publication bias; Egger test.