# Paper No. 07-1

**JB Copas and C Lozada**

**Asymptotic Approximations for the Radial Plot in Meta Analysis, and a Bias Correction to the Egger Test**

**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 Egger test for publication bias is equivalent to testing the signi¯cance of the intercept in linear regression. In practice, both x- and y-coordinates of the points in a radial plot are subject to sampling error, which may be correlated, and so the standard theory of least squares does not apply. For the Egger test, the actual significance levels are intercept, sometimes substantially so. We derive approximations to the sampling properties of the radial plot, assuming that the within-study sample sizes are large. This leads to an asymptotic bias correction for the Egger test. A simulation study suggests that the bias correction controls the signi¯cance level of the Egger test without any appreciable loss of power in detecting non-random study selection. A clinical trials example is used as an illustration.

**Keywords****:** Radial plot; Publication bias; Small study effects; Egger test; Asymptotic bias correction.