A test for funnel plot asymmetry (small study effects) formally examines whether the association between estimated intervention effects and a measure of study size (such as the standard error of the intervention effect) is greater than might be expected to occur by chance. For outcomes measured on a continuous (numerical) scale this is reasonably straightforward. Using an approach proposed by Egger et al. (Egger 1997a), we can perform a linear regression of the intervention effect estimates on their standard errors, weighting by 1/(variance of the intervention effect estimate). This looks for a straight-line relationship between intervention effect and its standard error. Under the null hypothesis of no small study effects (e.g. Panel A in Figure 10.4.a) such a line would be vertical. The greater the association between intervention effect and standard error (e.g. as in Panel B in Figure 10.4.a), the more the slope would move away from vertical. Note that the weighting is important to ensure the regression estimates are not dominated by the smaller studies.

 

When outcomes are dichotomous, and intervention effects are expressed as odds ratios, the approach proposed by Egger et al. (Egger 1997a) corresponds to a linear regression of the log odds ratio on its standard error, weighted by the inverse of the variance of the log odds ratio (Sterne 2000). This has been by far the most widely used and cited approach to testing for funnel plot asymmetry. Unfortunately, there are statistical problems with this approach, because the standard error of the log odds ratio is mathematically linked to the size of the odds ratio, even in the absence of small study effects (Irwig 1998) (see Deeks et al. for an algebraic explanation of this phenomenon (Deeks 2005)). This can cause funnel plots plotted using log odds ratios (or odds ratios on a log scale) to appear asymmetric and can mean that P values from the test of Egger et al. are too small, leading to false-positive test results. These problems are especially prone to occur when the intervention has a large effect, there is substantial between-study heterogeneity, there are few events per study, or when all studies are of similar sizes.

 

A number of authors have therefore proposed alternative tests for funnel plot asymmetry: these are summarized in Table 10.4.b. Because it is impossible to know the precise mechanism for publication bias, simulation studies (in which the tests are evaluated on a large number of computer-generated datasets) are required to evaluate the characteristics of the tests under a range of assumptions about the mechanism for publication bias (Sterne 2000, Macaskill 2001, Harbord 2006, Peters 2006, Schwarzer 2007). The most comprehensive study (in terms of scenarios examined, simulations carried out and the range of tests compared) was reported by Rücker et al. (Rücker 2008). Results of this and the other published simulation studies inform the recommendations on testing for funnel plot asymmetry below. Although simulation studies provide useful insights, they inevitably evaluate circumstances that differ from a particular meta-analysis of interest, so their results must be interpreted carefully.

 

Most of this methodological work has focused on intervention effects measured as odds ratios. While it seems plausible to expect that corresponding problems will arise for intervention effects measured as risk ratios or standardized mean differences, further investigations of these situations are required.

 

There is ongoing debate over the representativeness of the parameter values used in the simulation studies, and the mechanisms used to simulate publication bias and small study effects, which are often chosen with little explicit justification. Some potentially useful variations on the different tests remain unexamined. Therefore it is not possible to make definitive recommendations on choice of tests for funnel plot asymmetry. Nevertheless, we can identify three tests that should be considered by review authors wishing to test for funnel plot asymmetry.

 

None of the tests described here is implemented in RevMan, and consultation with a statistician is recommended for their implementation.