10.4.1 Funnel plots

A funnel plot is a simple scatter plot of the intervention effect estimates from individual studies against some measure of each study’s size or precision. In common with forest plots, it is most common to plot the effect estimates on the horizontal scale, and thus the measure of study size on the vertical axis. This is the opposite of conventional graphical displays for scatter plots, in which the outcome (e.g. intervention effect) is plotted on the vertical axis and the covariate (e.g. study size) is plotted on the horizontal axis.

The name ‘funnel plot’ arises from the fact that precision of the estimated intervention effect increases as the size of the study increases. Effect estimates from small studies will therefore scatter more widely at the bottom of the graph, with the spread narrowing among larger studies. In the absence of bias the plot should approximately resemble a symmetrical (inverted) funnel. This is illustrated in Panel A ofFigure 10.4.a, in which the effect estimates in the larger studies are close to the true intervention odds ratio of 0.4.

If there is bias, for example because smaller studies without statistically significant effects (shown as open circles in Figure 10.4.a, Panel A) remain unpublished, this will lead to an asymmetrical appearance of the funnel plot with a gap in a bottom corner of the graph (Panel B). In this situation the effect calculated in a meta-analysis will tend to overestimate the intervention effect (Egger 1997a, Villar 1997). The more pronounced the asymmetry, the more likely it is that the amount of bias will be substantial.

Funnel plots were first used in educational research and psychology, with effect estimates plotted against total sample size (Light 1984).It is now usually recommended that the standard error of the intervention effect estimate be plotted, rather than the total sample size, on the vertical axis (Sterne 2001). This is because statistical power of a trial is determined by factors in addition to sample size, such as the number of participants experiencing the event for dichotomous outcomes, and the standard deviation of responses for continuous outcomes. For example, a study with 100,000 participants and 10 events is less likely to show a statistically significant intervention effect than a study with 1000 participants and 100 events. The standard error summarizes these other factors. Plotting standard errors on a reversed scale places the larger, or most powerful, studies towards the top of the plot. Another potential advantage of using standard errors is that a simple triangular region can be plotted, within which 95% of studies would be expected to lie in the absence of both biases and heterogeneity. These regions are included in Figure 10.4.a. Funnel plots of effect estimates against their standard errors (on a reversed scale) can be created using RevMan. A triangular 95% confidence region based on a fixed-effect meta-analysis can be included in the plot, and different plotting symbols allow studies in different subgroups to be identified.

Publication bias need not lead to asymmetry in funnel plots. In the absence of any intervention effect, selective publication based on the P value alone will lead to a symmetrical funnel plot in which studies on the extreme left or right are more likely to be published than those in the middle. This could bias the estimated between-study heterogeneity variance.

Ratio measures of intervention effect (such as odds ratios and risk ratios) should be plotted on a logarithmic scale. This ensures that effects of the same magnitude but opposite directions (for example odds ratios of 0.5 and 2) are equidistant from 1.0. For outcomes measured on a continuous (numerical) scale (e.g. blood pressure, depression score) intervention effects are measured as mean differences or standardized mean differences, which should therefore be used as the horizontal axis in funnel plots. So far as we are aware, no empirical investigations have examined choice of axes for funnel plots for continuous outcomes. For mean differences, the standard error is approximately proportional to the inverse of the square root of the number of participants, and therefore seems an uncontroversial choice for the vertical axis.

Some authors have argued that visual interpretation of funnel plots is too subjective to be useful. In particular, Terrin et al. found that researchers had only a limited ability to correctly identify funnel plots from meta-analyses subject to publication bias (Terrin 2005).

A further, important, problem with funnel plots is that some effect estimates (e.g. odds ratios and standardized mean differences) are naturally correlated with their standard errors, and can produce spurious asymmetry in a funnel plot. We discuss this problem in more detail in Section 10.4.3.