Comparing fixed and random-effects estimates

In the presence of heterogeneity, a random-effects meta-analysis weights the studies relatively more equally than a fixed-effect analysis. It follows that in the presence of small-study effects such as those displayed in Figure 10.2.a, in which the intervention effect is more beneficial in the smaller studies, the random-effects estimate of the intervention effect will be more beneficial than the fixed-effect estimate. Poole and Greenland summarized this by noting that “random-effects meta-analyses are not always conservative” (Poole 1999). This issue is also discussed in Chapter 9 (Section 9.5.4).


An extreme example of the differences between fixed- and random-effects analyses that can arise in the presence of small-study effects is shown in Figure 10.4.c, which displays both fixed- and random-effects estimates of the effect of intravenous magnesium on mortality following myocardial infarction. This is a well-known example in which beneficial effects of intervention were found in a meta-analysis of small studies, subsequently contradicted when the very large ISIS-4 study found no evidence that magnesium affected mortality.


Because there is substantial between-trial heterogeneity, the studies are weighted much more equally in the random-effects analysis than in the fixed-effect analysis. In the fixed-effect analysis the ISIS-4 trial gets 90% of the weight and so there is no evidence of a beneficial intervention effect. In the random-effects analysis the small studies dominate, and there appears to be clear evidence of a beneficial effect of intervention. To interpret the accumulated evidence, it is necessary to make a judgement about the likely validity of the combined evidence from the smaller studies, compared with that from the ISIS-4 trial.


We recommend that when review authors are concerned about the influence of small-study effects on the results of a meta-analysis in which there is evidence of between-study heterogeneity (I2>0), they compare the fixed- and random-effects estimates of the intervention effect. If the estimates are similar, then any small-study effects have little effect on the intervention effect estimate. If the random-effects estimate is more beneficial, review authors should consider whether it is reasonable to conclude that the intervention was more effective in the smaller studies. If the larger studies tend to be those conducted with more methodological rigour, or conducted in circumstances more typical of the use of the intervention in practice, then review authors should consider reporting the results of meta-analyses restricted to the larger, more rigorous studies. Formal evaluation of such strategies in simulation studies would be desirable. Note that formal statistical comparisons of the fixed- and random-effects estimates of intervention effect are not possible, and that it is still possible for small-study effects to bias the results of a meta-analysis in which there is no evidence of heterogeneity, even though the fixed- and random-effects estimates of intervention effect will be identical in this situation.