9.5.4  Incorporating heterogeneity into random-effects models

A fixed-effect meta-analysis provides a result that may be viewed as a ‘typical intervention effect’ from the studies included in the analysis. In order to calculate a confidence interval for a fixed-effect meta-analysis the assumption is made that the true effect of intervention (in both magnitude and direction) is the same value in every study (that is, fixed across studies). This assumption implies that the observed differences among study results are due solely to the play of chance, i.e. that there is no statistical heterogeneity.

 

When there is heterogeneity that cannot readily be explained, one analytical approach is to incorporate it into a random-effects model. A random-effects meta-analysis model involves an assumption that the effects being estimated in the different studies are not identical, but follow some distribution. The model represents our lack of knowledge about why real, or apparent, intervention effects differ by considering the differences as if they were random. The centre of this distribution describes the average of the effects, while its width describes the degree of heterogeneity. The conventional choice of distribution is a normal distribution. It is difficult to establish the validity of any distributional assumption, and this is a common criticism of random-effects meta-analyses. The importance of the particular assumed shape for this distribution is not known.

 

Note that a random-effects model does not ‘take account’ of the heterogeneity, in the sense that it is no longer an issue. It is always advisable to explore possible causes of heterogeneity, although there may be too few studies to do this adequately (see Section 9.6).

 

For random-effects analyses in RevMan, the pooled estimate and confidence interval refer to the centre of the distribution of intervention effects, but do not describe the width of the distribution. Often the pooled estimate and its confidence interval are quoted in isolation as an alternative estimate of the quantity evaluated in a fixed-effect meta-analysis, which is inappropriate. The confidence interval from a random-effects meta-analysis describes uncertainty in the location of the mean of systematically different effects in the different studies. It does not describe the degree of heterogeneity among studies as may be commonly believed. For example, when there are many studies in a meta-analysis, one may obtain a tight confidence interval around the random-effects estimate of the mean effect even when there is a large amount of heterogeneity.

 

In common with other meta-analysis software, RevMan presents an estimate of the between-study variance in a random-effects meta-analysis (known as tau-squared (τ2 or Tau2)). The square root of this number (i.e. tau) is the estimated standard deviation of underlying effects across studies. For absolute measures of effect (e.g. risk difference, mean difference, standardized mean difference), an approximate 95% range of underlying effects can be obtained by creating an interval from 2×tau below the random-effects pooled estimate, to 2×tau above it. For relative measures (e.g. odds ratio, risk ratio), the interval needs to be centred on the natural logarithm of the pooled estimate, and the limits anti-logged (exponentiated) to obtain an interval on the ratio scale. Alternative intervals, for the predicted effect in a new study, have been proposed (Higgins 2008b). The range of the intervention effects observed in the studies may be thought to give a rough idea of the spread of the distribution of true intervention effects, but in fact it will be slightly too wide as it also describes the random error in the observed effect estimates.

 

If variation in effects (statistical heterogeneity) is believed to be due to clinical diversity, the random-effects pooled estimate should be interpreted differently from the fixed-effect estimate since it relates to a different question. The random-effects estimate and its confidence interval address the question ‘what is the average intervention effect?’ while the fixed-effect estimate and its confidence interval addresses the question ‘what is the best estimate of the intervention effect?’ The answers to these questions coincide either when no heterogeneity is present, or when the distribution of the intervention effects is roughly symmetrical. When the answers do not coincide, the random-effects estimate may not reflect the actual effect in any particular population being studied.

 

Methodological diversity creates heterogeneity through biases variably affecting the results of different studies. The random-effects pooled estimate will only estimate the average treatment effect if the biases are symmetrically distributed, leading to a mixture of over- and under-estimates of effect, which is unlikely to be the case. In practice it can be very difficult to distinguish whether heterogeneity results from clinical or methodological diversity, and in most cases it is likely to be due to both, so these distinctions in the interpretation are hard to draw.

 

For any particular set of studies in which heterogeneity is present, a confidence interval around the random-effects pooled estimate is wider than a confidence interval around a fixed-effect pooled estimate. This will happen if the I2 statistic is greater than zero, even if the heterogeneity is not detected by the chi-squared test for heterogeneity (Higgins 2003) (see Section 9.5.2). The choice between a fixed-effect and a random-effects meta-analysis should never be made on the basis of a statistical test for heterogeneity.

 

In a heterogeneous set of studies, a random-effects meta-analysis will award relatively more weight to smaller studies than such studies would receive in a fixed-effect meta-analysis. This is because small studies are more informative for learning about the distribution of effects across studies than for learning about an assumed common intervention effect. Care must be taken that random-effects analyses are applied only when the idea of a ‘random’ distribution of intervention effects can be justified. In particular, if results of smaller studies are systematically different from results of larger ones, which can happen as a result of publication bias or within-study bias in smaller studies (Egger 1997, Poole 1999, Kjaergard 2001), then a random-effects meta-analysis will exacerbate the effects of the bias (see also Chapter 10, Section 10.4.4.1). A fixed-effect analysis will be affected less, although strictly it will also be inappropriate. In this situation it may be wise to present neither type of meta-analysis, or to perform a sensitivity analysis in which small studies are excluded.

 

Similarly, when there is little information, either because there are few studies or if the studies are small, a random-effects analysis will provide poor estimates of the width of the distribution of intervention effects.

 

RevMan implements a version of random-effects meta-analysis that is described by DerSimonian and Laird (DerSimonian 1986). The attraction of this method is that the calculations are straightforward, but it has a theoretical disadvantage that the confidence intervals are slightly too narrow to encompass full uncertainty resulting from having estimated the degree of heterogeneity. Alternative methods exist that encompass full uncertainty, but they require more advanced statistical software (see also Chapter 16, Section 16.8). In practice, the difference in the results is likely to be small unless there are few studies. For dichotomous data, RevMan implements two versions of the DerSimonian and Laird random-effects model (see Section 9.4.4.3).