ABSTRACT
In this chapter the focus is on coping with unobserved heterogeneity. This problem has received considerable attention in the literature. There appears to be a common understanding that unobserved heterogeneity is abundant in most meta-analyses. Engels et al. (2000) investigate 125 meta-analyses and conclude that
Approaches differ in the way they cope with this important problem. In the simplest approach, the variance of the pooled estimator is supplemented by an additional variance term, the heterogeneity variance. This heterogeneity variance can be estimated in various ways such as the moment approach (DerSimonian and Laird (1986), Malzahn et al. (2000), Bo¨hning et al. (2002), Bo¨hning et al. (2002) or Sidik and Jonkman (2005) or the maximum likelihood method (Hardy and Thompson (1996) or Whitehead and Whitehead (1991)). After the heterogeneity variance has been estimated the pooled estimator is recomputed using weights that incorporate the heterogeneity variance and a - usually enlarged - confidence interval is computed on basis of the incorporated heterogeneity variance. In addition, the random effects approach might be supplemented by a more complete modeling (see also Hardy and Thompson (1998) and Hedges and Vevea (1998)). A latent variable, an unobserved covariate might be supposed to be the source of this form of heterogeneity. Parametric approaches assume a parametric distribution for the latent variable (see Hardy and Thompson (1998), Martuzzi and Hills (1995)), and the approaches differ usually in the kind of parametric distribution that is assumed for the latent variable such as normal or Gamma. This latent variable might be a missing covariate (see Section 4) such as a treatment modification or different patient population, although other sources such as correlation of observations might be a source for this kind of heterogeneity. It is also discussed frequently whether the underlying risk is a source of heterogeneity for the relative risk (see Sharp et al. (1996), Sharp and Thompson (2000), Brensen et al. (1999), Egger and Smith (1995), van Houwelingen and Senn (1999), Thompson (1994), Arends et al. (2000)). It is pointed out here that the profile likelihood approach has the advantage of eliminating the effect of the baseline parameter before considering heterogeneity in the relative risk. In
the following we develop a nonparametric random effects approach for modeling unobserved heterogeneity.
