Using a normal distribution for the underlying effects in different studies is a strong assumption, which is usually made in the absence of convincing evidence in its favour. The particular form that is assumed for the distribution can impact on the conclusions of the meta-analysis. Alternative parametric distributions have been suggested for f
(Φ). For example, a t
-distribution or skewed distribution might be used to reduce the effect of outlying studies (Smith et al., 1995
; Lee and Thompson, 2007
), and mixture distributions have been advocated to account for studies belonging to unknown groupings (Böhning, 2000
). Using MCMC methods in a Bayesian framework, parametric distributions for the random effects offer the same opportunities for inference as a normal distribution. However, some parametric distributions may not have parameters that naturally describe an overall mean, or the heterogeneity across studies. In a classical framework, the major limitation in applying non-normal parametric random-effects distributions is the computational complexity in performing the inference (Aitkin, 1999a
Meta-analyses of a large number of large studies lend themselves to relatively complex models, since there may be strong evidence against a normal distribution assumption for the random effect. At the extreme, models may be fitted that are so flexible that the observed data determine the shape of the random-effects distribution. Below we discuss two examples of these flexible approaches that have received particular attention: in a classical framework, non-parametric maximum likelihood (NPML) procedures and, in a Bayesian framework, semiparametric random-effects distributions.
NPML procedures provide a discrete distribution that is based on a finite number of mass points (Laird, 1978
; Böhning, 2005
). Estimation can be achieved via the EM algorithm, and the number of mass points, along with compatibility of the data with different parametric or common effect models, can be determined by comparing deviances (Aitkin, 1999a
, b). Estimates of the overall mean and the heterogeneity variance τ2
are available (Van Houwelingen et al.
, 1993; Aitkin, 1999b
), as are empirical Bayes estimates for the individual studies (Stijnen and Van Houwelingen, 1990
). A particular advantage of the NPML approach is its ability to detect and incorporate outliers. However, NPML has been noted to be unstable (Van Houwelingen et al., 1993
) and has not been widely adopted. Furthermore, the key difficulty with NPML with regard to our important objectives is that realistic predictions are unlikely to follow from the discrete distribution for the random effects, especially when the studies include one or more outliers.
Bayesian semiparametric random-effects distributions have been proposed (Burr et al., 2003
; Burr and Doss, 2005
; Ohlssen et al., 2007
) based on Dirichlet process priors. These comprise discrete mass points that are drawn from a baseline distribution, which might be normal, weighted in such a way that a single parameter α
controls how close the discrete distribution is to the baseline: high values of α
correspond to a random-effects distribution that is close to the baseline; low α
to an arbitrary shape. We can in principle use the observed data to learn about α
, although the small or moderate number of studies in a typical meta-analysis would carry little information. A truncated Dirichlet process offers a more convenient way to implement these models, in which a specified number of mass points is assumed (Ohlssen et al., 2007
), and a mixture of Dirichlet processes allows the discrete points to be replaced by normal distributions, so that the fitted random-effects distribution is a continuous, highly adaptable, mixture of normal distributions. These models can be fitted in WinBUGS, offering a variety of statistical summaries of the posterior distribution for the random effect: nevertheless it must be acknowledged that the analysis becomes considerably more complex. In common with classical non-parametric methods, predictions arising from Dirichlet process priors may be unhelpful, as they can have unconventional shapes that depend heavily on the particular studies that are included in the meta-analysis, although stronger assumptions about α
can constrain the distribution to be reasonably ‘close’ to a baseline distribution with a specified shape.