In this method of moments estimator, only a basic random-effects structure is allowed, where each group (usually corresponding to an independent study) provides a single estimate (unit of analysis) for one or multiple outcomes. This implies that the number of groups (i.e., the length of the lists) is identical to the number of units (m=n). In addition, only an unstructured form for the(co)variance matrix of the single level of random effects is permitted. Therefore, the estimation involves \(kp\) fixed-effects coefficients and \(k(k+1)/2\) random-effects parameters, corresponding to the lower triangular entries of the between-study (co)variance matrix.
The method of moment estimator implemented here represents a multivariate extension of the traditional estimator proposed by DerSimonian and Laird (1986), and simplifies to the standard method in the univariate case. The estimator used here is described in Jackson and collaborators (2013) as a generalization of that developed by Chen and collaborators (2012). However, this general version is computationally more intensive, and may turn out to be slow when applied to meta-analysis of a relatively high number of studies. An alternative and computationally faster method of moment estimator was previously proposed by Jackson and collaborators (2010), although it is not invariant to reparameterization. This latter estimator is currently not implemented in mixmeta. See references below.
This method of moments estimator is not bounded to provide a positive semi-definite random-effects (co)variance matrix, as shown in the simulation study by Liu and colleagues (2009). Here positive semi-definiteness is forced by setting the negative eigenvalues of the estimated matrix to a positive value close to zero at each iteration (see control). Little is known about the impact of such constraint.