A bivariate generalised linear mixed model is often used for meta-analysis of test accuracy studies. The model is complex and although packages have emerged to estimate the model’s five parameters they remain opaque to many. Here we present two standard approaches to deriving maximum likelihood estimates (MLE), which have received little attention in the literature in relation to the bivariate model.
To demonstrate from first principles how MLE for the parameters may be obtained using two methods based on Newton-Raphson (NR) iteration.
The first method uses the Observed Fisher Information (OFI), which reduces the problem to finding the roots of the score statistic (derivative of the likelihood function) and these may be estimated iteratively using the NR algorithm. The second uses the profile likelihood (PL) – this fixes all but one of the parameters and maximises over the one parameter that is not fixed using the NR algorithm. For different combinations of values for the fixed parameters this maps out a profile of maxima, and the largest is chosen. As convergence may depend on the proximity of the initial estimates to the global maximum, we provide a method for obtaining robust initial estimates. The methods are applied to two meta-analyses from the literature and the results are compared with those obtained using a standard package in R.
In general, without robust initial estimates, the OFI method fails to converge when all five parameters need to be estimated. However, with robust initial estimates, both the OFI and PL method converge and provide estimates of the parameters that are comparable with the reference package. Both algorithms changed smoothly between iterations without wild fluctuations. The PL method executed fewer iterations than the OFI method in the first case (10 versus 13) but more in the second (8 versus 7), but took longer per iteration.
The parameters to the bivariate model may be estimated effectively using two standard methods based on the NR algorithm that have, in general, not been applied to this model.