  Standard methods for the meta-analysis of medical tests without a gold
standard are limited to dichotomous data. Multivariate probit models are used
to analyze correlated binary data, and can be extended to multivariate ordered
probit models to model ordinal data. Within the context of an imperfect gold
standard, they have previously been used for the analysis of dichotomous and
ordinal tests in a single study, and for the meta-analysis of dichotomous
tests. In this paper, we developed a hierarchical, latent class multivariate
probit model for the simultaneous meta-analysis of ordinal and dichotomous
tests without assuming a gold standard. The model can accommodate a
hierarchical partial pooling model on the conditional within-study
correlations, enabling one to obtain summary estimates of joint test accuracy.
Dichotomous tests use probit regression likelihoods and ordinal tests use
ordered probit regression likelihoods. We fitted the models using Stan, which
uses a state-of-the-art Hamiltonian Monte Carlo algorithm. We applied the
models to a dataset in which studies evaluated the accuracy of tests, and test
combinations, for deep vein thrombosis. We first demonstrated the issues with
dichotomising test accuracy data a priori without a gold standard by fitting
models which dichotomised the ordinal test data, and then we applied models
which do not dichotomise the data. Furthermore, we fitted and compared a
variety of other models, including those which assumed conditional independence
and dependence between tests, and those assuming perfect and an imperfect gold
standard.
