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.
