As meta-analysis of multiple diagnostic tests impacts clinical decision making and patient health, there is an increasing body of research in models and methods for meta-analysis of studies comparing multiple diagnostic tests. The application of the existingmodels to compare the accuracy of three or more tests suffers from the curse of multi-dimensionality, that is, either the number of model parameters increases rapidly or high dimensional integration is required. To overcome these issues in joint meta-analysis of studies comparing T > 2 diagnostic tests in a multiple tests design with a gold standard, we propose a model that assumes the true positives and true negatives for each test are conditionally independent and binomially distributed given the 2T-variate latent vector of sensitivities and specificities. For the random effects distribution, we employ a one-factor copula that provides tail dependence or tail asymmetry. Maximum likelihood estimation of the model is straightforward as the derivation of the likelihood requires bi-dimensional instead of 2T-dimensional integration. Our methodology is demonstrated with an extensive simulation study and an application example that determines which is the best test for the diagnosis of rheumatoid arthritis.