Estimating diagnostic accuracy without a gold standard is an important problem in medical testing. Although there is a fairly large literature on this problem for the case of repeated binary tests, there is substantially less work for the case of ordinal tests. A noted exception is the work by Zhou, Castelluccio, and Zhou (2005, Biometrics 61, 600-609), which proposed a methodology for estimating receiver operating characteristic (ROC) curves without a gold standard from multiple ordinal tests. A key assumption in their work was that the test results are independent conditional on the true test result. I propose random effects modeling approaches that incorporate dependence between the ordinal tests, and I show through asymptotic results and simulations the importance of correctly accounting for the dependence between tests. These modeling approaches, along with the importance of accounting for the dependence between tests, are illustrated by analyzing the uterine cancer pathology data analyzed by Zhou et al. (2005).