We discuss the issue of identifiability of latent-class models for dichotomous diagnostic tests in the absence of a gold standard. Models are generally posited in terms of disease prevalences, sensitivities, specificities, and conditional dependence terms. Goodman's (1974, Biometrika) method is used to check identifiability via computing packages that allow symbolic matrix compuations such as Mathematica and Maple, and further developed to find model parameters that are actually identifiable in an otherwise non-identified model using an approach due to Rothenberg (1971, Econometrika). The method for dichotomous tests is then generalized to ordinal and continuous outcomes and applied to Bayesian semiparametric models for obtaining ROC curves without a gold-standard when covariates are available. We present illustrations using simulated and real data.