In studies of the accuracy of diagnostic tests, it is common that both the diagnostic test itself and the reference test are imperfect. This is the case for the microsatellite instability test, which is routinely used as a prescreening procedure to identify individuals with Lynch syndrome, the most common hereditary colorectal cancer syndrome. The microsatellite instability test is known to have imperfect sensitivity and specificity. Meanwhile, the reference test, mutation analysis, is also imperfect. We evaluate this test via a random effects meta-analysis of 17 studies. Study-specific random effects account for between-study heterogeneity in mutation prevalence, test sensitivities and specificities under a nonlinear mixed effects model and a Bayesian hierarchical model. Using model selection techniques, we explore a range of random effects models to identify a best-fitting model. We also evaluate sensitivity to the conditional independence assumption between the microsatellite instability test and the mutation analysis by allowing for correlation between them. Finally, we use simulations to illustrate the importance of including appropriate random effects and the impact of overfitting, underfitting, and misfitting on model performance. Our approach can be used to estimate the accuracy of two imperfect diagnostic tests from a meta-analysis of multiple studies or a multicenter study when the prevalence of disease, test sensitivities and/or specificities may be heterogeneous among studies or centers.