We present techniques for joint modeling of binomial and rank response data using the Bayesian paradigm for inference. The motivating application consists of results from a series of assessments on several primate species. Among 20 assessments representing 6 paradigms, 6 assessments are considered to produce a rank response and the remaining 14 are considered to have a binomial response. In order to model each of the 20 assessments simultaneously, we use the popular technique of data augmentation so that the observed responses are based on latent variables. The modeling uses Bayesian techniques for modeling the latent variables using random effects models. Competing models are specified in a consistent fashion which easily allows comparisons across assessments and across models. Non-local priors are readily admitted to enable more effective testing of random effects should Bayes factors be used for model comparison. The model is also extended to allow assessment-specific conditional error variances for the latent variables. Due to potential difficulties in calculating Bayes factors, discrepancy measures based on pivotal quantities are adapted to test for the presence of random effects and for the need to allow assessment-specific conditional error variances. In order to facilitate implementation, we describe in detail the joint prior distribution and a Markov chain Monte Carlo (MCMC) algorithm for posterior sampling. Results from the primate intelligence data are presented to illustrate the methodology. The results indicate substantial paradigm-specific differences between species. These differences are supported by the discrepancy measures as well as model posterior summaries. Furthermore, the results suggest that meaningful and parsimonious inferences can be made using the proposed techniques and that the discrepancy measures can effectively differentiate between necessary and unnecessary random effects. The contributions should be particularly useful when binomial and rank data are to be jointly analyzed in a parsimonious fashion.