  Rankings and scores are two common data types used by judges to express
preferences and/or perceptions of quality in a collection of objects. Numerous
models exist to study data of each type separately, but no unified statistical
model captures both data types simultaneously without first performing data
conversion. We propose the Mallows-Binomial model to close this gap, which
combines a Mallows' $\phi$ ranking model with Binomial score models through
shared parameters that quantify object quality, a consensus ranking, and the
level of consensus between judges. We propose an efficient tree-search
algorithm to calculate the exact MLE of model parameters, study statistical
properties of the model both analytically and through simulation, and apply our
model to real data from an instance of grant panel review that collected both
scores and partial rankings. Furthermore, we demonstrate how model outputs can
be used to rank objects with confidence. The proposed model is shown to
sensibly combine information from both scores and rankings to quantify object
quality and measure consensus with appropriate levels of statistical
uncertainty.
