Abstract
Suppose the same contestants play in tournaments of chess, shogi, and Go. Per-tournament rankings can be estimated. We may also try to recover a latent board game skill that accounts for some proportion of the variance in per-board game rankings. To accomplish this, a factor model is introduced. Identification issues with the ordinal paired item model are discussed. Simulation studies are presented to provide some guidance about sample size requirements. Both single item and multivariate correlation and factor model are validated using simulation-based calibration. We recommend leave-one-out cross-validation to assess model fit. To ease application of the methods described, an open-source companion R extension, pcFactorStan, is published on the Comprehensive R Archive Network. Application of pcFactorStan is demonstrated by analysis of a real-world dataset.