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.
Highlights
The notion that latent constructs such as general intelligence can partially account for more specific measurable attributes like reading comprehension has spawned a whole branch of productive research known as factor analysis [1]
Factor analysis has generally been applied to items that record absolute as opposed to relative judgments
We have reviewed some examples of paired comparison factor models from prior literature
Summary
The notion that latent constructs such as general intelligence can partially account for more specific measurable attributes like reading comprehension has spawned a whole branch of productive research known as factor analysis [1]. The worth of each object can be modeled as a latent variable and a factor model built on top, treating the latent worths as indicators [e.g., 4] This application of factor analysis can discern subtle relationships among objects, but it remains a single item analysis in terms of object comparison. The first item is associated with Neuroticism, the second with Extraversion, and the third with Openness This kind of survey is multidimensional, but researchers have no intention to posit a latent Personality factor that would account for some of the variance in the five traits. We have reviewed some examples of paired comparison factor models from prior literature These analytic methods do not try to assess an inaccessible, latent quality of objects by measuring multiple accessible facets of these objects. Matrix Cholesky factors are lower (not upper) triangular, and for the univariate case of the normal distribution , the second argument is a standard deviation not a variance
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