Inference in Thurstonian models for paired comparison data

Paired comparison data arise in many contexts, including genetics, sensory testing, psychometrics and sport data. Psychometricians commonly employ Thurstonian models to analyze paired comparison data. This type of models assumes that each object under comparison provides a stimulus to the person that compares them, and the stimuli follow a multivariate normal distribution. Different covariance structures of the multivariate normal distribution can be assumed, thus originating different specifications of Thurstonian models. In each paired comparison, a person prefers an object if the stimulus provided by that object is higher than the stimulus given by the other object. The assumption of a multivariate normal distribution for the stimuli implies that the paired comparisons made by the same person are dependent, while paired comparisons performed by different people are assumed independent. In the psychometric literature, Thurstonian models are estimated by means of a type of limited information estimation method that minimizes a Mahalanobis distance between the empirical univariate and bivariate proportions and the model-based moments. Simulation studies show that, in some specifications of Thurstonian models, the limited information estimation method yields estimates with a noticeable empirical bias, especially for the covariance parameters, and large variability. We investigate the performance of other estimating methods and compare it with the performance of limited information estimation. Thurstonian models are a particular instance of the multivariate probit model. In these situations, the integral has dimension equal to the number of paired comparisons, which may be quite large, therefore many of the usual algorithms for approximation of multivariate normal probabilities either cannot be employed or require a long computational time or have a relatively large approximation error. An alternative estimation method, typically used in choice models, is maximum simulated likelihood through the Geweke-Hajivassiliou-Keane algorithm. Another possibility is to employ a Laplace approximation. These methods have the advantage of being applicable also when the number of people that perform the paired comparisons is low, or even one, while the limited information estimation cannot be employed in this instance. We compare the proposed estimating methods with the limited information estimation commonly employed by psychometricians and consider an application to a real data set