Using multilevel random coefficient models to analyze ordinal responses

Abstract

The origin of gambling as an entertainment dates back several centuries. Probability originated from a gambler’s dispute in 1654 concerning the division of a stake between two players. The problem was proposed by a gambler to Pascal and Fermat and the correspondence which followed between the two mathematicians was fundamental in the development of modern concepts of probability. Using probability theory, it is possible to estimate the probability of a gaming event. In fact, the mathematics of gambling is a pool of probability applications encountered in games of chance. Throughout the centuries, gambling developed from real life casinos with level slot machines to electronic slot machines to online gambling. The online gambling market is expanding all over the world, due to a huge increase of players, who bet on a large variety of different products, mainly including games of chance, sports and poker on a regular basis. This paper presents a model that relates the profit made by the company on each gamer to a number of predictors. Since the gamers are nested in countries and the profits have an ordinal categorical scale, then the two-level cumulative logit model was fitted, assuming both random intercept and random slope. Section 1 describes the dataset and identifies the response variable, the predictors and the nesting structure. Section 2 presents the two-level random coefficient cumulative logit model. Section 3 describes the estimation techniques used to evaluate the marginal likelihood numerically using adaptive Gaussian quadrature, estimate the model parameters using the Newton-Raphson algorithm and predict the random effects using the empirical Bayes method. Section 4 describes the main results of the study.