A State-Space Model for National Football League Scores
Journal article by Mark E. Glickman, Hal S. Stern; Journal of the American Statistical Association, Vol. 93, 1998
Journal Article Excerpt
A State-Space Model for National Football League Scores. by Mark E. Glickman , Hal S. Stern 1. INTRODUCTION Prediction problems in many settings (e.g., finance, political elections, and in this article, football) are complicated by the presence of several sources of variation for which a predictive model must account. For National Football League (NFL) games, team abilities may vary from year to year due to changes in personnel and overall strategy. In addition, team abilities may vary within a season due to injuries, team psychology, and promotion/demotion of players. Team performance may also vary depending on the site of a game. This article describes an approach to modeling NFL scores using a normal linear state-space model that accounts for these important sources of variability. The state-space framework for modeling a system over time incorporates two different random processes. The distribution of the data at each point in time is specified conditional on a set of time-indexed parameters. A second process describes the evolution of the parameters over time. For many specific state-space models, including the model developed in this article, posterior inferences about parameters cannot be obtained analytically. We thus use Markov chain Monte Carlo (MCMC) methods, namely Gibbs sampling (Gelfand and Smith 1990; Geman and Geman 1984), as a computational tool for studying the posterior distribution of the parameters of our model. Pre-MCMC approaches to the analysis of linear state-space models include those of Harrison and Stevens (1976) and West and Harrison (1990). More recent work on MCMC methods has been done by Carter and Kohn (1994), Fruhwirth-Schnatter (1994), and Glickman (1993), who have developed efficient procedures for fitting normal linear state-space models. Carlin, Polson, and Stoffer (1992), de Jong and Shephard (1995), and Shephard (1994) are only a few of the recent contributors to the growing literature on MCMC approaches to non-linear and non-Gaussian state-space models. The Las Vegas "point spread" or "betting line" of a game, provided by Las Vegas oddsmakers, can be viewed as the "experts" prior predictive estimate of the difference in game scores. A number of authors have examined the point spread as a predictor of game outcomes, including Amoako-Adu, Marmer, and Yagil (1985), Stern (1991), and Zuber, Gandar, and Bowers (1985). Stern, in particular, showed that modeling the score difference of a game to have a mean equal to the point spread is empirically justifiable. We demonstrate that our model performs at least as well as the Las Vegas line for predicting game outcomes for the latter half of the 1993 season. Other work on modeling NFL football outcomes (Stefani 1977, 1980; Stern 1992; Thompson 1975) has not incorporated the stochastic nature of team strengths. Our model is closely related to one examined by Harville (1977, 1980) and Sallas and Harville (1988), though the analysis that we perform differs in a number of ways. We create prediction inferences by sampling from the joint posterior distribution of all model parameters rather than fixing some parameters at point estimates prior to prediction. Our model also describes a richer structure in the data, accounting for the possibility of shrinkage towards the mean of team strengths over time. Finally, the analysis presented here incorporates model checking and sensitivity analysis aimed at assessing the propriety of the state-space model. 2. A MODEL FOR FOOTBALL GAME OUTCOMESLet [y.sub.ii[prime]], denote the outcome of a football game between team i and team i[prime] where teams are indexed by the integers from 1 to p. For our dataset, p = 28. We take [y.sub.ii[prime]] to be the difference between the score of team i and the score of team i[prime]. The NFL game outcomes can be modeled as approximately normally distributed with a mean that depends on the relative strength of the teams involved in the game and the site of the game. We assume that at week j of season k, the strength or ability of team i can be summarized by a parameter [[Theta].sub.(k,j)i]. We let [[Theta].sub.(k,j)] denote the vector of p team-ability parameters for week j of season k. An additional set of parameters, [[Alpha].sub.i], i = 1, . . ., p, measures the magnitude of team i's advantage when playing at its home stadium rather than at a neutral site. These home-field advantage (HFA) parameters are assumed to be independent of time but may vary across teams. We let [Alpha] denote the vector of p HFA parameters. The mean outcome for a game between team i and team i' played at the site of team i during ... |
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