Volume 8, Issue 3

Several models are proposed for the probability of converting a third down attempt in the National Football League. The probability, which can depend on the number of yards to go, the strength of the offense, and the strength of the defense, leads to a logistic regression. We approach the problem through a hierarchical Bayes model and estimate parameters by using Markov chain Monte Carlo (MCMC). This MCMC estimation in the context of a hierarchical Bayes model may be relevant in other sports situations where a probability depends on the difference of strengths of the two teams. We find that the statistic "third-down conversion rate" to be a nearly meaningless measure of the efficiency of an offense. Even when this is adjusted for yards to go for a first down, there is little evidence that teams differ in their ability to achieve a first down on a third down conversion.

Professional sports leagues are motivated to promote competitive balance in order to maintain fan interest. The National Football League (NFL) has taken several steps to promote competitive balance, most notably free agency and the salary cap, which were instituted prior to the 1994 season. Previous research into competitive balance in sports focused on the variability of team strengths but ignored the year-to-year autocorrelation in team strengths. We present a Bayesian state-space model for paired comparisons that allows regression on the variance parameters. By modeling the variance parameters in a regression framework, we are able to simultaneously compare the variance and autocorrelation in team strengths over time. The autocorrelation of NFL team strengths has decreased over time while there has been little change in the variance in teams strengths since the 1970s.

This paper proposes a multiple-membership generalized linear mixed model for ranking college football teams using only their win/loss records. The model results in an intractable, high-dimensional integral due to the random effects structure and nonlinear link function. We use recent data sets to explore the effect of the choice of integral approximation and other modeling assumptions on the rankings. Varying the modeling assumptions sometimes leads to changes in the team rankings that could affect bowl assignments.

We use three-dimensional optical tracking data on the 25-frames-per-second positional data of 2,400 free throw shots by the twenty players with at least twelve tracked makes and twelve tracked misses over the course of the 2010-2011 NBA season, fit each trajectory to a comprehensive physics model to find the implied backspin, initial launch height, velocity, angle, and left-right deviation, and examine the differences of those five factors between makes and misses for each player with sufficient attempts in our sample. We find that usually one or two factors are most responsible for a given player’s misses, but the particular factors at fault differ across players. Thus, the causes of successes and failures in free throw shooting are idiosyncratic. This framework may also be useful in analyzing jump shots taken during the game.

This analysis revises and strengthens a study by Anderson and Pierce (2009) on referee bias in NCAA basketball. Using a logistic regression model, they determined that referees display a statistically significant tendency to even out the foul count between the two teams—every additional increase in the foul differential (home team fouls minus visiting team fouls) raises the odds of a foul on the home team. This study analyzes Anderson and Pierce’s data on the 2004-2005 season from the Big Ten, Big East and ACC conferences, as well as additional data on the 2009-2010 season. Generalized linear mixed modeling, which takes into account the correlation in the data by game and by home and visiting teams, was used to consider the effect of several variables on the odds of a foul on the home team. These included the same variables used by Anderson and Pierce as well as additional terms including the timing and type of the foul. We also used estimates of the random effects in our models to study the relative proneness of different teams to fouls.In Anderson and Pierce’s logistic regression model, every additional unit of foul differential was found to raise the odds of a foul on the home team by 12.5% in 2004-2005. Using a generalized linear mixed model with the same terms, along with random effects for game, home team, and visiting team, raised that estimate to 19.9% and improved the quality of the model. A more in-depth analysis of this data also found that a foul on the home team becomes less likely as the game progresses, particularly when the home team is winning. Similar results were found in an analysis of data from the 2009-2010 season. The 2009-2010 analysis also found evidence that the odds of more subjective offensive fouls were more affected by foul differential than personal or shooting fouls. The effect of individual referees on the amount of bias in each foul call was explored through a preliminary analysis but no significant results were found.A tendency to even out the number of foul calls on each team has the potential to lead to increased physicality in NCAA basketball if the referee tries to keep the foul count close even when one team is clearly playing more physically. These results strengthen the evidence that referees display this propensity significantly, consciously or unconsciously.

In a recent paper, Fair (2008) utilizes monthly data on on-base percentage plus slugging percentage (OPS) to estimate the expected age profile of batting performance in Major League Baseball (MLB) and finds a peak performance at 27.6 years. However, he notes that a small group of 18 players have age-performance profiles that deviate significantly from the expected profile, most notably Barry Bonds. In this paper, we extend the work of Fair (2008) by investigating the time series properties of Bonds’s OPS to test for a deterministic or stochastic trend and to search for structural breaks. While Bonds’s performance is above average, we should not expect that deviations in his age-performance profile from that of the typical batter should contain a deterministic trend. In our investigation, we utilize unit root tests that estimate breaks using monthly data from 1986 to 2007. We find that Bonds’s OPS deviations follow a deterministic trend with two structural breaks. In particular, we find that Bonds’s OPS follows a positive trend to the age of 28.9 (June 1993), which coincides closely with the expected peak performance age. Following this, we find that Bonds’s OPS was on a plateau until a second break in September 2000. At this point, at the age of 36.1, Bonds’s OPS increases unexpectedly and then declines thereafter until his retirement in September 2007 at age 43.

Quantitative evaluation of fielding ability in baseball has been an ongoing challenge for statisticians. Detailed recording of ball-in-play data in recent years has spurred the development of sophisticated fielding models. Foremost among these approaches, Jensen et al. (2009) used a hierarchical Bayesian model to estimate spatial fielding curves for individual players. These previous efforts have not addressed evolution in a player’s fielding ability over time. We expand the work of Jensen et al. (2009) to model the fielding ability of individual players over multiple seasons. Several different models are implemented and compared via posterior predictive validation on hold-out data. Among our choices, we find that a model which imposes shrinkage towards an age-specific average gives the best performance. Our temporal models allow us to delineate the performance of a fielder on a season-to-season basis versus their entire career.

Regression-based adjusted plus-minus statistics were developed in basketball and have recently come to hockey. The purpose of these statistics is to provide an estimate of each player's contribution to his team, independent of the strength of his teammates, the strength of his opponents, and other variables that are out of his control. One of the main downsides of the ordinary least squares regression models is that the estimates have large error bounds. Since certain pairs of teammates play together frequently, collinearity is present in the data and is one reason for the large errors. In hockey, the relative lack of scoring compared to basketball is another reason. To deal with these issues, we use ridge regression, a method that is commonly used in lieu of ordinary least squares regression when collinearity is present in the data. We also create models that use not only goals, but also shots, Fenwick rating (shots plus missed shots), and Corsi rating (shots, missed shots, and blocked shots). One benefit of using these statistics is that there are roughly ten times as many shots as goals, so there is much more data when using these statistics and the resulting estimates have smaller error bounds. The results of our ridge regression models are estimates of the offensive and defensive contributions of forwards and defensemen during even strength, power play, and short handed situations, in terms of goals per 60 minutes. The estimates are independent of strength of teammates, strength of opponents, and the zone in which a player's shift begins.

This paper addresses two important strategic questions in the sport of curling, namely (i) when it is best to blank rather than take one point, and (ii) when entering the final end, is it preferable to be winning by one point without last rock advantage or be losing by one point with last rock advantage. A multinomial logistic regression for end score probabilities and a Markov model for game win probabilities are introduced to answer both of the questions of interest. An asymptotic sampling procedure is used for statistical inference. The paper also updates and expands the data used in the previously published empirical curling analyses to take account of an important rule change.