Volume 6, Issue 2

In 1980, Bill James created the Pythagorean win expectation formula with a somewhat counterintuitive idea in mind. James believed, and his formula proved, that a baseball team's current runs scored to runs allowed ratio was better than a team's current record at predicting a team's future winning percentage. The rationale was that the outcomes of close games, which factor prominently in a record but not in a runs ratio, are subject to luck and randomness. The win expectation formula was referred to as Pythagorean because the exponents of two made it resemble the Pythagorean Theorem. James' idea has been extended to other major sports through a generalized Pythagorean win expectation formula, with different exponents—which we call "alphas"—emerging for each sport. In this paper, we estimate the alphas for the win expectation formulas for both full-length and overtime games in the National Basketball Association (NBA), National Football League (NFL), and Major League Baseball (MLB), based on games over the past 10-20 seasons. While our results for full-length games are similar to the generally-accepted win expectation formulas, we believe this is the first attempt to measure how teams' runs scored to runs allowed ratios—which we term "strength"—influence overtime games. We find through logistic regression that the overtime alphas for the NBA, NFL, and MLB are 9.22, 1.11, and .94, respectively. Comparing the full-length game win expectation formulas to the overtime formulas allows one to see how the impact of strength changes from full-length games to overtime games. It is discovered that the impact of strength on win probability decreases least in NBA overtime and most in NFL overtime. Therefore, NBA overtime games are most likely to be won by the team that would win a full-length game and NFL overtime games are most random relative to full-length games. If a team has a 75 percent chance of winning a full-length game, its chances of winning an overtime game are 67.28, 63.00, and 61.56 percent for the NBA, MLB, and NFL, respectively.

This paper analyzes the situation in a best-of-N set match, where both players/teams are given the opportunity to increase their probability of winning a set (increase in effort) on one particular set. To gain insight to the problem, a best-of-3 set match (as typically used in tennis) is analyzed. Using game theory to obtain an optimal solution, the results indicate that both players should use a mixed strategy, by applying their increase in effort at each set with a probability of one third. A conjecture is devised to obtain an optimal solution for a best-of-N set match. Some applications are given to the theoretical results, which could be used by coaches and players to optimize performance.

The sports industry is one of the ten largest business sectors in the U.S., and its primary source of revenue is derived from sports fans. Yet, little has been published in the academic journals about fans' allocation of time, effort, and/or financial expenditures in regard to the sports they care so desperately about. The objective of this research is to uncover the multidimensional aspects of the observed manifestations of fan avidity, and explore the nature of the heterogeneity of such expressions. Data were collected from a student sample of football fans from a well-known university. Thirty-five different expressions of fan avidity were uncovered related to how these fans follow and support their football team. A spatial choice multidimensional unfolding model is developed and reveals four latent dimensions of fan avidity expression. The managerial aspects of these empirical findings are provided, as well as several directions for future research.

Tournament design is of crucial importance in competitive sports. The primary goal of effective tournament design is to provide incentives for the participants to maximize their performance both during the tournament and in the time period leading up to the tournament. In spectator sports, a secondary goal of tournament design is to also promote interesting match ups that generate fan interest. Seeded tournaments, in general, promote both goals. Teams or individuals with strong performances leading up to a tournament receive higher seeds which increase their chances of progressing further in the tournament. Furthermore, seeding ensures that the strongest teams or players are most likely to meet in the final rounds of the tournament when fan interest is at its peak. Under some distributions of team or player skill, however, a seeding system can introduce anomalies that could affect incentives.Our analysis of the NCAA men's basketball tournament uncovers such an anomaly. The seeding system in this tournament gives teams with better success in the regular season more favorable first round match ups, but the tournament is not reseeded as the games progress. Therefore, while higher seeds progress to the 2nd round of the tournament at uniformly higher rates than lower seeds, this relationship breaks down in later rounds. We find that 10th and 11th seeds average more wins and typically progress farther in the tournament than 8th and 9th seeds. This finding violates the intended incentive structure of seeded tournaments.

The purpose of this paper is to demonstrate two methods to quantify skill importance for teams in general, and women's volleyball in particular. A division I women's volleyball team rated each skill (serve, pass, set, etc.) and recorded rally outcomes during all home games in a competitive season. The skills were only rated when the ball was on the home team's side of the net. Events followed one of these three patterns: serve-outcome, pass-set-attack outcome, or block-dig-set-attack-outcome. These sequences of events were assumed to be first-order Markov chains, meaning the quality of the performance of the current skill only depended on the quality of the performance of the previous skill. We analyze the volleyball data using two different techniques: one uses a Markovian transition matrix, while the other is an implementation of logistic regression. To estimate the Markovian transition matrix, we assumed a multinomial likelihood with a Dirichlet prior on the transition probabilities. The logistic regression model also uses a Bayesian approach. The posterior distributions of parameters associated with skill performance are used to calculate importance scores. Importance scores produced by the two methods are reasonably consistent across skills. The importance scores indicate, among other things, that the team would have been well rewarded by improving transition offense. Importance scores can be used to assist coaches in allocating practice time, developing new strategies, and optimizing team performance relative to player selection.

Since 2005, Major League Baseball (MLB) has suspended 258 players under its Drug Prevention and Treatment Program. Moreover, the Mitchell Report yielded the names of 89 alleged users of performance-enhancing drugs (PEDs). This documentation enables quantification of the impact of PEDs on player performance. Literature with this goal is limited, and has focused primarily on batters. Some authors have examined Roger Clemens, but there has been no previous work assessing the influence of PEDs on pitchers more generally. We gathered average fastball velocity from Fangraphs.com for all MLB pitchers who threw at least 10 innings in a month between 2002 and 2008 (11,860 player months). Pitchers were deemed to be PED users if they were named as such in the Mitchell Report or suspended by MLB for a positive PED test. Human growth hormone (HGH) usage was tracked separately. We modeled fastball velocity by PED and HGH usage, age, a Starter/Reliever indicator, and several control variables. Using PEDs significantly increased average fastball velocity by 1.074 MPH overall. When PED impact was allowed to vary by pitcher type (Starter/Reliever) and age, its benefits were most substantial later in a player's career. For example, at age 35, the effect of PEDs was 1.437 MPH for relievers and 0.988 MPH for starters. HGH use was significantly negatively correlated with fastball velocity. This suggests disproportional HGH use by injured players hoping to hasten their recoveries, and is consistent with frequent explanations provided in the Mitchell Report.

This paper introduces a new iterative model for ranking college football teams. It is first presented as a general model with a number of parameters. We then introduce two learning methods that use past data to predict the optimal values of the parameters for the model. Our learning algorithms are then implemented using data from 1998-2008. We analyze the accuracy of our rankings by considering bowl game outcomes for each season. We also compare our results with the Bowl Championship Series computer ranking system. We close with a discussion of possible directions for future work.

A receiver operating characteristic (ROC) curve visually demonstrates the tradeoff between sensitivity and specificity as a function of varying a classification threshold. It is a common practice to use ROC curves to measure the accuracy of predictions by different methods. Although this method has been used primarily in medical and engineering fields, it could be used effectively in sports as well. More precisely, an ROC plots the sensitivity versus (1 - specificity), and the area under the curve gives a measure of the prediction. So, the ideal best prediction should have one square unit of area under the ROC curve, where it achieves both 100% sensitivity and 100% specificity (which, in reality, rarely happens). Consequently, when we compare two methods, the one with the greater area under its ROC is judged best. This paper shows the effectiveness of using the ROC curves in analyzing cricket data. In particular, the quality of the decisions made by umpires is investigated. Also, the comparison of the accuracy of the methods for revising the target for matches shortening due to weather interruptions is the key interest in our investigation.

Crew race strategy is typically formulated by coaches based on rowing tradition and years of experience. However, coaching strategies are not generally supported by empirical evidence and decision-support models. Previous models of crew race strategy have been constrained by the sparse information published on crew race performance (quarterly 500-meter splits). Empirical research has merely summarized which quarterly splits averaged the fastest and slowest relative to the other splits and relative to the average speed of the other competitors.Video records of crew race world championships provide a rich source of data for those capable and patient enough to mine this level of detail. This paper is based on a precise frame-by-frame video analysis of five world championship rowing finals. With six competing crews per race, a database of 75 race-pair duels was compiled that summarizes race positioning, competitive drives, and relative stroke rates at 10-meter intervals recorded with photo-finish precision (30 frames per second). The drive-based research pioneered in this paper makes several contributions to visualizing the dynamics of crew race strategy and performance:1. A generic drive model that decomposes how pairs of crews duel in a race.2. Graphical summaries of the rates and locations of successful and unsuccessful drives.3. Contour lines of the margins that winning crews hold over the course of the race.4. Trend lines for what constitutes a probabilistically decisive lead as a function of position along the course, seconds behind the leader, and whether the trailing crew is driving.This research defines a new drive-based vocabulary for evaluating crew race performance for use by coaches, competitors and race analysts. The research graphically illustrates situational parameters helpful in formulating race strategy and guiding real-time decision-making by competitors.

Based upon contingent claims methodology and standard techniques in statistical modeling and stochastic calculus, we develop a framework for determining the financial value of professional soccer players to their existing and potential new clubs. The model recognizes that a player's value is a product of a variety of factors, some of them more obvious (i.e. on-field performance, injuries, disciplinary record), and some of them less obvious (i.e. image rights or personal background). We provide numerical examples based upon historical statistical performance indicators that suggest the value of a soccer player is not the same for all potential clubs present in a market. In other words this is a special case where the law of one price for one asset does not function. Our modeling employs the vast database of soccer players' performance maintained by OPTA Sportsdata; the same database has been used by major clubs in the English Premiership such as Arsenal and Chelsea. From a statistical point of view, our model can be applied to identify the relative value of players with similar characteristics but different market valuations, to explore patterns of performance for individual star players and teams over a run of games, and to explore correlations or interactions between pairs of players or small groups of players on the team. Moreover, it offers a tool to value players from a financial point of view using their past performance; hence this model can be also used to inform contractual negotiations.

Coaches in the NFL make approximately 1000 offensive play calls during the regular season. These calls are the result of countless hours of preparation and analysis and the coach's own personal experience and each coach has their own measures of success and biases regarding types of play calls. What has not been utilized previously is a systematic analytical approach to measure a play's outcome in relation to the drive, and an evaluation of whether coaches' are irrationally biased in their playcalling. Using play by play data from the 2005 through 2008 NFL regular season, an evaluation system is built around the concept of expected points. Expected points have been used in baseball for over 40 years and have been applied occasionally in football (Romer 2003; Carroll et al 1988). This framework allows for a true calculation of risk for different play types. Risk for passing plays is found to be lower than risk for running plays in certain situations, while still yielding a higher expected value. These results confirm previous analysis (Alamar 2008) that teams underutilize the pass.

A popular topic of argument among baseball fans is the prospective Hall of Fame status of current and recently retired players. A player's probability of enshrinement is likely to be affected by a large number of different variables, and can be approached by machine learning methods. In particular, I consider the use of random forests for this purpose. A random forest may be considered a black-box method for predicting the probability of Hall of Fame induction, but a number of parameters must be chosen before the forest can be grown. These parameters include fundamental aspects of the nuts and bolts of the construction of the trees that make up the forest, as well as choices among possible predictor variables. For example, one predictor that may be considered is a measure of the player's having seasons with many home runs hit, and there are multiple competing ways of measuring this. Furthermore, certain deterministic methods of searching the parameter space are partially undermined by the randomness underlying the forest's construction and the fact that, by sheer luck, two forests constructed with the same parameters may have differing qualities of fit. Using simulated annealing, I move through the parameter space in a stochastic fashion, trying many forests and sometimes moving toward a set of parameters even though its fit is apparently not as good as preceding ones. Since probabilities defined based on the votes of terminal nodes of a random forest for classification tend to be too moderate, the results of each forest considered are fed into a logistic regression to produce final probability estimates. From among four simulated annealing runs, the forest with the smallest mean squared error was selected, and analysis of the forests near it in the simulated annealing run indicate that its selection was probably not due to extraordinary "luck." Predictions performed using the out-of-bag samples correctly identify 75% of Baseball Writers Association of America Hall of Fame selections, while misclassifying only 1% of non-selections. Results indicate a smaller mean squared error than a previous neural network approach, although the large number of forests tried and discarded raised concerns about overfitting in this case.

The utilization of statistical methods in sports is growing rapidly. Sports teams use statistical analyses to evaluate players and game strategies, and sports associations develop ranking and ratings systems of players and teams. The evolution of the application of statistics to sports continues to be enhanced with extensive collaboration and interaction between sports analysts and professional statisticians. Unfortunately, opportunities for this collaboration are still relatively uncommon, as academic statisticians often work in isolation developing statistical methods for sports applications, while sports organizations often do not have access to well-trained statistical expertise and cutting edge statistical tools for the analysis of sports data.