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In the realm of competitive sports, understanding the performance dynamics of athletes, represented by the age curve (showing progression, peak, and decline), is vital. Our research introduces a novel framework for quantifying age-specific treatment effects, enhancing the granularity of performance trajectory analysis. Firstly, we propose a methodology for estimating the age curve using game-level data, diverging from traditional season-level data approaches, and tackling its inherent complexities with a meta-learner framework that leverages advanced machine learning models. This approach uncovers intricate non-linear patterns missed by existing methods. Secondly, our framework enables the identification of causal effects, allowing for a detailed examination of age curves under various conditions. By defining the Age-Conditioned Treatment Effect (ACTE), we facilitate the exploration of causal relationships regarding treatment impacts at specific ages. Finally, applying this methodology to study the effects of rest days on performance metrics, particularly across different ages, offers valuable insights into load management strategies' effectiveness. Our findings underscore the importance of tailored rest periods, highlighting their positive impact on athlete performance and suggesting a reevaluation of current management practices for optimizing athlete performance.

The impact of player age on performance has received attention across sport. Most research has focused on the performance of players at each age, ignoring the reality that age likewise influences which players receive opportunities to perform. Our manuscript makes two contributions. First, we highlight how selection bias is linked to both (i) which players receive opportunity to perform in sport, and (ii) at which ages we observe these players perform. This approach is used to generate underlying distributions of how players move in and out of sport organizations. Second, motivated by methods for missing data, we propose novel estimation methods of age curves by using both observed and unobserved (imputed) data. We use simulations to compare several comparative approaches for estimating aging curves. Imputation-based methods, as well as models that account for individual player skill, tend to generate lower RMSE and age curve shapes that better match the truth. We implement our approach using data from the National Hockey League.

Recording of events in National Hockey League rinks is done through the Real Time Scoring System. This system records events such as hits, shots, faceoffs, etc., as part of the play-by-play files that are made publicly available. Several previous studies have found that there are inconsistencies in the recording of these events from rink to rink. In this paper, we propose a methodology for estimation of the rink effects for each of the rinks in the National Hockey League. Our aim is to build a model which accounts for the relative differences between rinks. We use log-linear regression to model counts of events per game with several predictors including team factors and average score differential. The estimated rink effects can be used to reweight recorded events so that can have comparable counts of events across rinks. Applying our methodology to data from six regular seasons, we find that there are some rinks with rink effects that are significant and consistent across these seasons for multiple events.

It is often said that a sign of a great player is that he makes the players around him better. The player may or may not score much himself, but his teammates perform better when he plays. One way a hockey player can improve his or her teammates' performance is to create goal scoring opportunities. Unfortunately, in hockey goal scoring is relatively infrequent, and statistics like assists can be unreliable as a measure of a player's playmaking ability. Assists also depend on playing time, power play usage, the strength of a player's linemates, and other factors. In this paper we develop a metric for quantifying playmaking ability that addresses these issues. Our playmaking metric has two benefits over assists for which we can provide statistical evidence: it is more consistent than assists, and it is better than assists at predicting future assists. Quantifying player contributions using this measure can assist decision-makers in identifying, acquiring, and integrating successful playmakers into their lineups., Comment: 21 pages, 4 figures, 4 tables

Sports leagues consist of conferences subdivided into divisions. Teams play a number of games within their divisions and fewer games against teams in different divisions and conferences. Usually, a league structure remains stable from one season to the next. However, structures change when growth or contraction occurs, and realignment of the four major professional sports leagues in North America has occurred more than twenty-five times since 1967. In this paper, we describe a method for realigning sports leagues that is flexible, adaptive, and that enables construction of schedules that minimize travel while satisfying other criteria. We do not build schedules; we develop league structures which support the subsequent construction of efficient schedules. Our initial focus is the NHL, which has an urgent need for realignment following the recent move of the Atlanta Thrashers to Winnipeg, but our methods can be adapted to virtually any situation. We examine a variety of scenarios for the NHL, and apply our methods to the NBA, MLB, and NFL. We find the biggest improvements for MLB and the NFL, where adopting the best solutions would reduce league travel by about 20%., Comment: 20 figures, 1 table

We use the susceptible-infected-recovered (SIR) model for disease spread over a network, and empirically study how well various centrality measures perform at identifying which nodes in a network will be the best spreaders of disease on 10 real-world networks. We find that the relative performance of degree, shell number and other centrality measures can be sensitive to B, the probability that an infected node will transmit the disease to a susceptible node. We also find that eigenvector centrality performs very well in general for values of B above the epidemic threshold.

In this paper, we develop a logistic regression model to estimate the probability that a particular shot in an NHL game will result in a goal, and use the results to evaluate the performance of NHL skaters, goalies, and teams. We weight each shot based on the estimated probabilities obtained from our model, call this statistic "weighted shots", and use advanced statistics based on weighted shots as the basis of our evaluation. We also analyze whether advanced statistics based on weighted shots outperform traditional statistics as an indicator of future performance of skaters, goalies, and teams. In general, statistics based on weighted shots perform well, but not better than traditional statistics. We conclude that weighted shots should not be viewed as a replacement for those statistics, but can be used in conjunction with those statistics. Finally, we use weighted shots as the dependent variable in an adjusted plus-minus model. The results are estimates of each player's offensive and defensive contribution to his team's weighted shots during even strength, power play, and short handed situations, independent of the strength of his teammates, the strength of his opponents, and the zone in which his shifts begin., Comment: 19 pages, 10 figures, 9 tables

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., Comment: 24 pages, 5 figures, 7 tables

We study the density of complex critical points of a real random SO(m+1) polynomial in m variables. In a previous paper [Mac09], the author used the Poincare- Lelong formula to show that the density of complex zeros of a system of these real random polynomials rapidly approaches the density of complex zeros of a system of the corresponding complex random polynomials, the SU(m+1) polynomials. In this paper, we use the Kac- Rice formula to prove an analogous result: the density of complex critical points of one of these real random polynomials rapidly approaches the density of complex critical points of the corresponding complex random polynomial. In one variable, we give an exact formula and a scaling limit formula for the density of critical points of the real random SO(2) polynomial as well as for the density of critical points of the corresponding complex random SU(2) polynomial., Comment: 15 pages

The goal of this paper is to develop an adjusted plus-minus statistic for NHL players that is independent of both teammates and opponents. We use data from the shift reports on NHL.com in a weighted least squares regression to estimate an NHL player's effect on his team's success in scoring and preventing goals at even strength. Both offensive and defensive components of adjusted plus-minus are given, estimates in terms of goals per 60 minutes and goals per season are given, and estimates for forwards, defensemen, and goalies are given., Comment: 39 pages, 4 figures, 25 tables. Version 3: Typos fixed. Table of contents, list of tables, and list of figures added. Two paragraphs in discussion of goalies at the end of Section 3.3 were added

We study the density of complex zeros of a system of real random SO($m+1$) polynomials in several variables. We show that the density of complex zeros of this random polynomial system with real coefficients rapidly approaches the density of complex zeros in the complex coefficients case. We also show that the behavior the scaled density of complex zeros near $\mathbb{R}^m$ of the system of real random polynomials is different in the $m\geq 2$ case than in the $m=1$ case: the density goes to infinity instead of tending linearly to zero., Comment: 25 pages, 4 figures

Describes a community intervention for a population of youth who are often distrustful of mainstream mental health services. Program focuses on not viewing youth as patients but as experts in working with adults to design youth-friendly interventions. Outlines the development and implementation of a support-group model tailored specifically to the needs of homeless youth. (Author/GCP)