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  • This article was downloaded by: [Northeastern University]On: 04 November 2014, At: 15:18Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41Mortimer Street, London W1T 3JH, UK

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    Competing Risks in Basketball Competing Risks inBasketball Competing Risks in Basketball Laura Taylor aa department of mathematics and statistics , Elon University in North CarolinaPublished online: 27 Apr 2012.

    To cite this article: Laura Taylor (2012) Competing Risks in Basketball Competing Risks in Basketball Competing Risks in Basketball, CHANCE, 25:2, 31-36

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    competing risks in basketball competing risks in basketball competing risks in basketballLaura Taylor

    My husband has always encouraged me to use my super powers of statis-tics for the greater goodanalyz-ing sports data, that is. Therefore, it was not surprising that I found myself watching the 2011 NCAA Mens Basketball Tournament championship game that pitted But-ler University against the University of Connecticut on April 4, 2011. While watching this game, I began to think about how scoring in bas-ketball is a competing risk of points scored and points allowed where each is garnered from either a free throw, two-point shot, or three-point shot. Not only are these outcomes competing against each other, they are recurrent throughout the entire game. Thus, using my super power to model the scoring of Butler and UCONN as recurrent competing risks became more interesting to me.

    To this end, I observed that the most recent common opponent was the University of Pittsburgh. UCONN met Pittsburgh during the Big East Tournament; Butler took on Pittsburgh during the third round of the NCAA tournament. Nineteenth-ranked UCONN beat third-ranked Pittsburgh 7674

    on March 10, 2011, and eighth-seed Butler conquered first-seed Pittsburgh on March 19, 2011, 7170. This article seeks to model the points scored and the points allowed by both championship con-tender teams based on their perfor-mance against Pittsburgh using a combination of competing risks and recurrent events.

    Competing risks and recurrent events are both fields of survival analysis. Competing risks garners its name from the way the data are observed. A unit is subjected to risks that are competing to be the first and only cause of failure (or success). The time and cause of failure are both recorded. For example, a pace-maker can be subjected to either mechanical or electrical failures, both of which are competing to be the cause of failure for the pacemaker.

    It is common to observe competing risks in biomedical studies and engineering. For a basketball team, each play of the game can result in one of the two teams scoring so there are six out-comes of interestsuccessful points made from free throws, two-point baskets, and three-point baskets and

    points allowed by the opponent from free throws, two-point baskets, and three-point baskets.

    In traditional competing risks analysis, an observation ends when the first success occurs and the time to event for the remaining risks are not observed. However, after the first score in a basketball game, the game goes on! After a point is scored or allowed, the stage is set for the team to score or allow points again.

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    Recurrent events analysis mod-els data for a single type of failure (or success) repeated over time. After each failure (or success), the unit is repaired using either a com-pletely perfect, minimal, or imper-fect repair strategy and observation of the unit continues for further events. If we consider monitoring a string of Christmas lights for a failed light bulb, a completely per-fect repair would replace all the light bulbs when one bulb fails. A mini-mal repair strategy would replace only the blown light bulb with a light bulb with the same age as the previously blown light bulb while all remaining light bulbs would con-tinue to age. An imperfect repair strategy would also only replace the blown light bulb. However, it could be replaced with either a new or aged light bulb. Recurrent events are common in biomedical stud-ies, engineering, and economics. As another example, a patient can be

    monitored over time for the recur-rence of nonmalignant tumors.

    In a basketball game, we can record the times of each free throw, two-point score, and three-point score for each team. From the perspective of one of the teams, there are six types of recurrent competing risks corre-sponding to offensive and defen-sive success or failurefree throw scored, two-pointer scored, three-pointer scored, free throw allowed, two-pointer allowed, or three-pointer allowed. We will model the offense and defense of Butler and UCONN based on their per-formance against Pittsburgh, their most recent common opponent.

    competing risks and recurrent Event dataPlay-by-play data were collected from ESPN.com for each of the games of interest. Both of the

    Figure 1. Time between points scored and points allowed for Butler during their game against Pittsburgh for 20 periods of game time. Butler is represented by the solid green shapes, and Pittsburgh is represented by the red outlined shapes.

    games occurred during tourna-ment play at the end of the season, so we would expect that Butler and UCONN were playing with a high level of commitment. Also, neither game went into overtime, so there is a total of 40 minutes of game time for each team ver-sus Pittsburgh. The time of occur-rence for free throws, two-pointers, three pointers, and time-outs was observed for both competitors in each game. Time-outs were con-sidered censoring since the game clock was stopped and the ball had to be thrown back in-bounds by the team in possession. Therefore, the data were considered to be compet-ing risks with six successful events: points scored and allowed from free throws, two-pointers, and three-pointers. After a successful two-pointer or three-pointer, the ball is returned to the opposing team and the scoring cycle repeats. Typically speaking, after a free throw, the opposing team generally reclaims possession of the ball. In this scenario, the basketball scoring data will be considered to operate under a perfect repair strategy.

    A point that should be discussed is how free throws are considered. First, scoring one-for-one, one-for-two, two-for-two, or any other vari-ation of at least one successful free throw were all recorded as one type of eventa successful free throw. For free throws that are recorded concurrently with a time-out, their occurrences were recorded as one-tenth of a second before the time-out, since those points are resulting from some action in the game prior to the time-out. Any other points recorded simultaneously in the official play-by-play were adjusted using the same technique.

    The data are presented in Figures 1 and 2 for Butler and UCONN, respectively. Each unit in this sce-nario as depicted in the graph consists of continuous game time between time-outs or the end of a half. For example, there were 20 units of game time for the Butler versus Pittsburgh game due to time-outs, TV time-outs, and the end of both halves. On the horizontal axis,

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    the calendar time of each successful score is recorded. After each time-out or the end of a half, calendar time is reset to zero and monitoring of continuous game time restarts for events of interest.

    The first striking characteris-tic in the data is that Butler does not make its first free throw until less than 10.2 minutes left in the second half of the game, as rep-resented during the 12th unit of game time. In total, Butler scored seven successful free throws. Comparatively, UCONN scored 12 free throws spread throughout the game. Typically, this would indicate that UCONNs players are more aggressive in seeking out or being able to obtain shots in the paint.

    Butler and UCONN also dif-fer in their scoring performance based on the type of successful shots made. UCONN scored a total of 23 two-pointers and 3 three-pointers compared to But-lers 12 two-pointers and 12 three-pointers. Butler and UCONN performed similarly in terms of points allowed by Pittsburgh, with each team allowing 7 free throws and 19 two-point baskets. Both teams allowed similar three-point baskets, as well, with Butler allowing 6 and UCONN allowing 8. Butler and UCONN both fairly consistently allowed free throws by Pittsburgh throughout the course of the games.

    modeling inter-Event times for points scored and points AllowedCompeting risks can focus on mod-eling the time to an event, which progresses into modeling the time between events or inter-event times for recurrent event analy-sis. The inter-event time distribu-tion can be modeled by obtaining maximum likelihood estimates for the parameters of the hazard func-tion, the instantaneous rate of fail-ure for a unit that has survived up to a specified point in time. That is, the hazard function gives the

    probability of a failure occurring at time t, given that it has not yet occurred, and mathematically, the hazard function is the ratio of the probability density function over the survival function. The probabil-ity density function and the sur-vival function can both be defined in terms of the cumulative distribu-tion function, F(t) P(T t). The probability density function, f(t), is the derivative with respect to t of F(t), whereas the survival func-tion is given by S(t) 1 F(t) P(T t). Assume that the hazard function associated with the inter-event time distribution for the qth type of scoring is given by the function q (t;q), where q is a vector of parameters associated with the qth type of scoring. For the basketball data, q is the type of basket made with a 1, 2, or 3 denoting a team scoring a free throw, two-pointer, or three-pointer, respectively, and 4, 5,

    Figure 2. Time between points scored and points allowed for UCONN dur-ing their game against Pittsburgh for 19 periods of game time. UCONN is represented by the solid green shapes, and Pittsburgh is represented by the red outlined shapes.

    and 6 denoting a team allowing each of these, respectively. We choose to model the data using a Weibull distribution with the following parameterization of the hazard and probability density functions:

    The Weibull distribution is a popular choice when modeling inter-event times (or time to event data), since the probability density function can flexibly take on a wide variety of shapes. When 1, the probability density function associ-ated with the Weibull distribution is a convex curve. For 1, the curve follows a more traditional right-skewed shape, shifting away

    t t; ,1

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    f t t e; , .t1

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    from 0 until it quickly begins to resemble a left-truncated unimodal curve. Each of these shapes is found in Figure 3. Interestingly, the expo-nential distribution, which is also commonly used to model waiting times, is a special case of the Weibull distribution when 1.

    To estimate the parameters asso-ciated with the hazard functions for each event type (and consequently the parameters associated with the distribution) for a basketball team, first consider the individual data for each team in the context of calen-dar times as opposed to inter-event times. Denote the calendar time of the jth recurrence or score for the ith unit of game time by Sij. For each

    calendar time, we also record the type of event, ij, where ij is a 1, 2, 3, 4, 5, or 6. Table 1 displays the data col-lected for the first two units of con-secutive game time (i 1 and 2) for Butler versus Pittsburgh. These are illustrated in the first two horizontal lines from the bottom of the data in Figure 1. Additionally, the recur-rent inter-event times are censored when a time out is called or the end of a half is reached. We indicate the time of censoring for the ith time period as i. Since the first time out occurred 285 seconds into the game for Butler versus Pittsburgh, 1 = 285 seconds as seen in Table 1 and the first horizontal row of data in Figure 1, which ends at 285 seconds.

    resultsFrom this data, we fit a Weibull hazard function by using the maxi-mum likelihood approach to model the inter-event time distributions associated with each of the six com-peting risks. The estimates for the Weibull shape and scale parameters, and , for the time between events for each type of scoring and team versus Pittsburgh, as well as the mean and mode of the estimated inter-event time distributions, are given in the Table 2.

    The estimated hazard function and inter-event time distribution function for free throws scored by Butler versus Pittsburgh are given by

    The estimated inter-event time distributions for all events are dis-played pictorially in Figure 3, which compares the inter-event time dis-tributions for Butler and UCONN (with Pittsburgh as their opponent) for each of the six events.

    From Table 2 and Figure 3, sev-eral differences between the scoring abilities for Butler and UCONN are visible. In particular, UCONN is able to generate free throws...