# The Role of Rest in the NBA Home-Court Advantage

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1 The Role of Rest in the NBA Home-Court Advantage Oliver Entine Department of Statistics, The Wharton School of the University of Pennsylvania, 3730 Walnut St., Philadelphia, PA 19104, U.S.A. Dylan S. Small Department of Statistics, The Wharton School of the University of Pennsylvania, 3730 Walnut St., Philadelphia, PA 19104, U.S.A. November 30, 2007 Abstract To date, the factors which lead to the very large home court advantage characteristic of the NBA have not yet been well isolated. This study analyzes the relationship between that home court advantage and the comparatively fewer days of rest between games that the NBA schedule imposes on visiting teams. A statistical model has been developed and applied to the NBA data for the and seasons to estimate the importance of the effect of rest on the magnitude of the home court advantage. The results indicate that lack of rest for the road team, while not a dominant factor, is an important contributor to the home court advantage in the NBA. 1

4 of data shows that the average margin of victory experienced by the home teams over the visitors was 3.6 points. The analysis below is designed to determine how many, of those points, if any, arise from the lack of rest of the visiting teams. Our starting model relates the home team s margin of victory in points to several variables. These include the strengths of each team for that given year, the home court advantage for the host team, and the amount of rest each team has coming into the game as measured by number of days off in between contests. While Harville and Smith (1994) use a random effects model to describe the individual strengths of each team, here we just consider all of the factors, including team strength, as fixed effects. Y ij = θ i θ j + δ i + β 1 I[Rest i = 0] + β 2 I[Rest i = 1] + β 3 I[Rest i = 2] β 1 I[Rest j = 0] β 2 I[Rest j = 1] β 3 I[Rest j = 2] + ɛ (1) Y ij = Point Margin for home team i over road team j θ i = Strength of home team i θ j = Strength of road team j δ i = Home-court advantage for team i if both teams have equal rest β 1, β 2, β 3 = Effect of rests of zero, one, and two days respectively compared to rest of three or more days. ɛ N(0, σ 2 ) Notice that in this particular model, there is an underlying assumption that the effect of 0, 1, or 2 days rest is the same for both the home and visiting teams, an assumption we will justify shortly. Applying this model to the data from the and seasons, one gets the point estimates for the coefficients β 1, β 2, and β 3 shown in Table 2. Thus, all things being equal, a team playing a game the day immediately following another game should be expected to score roughly 1.77 points (β 1 ) less than if that same team has more than three days of rest between contests. In contrast, playing a game after only one day off reduces the team s scoring ability by only 0.13 points (β 2 ) compared to if the team had three or more days off. Oddly enough, the data also indicates that a team with two days 4

5 Rest Coefficient Estimate SE p-value β β β Table 2: Point-Estimates, Standard Errors, and P-Values for the Rest Coefficients of rest actually seems to improve its scoring margin by about 0.32 points (β 3 ). However, the analysis indicates that only the β 1 coefficient is significant. That is, there seems to be a significant (albeit a small) detrimental effect to playing games on consecutive nights, but no further benefit of rest once the team has one or more days of rest. 3 Does Rest Have a Different Effect for the Home and Road Teams? A more general model than (1) that we could use is to have different parameters for the rest coefficients of the home team versus those for the visiting team. Y ij = θ i θ j + δ i + β 1 I[Rest i = 0] + β 2 I[Rest i = 1] + β 3 I[Rest i = 2] β 4 I[Rest j = 0] β 5 I[Rest j = 1] β 6 I[Rest j = 1] + ɛ (2) Does this model represent a significant improvement over (1) which has the constraints β 1 = β 4, β 2 = β 5, and β 3 = β 6? Table 3 below shows a comparison of the results obtained with both the unconstrained and constrained models. Model Degrees of Freedom Error Sum of Squares F-Statistic P-Value Unconstrained Constrained Table 3: Sum of Squares Comparison for Unconstrained and Constrained Model The comparison shows that the removal of the constraints on the β s leaves the error sum of squares essentially unchanged, thereby indicating that the assumption that the effect 5

7 Length of Road-Trip Indicator Estimate SE p-value 1st Game of trip 0 NA NA 2nd Game of trip rd Game of trip th Game of trip or longer Table 4: Point-Estimates, Standard Errors, and P-Values for the Length of Road-trip Coefficients playing its second consecutive game on the road, with a significance-level of 7%. For road trips longer than that, there is no longer much indication of a contribution of home court advantage associated with the length of such trip. 6 Effect of Rest on Winning So far, we have concluded that having three or more days of rest can improve a team s score margin by as much as 1.77 points per game. However, in most NBA games, the eventual margin of victory tends to be much larger than 1 or 2 points. Thus, one might ask whether the number of days of rest variable, though significant in altering final point margins, has a significant effect on a game s final outcome (victory versus defeat). Here, we construct a model with the same variables as before, only now, we use a logistic regression framework to predict the effect of rest on the probability of a team winning the game. 7

8 Y ij = 1 if home team i wins, 0 if visiting team j wins P (Y ij = 1) = expit{θ i θ j + δ i + β 1 I[ Home Rest = 0 ] + β 2 I[ Home Rest = 1 ] +β 3 I[ Home Rest = 2] β 1 I[ Road Rest = 0 ] β 2 I[ Road Rest = 1 ] β 3 I[ Road Rest = 2]} where expit(x) = ex 1 + e x θ i = Strength of home team θ j = Strength of away team δ i = Home-court advantage for team i if both teams have equal rest β 1, β 2, β 3 = Effect of rests of zero, one, and two days respectively compared to rest of three or more days Estimate 95% CI exp(β 1 ) 0.75 (0.53,1.06) exp(β 2 ) 1.02 (0.73,1.41) exp(β 3 ) 1.14 (0.80, 1.62) This data indicates that playing back to back games compared to having rest of three or more days multiplies the odds of winning by 0.75 (β 1 ). While not quite significant at the 5% level, this variable is significant at the 10% level, making it worthy of further consideration. β 2 and β 3 are actually slightly larger than one, which would mean that the odds of a team winning actually improves when the team has only one or two days of rest as opposed to three or more. However, neither of these coefficients are statistically significant. 7 Conclusions The fact that home teams in the NBA win so large a fraction of the time is quite a fascinating observation since it implies that factors other than the skill of the competing teams play critical roles in the outcome of games in professional basketball. We have examined one of 8

9 these factors, the travel factor, from two points of view, namely the effect of the fact that, on balance, the traveling team has a schedule which provides fewer days of rests between games, and that just being on the road an extended number of days could lead to a decrease in athletic performance. Our analysis of the 2,415 games which took place in the and seasons of the NBA indicates that the travel schedule does seem to be a real, although not dramatic factor contributing to the NBA s home team advantage. On average, for these two seasons, the home teams scored 3.24 more points than the visitors, of which 0.31 points arise from the smaller amount of rest that the NBA schedule provides the traveling team and 2.93 points are associated with other, non-related factors. The travel schedule effect is most clearly illustrated by the fact that visiting teams with back to back games are estimated to be 1.77 points worse off on average than visiting teams that are fully rested. The data also suggest that traveling teams score one point less during their second game on each trip, but this observation is only weakly significant. When the issue of the home team advantage was studied with respect to the number of games won or lost, as opposed to the number of points scored, the data once again showed the importance of the tight schedule faced by the traveling teams. As with the margin of victory measured in points, the condition indicating the highest effect was when the visiting team played the second of a back to back pair of games. In this case, the odds of the visiting team winning were decreased to an estimated 75% of those corresponding to the fully rested state, although the level of significance was weak. Based on these analyses, we conclude that the extraordinary high home court advantage enjoyed by NBA teams is partially explained by the tendency of the NBA schedules for the traveling teams to have reduced rest, but that the bulk of the advantage arises from other, non-related factors. References Courneya, K.S. and Carron, A.V. (1992), The Home Advantage in Sport Competitions: a Literature Review, Journal of Sport and Exercise Psychology, 14, Efron, B. and Tibshirani, R.J. (1986), Bootstrap Methods for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy, Statistical Science, 1, Harville, D.A., Smith, M.H. and Rubin, D.R. (1994), The Home-Court Advantage: How 9

10 Large Is It and Does it Vary from Team to Team?, The American Statistician, 48, Jones, M.B. (2007), Home Advantage in the NBA as a Game-Long Process, Journal of Quantitative Analysis in Sports, 3, Nevill, A.M. and Holder, R.L. (1999), Home Advantage in Sport: An Overview of Studies on the Advantage of Playing at Home, Sports Med, 4, Simon, A.S. and Simonoff, J.S. (2006), Last Licks : Do They Really Help?, The American Statistician, 60,

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