The NBA and Fatigue

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The National Basketball Association and Fatigue:

How Days of Rest Affect NBA Outcome and Game Play

Adam Yudelman

Wake Forest University Department of Economics

Under the Direction of Dr. Todd McFall and Dr. Amanda Griffith

Fall 2015

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Abstract

The nature of the complicated National Basketball Association (henceforth referred to as

NBA) schedule necessitates imbalanced schedules, particularly in regard to days of rest between

games. Research in the past shows teams with the rest advantage over their opponent win more

often. Consequently, both the NBA League Office and individual teams are investing heavily in

fatigue analysis and prevention. The League Office has installed SportVU optical cameras to

track players’ speeds while teams are using GPS tracking, sleep analysis, and even full body

scanners to track the impacts of a demanding schedule. Recent schedule changes to limit games

on consecutive days and build in a longer mid-season break attempt to mitigate fatigue, yet a

fully rest optimized schedule is impossible. Using data from a variety of sources, this paper seeks

to determine the extent to which fatigue affects both game outcome and in game metrics. I

accomplish this using models that incorporate rest to predict outcome and shot making while also

conducting means testing to look for differences between zero rest games and one-plus rest

games. I find that teams which are more rested are more likely to win, particularly when the team

holds a rest advantage over the opponent. Searching for in game effects, we learn that scoring,

particularly in the paint and on fast breaks, significantly diminishes for games of zero days of

rest. With the shot making model, this paper finds that the shots themselves are not affected

significantly by fatigue but rather it is poorer shot selection and fewer assists that associate with

the lower scoring for teams with zero days of rest. The research sets the stage for future

investigations into optimal game strategies to limit the effects of fatigue on team success.

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Introduction

The constraints of a National Basketball Association (NBA) season creates unfavorable

traveling conditions. Fitting 82 games into a season requires teams to play two games in two

days, often which requires overnight travel if the team is made to travel to the second of two

games on consecutive days (otherwise known as the latter half of a back-to-back). Looking only

at games from the 2013-14 season, a startling 96.2% of back-to-backs required overnight

traveling. This is significant, for teams historically win fewer games in these settings

(Haberstroh, 2014). Attempts to build more rest into the schedule, such as stretching the 82

games over a longer time period, would put the NBA at further economic competition with other

major sports such as the National Football League and Major League Baseball. Moreover, this

would shorten the much needed offseason where players both rest and play for their national

teams in worldwide competitions. Hence, fatigue will remain a factor in the NBA for years to

come.

In the early days of the league, transport between games proved to be difficult. Teams

traveled on mostly trains in order to limit travel expenses. In the very first seasons, the NBA

even experimented with a type of revenue sharing system that supported teams who were

burdened with an above average travel cost. In The Rise of the National Basketball Association,

David Surdam (2012) recounts one story of these expeditions from the 1950s for the New York

Knicks, who had just traveled by train to a quiet stop in Indiana:

The only thing there was an uncovered wooden platform. Traveling with the Knicks, I

knew that our instructions were to walk on a two-lane blacktop road toward a blinking

yellow light a half-mile away. That turned out to be the only light at a crossroads where

there were ten or twelve buildings, nothing taller than two stories. Then we were to look

for the plate glass window with the sign of the Green Parrot Café. Carl Braun was our

designated shooter of the pebbles up to the second-floor window, a frowsy-haired woman

would look out and say, ‘Oh the Knicks.’ She’d get on the phone, and in a little while

four or five cars would gather and drive us the forty miles into Fort Wayne. We’d go

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right to bed get up for the game that day or night. That was the only way to get to Fort

Wayne from Rochester.

The NBA has come a long way since the days of trains and busses and rock throwing, yet an ever

busy schedule combined with exogenous variables account for significant player fatigue.

In this paper, I examine the effects of fatigue, defined by the days of rest between games,

on both the outcome of the game and game play itself. I first look at the historical NBA schedule,

going back to the 1940s, to examine the role fatigue has played in the outcome of games over the

decades. Then, using data from the 2014-15 season, I analyze both box score statistics and

publicly available SportVU data in an attempt to identify which aspects of game play are most

affected by fatigue. Next, I look at history of NBA game outcomes to search whether fatigue is

becoming more or less of an advantage. Finally, using SportVU shot log data and Krishna

Narsu’s shot difficulty metric, I search for which type of player and what type of shot is most

affected by fatigue. My hope is that through this analysis, teams are able to plan around the

aspects of the game most affected by exhaustion.

The Professional Sports Market and the National Basketball Association

For the National Basketball Association, the quality of gameplay and the competitive

balance (or lack thereof) is of the utmost importance for both individual teams and the League

Offices. Teams strive to improve their roster and coaching to gain a competitive advantage while

the League as a whole must compete successfully against other professional leagues to maintain

profitability. There are natural barriers to this competitive balance, such as unbalanced markets

leading to more revenue for one team than another. To combat this, the NBA uses methods such

as amateur drafts, salary caps, and luxury taxes to an attempt to level the playing field.

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Competitive balance is necessary to create uncertainty in the outcome of the game, or there

would be little incentive to watch and support an inferior team. (McFall, 2014).

Despite the tactics of the NBA, there are still artificial competitive imbalances that come

from an 82 game schedule. Unbalanced schedules, related to talent imbalances between

Conferences, punish certain teams in more difficult conferences. Along with this, unlike in the

National Football League, teams do not have uniform time to prepare for games1. The NBA

schedules is equally split between home and away to try to mitigate home court advantage, yet

just over 50% of the games in the most recent 2014-2015 season do not offer equal rest, and the

resulting win percentages illustrate the advantage of each additional day of rest (See Table 1 and

Table 2 in appendix). Fitting a full season’s worth of games into a six month schedule creates

travel strains on players, creating an advantage for the better rested team. In the past ten years,

franchises have recognized that combatting fatigue can provide an edge and investing in staff,

research and improved travel to mitigate the effects.

Law of Increasing Opportunity Cost

Fatigue in the NBA can best be associated with the economic principle of the Law of

Increasing Opportunity Cost. Opportunity cost is defined as, “the best alternative that we give

up, or forgo, when we make a choice or decision” (Case et al, 2009). The demand for the NBA is

as high as ever, as evidenced by unparalleled increases in revenue2 the tradeoff between

maintaining a full schedule and keeping a rested product on the floor must be examined (Draper,

1 For the NFL, teams historically have one week to prepare for each game. Occasional exceptions

occur throughout a season, including Bye Weeks, Monday Night games, and more recently

Thursday night games. 2 This is more commonly known as Basketball Related Income.

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2015). The NBA must balance the health of players, who are adversely affected by fatigue by

increasing the chances of injury or performing below peak levels, with their revenue maximizing

objectives. On a team level, players are assets to the franchise. Teams must find an equilibrium

between the successes of today, perhaps by playing a player 40 minutes a game, with the future

successes of tomorrow3. Often, however, there are mixed incentives, for a team with a player on

a one year contract may only be concerned about the present while the player is concerned with

his long term health and earning potential.

So far, the League Office, which builds the schedule, has decided to sacrifice player

fatigue rather than limit the number of games played. While the League has responded to

criticism and reduced certain types of games, such as four games in five days and long haul

back-to-backs, the schedule remains treacherous for those expected to play significant minutes in

at least 82 games (Partnow, 2015).

General Research on the Effects of Fatigue

Fatigue affects workers in all workplaces. A study conducted by Shahrokh-Shahraki and

Nooh –Bin Abu Baker on the effects of fatigue on workplace productivity identified two types of

fatigue: physical and nervous. The researchers find that efficiency is linked to fatigue, mild

rhythm of the work volume and sufficient sleep. Focusing on sleep deprivation, the study

discovers that this fatigue increases the probability of accidents in taxi drivers as well mental

distress, physical disorders, and cardiovascular diseases (Shahraki and Baker, 2011). Beyond

reaction type, sleep deprivation has negative cognitive effects. Studies show that even partial

3 This was famously seen in Major League Baseball with the Washington Nationals shutting

down Stephen Strasburg in the midst of the 2012 playoff race.

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sleep deprivation impairs attention, working memory, long term memory, decision making, and

vigilance (Alhoa and Polo-Kantola). Extrapolating this to the NBA, sleep deprivation and an

inconsistent substitution pattern decrease the efficiency of players.

Investments in Fatigue Mitigation in the NBA

Given the evidence of the effects of fatigue on the general workplace, it is no surprise

that NBA teams are racing to understand fatigue in their own workplace. Over the past handful

of years, the NBA has invested in fatigue analysis through player monitoring. On a league wide

level, the NBA introduced SportVU optical tracking technology in every arena at beginning of

the 2013-14 season to measure the movement of every player on the court 25 times per second.

While these cameras report measurements like player shot location and how well a player

defends at the basket, algorithms can also detect when a player is running up the court at a

significantly below average speed. These algorithms can hint at problems players are not

disclosing. Teams then can take preventive measures to prevent extreme fatigue or even mitigate

serious injury.

On a team level, organizations are taking advantage of wearable technology offered by

companies such as Catapult, an Australian firm that works with roughly 20 NBA teams (up from

eight at the beginning of 2013) and countless others in the NFL, English Premier League, and

NCAA. These devices can track force exertion on player movements, further adding more data

that can track for anomalies. While teams and independent peer-reviewed analysis will swear by

the benefits of the wearable technologies, the League and NBA Players Association has not yet

agree on the need for players to wear the technology on court during games (Lowe, 2015).

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Off the court, the Dallas Mavericks began to analyze the sleep patterns of players during

the 2013-14 season. Using a technology named Readiband provided by the Canadian Company

Fatigue Analysis, the watch-like wearable measures how long and how well a player sleeps

(Caplan, 2013). Mirroring this data with on court performance can convince a player to sleep an

hour more to improve his own game play, and more teams have contacted Fatigue Analysis to

also acquire the product (Caplan, 2013). In attempts to keep up, the Los Angeles Lakers recently

bought a full body scanner from the German company Human Solutions, which historically has

worked with the fashion industry. The Lakers thinking goes that if the scanner can detect

deviations in a player’s posture, there may be an injury that has yet to manifest itself in a

noticeable manner. The Lakers do not stop there, for they also use technology to measure

hydration and oxygen levels, and are in talks with a company to install microchips into shoes to

gather even more detail (Holmes, 2015). As shown, teams will search around world for any type

of fatigue analysis technology to try to provide a competitive advantage.

The most recent NBA champion, the Golden State Warriors, are at the forefront of rest

analysis. The Warriors lost 1,252 minutes to injury during the 2014-15 season, the lowest in the

NBA. While there may be luck involved, the franchise has invested heavily in not only Catapult

technology but also data analysis to create a readiness metric for players for each game based on

a variety of fatigue related resources ranging from SportVU metrics, heart variability data, and

even player surveys (Haberstroh, 2015c).

While teams, players, and the League have incentive for maximizing the output of

players and prolonging careers, such biometric information is very personal by nature. Players

and their agents are hesitant that such fatigue analysis, which goes hand-in-hand with injury

prevention analysis, will be used against players in negotiations (Lowe, 2015). As teams push

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further for more information about players, they may even run into Genetic Information

Nondiscrimination Act, passed in 2008, which make it illegal for employers to discriminate

based on genetic information (Torre & Haberstroh, 2014).

Current Discussion

Apart from formal academic research, the discussion regarding the effects of fatigue has

picked up in recent years through the mainstream media. At the forefront of this has been Tom

Haberstroh, a graduate of Wake Forest University in 2008 and a prominent member of ESPN’s

NBA Insider team. Summarizing Haberstroh’s research provides the impetus and necessity for

my own research.

Citing the Dr. Charles Czeisler, the director of sleep medicine at Harvard Medical

School, Haberstroh notes that going 24 hours without sleep is equivalent to being legally drunk.

Moreover, the same sleep deprivation lowers testosterone levels to an equivalent level of 11

years of aging as well as significantly slowing reaction time. This is troublesome for back-to-

back game scenarios, where the NBA requires teams to fly overnight for away games rather than

traveling on the day of the game, and charter planes are not yet a perfect substitute for hotel

accommodations (Haberstroh, 2015d).

Shortening the season would in theory create more uncertainty by limiting the sample

size of games, yet research suggests that the relative team quality is evident in the standings after

only 22 games (Paine, 2012). While no one has yet seriously suggested a 22 game season,

proposals for seasons as short as 44 games have been published. Yet, the NBA is a revenue

producing entity, and limiting games limits ticket sales. Hence, there is push back for any

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suggested change4. Other suggestions include taking a Major League Baseball approach to

traveling by having teams play multiple games in a city at a time rather than having return trips.

Haberstroh himself champions the idea of a 60 game season that eliminates back-to-backs and

limits each week to two or three games. All suggestions intend to increase player safety and the

quality of competition, yet the prospect of decreasing revenue proves to have too much inertia to

create much of a challenge to the status quo (Haberstroh, 2014).

The 2014-15 schedule added a bit more rest to the schedule by adding several extra days

of rest to the All-Star break while maintaining a full schedule, yet this added 18 additional back-

to-backs on the schedule. Players are, understandably, beginning to take action. Kobe Bryant,

LeBron James, Derrick Rose, a the entire core of the San Antonio Spurs are taking full games off

to rest to limit injury and prolong careers, all with the blessing of their organizations. This is

much to the dismay of the fans, which pushed the League to fine the Spurs $250,000 for resting

key players on a nationally televised second game of a back-to-back. The Denver Nuggets even

experimented with cancelling morning shootarounds to give players more sleep (Haberstroh,

2015a). Coaches are limiting minutes in an unprecedented manner, where averaging 35 minutes

per game for a player is rare. Much of this is spurred by the success of the San Antonio Spurs,

which won the 2014 Finals without having any player play more than 30 minutes a game. This

emphasizes depth on a roster, yet also takes the best (and most marketable) players off the court.

This again is a tradeoff teams and the League must consider, particularly with fans interest and

money at stake (Haberstroh, 2015b).

4 For all intents and purposes, the NBA acts as a monopoly. Monopolies judge marginal cost

against marginal revenue when deciding the quantity of the product, for monopolies want to

produce in the elastic portion of demand. The NBA’s refusal to shorten the season (in addition to

the extension of first round playoff series from five games to seven) may say something relative

elasticity of the NBA.

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Literature Review

Prior to delving into my own data, methodology, and results, it is necessary to take an

extensive look at previous academic research. Research by Bean and Birge (2980) looks at how

the NBA could optimize the schedule to reduce miles traveled while Kelly (2010) searches for

potential bias in the scheduling of back-to-back games. Further research by Nutting (2010),

Ashman et al. (2010), Entine and Small (2007), Silver, and Yudelman (2015) examines the

effects of days of rest and traveling across time zones. Finally, a paper by Mah et al. (2011)

investigates how sleep extension affects shot making.

Schedule Discussion

Bean and Birge (1980) looks at the travel requirements for NBA teams in the 1978-79

and 1979-1980 season, focusing on the financial impact of extraneous travel mileage. Through

heuristic algorithms that limited road trips to a maximum of five games, the authors are able to

reduce travel by over 20% while also creating more uniformity in travel mileage among teams.

The study acknowledges that the most difficult constraint in scheduling is arena availability,

which is estimated at 30% of the days during the season; however, more recent constraints, such

as placing high leverage games and popular teams in national TV slots, have further complicated

schedule making (Birge and Bean, 1980).

A more recent study by Kelly (2010) into NBA scheduling looks exclusively at back-to-

back games. This paper reviews five years of scheduling data with the intent to find whether

there was scheduling bias against certain teams with the assignment of the more difficulty back-

to-back games. From 1946 to 2004, teams playing back-to-back games had a win percentage of

just 46.1% in such games When isolating just road back-to-backs, the win percentage from 1999-

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2003 was only 38.3%. The research notes these results as a competitive disadvantage. Using a

Chi-Squared test, Kelly finds no significant bias against teams when deciding back-to-backs.

However, he notes that the randomness of assignment did in fact impact playoff seeding during

his five year sample. In an effort to explain why certain back-to-backs are scheduled, the study

discusses the cost-saving externality, in which arena availability and the cost of away

accommodation are accounted for. For example, playing a three game road trip within four days

rather than a five days where there is uniform rest can save the NBA over $3 million (Kelly,

2010).

Win-Loss Impact of Rest Imbalance

While the marginal cost of creating a more efficient 82 game schedule is high, the debate

must be balanced with the costs of restrictive travel on the outcome of games. A study by

Nutting (2010) of the University of Idaho investigates the effects of both travel mileage and days

of rest on wins, offense, and defense. Looking at 16 full NBA seasons and splitting the data set

into first and second halves of the season, Nutting finds that mileage between games is

insignificant; however, days of rest significantly affect team gameplay. Teams with fewer days

of rest, particularly visiting teams, produce fewer wins. This effect is exasperated in the second

half of the season. Additionally in the second half, time zone factors become significant. Due to

the wide spread of teams, teams located in the western time zones win more often than their

eastern counterparts. The traveling of time zones eastward proves to be an advantage as western

teams are never forced to play late night games on the road (late night being defined as the local

time in the visiting team’s home time zone). Nutting notes that a similar study on the NFL

conducted by Jehue, Street, and Huizenga (1993) found supporting results for traveling to night

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games. The importance that time zone differences are only significant in the second half of the

season suggests that fatigue compounds itself over the arduous schedule (Nutting, 2010).

A study on NBA wagering markets supports Nutting. By using point spreads as a

measure of expected point differential, Ashman et al. (2010) looks at the performance of home

teams against the expected. Using 19 seasons of data beginning in 1990-1991 season, the authors

identify the specific scenario of back-to-back games as a potentially inefficient game. They

hypothesize that point spreads were not accurately depicting the effects of fatigue on the teams.

By looking at two extraneous factors, days of rest and time zone travel, the research finds that

home teams did poorly against the spread when teams playing at for the second of a back-to-back

played against a visiting team had one or two days of rest. Traveling from a western time zone to

the home game further exasperated the inefficiency. Interestingly, the point spreads were

particularly ineffective for this scenario when the home team was the underdog. The authors

suggest that this may be because of a betting bias towards teams on an extended home stand as

well as the historical performance of home underdogs in general (Ashman et al, 2010).

A study by Entine and Small (2007) also exams fatigue and home court; however, this

paper focuses on the fatigue of the road team. Looking at games from 2004-05 and 2005-06

seasons, the researchers want to see whether home court advantage (the home team won 60.5%

of all games in this sample) can be explained by fatigue imbalance. The study finds that of the

average 3.24 point spread for games, .31 points can be attributed to rest factors. However, teams

on the second of consecutive games are 1.77 points worse off than those that are fully rested.

When looking at outcome, the odds of the visiting team decreased by 75% when on a back-to-

back compared to fully rested teams. The study concludes that home court advantage can only

minimally be explained by fatigue (Entine and Small, 2007).

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While the previously discussed studies focus on regular season data where rest is not

uniform, Nate Silver (2014) of FiveThirtyEight looks at how a prolonged playoff series affects a

team the rest of the postseason. Inspecting only teams seeded first or second in their conference

to ensure team quality consistency, Silver calculates that teams which played more games in

their first series more often loss in the following series. Improving his study by building a model

that uses Basketball-Reference’s Simple Rating System (SRS) to measure team quality and

looking at the expected point differential, Silver finds that teams that swept the first round

outscored the SRS projected score margin by 3.0 points per game while those teams who took

seven games to dispatch their first round opponent underperformed by an average of 5.7 points

per game. The study admits that a long playoff series may hint that a team may be struggling

compared to its peak regular season performance; however, it does suggest that fatigue can be a

factor even in the playoffs where no back-to-back games occur (Silver, 2014).

Fatigue and Performance

Fatigue is a factor in the outcome of games. However, none of the previously discussed

studies look for how the fatigue manifests itself in game. A combined study by Stanford

University and the University of California, San Francisco studies how sleep affects the

performance of collegiate basketball players. Using a sample of 26 University of Stanford varsity

basketball players where 11 participated and 15 were used as the control group, the research

measures speed (through a 282 foot sprint), free throw accuracy, three point accuracy, and

subjective mental and physical surveying. For a two to four week period, the subjects slept

within their normal constraints of about six to nine hours of sleep a night. The measurements

taken in this period acted as the base case for the study (Mah et al., 2011). The participants then

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entered a five to seven week extended sleep period where ten hours of sleep were suggested. As

anticipated, all measurements significantly improved in the extended sleep period. Sprint time

decreased from 16.2 seconds to 15.5 and shooting accuracy increased 9.0% and 9.2% for free

throws and three point shots respectively. The players also reported, in a statistically significant

measure, to be in better mental and physical shape (Mah et al., 2011).

In my previous research on the topic, I look at how days of rest affect the context in

which shots are taken by examining the means of certain metrics. I find that field goal percentage

is drastically lower of days of zero rest than for other games. Shots are taken earlier in the shot

clock, further from the basket, less distance from the closest defender, and after more dribbles.

Splitting the data into just two point and three point shots show that it is two point shots that are

most affected by the fatigue (Yudelman, 2015). This suggests that more difficult shots are being

taken on days at which players should feel the effects of fatigue. My research intends to build

upon this.

Data and Methodology

To examine the effects of fatigue on outcome and game play, I look at the following data

sets:

Historical Basketball Elo Ratings and Schedule from 1947 to 2014-15 from

FiveThirtyEight

2014-15 Individual game box scores from NBA.com

2014-15 SportVU box scores from NBA.com

2014-15 SportVU shot logs from NBA.com

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The Elo Rating and schedule data are used to see whether rest is a significant factor in

determining game outcome. The box scores allow for means testing to see which statistics are

most affected by rest. Finally, the shot logs allow me to build a model to see whether shot

making is independent of fatigue. With the exception of the schedule and Elo rating data, all data

is pulled from an online database provided by Darryl Blackport, who collected the data from

NBA.com.

The FiveThirtyEight data are a collection of almost every American professional

basketball game, beginning in 1947 all the way up to end of the 2014-15 season. From this

sample of 63,157 individual games, I am able to calculate the days of rest between games for

each team. The dataset itself includes the location and date of the game, the Elo Rating for each

team entering and leaving the game, and the final score. Focusing first on the days of rest, the

data show how the travel requirements in the NBA have changed for the decade. Looking at

Table 3, we see that the 1950s and 1960s had more than 40% of the games featuring zero days of

rest while recent decades have seen the percentage of back-to-backs stabilize around 23%.

Breaking down the schedule data even further into just away games in Table 4, we can

see that visitors are forced to play the majority of the zero rest games in every decade.

Amazingly, teams in the 1950s and 1960s played almost half of their road games on zero rest.

Looking to later decades, we can see that the advent of more efficient travel in the 1970s and

1980s led to fewer demanding travel requirements, and the schedule adjusted accordingly by

significantly lowering the number of away back-to-backs to 42.56% and 33.49% for each decade

respectively. Yet since the 1990s, there has been little change as away teams play on zero days of

rest roughly 30% of the time.

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This paper intends to search for whether rest has been a significant factor in the outcome

of games. To accomplish this, I use a model that includes fatigue factors while also adjusting for

team quality using Elo Rating. Elo Rating intends to measure the quality of the team at any given

time relative to the rest of the league. It is calculated by using only the final score of the game,

the location of the game, and the Elo Rating of each team coming into the game. Upsets and

large score margins lead to a higher Elo increase (Silver and Fischer-Baum, 2015). One

disadvantage of depending on the result of the game is that team strength can change suddenly in

an offseason or through an in-season trade or injury, and this is not captured by entirely Elo

Rating. However, given the large number of observations, the effects of such a drastic change

should be mitigated.

The first important question to be asked is if fatigue is a significant factor in determining

the outcome of the game. Previous studies have concluded so, yet these data give the most

thorough and extensive history of the NBA. To investigate the significance, I use the two models

to estimate the effect of fatigue on the outcome of games. The null hypothesis is that any level of

team rest and opponent rest have coefficients equal to zero. I anticipate that the null hypothesis

will be rejected, for previous literature supports the idea that rest is a significant factor in

determining game outcome.

To add a further level to the research, the models search for how the effect of rest has

changed over time. Each logit model is run for every decade of data available, which ranges from

the 1940s to the 2010s. The results will show when and if rest became a significant factor in

determining the outcome of games. The following shows the two models:

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(1) 𝑂𝑢𝑡𝑐𝑜𝑚𝑒𝑖𝑠 = 𝛽𝑿𝑖𝑠 + 𝛼1(𝑇𝑒𝑎𝑚𝑅𝑒𝑠𝑡1𝐷𝑎𝑦𝑖𝑠) +

𝛼2(𝑇𝑒𝑎𝑚𝑅𝑒𝑠𝑡2𝐷𝑎𝑦𝑠𝑖𝑠) + 𝛼3(𝑇𝑒𝑎𝑚𝑅𝑒𝑠𝑡3𝑃𝑙𝑢𝑠𝐷𝑎𝑦𝑠𝑖𝑠) +

𝛾1(𝑂𝑝𝑝𝑜𝑛𝑒𝑛𝑡𝑅𝑒𝑠𝑡1𝐷𝑎𝑦𝑖𝑠) + 𝛾2(𝑂𝑝𝑝𝑜𝑛𝑒𝑛𝑇𝑒𝑎𝑚𝑅𝑒𝑠𝑡2𝐷𝑎𝑦𝑠𝑖𝑠) +

𝛾3(𝑂𝑝𝑝𝑜𝑛𝑒𝑛𝑡𝑇𝑒𝑎𝑚𝑅𝑒𝑠𝑡3𝑃𝑙𝑢𝑠𝐷𝑎𝑦𝑠𝑖𝑠) + 𝜀𝑖𝑠

(2) 𝑂𝑢𝑡𝑐𝑜𝑚𝑒𝑖𝑠 = 𝛽𝑿𝑖𝑠 + 𝛼(𝑇𝑒𝑎𝑚𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙𝑖𝑠) + 𝜀𝑖𝑠

In both models, Outcome is categorized into wins, assigned as one, and losses, assigned as zero.

X contains controls for team quality (the Elo Ratings) and the location of the game. For model

(1) 𝛼𝑛 represents the effect of n days of team rest on the win probability while 𝛾𝑖 represents the

effect of n days of opponent rest. Days of rest greater than or equal to three days are grouped

together. The base case for these indicator variables is when a team has no rest between games.

Model (2) simply takes (team rest – opponent rest) as the variable of interest for each game. The

subscripts i and s indicate the team and date of game.

Assuming that fatigue is in fact significant for determining the outcome of the game, the

next logical question to be asked is how does the effect manifest itself in the actual gameplay. To

look at this, this paper uses data from game-by-game box score statistics for the 2014-15 regular

season to search for which statistics are significantly affected by fatigue.

Given that the NBA regular season has 1,230 games, the study looks at 2,460 team-level

box scores matched to the corresponding days of rest for the respective team. These data include

traditional box scores, advanced box scores, miscellaneous box score statistics, player tracking,

and possession type. The intent is to find which statistics are most significantly affected by

fatigue. Using the null hypothesis that there is no difference in means between zero days of rest

games and one-plus days of rest, I will be able to see which, if any, box score statistics are

affected by fatigue.

Yudelman 19

Following this testing strategy, the paper looks at how shot making is affected by days of

rest. Because I want to be able to look at the effects of fatigue beyond the scope of just field goal

percentages, this paper builds a shot making model that incorporates fatigue to search for

significance. Since 2013, the NBA has published SportVU shot logs with detailed data of almost

every shot taken. To search for which factors help determine whether a shot will be made, this

paper uses 2014-15 shot log data of 202,946 shots. To build my model, I will adopt a version of

Krishna Narsu’s KOBE shot difficulty metric. Narsu used the SportVU data and concatenated it

with player heights to control for height difference (Narsu, 2015). The model’s goal is to identify

the expected number of points for a shot based on the scenario under which the shot is taken5.

The model uses Narsu’s model as a base, but adds indicator variables for days of rest.

Similarly to the game outcome regressions, this model is a logit regression where the null

hypothesis is that all levels of days of rest have a non-significant effect on shot outcome. I

anticipate rest will be significant, for the players with more fatigue should have more tired legs.

The model is as follows:

(3) 𝑆ℎ𝑜𝑡𝑂𝑢𝑡𝑐𝑜𝑚𝑒𝑗𝑖𝑠 = 𝛽𝑿𝑗𝑖𝑠 + 𝛼1(𝑇𝑒𝑎𝑚𝑅𝑒𝑠𝑡1𝐷𝑎𝑦𝑖𝑠) +

𝛼2(𝑇𝑒𝑎𝑚𝑅𝑒𝑠𝑡2𝐷𝑎𝑦𝑠𝑗𝑠) + 𝛼3(𝑇𝑒𝑎𝑚𝑅𝑒𝑠𝑡3𝑃𝑙𝑢𝑠𝐷𝑎𝑦𝑠𝑗𝑠) + 𝜀

In this model, the dependent variable ShotOutcome is whether a shot is made or missed. Similar

to models (1) and (2), 𝛼𝑖 acts as the coefficient for the days of rest for the shooter. Subscript j

represents the number shot for player on team i at date s.

5 One can read more about Narsu’s model here:

http://nyloncalculus.com/2015/09/28/introducing-kobe-a-measure-of-shot-quality/

Yudelman 20

In order for me to accurately search for the effect of the exogenous days of rest, the

model controls in X for the following endogenous variables gathered from the SportVU dataset.

Shot distance is the distance the shooter is away from the basket at the time of the shot. Closest

defender distance6 is the distance between the shooter and the closest opposing player. Dribbles

are divided into three groups: no dribbles, one dribble, and two-plus dribbles. Location is divided

into home and away. Shots are divided into open or contested, defined by whether a defender is

within 3.5ft of the shooter. Shots are divided into open or contested, defined by whether a

defender is within 3.5ft of the shooter. The shot clock is divided into six groups: 24-22 seconds,

very early (21.9 to 18 seconds), early (17.9 to 15 seconds), average (14.9 to seven seconds), late

(6.9 to four seconds), very late (3.9 to zero seconds), and none (no shot clock). Touch time is a

continuous variable. Rest again is divided into zero days, one day, two days, and three-plus days.

Finally, the model also includes an interaction term between the shot clock and touch time.

It is also important to note that not every player is equal nor is every shot privy to the

same factors, so the data are divided into nine separate groups based on two levels: The height of

the player to estimate position and the distance of the shot. The three player groups were defined

as guards who are less than or equal to 6 foot 3 inches, wings who are greater than 6 foot 3

inches and smaller than or equal to 6 foot 9 inches, and big men who are taller than 6 foot 9

inches. Shots are then divided up by the distance of the shot from the basket. These categories

are less than or equal to 5 feet, greater than 5 feet and less than or equal to 10 feet, and greater

6 It is necessary to note that while these data are optical and computer processed, they are not

perfect. A recent article by Vantage Sports detailed the some of the difficulties for interpreting

the data. I am going to assume that the large sample size will be able to negate out some of the

noise caused by faulty readings. See more:

http://www1.vantagesports.com/Articles/archive_article_view/Vg1ZYyMAALIAoMzK

Yudelman 21

than 10 feet. The first group is intended to estimate shots near or at the basket, the second are in

between shots in the paint but not at the basket, and the third group is for jump shots.

Results

Tables 13 and 14 show the results for model (1), the historical season model, where each

decade has its own regression. This model attempts to measure the effects length of rest has on

outcome. Despite having a schedule with many back-to-backs and poor travel conditions,

outcome was not affected by a fatigue effect in the 1950s. As travel improved and the number

back-to-back games decreased, days of rest became an economically and statistically significant

factor for some of the rest levels tested. Furthermore, in recent decades, the different levels of

rest have become even more significant at all indicator levels in determining the outcome of the

games. While the coefficients of a logit model are difficult to interpret, the results show whether

there is a difference between zero days of rest and the other levels, and in the 2010 decade, the

coefficient estimate on the one day indicator variable is 0.235, two days is 0.235, and three-plus

days is 0.225. This pattern of near uniformity is similarly found in 1990 and 2000 decades. With

zero days of rest as the base case, this model suggests that in these decades there is not much a

relative advantage of having two days of rest against just one while that there is a key distinction

between zero days rest versus against one-plus days rest. Interestingly, the coefficients for the

1970s and 1980s are not as uniform, so perhaps improvements in travel has had a significant

effect. Given the null hypothesis that rest does not affect the outcome of games, I can say with

great confidence that I can reject the hypothesis, but only for games played in more recent

decades. It is truly surprising that the 1940 and 1950 model showed little effect.

Yudelman 22

Tables 15 and 16 show the results model (2), which support the findings in Tables 13 and

14. The difference in rest have a positive effect on the outcome. However, the difference is only

significant (and gaining magnitude) in recent decades, particularly 2000 and 2010, suggesting

that an edge in rest is becoming more and more of a competitive advantage. This may be due to

greater investment in recovery methods and technology.

One would assume that as travel improved, the effects of fatigue would be mitigated, yet

this does not seem to be true. I attest this to the demanding NBA season, where teams are spread

about the continental United States and travel on charter planes is not a perfect substitute for

sleeping at home. Moreover, as the NBA is a physical game by nature, the increasing cost of

playing games in a short time span lends itself to fatigue; this is a factor where the only solution

may be limiting the minutes of players. Travel and recovery improvements can only have a

marginal effect for players that are forced to play significant minutes continually over a season.

Given that rest effects are indeed significant, I can look critically at the results of the

means testing. Tables 5 through 10 in the Appendix show the results of means testing for zero

days of rest and one-plus days of rest. (Note that some statistics are repeated under different

tables). The results are summarized as follows:

Offensive rating drops from 103.355 to 102.301.

Team field goal percentage decreases significantly from 44.5% to 44.2%. This is

accompanied by 2.21 fewer points in the paint.

Teams average almost one less possession per game in zero rest games. Given the slower

pace, fast break points per game fall from about 13.4 to 12.3.

Ball movement is significantly affected. There are 0.776 fewer assists and 0.314 fewer

secondary assists, and an assist ratio that falls about .5. This is despite an insignificant

Yudelman 23

difference in passes and touches between the two types of games, which suggests a

hidden effect on player movement.

Defensive rating increases from 102.731 to 104.321 in zero rest games compared to one-

plus rest games.

Points off turnovers fall from 16.356 to 15.508, steals drop from 7.810 to 7.494, and

blocks decrease from 4.888 to 4.497.

Overall, there is a net rating change from 0.624 to -2.018.

Interestingly, teams pass at the same rate regardless of rest, yet it is the quality of pass

that suffers as represented by the decrease in assists and secondary assists. Quality passing

requires split second decision making, so more fatigue seems to slow this reaction time. Similar

reaction statistics like steals and blocks also significantly decrease. These results are supported

by the previously discussed Shahrokh-Shahraki and Nooh –Bin Abu Baker study regarding taxi

drivers and the increase probability of accidents. Moreover, teams are running the same distance

in the games, yet with significantly slower pace and fewer fast break points. This suggests effort,

or at least willingness to exert excessive effort, is affected. The Alhoa and Polo-Kantola study

regarding the cognitive effects perhaps explains the decrease in quality of ball movement, for

that requires quick thinking from all players. Most importantly to game outcome, however, is the

decrease in net rating and field goal percentage. To investigate this thoroughly, we look towards

the shot making models.

The results of model (3) seen in Tables 15, 16, and 17 for shot making fail to reject the

null hypothesis that rest has no effect. While some of the coefficients for the days of rest are

significant, there is no discernable pattern among any of the model groups. I will attest this minor

significance to noise in the data. Moreover, the coefficients for the rest variables hover around

Yudelman 24

0.05, so there is a very minor effect if any. This is a very surprising results. Conventional

thinking suggests that tired legs hurt shot making, yet the models suggest otherwise. Looking at

the other variable, the models are consistent in that shot distance, defender distance, height

differential, and openness are significant in shot making. For jumpers in particular, taking a

dribble and shot clock have negative effects on shot making. This supports the thinking that

isolation basketball late in the shot clock is not advantageous for shot making.

Further investigating why field goal percentage and offensive rating decrease with no

effects of rest on the actual shots, I performed means testing for certain continuous variables and

shot location. These results are available in Tables 11 and 12. The data shows that teams shoot

0.358 fewer shots at the basket in zero rest games. Moreover, the shots are taken 0.167 seconds

earlier in the shot clock, after 0.06 more dribbles, and 0.115 feet further away from the basket, all

of which are significant trips. Previous preliminary research suggests that this is most significant

in two point shots rather than three point shots (Yudelman, 2010). Thus, teams are shooting more

difficult two point shots on zero days of rest. Given that mid-range two point shots are inefficient

to begin with, fatigued teams are taking more inefficient shots. Rather than tired legs affecting a

player’s elevation on a shot or the angle of the arm of the shot, it is the shot selection that is

adversely affect. Simply, offenses are taking more difficult shots that have a lower probability of

being successful.

Conclusion

This paper looks at the effects of fatigue, defined as days of rest between games, on the

outcome of games, in-game metrics, and shot making. Using two logit models to search for

fatigue effects on outcome, I find that fatigue was not a significant factors in the 1950s, but it has

been increasingly significant since then. Furthermore, I found that the difference in days of rest

Yudelman 25

for the two teams playing is also significant in more recent decades. Using means testing on box

score statistics, I found that ball movement, pace, net rating, defensive movement, and shot

selection is significantly hurt by zero days of rest compared to all other game. Finally, using a

shot making metric, I found that shots of similar context are not affected by rest. This suggests

that teams on zero days of rest are struggling to find good shots rather than missing shots the

players usually would make.

In future research, I would like to look at the effects of traveling across time zones for

back-to-back games. Previous literature suggests that this is significant in determining game

outcome, and I would like to see how that affects both the game statistics and shots.

Furthermore, I would like to test my hypothesis that teams that rely more on threes are less

affected by fatigue. Chart 1 in the appendix looks at total three point field goal attempts in the

2014-15 season and the win percentage in zero rest games. This can be looked at in a future

study that builds an empirical model to investigate how a team’s reliance on three point shots are

related to the ability to compete in games with a fatigue disadvantage.

Yudelman 26

Works Cited

Alhola, P. & Polo-Kantola, P. (2007). Sleep deprivation: Impact on cognitive performance.

Neuropsychiatric Disease and Treatement¸ 3(5), 553-567.

Ashman, T., Bowman, A., & Lambrinos, J. (2010). The role of fatigue in NBA wagering

markets: The surprising "home disadvantage situation". Journal of Sports Economics,

11(6), 602-613. http://dx.doi.org/10.1177/1527002509357545

Bean, J. C., & Birge, J. R.. (1980). Reducing Travelling Costs and Player Fatigue in the National

Basketball Association. Interfaces, 10(3), 98–102. Retrieved from

http://www.jstor.org/stable/25059927

Caplan, J. (2013, October 16). Mavs first to dive into fatigue analysis [Blog post]. Retrieved

from Hang Time website: http://hangtime.blogs.nba.com/2013/10/16/mavs-first-to-dive-

into-fatigue-analysis/

Case, K., Fair, R., & Oster, S. (2009). Principles of Economics (9th ed.). Upp Saddle River, NJ:

Prentice Hall.

Draper, K. (2015, July 9). The NBA's salary cap and tax level just went up: Here's what it means.

Retrieved November 28, 2015, from http://deadspin.com/the-nbas-salary-cap-and-tax-

level-just-went-up-heres-w-1716694473

Entine, O., & Small, D. (2007, November). The role of rest in the NBA home-court advantage.

Retrieved from http://www-stat.wharton.upenn.edu/~dsmall/publications.html

Haberstroh, T. (2014, September 11). Fixing NBA's unhealthy schedule. Retrieved November

12, 2015, from http://insider.espn.go.com/nba/story/_/id/11506366/how-fix-league-back-

back-problem-nba

Haberstroh, T. (2015a, January 9). Relief for LeBron James. Retrieved November 12, 2015, from

http://insider.espn.go.com/nba/story/_/id/12141819/relief-lebron-james-nba

Yudelman 27

Haberstroh, T. (2015b, February 24). Measuring "The Spurs Effect". Retrieved November 12,

2015, from http://insider.espn.go.com/nba/story/_/id/12380337/nba-spurs-effect-35-

minute-players-becoming-extinct

Haberstroh, T. (2015c, June 17). Biggest winner of the Finals? Rest! Retrieved November 12,

2015, from http://espn.go.com/nba/playoffs/2015/story/_/id/13098001/golden-state-

warriors-show-rest-best

Haberstroh, T. (2015d, September 10). The NBA's back-to-back problem. Retrieved November

12, 2015, from http://insider.espn.go.com/nba/story/_/id/11498509/the-negative-effects-

playing-more-back-back-games-nba

Holmes, B. (2015, October 23). Lakers trainer searches for an injury solution. Retrieved

November 12, 2015, from http://espn.go.com/nba/story/_/id/13947039/los-angeles-

lakers-trainer- searches-injury-solution

Kelly, Y. (2010). The myth of scheduling bias with back-to-back games in the NBA. Journal of

Sports Economics, 11(1), 100-105. http://dx.doi.org/10.1177/1527002509337497

Lowe, Z. (2015, September 17). From BMI to TMI: The NBA is learning toward wearable tech.

Retrieved November 12, 2015, from http://grantland.com/the-triangle/from-bmi-to-tmi-

the-nba-is-leaning-toward-wearable-tech/

Mah, C., Mah, K., Kezirian, E., & Dement, W. (2011). The effects of sleep extension on the

athletic performance of collegiate basketball players. SLEEP, 34(7), 943-949.

http://dx.doi.org/10.5665/SLEEP.1132

McFall, T. A. (2014). The (Peculiar) Economics of NCAA Basketball. New York: Palgrave

Macmillan.

Yudelman 28

Narsu, K. (2015, September 28). Introducing KOBE: A measure of shot quality. Retrieved

December 1, 2015, from http://nyloncalculus.com/2015/09/28/introducing-kobe-a-

measure-of-shot-quality/

Nutting, A. (2010). Travel costs in the NBA production function. Journal of Sports Economics,

11(5). http://dx.doi.org/10.1177/1527002509355637

Paine, N. (2012, October 17). Do we really need 82 games? Retrieved November 28, 2015, from

http://espn.go.com/nba/insider/story/_/id/8515058/nba-why-82-game-season-

unnecessary-nba

Partnow, S. (2015, August 13). This season’s NBA schedule benefits the Knicks the most – and

the Timberwolves the least. Washington Post. Retrieved from

https://www.washingtonpost.com/news/fancy-stats/wp/2015/08/13/this-seasons-nba-

schedule-benefits-the-knicks-the-most-and-the-timberwolves-the-least/

Torre, P. S., & Haberstroh, T. (2014, October 10). New biometric tests invade the NBA.

Retrieved November 12, 2015, from http://espn.go.com/nba/story/_/id/11629773/new-

nba-biometric-testing-less-michael-lewis-more-george-orwell

Shahraki, S., & Abu Bakar, N. -. B. (2011). Review of relationship between knowledge of

management, workforce productivity and nervous fatigue. African Journal of Business

Management, 5(34).

Silver, N. (2014, May 5). Good news: You won game 7; Bad news: You're Less Likely to Win

Round 2. Retrieved September 28, 2015, from FiveThiryEight website:

http://fivethirtyeight.com/features/good-news-you-won-game-7-bad-news-youre-less-

likely-to-win-round-2/

Yudelman 29

Silver, N., & Fischer-Baum, R. (2015, May 21). How we calculate NBA Elo Ratings. Retrieved

November 28, 2015, from http://fivethirtyeight.com/features/how-we-calculate-nba-elo-

ratings/

Surdam, D. G. (2012). The rise of the National Basketball Association. Urbana, IL: University of

Illinois Press.

Yudelman, A. (2015, July 27). NBA players need their rest: A look at shot difficulty. Retrieved

November 27, 2015, from http://nyloncalculus.com/2015/07/27/nba-players-need-their-

rest-a-look-at-shot-difficulty/

Yudelman 30

Appendix

Significance codes for the tables:

“*” = .10 level

“**” = .05 level

“***” = .01 level

Table 1: Days of Rest for the 2014-15 NBA Season

Opponent Days of Rest

0 1 2 3+ NA Grand Total

Team 0 94 54 18 5 171

Days 1 222 457 87 18 784

Of 2 62 72 35 10 179

Rest 3+ 27 25 7 20 79

NA 4 13 17

Grand Total 409 608 147 53 13 1230

*All days with 3 or more days of rest are grouped as one, and “NA” represents opening day

games. Team days of rest are from the home team’s perspective. Includes two neutral games.

Table 2: Win Percentage by Days of Rest for the 2014-15 NBA Season

Opponent Days of Rest

0 1 2 3+ NA Grand Total

Team 0 53.19% 48.15% 61.11% 80.00% 53.22%

Days 1 54.95% 58.64% 54.02% 55.56% 57.02%

Of 2 62.90% 66.67% 45.71% 60.00% 60.89%

Rest 3+ 62.96% 60.00% 57.14% 60.00% 60.76%

NA 50.00% 76.92% 70.59%

Grand Total 56.23% 58.72% 53.06% 60.38% 76.92% 57.48%

*All days with 3 or more days of rest are grouped as one, and “NA” represents opening day

games. Team days of rest are from the home team’s perspective. Includes two neutral games.

Yudelman 31

Table 3: NBA Schedule and Days of Rest

Days of Rest 1940s 1950s 1960s 1970s 1980s 1990s 2000s 2010s Grand Total

0 25.82% 41.86% 42.06% 34.32% 27.10% 24.46% 22.95% 23.13% 28.60%

1 33.23% 30.97% 32.71% 37.03% 44.70% 50.82% 52.66% 55.06% 45.42%

2 20.42% 16.01% 15.13% 17.43% 18.89% 15.98% 16.20% 14.53% 16.57%

3+ 18.89% 9.81% 8.94% 10.08% 8.16% 7.55% 7.04% 6.11% 8.23%

NA 1.64% 1.35% 1.16% 1.14% 1.14% 1.19% 1.14% 1.18% 1.18%

Table 4: NBA Schedule and Visitor Days of Rest

Away Days of

Rest 1940s 1950s 1960s 1970s 1980s 1990s 2000s 2010s Grand Total

0 34.18% 47.52% 49.24% 42.56% 33.94% 32.38% 31.36% 31.30% 36.35%

1 34.18% 32.80% 30.79% 34.60% 41.26% 46.11% 48.19% 50.48% 42.07%

2 16.30% 12.13% 12.23% 14.47% 17.05% 14.31% 13.81% 12.30% 14.17%

3+ 13.97% 6.40% 6.76% 7.39% 6.69% 6.07% 5.62% 4.85% 6.36%

NA 1.38% 1.14% 0.98% 0.98% 1.06% 1.13% 1.02% 1.07% 1.05%

Yudelman 32

Table 5: Means Testing for Advanced Box Scores Statistics

Advanced Box Score Statistic 0 rest 1+ rest p-value Significance Code

Offensive Rating 102.301 103.355 0.050 *

Defensive Rating 104.321 102.731 0.003 ***

Net Rating -2.018 0.624 0.000 ***

Assist Percentage 0.580 0.589 0.107

Assist to Turnover 1.646 1.697 0.133

Assist Ratio 16.549 16.967 0.008 ***

Offensive Rebounding Percentage 0.247 0.250 0.367

Defensive Rebounding Percentage 0.748 0.752 0.307

Rebound Percentage 0.497 0.501 0.066 *

Team Turnover Percentage 14.820 14.732 0.651

Efficient Field Goal Percentage 0.493 0.499 0.065 *

True Shooting Percentage 0.532 0.536 0.139

Usage Percentage 0.199 0.199 0.030 **

Number of Possessions Per Game 95.763 96.480 0.001 ***

Player Impact Estimate 0.486 0.504 0.000 ***

*

Table 6: Means Testing for Four Factors Statistics

Four Factors Statistics 0 rest 1+ rest p-value Significance Code

Efficient Field Goal Percentage 0.493 0.499 0.065 *

Free Throw Rate 0.275 0.277 0.606

Team Turnover Percentage 0.148 0.147 0.655

Offensive Rebounding Percentage 0.247 0.250 0.367

Opponent Efficient Field Goal Percentage 0.499 0.497 0.402

Opponent Free Throw Rate 0.282 0.275 0.167

Opponent Team Turnover Percentage 0.144 0.149 0.006 ***

Opponent Offensive Rebounding Percentage 0.252 0.248 0.308

*All numbers are rounded to 3 decimal points

Yudelman 33

Table 7: Means Testing for Traditional Box Score Statistics

Traditional Box Score Statistic 0 rest 1+ rest p-value Significance Code

Field Goals Made 36.934 37.704 0.001 ***

Field Goals Attempted 83.188 83.700 0.135

Field Goal Percentage 0.445 0.452 0.009 ***

3 Point Field Goals Made 7.947 7.814 0.401

3 Point Field Goals Attempted 22.402 22.421 0.952

3 Point Field Goal Percentage 0.353 0.346 0.171

Free Throws Made 17.066 17.153 0.762

Free Throws Attempted 22.513 22.934 0.235

Free Throw Percentage 0.758 0.748 0.038 **

Offensive Rebounds 10.744 10.931 0.301

Defensive Rebounds 32.014 32.539 0.032 **

Rebounds 42.758 43.470 0.018 **

Assists 21.440 22.216 0.001 ***

Steals 7.494 7.810 0.026 **

Blocks 4.497 4.888 0.001 ***

Turnovers 13.651 13.725 0.702

Personal Fouls 20.449 20.134 0.138

Points 98.881 100.375 0.008 ***

Point Margin -1.620 0.500 0.009 ***


Yudelman 34

Table 8: Means Testing for Player Tracking Statistics

Player Tracking Statistic 0 rest 1+ rest p-value Significance Code

Distance 16.623 16.667 0.189

Offensive Rebounding Chances 20.484 20.785 0.314

Defensive Rebounding Chances 51.451 51.810 0.388

Rebounding Chances 71.934 72.595 0.193

Touches 422.055 422.606 0.760

Secondary Assists 5.174 5.489 0.009 ***

Free Throw Assists 2.212 2.265 0.476

Passes 301.439 301.280 0.967

Assist 21.440 22.216 0.001 ***

Contested Field Goals Made 20.865 21.527 0.001 ***

Contested Field Goals Attempted 46.257 46.423 0.065 *

Contested Field Goal Percentage 0.454 0.466 0.000 ***

Uncontested Field Goals Made 16.067 16.174 0.597

Uncontested Field Goals Attempted 36.926 37.271 0.268

Uncontested Field Goal Percentage 0.435 0.434 0.756

Field Goal Percentage 0.445 0.452 0.009 ***

Defended Field Goals Made 15.998 16.080 0.744

Defended Field Goals Attempted 30.718 31.249 0.223

Defended Field Goal Percentage 0.528 0.521 0.213


Table 9: Means Testing for Miscellaneous Box Score Statistics

Miscellaneous Box Score Statistic 0 rest 1+ rest p-value Significance Code

Points off Turnovers 15.508 16.356 0.003 ***

2nd Chance Points 13.038 13.299 0.298

Fast Break Points 12.307 13.357 0.000 ***

Points in the Paint 40.370 42.580 0.000 ***

Opponent Points off Turnovers 16.406 16.078 0.260

Opponent 2nd Chance Points 13.200 13.249 0.848

Opponent Fast Break Points 13.240 13.069 0.576

Opponent Points in the Paint 42.238 42.003 0.590

Blocks 4.497 4.888 0.001 ***

Block Attempts 5.016 4.728 0.021 **

Personal Fouls 20.449 20.134 0.138

Personal Fouls Drawn 20.003 20.272 0.199


Yudelman 35

Table 10: Means Testing for Possession Statistics

Possessions Statistic 0 rest 1+ rest p-value Significance Code

Touches 422.078 422.671 0.742

Front Court Touches 310.285 310.213 0.966

Elbow Touches 18.147 18.458 0.372

Post Touches 17.112 17.970 0.001 ***

Paint Touches 13.598 14.454 0.000 ***


Table 11: Means Testing for SportVU Shot Statistics

SportVU Shots Statistics 0 rest 1+ rest p-value Significance Code

Shot Clock 12.277 12.444 0.000

***

Dribbles 2.063 2.003 0.001

***

Touch Time 2.786 2.735 0.001

***

Shot Distance 13.728 13.613 0.011

**

Closest Defender Distance 4.123 3.139 0.249


Table 12: Means Testing for Shot Locations

Shot Location 0 rest 1+ rest p-value Significance Code

3-5 feet 10.401 10.759 0.035 **

5-10 feet 14.420 14.233 0.359

Midrange 24.378 23.985 0.269

Restricted Area 10.997 11.583 0.002 ***

Three 22.225 22.227 0.993


Yudelman 36

Table 13: Results for Logit Model (1)

Variables 1940s 1950s 1960s 1970s

Team Elo Rating 0.0068291 0.0061 0.0056505 0.0055128

[0.0007885]*** [.0004240]*** [0.0002853]*** [0.0002189]***

Opponent Elo Rating -0.005817 -0.0051235 -0.005534 -0.004851

[0.0007575]*** [.0004621]*** [0.0002839]*** [0.0002183]***

TeamRest1Day 0.4734918 -0.0647317 0.2173422 0.2436567

[0.2197404]* [0.1007432] [0.0768380]** [.0552178]***

TeamRest2Days 0.1377689 -0.0887972 0.2011357 0.2007692

[0.2323931] [0.1086083] [0.0927735]* [0.0640188]**

TeamRest3PlusDays 0.1227508 0.0736141 0.0347281 0.1122862

[0.2278218] [0.1254877] [0.1078086] [.0727447]

OpponentRest1Day -0.2138393 -0.1386232 -0.2404164 -0.2411256

[0.13785586] [0.0918297] [0.0735518]** [0.0502255]***

OpponentRest2Days -0.0032529 -0.2031811 -0.1331269 -0.1680821

[0.2239476] [0.1233789]. [0.1023840] [0.0659744]

OpponentRest3PlusDays -0.1994328 -0.0131388 -0.1610446 -0.0514392

[0.2307176] [0.1651788] [0.1281525] [0.0658744]

Intercept -1.0100286 -0.627956 0.3640409 -0.3831765

[1.1545816] [0.8891605] [0.4697546] [0.4087817]

Observations 945 3595 5118 11213

Yudelman 37

Table 14: Results for Logit Model (1) (cont.)

Variables 1980s 1990s 2000s 2010s

Team Elo Rating 0.0060904 0.0063635 0.0058844 0.006077

[0.0002299]*** [0.0001860]*** [0.0001950]*** [0.0002374]***


[0.002459]*** [0.0001901]*** [0.0001963]*** [0.0002381]***

TeamRest1Day 0.1944956 0.2252715 0.2443057 0.2351474

[0.0524653]** [0.0609799]*** [0.0583857]*** [0.0753050]**

TeamRest2Days 0.1352093 0.2033557 0.2711411 0.2353326

[0.0739424]. [0.0746953]** [0.0691612]*** [0.0919651]*

TeamRest3PlusDays 0.0971081 0.194525 0.2281229 0.2253814

[.0922217] [0.0909491]* [0.0872334]** [0.1180641].

OpponentRest1Day -0.3004156 -0.2611425 -0.280012 -0.2141618

[0.0552206]*** [0.0504368]*** [0.0464271]*** [0.0600568]***

OpponentRest2Days -0.2801865 -0.3300063 -0.22989502 -0.2557942

[0.0705125]*** [0.0695218]** [0.0643531]*** [0.0871872]**

OpponentRest3PlusDays -0.0038951 -0.2532485 -0.1592331 -0.3856768

[0.101241] [0.0989624]** [0.0938512]. [0.1265803]**

Intercept 0.6740144 -0.3336221 -0.0955431 -0.6082685

[0.4558443] [0.3538425] [0.3719570] [0.4516081]**

Observations 10090 11650 12905 7641

Table 13: Results for Logit Model (2)

Variables 1940s 1950s 1960s 1970s

Team Elo Rating 0.0066587 0.0060918 0.0055733 0.0054791

[.0007665]*** [0.0004228]*** [0.0002837]*** [0.0002179]***


[0.0007536]*** [0.0004610]*** [0.0002828]*** [0.0002175]***

Rest Differential 0.0036886 0.0232163 0.0346772 0.0274332

[0.0412187] [0.268983] [0.0213684] [0.0150650].

Intercept -0.7130596 -0.7395905 0.4494941 -0.3259048

[1.1169599] [0.8858126] [0.4652079] [0.4080794]

Observations 945 3595 5118 11213

Yudelman 38

Table 14: Results for Logit Model (2) (cont.)

Variables 1980s 1990s 2000s 2010s

Team Elo Rating 0.0060431 0.0063195 0.0058652 0.0060525

[0.0002290]*** [0.0001851]*** [0.0001938]*** [0.0002364]***


[0.0002353]*** [0.0001896]*** [0.000195]*** [0.0002476]**

Rest Differential 0.0479829 0.0796529 0.0728555 0.0805172

[0.0189666]* [0.0187290] [0.0174865]*** [0.0230656]***

Intercept 0.07074645 -0.2972687 -0.0739086 -0.55640508

[0.4548137] [0.3524260] [0.3712539] [0.4505961]

Observations 10090 11650 12905 7641

Table 15: Results for Logit Model (3) for Shots Close to the Basket

Variables Guards Close Wings Close Forwards Close

Rest1 -0.034364 0.06643 0.026826

[0.056411] [0.36408]. [0.038996]

Rest2 -0.02329 0.078014 -0.032055

[0.076909] [0.050452] [0.053105]

Rest3+ -0.213476 0.065982 0.066074

[0.102018]* [0.062469] [0.067763]

Shot Distance -0.09048 -0.085974 -0.115091

[0.018652]*** [0.012155]*** [0.012773]***

Log(Closest Defender Distance + 1) 0.686145 0.835745 0.710456

[0.061228]*** [0.039221]*** [0.041743]***

One Dribble -0.179655 -0.106572 0.028783

[0.090258]* [0.043197]* [0.048298]

2+ Dribbles -0.377398 -0.221379 0.095006

[0.081023]*** [0.048538]*** [0.073423]

Open 0.570641 0.623752 0.721938

[0.083852]*** [0.059799]*** [0.068822]***

Location Home 0.013418 0.036593 0.020493

[0.045459] [0.029615] [0.031687]

Shot Clock Average 0.076759 0.151767 0.416326

[0.152424] [0.060857]* [0.055091]***

Shot Clock Early 0.23159 0.227383 0.546929

[0.184565] [0.079869]** [0.078288]***

Yudelman 39

Shot Clock Late 0.089852 0.093185 0.12547

[0.192349] [0.084707] [0.084200]

Shot Clock None 0.27895 -0.043951 0.095379

[0.212803] [0.108200] [0.113442]

Shot Clock Very Early 0.36014 0.530293 0.794075

[0.165135]* [0.074722]*** [0.088581]***

Shot Clock Very Late -0.107091 -0.211033 0.220678

[0.222199] [0.106888]* [0.108801]*

Height Differential 0.048149 0.051447 0.038943

[0.005765]*** [0.004589]*** [0.005226]***

Touch Time 0.03367 -0.004358 -0.100641

[0.055692] [0.034399] [0.034363]**

Shot Clock Average: Touch Time -0.028848 -0.03786 -0.020113

[0.055757] [0.036080] [0.038113]

Shot Clock Early: Touch Time -0.044145 -0.043853 -0.016064

[0.058310] [0.038741] [0.048822]

Shot Clock Late: Touch Time -0.030934 -0.009046 0.53897

[0.057472] [0.038130] [0.043315]

Shot Clock None: Touch Time -0.047329 0.011988 0.045981

[0.059522] [0.042129] [0.074103]

Shot Clock Very Early: Touch Time -0.07493 -0.112652 -0.175776

[0.060676] [0.04314]** [0.053893]**

Shot Clock Very Late: Touch Time -0.075493 0.002521 -0.027223

[0.059416] [0.040466] [0.054893]

Intercept 0.236359 -0.304999 -0.228013

[0.167309] [0.074073]*** [0.071713]**

Observations 9200 25366 19715

Table 16: Results for Logit Model (3) for Shots 5 to 10 Feet from the Basket

Variables Guards 5-10 Wings 5-10 Forwards 5-10

Rest1 -0.041198 0.018633 0.026197

[0.064084] [0.042396] [0.46199]

Rest2 0.07013 0.091757 -0.051492

[0.087670] [0.058757] [0.065437]

Rest3+ 0.070173 0.069353 -0.082805

[0.116973] [0.074309] [0.085317]

Shot Distance -0.080298 -0.128653 -0.136197

[0.018313]*** [0.012490]*** [0.014063]***

Log(Closest Defender Distance + 1) 0.241213 0.379033 0.414974

Yudelman 40

[0.084290]** [0.054163]*** [0.062558]***

One Dribble -0.192977 0.067272 -0.144199

[0.133593] [0.054416] [0.050749]**

2+ Dribbles -0.160381 -0.121403 -0.070842

[0.122209] [0.055894]* [0.063844]

Open 0.340307 0.223593 0.265272

[0.084294]*** [0.061741]*** [0.071383]***

Location Home 0.080758 0.078829 0.0169446

[0.052330] [0.034731]* [0.037909]

Shot Clock Average -0.73666 -0.112532 -0.028445

[0.287919]* [0.141408] [0.131935]

Shot Clock Early -0.79145 0.078161 0.174115

[0.323363]* [0.158630] [0.154592]

Shot Clock Late -0.927511 -0.240327 -0.009317

[0.306444]** [0.158638] [0.154545]

Shot Clock None -1.185441 0.024398 0.107108

[0.353463]*** [0.195231] [0.215249]

Shot Clock Very Early -0.748957 0.538055 0.335789

[0.320805]* [0.157131]*** [0.170375]*

Shot Clock Very Late -1.192569 -0.325164 -0.193774

[0.317950]*** [0.173468]. [0.176715]

Height Differential 0.018329 0.036614 0.034826

[0.006349]** [0.005425]*** [0.006652]***

Touch Time -0.204806 0.010731 -0.034679

[0.108605]. [0.66491] [0.072665]

Shot Clock Average: Touch Time 0.195828 -0.006539 0.02671

[0.108624]. [0.066739] [0.073847]

Shot Clock Early: Touch Time 0.21434 -0.062234 -0.11066

[0.111072]. [0.068727] [0.083279]

Shot Clock Late: Touch Time 0.193664 -0.021117 0.027967

[0.109390]. [0.068281] 0.76230]

Shot Clock None: Touch Time 0.228556 -0.022998 -0.02329

[0.110917]* [0.070822] [0.093950]

Shot Clock Very Early: Touch Time 0.173737 -0.112824 -0.049622

[0.115199] [0.070672] [0.088848]

Shot Clock Very Late: Touch Time 0.1948333 -0.017572 0.028712

[0.109749] [0.069366] [0.080005]

Intercept 0.912028 0.246521 0.236847

[0.316651]** [0.167176] [0.167512]


Yudelman 41

Table 17: Results for Logit Model (3) for Shots 10 or More Feet from the Basket

Variables Guards Jumper Wings Jumper Forwards Jumper

Rest1 -0.0531669 -0.02651 -0.017879

[0.0310031]. [0.02241] [0.035213]

Rest2 -0.0286591 -0.01714 -0.009602

[0.0428431] [0.03088] [0.048078]

Rest3+ -0.0744415 -0.04511 -0.045135

[0.0560379] [0.03924] [0.064690]

Shot Distance -0.046272 -0.03824 -0.044451

[0.0032056]*** [0.002406]*** [0.003668]***

Log(Closest Defender Distance + 1) 0.2931293 -0.327 0.454431

[0.0499173]*** [0.03396]*** [0.0563161]***

One Dribble 0.0007084 -0.07214 -0.101426

[0.0441680] [0.02943]* [0.048579]*

2+ Dribbles -0.1082042 -0.09852 -0.177061

[0.0417845]** [0.03326]** [0.063956]**

Open 0.1083183 0.05683 0.016199

[0.0389176]** [0.02860]* [0.049663]

Location Home 0.034717 0.04512 0.022243

[0.0255765] [0.01837]* [0/028764]

Shot Clock Average -0.3439358 -0.2553 -0.358739

[0.1634405]* [0.1046]* [0.165135]*

Shot Clock Early -0.3949603 -0.2659 -0.332559

[0.1703330]* [0.1081]* [0.173120].

Shot Clock Late -0.3007779 -0.3447 -0.402075

[0.1680478]* [0.1089]** [0.137291]*

Shot Clock None -0.7012723 -0.5986 -0.670878

[0.1832306]*** [0.1197]*** [0.209844]**

Shot Clock Very Early -0.3862168 -0.2442 -0.272605

[0.1733800]* [0.1110]* [0.188680]

Shot Clock Very Late -0.5746368 -0.547 -0.620104

[0.1719110]*** [0.1120]*** [0.179296]***

Height Differential 0.0061555 -0.0000527 0.008738

[0.0034618]. [0.003029] [0.004666].

Touch Time -0.3205059 -0.1666 -0.102978

[0.1004491]** [0.06824]* [-.096982]

Shot Clock Average: Touch Time 0.3215842 0.167 0.111445

[0.1004009]** [0.06841]* [0.097689]

Shot Clock Early: Touch Time 0.3424758 0.1897 0.093262

[0.1007519]*** [0.06906]** [0.104081]

Yudelman 42

Shot Clock Late: Touch Time 0.3175461 0.1689 0.098068

[0.1006179]** [0.06877]* [0.099122]

Shot Clock None: Touch Time 0.32749 0.1779 -0.008126

[0.1009991]** [0.06967]* [0.122390]

Shot Clock Very Early: Touch Time 0.3586601 0.1743 0.092467

[0.1019578]*** [0.07069]* [0.111295]

Shot Clock Very Late: Touch Time 0.3080814 0.1668 0.103094

[0.1007491]** [0.06894]* [0.100285]

Intercept 0.3745494 -0.02146 -0.026067

[0.1890453]* [-0.1241] [0.193239]


Chart 1: How 3FGA And Zero Rest Win% are Related

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est

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Total 3FGA in 2014-15 Season

3FGA and Zero Rest Win %

Documents

The NBA and Fatigue