writing sample (6500 words)

January 2016 Ian Weaver

The Effects of Time Off Between Games on the FreeThrow Shooting of Professional Basketball Players

A Case Study of when Extra Rest is Detrimental to an Employee’sProductivity of a Particular Task

Ian Weaver

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0 Executive Summary

This paper looks at the free throw shooting success rates of NBA players from the

2009-10 to the 2013-14 season. The key question was what is the optimal number

of days off between in game free throws in terms of maximizing the success rate

of said free throws. To analyze this situation an orignal dataset was created by

collecting various factors from every regular season game log for the five seasons

from 2009-10 to 2013-14.

This aspect of success rates of NBA free throws has never been studied in

the academic literature, and so any type of result is automatically an addition to

the basketball free throw literature. To find these effects, multiple models were

created that incorporated all of the important variables in free throw shooting;

these variables were determined through reading other literature and through my

own ideas.

The results show that players have their highest success rates in converting free

throws when they have just had an in game free throw attempt the previous day.

This is surprising as the literature shows that basketball teams perform worse on

the latter of games on consecutive days. Nevertheless, these effects were not that

large in magnitude. There was essentially no difference in free throw performance

between having one day and two days off. Players with 3 to 4 days off between

in game throws were 0.5% less likely to score than those with one day off while

holding all other explanatory variables constant. This rose to 0.7% and 2.1% for

the groups of 5 to 9 days off and 10 days or more off, respectively. These results

were statistically significant with the latter two being significant at the 0.01 level.

While these effects may seem small, considering that our dataset observes a

simple one person game that is being played by the most talented basketball players

in the world who practice constantly, it is still noteworthy. It appears that in game

free throw skills atrophy quickly especially for layoffs of over 10 days. A possible

reason could be that it is not the time off that is causing the decreased performance

in scoring free throws but rather that players with long layoffs between free throws

could be battling injuries or other issues that cause them not to shoot as well.

When these factors are accounted for, the effects of days off became more muddled

but players with more days off were still less likely to score a free throw than if

they had shot one in a game the previous day.

Overall this suggests that NBA players may wish to practice free throws in a

setting that best replicates an in game atmosphere (for example they could cause

fan distractions and play crowd noise at the stadium during practice) if they are

to have long layoffs in shooting in game free throws. Also, if NBA players who are

highly talented and practice continually have significant decreased productivity in

performing a simple task after short layoffs, it provides incentive for other managers

to check the productivity of their own workers after periods of rest.

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1 Introduction

1.1 Motivation

A common thought is that rest is a good for the performance of an individual. This

is difficult to refute as one would likely not suggest staying up all night the day

before an exam. Likewise, however, one would not likely suggest to take a few days

off to rest and not study before an exam. This causes a grey area on what type of

rest is best for an individual when they are trying to maximize their productivity.

An interesting question is whether managers get higher performance out of

their employees on Monday after they have gotten time off from work (but have

had their skills atrophy) or on Friday when they are tired of work (but their skills

are sharp from continual practice). Unfortunately, this is a difficult and nuanced

problem to answer. One area that does have a plethora of well documented data

to take a stab at this problem is the world of professional sports. However, sports

are played as one team directly against another which means that the success of

one team is directly related to the success of other team; this means that even if

one performs badly as long as the other team is worse then it is a positive result.

This is not particularly representative of the problem that managers face in getting

the most out of their employees. While their employees may have opponents it

is more likely that their task could be modeled as a one person game under the

exogenous restrictions placed on them by the world. Fortunately, there is one part

of basketball known as the free throw which is a one person game. For this reason,

this paper makes a small step forward in answering how time off between work

affects performance by looking at how well professional basketball players perform

under varying circumstances.

1.2 Outline

More specifically, the question this paper investigates is how does the number of

days between games effect the performance of NBA players in scoring a free throw

shot.1. The free throw is a simple one person game as it has no opposition. It also

something that NBA players will have practiced their entire life on their way to an

NBA career. Considering these are the most skilled players in the world performing

a simple task that they practice regularly, it would not be surprising if an extra

day off between games had no effect on their performance; yet this is not the case.

The way this will be investigated is by creating a binary outcome model where

the result of the free throw is the binary independent variable. Next multiple

specific models will be created by using a plethora of control variables and by

altering the variables that consider the time off between games. The probit model

1A free throw in basketball is an unimpeded shot at the net from a set line 15 feet away andoccurs due to infractions committed by the other team

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will be used as it accommodates any pattern of correlation and heteroskedasticity

in the error term.

The result of this paper will be useful in two ways. First, it will be useful to those

interested in sports analytics as it will give a direct result on how the schedule of

the NBA season affects the abilities of an NBA player to score a free throw. Second,

it will provide an interesting case study in how employee performance is affected

due to time off between work even when the task is fairly simple, the employee has

large amounts of practice in the task and the employee is skilled at the task.

2 Literature Review

There is a small section of literature dedicated to the effects of travel and days rest

on the ability of basketball players to perform; a subsection of this is even focused

on the ability to shoot free throws. A technique used in the life sciences is to create

an experiment such as Mah et al. [1] where they considered the effects of sleep

extension on the performance of collegiate basketball players. They discovered

that when the average sleep per night was increased by 2 hours that free throw

shooting percentage increased by 9%. Such a technique is not really applicable

here as these shots are taken in non game situations, and it measures sleep per

night rather than days between free throws (though there may be a correlation

between these).

A more relevant area of literature are econometric approaches which attempt to

ascertain the ability of basketball players to perform given varying schedules dic-

tated to them by the NBA. To the author’s knowledge, there exists no econometric

paper in the literature that examines these effects on the rate of success of players’

in scoring free throws. There are, however, a couple papers that consider the effect

of the NBA schedule on the ability of teams to win [2,3]. Due to the similarity, the

models in these papers will be pertinent in shaping our own models.

In both of these papers regressions are run by using point differential between

the home and away team as the dependent variable; common independent variables

include the strengths of both teams, whether they were home or away team and the

number of days of rest teams had before the game. The Steenland and Deddens

(SD) article also includes independent variables for travel such as distance, time

zones and directions travelled between games [2]. Overall both articles came to the

conclusion that teams did significantly worse when there was no off day between

games as compared to games where they had at least one day off between games.

Further, SD concluded that it was optimal to have two full days of rest between

games. Additional conclusions were that days rest affected the home and away team

equally (ES), and that duration and direction of travel did not have a significant

effect except when travelling from one coast to the other (SD).

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ES also repeated the analysis using a binary dependent variable on whether the

team won or not; this method, though, did not lead to any significant results. Part

of the reason for this may be an econometric criticism that I have with that model.

Their proxy variable for team ability was just that season’s team winning percentage

which by itself has a direct linear relationship with the dependent variable. As

a result the effects of rest and home court advantage that they were trying to

determine were partially being included in their variable for ability. For example,

consider a team with a tough schedule in terms of rest days (perhaps they have

more back to back games and then long layoffs than other teams). Their winning

percentage (and thus proxy for ability) will be lower than their actual ability to

win as the effects of the tough schedule are already erroneously being included in

their proxy variable. As a result, when they determined the effects of rest, it was

diminished as it is already being included to a certain degree into the proxy variable

for ability. This is likely why their model resulted in no conclusive results. This is

also something that must be taken into account in determining the proxy variable

for ability in free throw shooting in our own models.

The one econometric/statistical paper that could be found on free throw shoot-

ing was a project on how crowd behaviour affected players’ ability to score free

throws [4]. They showed through some statistical models that the home team was

less likely to score a free throw in a “high pressure” situation in comparison to

the away team. Their reasoning was that fans cheer loudly to try and distract the

away team player when he is shooting and are silent when the home team player is

shooting; however, a stadium full of silent fans creates an environment that makes

it more difficult to score free throws then when everyone is cheering. Regardless,

“high pressure” situations occur almost solely during close games in the final quar-

ter. As a result this research suggests two new control variables for our regressions:

the quarter in which the shot occurs and the score differential at the time of the

shot.

3 Data

3.1 Data Source and Variables

The data for this project comes from NBA game logs that are available at Basket-

ball Reference [5]. Using these game logs, I wrote programs which stripped relevant

information about every free throw that was taken in the regular season in the NBA

between the 2009-2010 season and the 2013-2014 season. These relevant variables

include whether the shot was scored, the ability of the player shooting, the date the

shot was taken, the quarter the shot was taken, the score differential of the game

when the shot was taken, whether the player was home or away when taking the

shot, the days rest the player had before taking the shot, whether the player took

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a free throw last game, and the number of time zones the player travelled before

taking the shot. Most of these were inspired by the papers from the literature

section except for a few that I added myself. Variables like height, age, etc. were

not included even though they have an effect on scoring free throws as they are

all encompassed in the variable of ability. The variables are specifically described

below.

score - A binary variable that is 1 if the free throw is scored and 0 if the free throw

is missed

ftability - A measure of the ability of the free throw shooter (will be formulated

in Section 4).

home - A binary variable that is 0 if the game was played away and 1 if the game

was played at home

date - A binary variable that is 0 if the free throw was taken in the first half of the

season and 1 if it was taken in the latter half of the season

quarter - The quarter that the free throw was taken. 1 stands for first quarter, 2

for second quarter etc.

daysrest - The number of days between the game in which the current free throw

was shot and the last game in which he attempted a free throw

playedprev - A binary variable that is 1 if the free throw shooter had a free throw

attempt in the last game that his team played and 0 otherwise

absscorediff - The absolute score difference in the game at the time of the shot

timediff - The difference in time between the time zone that the current free throw

is being taken and the time zone that the team last played in (travelling east to

west is negative and travelling west to east positive)

abstimediff - The absolute value of timediff

As some of these definitions are nuanced, here is an example to illustrate what

is meant. Assume that a team plays on January 1st, 2nd and 4th. A certain player

attempts and scores exactly one free throw on both January 1st and January 4th;

he attempts none on January 2nd. Also, the games on January 1st and 4th are

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played in the Eastern Time Zone while the one on January 2nd is played in the

Central Time Zone. So for the shot on January 4th, daysrest = 3 as the last game

he attempted a free throw was January 1st. playedprev is 0 as he did not attempt

a free throw in the last game his team played. timediff is 1 as he is currently in

a time zone that is one hour ahead of the time zone that his team last played.

Here we have chosen daysrest to be the time between in game free throw at-

tempts and not the time between games played in. This is because here we are

interested in measuring the effect on performance of total time off from doing a

task in a work setting as opposed to the effect on performance of total time off

from work. Also we have denoted daysrest to be 5 for free throws shot in season

openers as on average there is 5 days between when a team last plays a pre-season

game and the start of their regular season.

The variable playedprev is of potential interest because it could be that those

players who often do not have a free throw attempt in a game are having their

ability to score affected by things other than the time between in game free throws.

For example they may be battling injuries and this may be affecting their usual free

throw performance in ways we can not observe. If we do not include playedprev

then we will be incorporating some of the effects of the injury into daysrest.

The variable timediff measures the time zone difference between the time zone

of the game in which he currently shot a free throw and the time zone of the game

that his team last played in. This is because it is assumed that the player travels

with the team even when he gets limited playing time, is benched or is injured

(sometimes for larger injuries they do not travel with the team, but there is no way

of tracking their movements so it will be assumed that they at least start travelling

with the team just before their return).

Further, daylight savings was included when applicable in the calculations of

timediff . Interestingly, the state of Arizona does not observe daylight savings

time while every other state with an NBA franchise does. This is only a minor

point for this paper, but may be of use to others who are trying to separate the

effects of distance travelled and time zones travelled.

3.2 Summary of Key Statistics

A summary of the key statistics is given in Table 7.1 in the appendix. The main

results were that over the 5 seasons from 2009-10 to 2013-14 there were 276642 free

throws taken and 75.7% of these were scored. Other interesting results are that

51.1% of free throws were for the home team (which suggests either referee bias or

that teams play better at home), more free throws occur in the second half of a

game than the first half (as mean of quarter = 2.67 > 2.5) and that 73.9% of free

throws are taken by players who attempted at least one free throw in the previous

game that their team played in.

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4 The Econometric Model

4.1 The Basic Model

To achieve the basic econometric model, all of the free throws from the 2009-10

to 2013-14 season were pooled together into one cross section. This can be done

as time should not be affecting any variables except for ability and we allow this

variable to change every season. Once the other factors in our model have been

accounted, then will use a probit model.

The basic econometric model is described by the following equation:

score = β0 + β1ability + β2daysrest+ control variables+ µ

In this equation the dependent variable is score which is a binary variable that is 1

if the free throw was scored and 0 if it was missed. The main variable is daysrest

as the goal of the paper is to see how the amount of time off between free throws

affects a player’s ability to shoot a free throw. There are also numerous control

variables as listed in section 3.1. Different uses of these variables and variations

of daysrest will provide all of the econometric regressions for this paper. Of the

control variables, ability is not actually observed, which is a problem that will be

dealt with shortly.

A main assumption in this model is that there is no correlation between the

unobservable error variables and the independent variables. This is a reasonable

assumption for two reasons. First there is no issue of simultaneity, as whether the

player scores his free throw is not affecting any of the explanatory variables. For

example, whether he scores or not does not explain which quarter he took the shot,

the score differential of the game, the amount of rest he had before the shot, his

ability to score free throws etc. Second, omitted variable bias will not be a problem

(with the exception of determining ability) as there is a wealth of data available on

any free throw attempt; any omitted variables would be our mistake in identifying

the model and not because we were unable to get data for a certain variable.

At this point, the regression for the basic model can run as soon as it is deter-

mined how to deal with ability which is not observed. Fortunately, a good proxy

variable exists and developing it will be the focus of the next section.

4.2 Finding a Proxy for Ability

4.2.1 Proposing a Proxy for Ability

A very natural proxy for ability would be to use the free throw percentages for

a particular season for the player of interest. Alternatively worded this is the

percentage of free throws that a particular player scored during a particular season.

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As an example, if the free throw was taken by Tim Duncan in 2011-12, then we

would use his free throw percentage from that year as a measure of his ability to

score that free throw. Intuitively, this should be a good estimator of the unobserved

ability of a player to score any given free throw.

Unfortunately there is one major problem with this estimator, which was alluded

to in Section 2. The issue is that a player’s free throw percentage for a given season

might not be completely representative of their actual unobserved ability. Each

season that player may have to take more 4th quarter free throws, more free throws

after long periods of not playing, more free throws in close games etc. as compared

to usual. However, we have hypothesized that these factors affect whether a player

will score a free throw, so by using the season free throw percentages as a proxy for

ability we are incorporating the exact effects we are trying to determine into our

proxy for ability.

To help ameliorate these issues, we will propose another proxy variable for play-

ers’ ability to score free throws. This variable is called ftability and it will be

the average free throw percentage of a player for all free throw attempts made by

that player during the season he took that free throw as well as the season before

and after that. This takes advantage of the fact that a player’s free throw ability

should change very little between consecutive seasons while giving more data so

that the estimates for ability will be less governed by randomness. Further, the

free throw percentage calculated over three seasons should be more accurate in

capturing a player’s true ability during the season that the free throw was taken

as it less prone to the particular characteristics of the free throws taken during the

season of interest.

This proxy does not solve all of the issues. Consider that a player may have

certain unobserved traits. For example, each season he may only play in lopsided

games. As a result he does not play often and this means he is placed in a dis-

proportionate amount of situations where he has a lot of time off between in game

free throws. Thus, he is regularly placed in tougher than normal situations to score

a free throw, which means that ftability will be lower than ability. Nevertheless,

these differences should be random among the entire sample and they should also

be very small now that the proxy includes three seasons instead of one.

The reason that we chose only 3 seasons instead of more seasons is because that

a player’s free throw ability can change over time. As evidence for this, consider

the case of Tim Duncan. Over the first 10 seasons of his career he had free throw

percentage of 67.75% while over the last 9 seasons he had a free throw percentage of

73.66%. By the t-test assuming unequal variances we reject the null hypothesis that

the two means are equal at the 5% significance level (p-value of 0.028). Whether

using three seasons is optimal is not known. Solving this problem would require

somehow balancing the issues with changing player ability over time, with the added

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benefit of gaining more seasons of data. Above we have intuitively argued for why

three seasons will make a very good estimator.

On a technical note when the player has no free throw information for the season

before or season after the season in which the free throw was taken then only the

data available is used. For example, if a player only played one year in the NBA,

then his ftability is just his free throw percentage for that one season.

4.2.2 Showing that ftability is a Good Proxy

According to Wooldridge [6], any good proxy variable should have certain prop-

erties. First, the assumption that µ is uncorrelated with ftability was already

discussed in section 4.1. The more controversial assumption needed is that

E(ability | explanatory variables) = E(ability | ftability)

This is the case if the average level of ability only changes with ftability and not

home, date, quarter, absscorediff , daysrest and timediff . This also seems a

reasonable assumption as why would a player’s natural ability to score free throws

be affected by variables determining where and when they were playing. Thus, our

proxy variable satisfies the assumptions to be a good proxy.

5 Empirical Results

5.1 Results of Basic Model by Probit Regression

By incorporating the proxy from section 4 into the basic model we get the following

equation, which we will estimate by probit:

score = β0 + β1ftability + β2home+ β3date+ β4absscorediff

+β5abstimediff + β6quarter + β7daysrest+ µ(1)

Note that this model has no interaction between dummy variables and other ex-

planatory variables, which means that it does not allow for a difference of slopes

for the different binary choices. For example, this means that it is assumed that

the return of an extra day off between free throw attempts is the same whether the

player was at home or away.

The results are found in Table 2, section 7.2 of the appendix. Using the hy-

pothesis test of H0 : Bi = 0 against H1 : Bi 6= 0 for any i ∈ {1, 2, 3, 4, 5, 6, 7},

almost all of our variables were significant at very small significance levels. The

other two variables (abstimediff and home had p-values under 0.15 which suggest

that they should not be dropped from the regression even if they provide results of

little interest on their own.

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The directions of the results are what was expected for almost of these variables.

Players score free throws with higher success rates when they have higher abilities,

when they play at home (home court advantage), when it is the latter half of the

season (more warmed up), when it is in the earlier quarters of a game (less tired),

when the score differential in the game is higher (less pressure), and when they

travel fewer timezones between games (less tired). The one exception is that of

rest between games. If the hypothesis that players do worse when they travel more

because travel is tiring is accepted then one would expect that more days between

free throw shots would improve scores. Nevertheless, it makes it more difficult with

a p-value less than 0.001. There could be a few reasons for this, which will be

explored shortly. For now, the takeaway is that the control variables are all very

reasonable.

When the average marginal effects are calculated (Table 3 in the appendix) then

the results turn out not to be that drastic. With the exception of ftability all of the

independent variables have very small effects. For example, if the number of days

between games with a free throw is increased by 1 (while holding other independent

variables constant), then the average player is 0.09% less likely to score that free

throw. Despite being significant even at very low p-values, this is a very small effect

that will be of little interest to those with stakes in the game of basketball.

5.2 Probit Results using Dummy Variables for Rest

Part of the reason for the results in the previous section may be that the effects of

the number of days between free throws in games is not linear. To figure this out, a

method similar to Entine and Small (ES) [3] will be used, which is to partition the

daysrest variable into multiple dummy variables. With regards to notation, let us

denote daysrestij as a dummy variable that is 1 if the player had between i and

j days (including boundaries) between the game in which the current free throw

was shot and the last game in which the same player attempted a free throw; it is

0 otherwise.

At this time the basic model will also be altered to consider the effects of each

quarter separately by creating dummy variables. That is quarterk is a dummy

variable that is 1 if the shot was taken in the kth quarter and 0 otherwise.

Two regressions will be run. The first will partition the set of number of

days between in game free throws as {{1}, {2}, {3}, {4}, {5, ...135}} while the sec-

ond will partition the set of number of days between in game free throws as

{{1}, {2}, {3, 4}, {5, 6, 7, 8, 9}, {10, ..., 135}}. This covers the entire data set as daysrest

has a value between 1 and 135 for every observation in our data set. In terms of

equations these are written as follows with the first partition being represented

in equation (2) and the second partition in equation (3). Here our base group is

daysrest11 , so it is not included in the regression equation.

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score = β0 + β1ftability + β2home+ β3date+ β4absscorediff+

β5abstimediff +

4∑k=2

αkquarterk +

4∑i=2

γidaysrestii + γ5daysrest5135 + µ(2)


β5abstimediff +

4∑k=2

αkquarterk + γ2daysrest22 + γ3daysrest34+

γ4daysrest59 + γ5daysrest10135 + µ

(3)

The only difference between our model in equation (1) and those in (2) and (3)

is how quarter and daysrest are partitioned in dummy variables. In both models

(2) and (3), both quarter2 and quarter4 are negative. This means that players

are less likely to score a free throw during the 2nd or 4th quarter than in the 1st

quarter (holding all of the other independent variables constant). As quarter3 is

not statistically significant then overall the negative coefficients of quarter2 and

quarter4 may be because players get tired throughout the game. However, this is

likely not the whole situation as quarter4 is much more significant then quarter2

and has a much larger average marginal effect. Indeed in model (2) the average

player is 0.67 percentage points (0.65 percentage points in model (3)) less likely to

score a free throw in the fourth quarter as compared to the first quarter. While this

is still a small effect, considering how consistent NBA players are this is noteworthy.

The main question is what are the effects of the dummy variables for daysrest.

In the model from equation (2), it is obtained that there is no significant difference

between two days between in game free throws and one day. For the groups of

three days and four days, these both have small negative effects with p-values

around 0.10. In fact, when a regression was run that gave every amount of days

off between in game free throws a separate dummy variable, similar effects and p-

values were obtained for each variable. Part of the problem is a lack of observations

for many of these dummy variables, which is why different grouping is necessary.

Using the results of model (2) as motivation, we obtain the groupings for equa-

tion (3). The difference here is that days off of three and four days have been

grouped into one dummy. Further instead of having 5 or more days off as a one

dummy this has been separated into two dummy variables (5 to 9 days and 10 days

or more). The results were that the average player is 0.5 percentage points less

likely to score a free throw with three or four days between in game free throws

as compared to having one day off between in game free throws (while holding all

other independent variables constant). For the groups of 5 to 9 days and 10 days

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or more, the average marginal effects were 0.7% and 2.1% respectively. The latter

of these has a p-value less than 0.001 and 2.1% is actually a fairly large effect.

Nevertheless, it may be expected that after such a large layoff that a decrease in

productivity would be expected.

Perhaps the more interesting result is that even for periods of time of three or

four days (which are fairly normal time periods between games), that players were

0.5 percentage points less likely to score a free throw than if it had only been one

day off. The results in the academic literature [2,3,7] as well as conventional wisdom

suggest that teams play worse on the second of consecutive games as compared to

a larger (though standard) amount of days between games. However, free throws

provide an example where players perform significantly worse and by an appreciable

amount (considering how consistent NBA players are in performing this simple

task).

5.3 How Does Being In and Out of the Lineup Affect Free

Throw Shooting

Possibly the biggest concern to the previous models is that it treats all layoffs

between in game free throws as the same even this may not be the case. For

example, a player coming back from an injury may shoot worse than if he had

not been coming back from an injury (holding days between free throws constant).

Another example is that a coach observes that his player is not playing well and so

he gets less playing time and often goes games without a free throw. He may shoot

worse during these periods than periods where he is playing well and regularly

scoring free throws. Whatever the case, to account for this the variable playedprev

will be added to the model given by equation (3). It is a binary variable that is 1 if

the free throw shooter had a free throw attempt in the last game his team played

and 0 otherwise. The updated model is represented by the following equation:


β5abstimediff + β6playedprev + β7quarter + γ2daysrest22+

γ3daysrest34 + γ4daysrest59 + γ5daysrest10135 + µ

(4)

As predicted, playedprev is positive which means that a player that had a free

throw attempt in his team’s last game is more likely to score his free throw than if

he did not have a free throw attempt in his team’s last game (while holding days

off and other independent variables constant). The introduction of playedprev also

decreased the average marginal effects of our dummy variables for daysrest by

about 30%.

Unfortunately, playedprev is not statistically significant and the p-values for

the days off dummy variables have grown greatly rendering many of those results

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not statistically significant. This is not that surprising as there is a strong correla-

tion between playedprev and the daysrest. Indeed due to NBA scheduling, when

dayrest is less than or equal to 2 then playedprev must be equal to 1. If daysrest

is greater than or equal to 10 then playedprev must be equal to 0.

To further investigate, two more regressions will be run on models which are

restricted to only observations with 3 or 4 days off between in game free throws or

only observations with 5 to 9 days off separately. In terms of model specification,

both (5) and (6) follow the restricted model represented by the equation below.

The difference is that (5) is restricted to those observations where daysrest34 = 1

and (6) is restricted to those observations where daysrest59 = 1.


β5abstimediff + β6playedprev + β7quarter + µ

When the data set is restricted to observations where daysrest34 = 1 , playedprev

was statistically significant with a p-value of 0.053. Further, the average player is

0.65% more likely to score a free throw if he had a free throw attempt in his team’s

last game than if he did not have a free throw attempt in his team’s last game

(holding all other independent variables constant). However, when the data set is

restricted to those observations with daysrest59 then playedprev is not statistically

significant (p-value of 0.845) and has an average marginal effect of 0.09%. In other

words, there is no discernible difference in free throw performance between those

who had an attempt in their team’s last game and those who did not.

Overall, the effects of playedprev are still unclear. I believe the problem could

be fixed with an even larger data set to overcome the collinearity issues.2 The

reason I believe this would work is because when the data set is restricted as it was

in (5) and (6), then date, absscorediff and quarter all had large changes in effects

as well as losing significance. The reason they all lost significance is likely due to

the lack of sample size; our trends are so subtle that even sample sizes of 50000 are

not enough. This is further supported by the fact that over the 5 season data set,

free throws are significantly harder to score in the 4th quarter than the 1st with

a very small p-value of 0.005. Yet when the calculations are restricted by season

there was one season where players actually performed better in the 4th quarter.

Thus, data sets of 50000 are not large enough to accurately the effects that we are

interested in.

2This is unfortunately something that I did not have time to do under the time constraintsof this course.

14


6 Conclusion

This paper looks at the effects of time off between games on the free throw shooting

of professional basketball players. It concludes that players are their best at free

throw shooting when they had an in game free throw attempt the previous day.

This is an interesting result as the current literature and wisdom supports the

notion that players perform worse when they have to play two days in a row and

yet this is one aspect of the game where they perform better.

For the most part, the advantages to having attempted an in game free throw

the previous day are quite small and thus not of much use to those with a stake

in basketball games. For layoffs of over 10 days, though, these effects are over 2%,

which suggest that NBA players with long layoffs may wish to practice free throws

in a setting that best replicates the game atmosphere before they return .

Ignoring the effects on the game of basketball itself, it is a noteworthy result

in that even small amounts of time off from performing this task during a game

can lead to decreased productivity. Considering this is a simple one player task

that is being completed by the most talented players in the world who practice

the task continually in non-game scenarios under the best coaches, it is slightly

surprising that even a few days off between in game performance of the task could

lead to a significant decrease in the ability to perform it. This provides incentive

for other managers to check the productivity of their own workers after periods of

time off as a further research project. They likely have employees with less training

and less dedication, performing tasks that are more complicated than the situation

described in this paper, and so their decrease in productivity may be even larger.

15


References

[1] Mah, Cheri D. et al. ’The Effects Of Sleep Extension On The Athletic Perfor-

mance Of Collegiate Basketball Players’. SLEEP (2011): n. pag. Web.

[2] Steenland, Kyle, and James Deddens. ’Effect Of Travel And Rest On Perfor-

mance Of Professional Basketball Players’. Sleep 20.5 (1997): 366-369. Print.

[3] Entine, Oliver A, and Dylan S Small. ’The Role Of Rest In The NBA Home-

Court Advantage’. Journal of Quantitative Analysis in Sports 4.2 (2008): n. pag.

Web.

[4] Goldman, Matt and Justin Rao. ’Effort vs. Concentration: The Asymmetric

Impact of Pressure on NBA Performance’. MIT Sloan Sports Analytics Conference

(2012): n. pag. Web.

[5] Basketball-Reference.com,. ’Basketball-Reference.Com’. N.p., 2015. Web. 8

Dec. 2015.

[6] Wooldridge, Jeffrey M. Introductory Econometrics. Australia: South-Western

College Pub., 2003. Print.

[7] Sampaio, Jaime, Eric J. Drinkwater, and Nuno M. Leite. ’Effects Of Season

Period, Team Quality, And Playing Time On Basketball Players’ Game-Related

Statistics’. European Journal of Sport Science 10.2 (2010): 141-149. Web.

16


7 Appendix

7.1 Summary Statistics for Key Variables

Table 1: Summary Statistics

Variable Mean Std. Dev. Min Max

score .75710 .42883 0 1ftability .75797 .09532 0 1date .49435 .49997 0 1home .51143 .49987 0 1quarter 2.6735 1.1035 1 4playedprev .73873 .43933 0 1absscorediff 8.1845 6.9521 0 54daysrest 3.5380 4.9129 1 135abstimediff .45028 .70622 0 3

Number of Observations = 276642

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7.2 Probit Results for the Basic Model and Dummy Variable

Extensions

Table 2: Probit Results, Dependent Variable: score

Independent Variables (1) (2) (3)

ftability2.9936***

(.0272)2.9948***

(.0272)2.9912***

(.0272)

date0.0127**(.0053)

0.0109**(.0053)

0.0117**(.0053)

home0.0077(.0055)

0.0083(.0055)

0.0082(.0055)

absscorediff0.0012***

(.0004)0.0010***

(.0004)0.0011***

(.0004)

abstimediff-0.0055(.0038)

-0.0053(.0038)

-0.0052(.0038)

quarter-0.0052**(0.0025)

— —

quarter2 —-0.0122(.0081)

-0.121(.0081)

quarter3 —0.0086(.0083)

0.0085(.0083)

quarter4 —-0.0222***

(.0080)-0.0218***

(.0080)

daysrest-0.0031***(0.0005)

— —

daysrest22 —-0.0022(.0078)

-0.0021(.0078)

daysrest33 —-0.0165*(.0095)

—

daysrest44 —-0.0171(.0116)

—

daysrest5135 —-0.0333***

(.0087)—

daysrest34 — —-0.0167*(.0086)

daysrest59 — —-0.0238***

(.0091)

daysrest10135 — —-0.0693***

(.0141)

Observations 276642 276642 276642Pseudo R-Squared 0.0411 0.0411 0.0411

Note: Numbers in brackets are the standard errors. *** represents P < 0.01,

** represents P < 0.05 and * represents P < 0.10.

18


7.3 Average Marignal Effects for Table 2

Table 3: Average Marignal Effects, Dependent Variable: score

Independent Variables(1)

dy/dx(2)

dy/dx(3)

dy/dx

ftability.8957***(.0077)

.8960***(.0077)

.8949***(.0077)

date.0038**(.0016)

.0033**(.0016)

.0035**(.0016)

home.0023

(.0016).0025

(.0016).0024

(.0016)

absscorediff.0004***(.0001)

.0003***(.0001)

.0003***(.0001)

abstimediff-.0017(.0011)

-.0016(.0011)

-.0016(.0011)

quarter-.0016**(.0007)

— —

quarter2 —-.0037(.0024)

-.0036(.0024)

quarter3 —.0026

(.0025).0025

(.0025)

quarter4 —-.0067***(.0024)

-.0065***(.0024)

daysrest-.0009***(.0002)

— —

daysrest22 —-.0006(.0023)

-.0006(.0023)

daysrest33 —-.0049*(.0028)

—

daysrest44 —-.0051(.0035)

—

daysrest5135 —-.0010***(.0026)

—

daysrest34 — —-.0050*(.0026)

daysrest59 — —-.0071***(.0027)

daysrest10135 — —-.0211***(.0043)

Observations 276642 276642 276642



19


7.4 Probit Results for Various Models with the Variable

playedprev

Table 4: Probit Results, Dependent Variable: score

Independent Variables (4) (5) (6)

ftability2.989***(.0273)

2.932***(.0579)

2.933***(.0621)

date.0120**(.0053)

.0257**(.0113)

.0029(.0129)

home.0079

(.0055).0122

(.0116)-.0165(.0131)


.0016*(.0008)

.0013(.0009)

playedprev.0110

(.0088).0214*(.0115)

.0028(.0143)

quarter-.0052**(.0025)

-.0111**(.0053)

-.0074(.0061)


-.0117(.0076)

.0051(.0089)

daysrest22-.0020(.0078)

— —

daysrest34-.0121(.0095)

— —

daysrest59-.0168(.0111)

— —

daysrest10135-.0605***(.0166)

— —

Observations 276642 61608 46613Pseudo R-Squared 0.0411 0.0385 0.0437



20


7.5 Average Marignal Effects for Table 4

Table 5: Average Marignal Effects, Dependent Variable: score

Independent Variables(4)

dy/dx(5)

dy/dx(6)

dy/dx

ftability.8943***(.0077)

.8852***(.0165)

.9024***(.0178)

date.0036**(.0016)

.0078**(.0034)

.0009(.0040)

home.0024

(.0016).0037

(.0035)-.0051(.0040)


.0005*(.0003)

.0004(.0003)

playedprev.0033

(.0026).0065*(.0035)

.0009(.0044)

quarter-.0016**(.0007)

-.0034**(.0016)

-.0023(.0019)


-.0035(.0023)

.0016(.0027)

daysrest22-.0006(.0023)

— —

daysrest34-.0036(.0028)

— —

daysrest59-.0050(.0033)

— —

daysrest10135-.0184***(.0051)

— —

Observations 276642 61608 46613



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