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## Randy Classen

on 18 April 2014

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Introduction
A Statistical Analysis
By Randy Classen

The sample for this study is 136 games played between teams ranked in the top 20.
The home team won 92 games (67%), in this sample
Home Court Advantage is a commonly known fact of team sports
This Study
Paired T-tests & Signed Rank Test
Checking for differences
H : µ - µ = 0
0 Home Away
H : µ - µ ≠ 0
1 Home Away
vs.
Paired T-test Hypotheses:
H : Md - Md = 0
0 Home Away
H : Md - Md ≠ 0
1 Home Away
vs.
Signed Rank Test Hypotheses:
Field Goal Pct: Mean = 0.0244, p-value = 0.005
Paired T-test Results:
Decision
: Reject Hₒ. The difference is not zero. Interpreting the mean indicates advantage
Decision
: Fail to reject Hₒ. The difference is not distinguishable from zero. There is no advantage.
Signed Rank Test Results:
Blocked Shots: Median = 1, p-value = 0.0068
Decision
: Reject Hₒ. The difference is not zero. Interpreting the median indicates advantage
Free Throw Percent
The home crowd does all they can to distract away players shooting free throws. Including,
Pictures waved in the crowd
Yelling, screaming, etc
These tactics have no apparent effect on the away team's free throw percent. It appears that the home fans need to try something new
Foul Difference
Why would the home team have fewer fouls?
End of game fouling strategy?
To check, I used a paired T-test on a sample of large point margin victories
Foul Difference
Does the away team have just as much of an advantage when they win?
To check this explanation, I ran a Kruskal-Wallis test analyzing the foul difference against the outcome, home win or away win.
H : Md = Md
0 Home Away
H : Md ≠ Md
1 Home Away
vs.
Result
: Median = -2 p-value = 0.0001

Kruskal-Wallis hypothesis:
T-test Results: Mean = -2.19, p-value = 0.0005
Decision: Reject Hₒ. There is still an advantage for the home team.
In a home win, the home team benefited in more than 71% of the games.
Summary So Far
The t-tests showed several significant differences
Correlation
Correlation does not indicate causality
But it is helpful for prioritizing variables for a linear model
The difference in blocked shots was shown significant but since r is essentially zero, it is not useful for a linear model, so we can disregard it for our purposes. Similarly for turnover difference.
Hₒ : r = 0 versus H : r ≠ 0
Correlation Hypothesis:
¹
Example of usefulness:
The p-value for blocks is greater than α = 0.05 so r cannot be determined to be different than zero.
The differences that are significant and related to winning are field goal percent, fouls, and free throws made
Multiple Linear Regression
Eliminated outliers by analyzing residuals
Built several models to balance simplicity with power
Full Model

Variables (Differences): Fouls, FTM , Steals, Blocks, Turnovers, FG%, FT%, and 3-Point %

PtDiff FGpctDif ThreeDif FTpctDif
FoulDif FTMDif StlDif BlkDif

TODif
2
2

Variables (Differences): Turnovers, FG%, FT%, and 3-Point%

PtDif FGpctDif ThreeDif FTpctDif TODif

2
2
Simple Model
Variables (Differences): FG%, Fouls, and FTM

PtDif FGpctDif FTMDif FoulDif
2
2
Model with Significant Variables
y = 2.2989 + 80.9879 (0.0244) + 0.4959 (2.10) + 0.4691 (2.51)
PtDiff
y = 6.49
PtDiff
As an example of using this model, let's plug in the average number for these variables.
Using Our Model
So this model says that in an average game, 6.5 points can be explained by variables that are advantageous for the home team
Statistical Methods
Signed Rank tests and a Kruskal-Wallis test are used on data that has no associated distribution.
The p-value is the lowest α at which we begin to reject Hₒ
Most of the data is normal and able to be tested with paired T-tests
Conclusion
The home team has an advantage in field goal percent, fouls called, turnovers, free throws made, and blocked shots.
Home does not have an advantage in FT%.
Fans need to find a new idea to help their team.

The refs are discernibly biased toward the home team by about 2 fouls per game.

Overall, the home team has an advantage of over 6 points per game.
Many of the likely causes are hard to measure
Home team's comfort in the facilities
Crowd noise
Biased referees
The intention is to find and observe differences between home and away performance to make conclusions about home court advantage in general
A conventional significance of α = 0.05 is used for hypothesis testing unless otherwise stated
Other significant differences:
Fouls, Turnovers, & FTM
Free Throw Pct: Mean = 0.0144, p-value = 0.368
Other insignificant differences:
Steals & Three Point Pct

Decision
: Reject Hₒ. The difference is biased in spite of winner.
It appears that the refs are biased toward the home team
So the foul difference was not simply a by-product of being the winning team.
In away wins, the away team benefitted from the refs less than 55% of the time.
The Kruskal-Wallis test indicated unbalance in spite of winner.
How much do they actually affect the outcome?
Checked for multicollinearity
This model sacrifices simplicity for the sake of a high R.
y = 1.0266 + 75.6038 x + 16.0627 x + 9.5149 x + 0.1682 x + 0.3354 x + 0.2871 x – 0.2488 x – 0.7028 x
R = 0.8031, PRESS = 6252.1754
This model has an R that is only 0.0607 less than the full model but with half as many variables.
It is a much simpler model but still powerful.
y = 1.2380 + 77.2159x + 14.6452x + 15.1905x – 0.7900x
R = 0.7424, PRESS = 3425.7035
The R is considerably lower than the other models
In this model, 63% of the variability of point difference is explained by variables that are significant between the home and away teams.
y = 2.2989 + 80.9879x + 0.4959x + 0.4691x
R = 0.6291, PRESS = 3805.4592
Key findings:
Of these, field goal percent, fouls, and free throws made are also notably correlated to point difference
The difference in fouls is not caused by end of game fouling.
Another t-test eliminated end of game fouling as a cause for the foul difference

PtDiff i i
y = a + b x
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