Thursday, January 21, 2016

Testing for a difference in average team home and road wins

Testing for a difference in average team home and road wins

Previous posts used tests on proportions to assess the importance of home field advantage in MLB.     This post examines and tests for a difference in average home and away wins across all MLB teams.

Question:   Below is data on home wins, road wins and the difference between home and road wins for all MLB teams during the 2015 regular season. 

·      Test the null hypothesis that the difference in the mean number of home wins and road wins is identical.

·      Barring some fart in the schedule each team plays 81 games on the road and 81 games at home.   What happens if we test the hypothesis that mean of the percent of games won at the home is equal to the mean percent of games on the road?

·      How do the tests on differences in mean home/away wins compare to tests on differences in proportion of home/away games?




Home and Road Wins 2015 Regular Season b.y Team
Team
Wins at Home
Wins on Road
Difference (Home - Road)
Toronto
53
40
13
New York
45
42
3
Baltimore
47
34
13
Tamp Bay
42
38
4
Boston
43
35
8
Kansoas City
51
44
7
Minnesota
46
37
9
Cleveland
39
42
-3
Chicago
40
36
4
Detroit
38
36
2
Texas
43
45
-2
Houston
53
33
20
Los Angeles
49
36
13
Seattle
36
40
-4
Oakland
34
34
0
New York
49
41
8
Washington
46
37
9
Miami
41
30
11
Atlanta
42
25
17
Philadelphia
37
26
11
Saint Louis
55
45
10
Pittsburgh
53
45
8
Chicago
49
48
1
Milwaukee
34
34
0
Cinncinatti
34
30
4
Los Angeles
55
37
18
San Francisco
47
37
10
Arizona
39
40
-1
San Diego
39
35
4
Colorado
36
32
4


Answer:  The table below contains the mean and standard deviation for wins at home, wins on the road and the paired differences of wins at home and on the road.



Summary Statistics on Wins at Home and on the Road
2015 Regular Season MLB
Wins at Home
Wins at Road
Difference
Average
43.833
37.133
6.700
STD
6.587
5.563
6.587


We need to consider whether the variances from the home and away samples are identical or different.   If we fail to reject the null hypothesis of identical variances we can use a standard error based on the pooled variance.

The ratio of the estimators of the variance is 1.4.


The critical value for tests on the ratio of variances is obtained from the
FINV function   =F INV(0.05,29, 29)=1.86.

We cannot reject the null hypothesis of equal variances so I will use a pooled variance estimator.


Pooled Variance = (43.4+30.9)/2   = 37.2

The tests statistic for the difference in two means is


Z = (AVG(HW) –AVG(RW))/((STD_pooled x 2/30)0.5



Here HW is home wins and RW is road wins.


Z   =  6.7/(37.2 x (1/15))0.5


=4.2.

We reject the null hypothesis mean wins on the road is equal to mean wins at home.

Results are even stronger when we use the paired difference procedure.

The test statistic for the paired mean test is

Z = AVG(Diff)/ (Var(Diff)/n)0.5


=5.57

Again we reject the null hypothesis that the difference in means is zero.


Would results differ if we divided the mean wins by 81 the number of home and away games?

No.

The test statistics for the numerator and denominator are both divided by 81.  The Z scores are unchanged.   The critical values are unchanged.

Test results on the mean of wins are identical to test results on means of proportions of wins.


How do the results presented here differ from results in previous posts that looked at proportion of games won?


The questions asked and answered differ as do the samples used to address the issue.

One post considers whether the proportion of all MLB games won by the home team is larger than the proportion of games won by the away team.   The sample covers all games played in the major leagues.  There are 81 x 30 regular season games in a season.



The second post tests for home field advantage for each team in MLB.   The sample size is the number of home games for each team.   Each team plays 81 games.


This post looks at average home and road wins.   The sample size is the number of teams in the league, 30 teams.

How would size of samples differ if we looked at home field advantage in college basketball or baseball?

Authors Note:  Please consider my book Statistical Applications of Baseball.




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