## 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?

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.