## Saturday, January 16, 2016

### Two tests for difference between home and road win proportions

Question:  Below we have data on the win-loss record at home and on the road for the Toronto Blue Jays.

 Toronto Blue Jay Record Home and Away 2015 Home Away Win 53 40 93 Loss 28 41 69 81 81 162

Use the large sample t-test comparing two proportions and the chi-square test on difference between actual and expected proportions to determine whether home field advantage was important to the Toronto Blue Jays in 2015.

Analysis:  The calculation of the t-test for a significant difference between the home-win and road win proportions is presented in the table below.

 Calculation of t-test for difference in home and road win percentages Label Number of formula Observed Win/Home 53 Observed Lose/Home 28 Observed Win Away 40 Observed Lose Away 41 Total Home 81 Total Away 81 Win Prob Home 0.654 Win Prob Away 0.494 Diff Win Probabilities 0.160 Pooled Variance 0.238 Standard Error 0.077 t-statistic 2.093 p-value 0.036

The calculation of the chi-squared test for a significant difference in the two proportions is presented in the table below.

 Calculation of chi square test for difference in home  versus road win percentages Team Toronto Blue Jays Observed Win/Home 53 Observed Lose/Home 28 Observed Win Away 40 Observed Lose Away 41 Total Home 81 Total Away 81 Total Observations 162 Pooled Win Prob 0.5741 Pooled Loss Prob 0.4259 Expected Win/Home 46.50 Expected Loss/Home 34.50 Expected Win/Away 46.50 Expected Loss/Away 34.50 (OWH-EWH)2/EWH 0.91 (OLH-ELH)2/ELH 1.22 (OWA-EWA)2/EWA 0.91 (OLA-ELA)2/ELA 1.22 Chi Square 4.27 P-value 0.039

Notes:

Note One:  I have arranged the calculators vertically.   The first four lines of each calculator is input.   The remaining lines are formula.   If you want to test home win proportions for some other team just put appropriate input into the first four lines of the calculator and it will update.

Note Two:  The chi-square test is commonly used for small as well as large samples.   The t-test is most commonly used for larger samples.

Note three:  The p-values from the two tests are close 0.039 for the chi-square test and 0.036 for the t-test.   The test results presented here are two-tailed.   We reject the null hypothesis that the home-win proportion is identical to the road-win proportion.

Note four:  The t-test for no difference in home versus road win proportions for all MLB teams in the 2015 regular season along with a nice statistical discussion on the meaning of hypothesis tests is presented in the post below.

Concluding thoughts:  The Toronto Blue Jays did a better job at home than on the road during the regular season in 2015.  I wonder if this is true of all teams.

Is home field advantage more important for some teams than other teams?  If so, why is it more important for some teams than other teams?   Is home field advantage as important in the post season?   Is home field advantage important in every sport?

Variants of this problem can be used to teach both baseball and statistics.   More will follow.

Authors Note: In 1996, I wrote a short book “Statistical Applications of Baseball”  The book is a bit informal but it has a lot of statistical problems and it got a good review from the academic journal Chance.   If you are interested the book is available on Kindle.

Go back to home-field advantage testing problems:

http://www.dailymathproblem.com/p/home-field-hypothesis-testing-problems.html