# Statistical test for one sample proportions

I am carrying out a study into the handedness of racehorses. There are clockwise and anti-clockwise racecourses. I want to test to see if 1) horses are more likely to win if they are left handed on anti-clockwise (left hand) courses than those who are right handed and 2) if they are more likely to win if they are right handed on clockwise (right hand) courses than those who are left handed. Would the one sample z - test for proportions be appropriate. I have looked at 280 races so there are 280 winners in all. Thanks, I look forward to hearing your thoughts.

• Could you expand on exactly what "proportions" you have? With its several explanatory variables (handedness of horse and handedness of course) this sounds like a more complex situation than a mere test of proportions. – whuber Jun 6 '16 at 13:39
• On right handed courses wins were L=84 R=60. On left hand courses wins were L=71 R=65. Thanks you – Inelberg Jun 6 '16 at 16:26
• I didn't want to know the values! (Those are meaningless in themselves.) I wanted to know the nature of your data and what they represent. As the answers are pointing out, summarizing the results in such a simple manner likely overlooks important information. – whuber Jun 6 '16 at 16:49
• Sorry but I don't understand what you are asking me for. – Inelberg Jun 6 '16 at 21:12
• At a minimum you ought to analyze a dataset whose records list the course, the horse, how it placed, and the handedness of both the course and the horse. Reducing it to counts of wins loses so much information that you might fail to detect what you are looking for. – whuber Jun 6 '16 at 22:55

As you are looking into the winners, the numbers lend themselves to be displayed in a 4 fields contingency table, which usually calls for a chi-square test for independence or a Fisher test.

However, you should consider, whether your data are i. i. d. enough. If there is one horse, that's left-handed and wins a lot of anti-clockWise courses than that's not the same, as if a lot of left-handed horses win a lot of anti-clockwise races.

edit: This may be your contingency table

            is left | is right
won clockw.    84       60
won antic.     71       65

• There are no duplicate horses among the winners – Inelberg Jun 6 '16 at 21:15
• So how do I use the 4 field contingency table please? – Inelberg Jun 6 '16 at 21:36
• I have added a contingency table to my answer. If you use SPSS, please search the web for how to apply Fisher's test on that. Or download R software from www.r-project.org and enter fisher.test(matrix(c(84,60,71,65), nrow=2)) or use one of many free online calculators for Fisher's test or Chi-square test of independence. p-values larger than 0.05 indicate, that differences may very well have appeared by chance. – Bernhard Jun 7 '16 at 5:54
• Thank you so much Bernhard that has helped me a lot. All the best – Inelberg Jun 7 '16 at 10:13

As opposed to restricting to just one handedness of the course, another possibility is to measure wins and losses where group A is horses of same-handedness of course versus group B being horses with different-handedness of course. But you may want to use mantel-haenszel statistic where you stratify upon the handedness of the course.

I highly caution against making any causal claims though: Handedness of horse is likely inherited from parents, as is some degree of speed or racing expertise. On top of that, stud horses are presumably chosen because they have good racing qualities, perhaps handedness being one of them. So this isn't a randomized trial where handedness of horse is randomly assigned to each horse.

I'm not sure how it works, but it might be interesting to see the handedness of participating horses by handedness of race track - is there a sizeable difference in the number of lefthanded horses running on clockwise versus counterclockwise tracks? That might indicate whether there could be motivation to submit one's horse to a track with handedness that would give (perceived) advantage.

Also, +1 to @bernhard for pointing out that unless each horse appears only once in the dataset then there might be some issues with i.i.d. data

• In my study there are twice as many left handed tracks as right handed ones. You are quite right that it is advantageous to rum a horse in a direction that it is not comfortable with so that is may well loose and have its handicap weight reduced. – Inelberg Jun 6 '16 at 21:18
• Also yes there are genetic reasons as to why the horse may race well, but there are anatomical and training issues which will make a difference also. Although it would be great to find out why an individual horse is right, left or ambidextrous this is outside the remit of my study. The study is there to add to the debate of laterality in horses i.e. does it exist and if so by how much. Then more studies can be carried out as to how this knowledge may contribute to reducing injury and deaths for these horses. – Inelberg Jun 6 '16 at 21:25