Measuring Correlation

If we want to know a correlation between a boxer winning a match and using red pants, we can mathematically determine it based on past records of those conditions. There are four possible combinations of the conditions.

The boxer loses and he doesn’t use red pants (00)
The boxer loses and he uses red pants (01)
The boxer wins and he doesn’t use red pants (10)
The boxer wins and he uses red pants (11)

For example, we have the following past records.

Condition “00” is 23 times
Condition “01” is 10 times
Condition “10” is 45 times
Condition “11” is 9 times

Then, we can use the phi coefficient formula. The result of this formula is between -1 and 1. If the result is close to 0, it means the conditions have no correlation. If it is close to 1, it means the conditions are strongly correlated. Meanwhile, if it is negative, the correlation is strong but in the opposite way. The formula is as follows.

phi = (n11 x n00) – (n01 x n10) / sqrt(n1X x n0X x nX1 x nX0)

n1X represents the condition where the boxer wins, regardless of the color of his pants. While nX1 represents the condition where the red pants are used, regardless result of the match.

phi = (9 x 23) – (45 x 10) / sqrt( 54 x 33 x 19 x 68 )

phi = -243 / 1517

phi = -0.16

It shows that winning a match and the red pants have no correlation. What if there are more other conditions that may have a correlation to a win. For example, on a winning day, sometimes he drinks coffee before the match, talks to his girlfriend, or plays with his buddies. We can perform the formula separately for each condition and the main condition which is winning a match. For example, the result is as follows.

phi coffee = 0.8
phi girlfriend = -0.7
phi buddies = 0.05

It shows that drinking coffee has a strong correlation. Meanwhile talking to a girlfriend has negative value therefore the assumed condition should become “not talking to girlfriend”. We can also perform an additional phase which is combining the two potential conditions into a single condition so that it becomes “drinking coffee and not talking to girlfriend”. Then, we need to tally again all records where the boxer wins or loses while he both drinks coffee and doesn’t talk to his girlfriend or not.

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