The similarity metric S_yule (#61) is S_yule = ad - bc/ad + bc Think about co-va

Question

The similarity metric S_yule (#61) is S_yule = ad - bc/ad + bc Think about co-variation (change in the same direction) and contra-variation (change in opposite directions). Generically, these may also be named positive-correlation and negative-correlation in other domains. IOW, things might occur-together, or they occur-oppositely. They are related to what are called compliments and substitutes in economics. If you cat bread (e.g. sandwich) you may not cat pasta or rice, or potatoes (substitutes e.g. carbs). But if you cat French fries you cat more ketchup (compliments). What is ad? What is be? What about ad+bc? Is this metric normalized in any sense? How?

Explanation / Answer

In probability theory two binary variables are called uncorrelated if they share zero covariance, that is, ad bc = 0. The covariance between two binary variables is defined as the determinant of the 2 × 2 contingency table. In addition to being uncorrelated, two variables may be statistically independent, which is in general a stronger requirement compared to uncorrelatedness. The two concepts are equivalent if both variables are normally distributed. Probability theory tells us that two binary variables satisfy statistical independence if the odds ratio equals unity, that is ad / bc = 1. The odds ratio is defined as the ratio of the odds of an event occurring in one group (a/b) to the odds of it occurring in another group (c/d). These groups might be any other dichotomous classification. An odds ratio of 1 indicates that the condition or event under study is equally likely in both groups. An odds ratio greater than 1 indicates that the condition or event is more likely in the first group. The value of the odds ratio lies between zero and infinity.

Yule proposed two measures SYule = ((ad /bc) 1)/ ((ad/ bc + 1) = (ad bc) /(ad + bc)

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