# Perfect separation of two groups but rs is not 1

I have a dichotomous variable (group 1 and 2) and an ordinal scaled variable. The values of the ordinal scaled variable for group 1 are always lower than those for group 2: As I understand it, this corresponds to a perfect association between the dichotomous variable and the ordinal variable.

I want to express the association between these two variables with an effect size measure, preferably some sort of correlation coefficient (for reasons of consistency). For the correlation of a dichotomous and a interval scaled variable, you'd go with the Pearson correlation coefficient. Based on comments on StackExchange, I understand that a similar rationale holds true for the Spearman rank correlation if you have an ordinal scaled variable (see e.g. here).

Thus I calculated the Spearman rank correlation coefficient (SPSS 21) and got the result rs = .87. N per group is 7 and there are some tied ranks.

My question is: Why is the correlation coefficient not equal to 1, as there are no intersections between the values of group 1 and group 2?

• Measures of correlation corresponding to the notion of "perfect separation" here would be nearer to say the quadrant correlation or to the phi coefficient. – Glen_b Aug 3 '15 at 1:14

## 1 Answer

It is easy to see from your graph that also Pearson's linear correlation is not perfectly 1 resp. -1: Due to ties, the regression line would not go through all points. Similar picture if you convert the numeric values to ranks. So it is only a matter of ties in the binary variable.

A different explanation is the following: if you know the value of the numeric value, you know the value of the binary one but not vice versa. Thus, a symmetric measure of association should not point to perfect association.