In statistics, the correlation ratio is a measure of the relationship between the statistical dispersion within individual categories and the dispersion across the whole population or sample. The measure is defined as the ratio of two standard deviations representing these types of variation. The context here is the same as that of the intraclass correlation coefficient, whose value is the square of the correlation ratio.
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Definition
Suppose each observation is yxi where x indicates the category that observation is in and i is the label of the particular observation. Let nx be the number of observations in category x and
where
which can be written as
i.e. the weighted variance of the category means divided by the variance of all samples.
It is worth noting that if the relationship between values of
Range
The correlation ratio
Example
Suppose there is a distribution of test scores in three topics (categories):
Then the subject averages are 36, 33 and 78, with an overall average of 52.
The sums of squares of the differences from the subject averages are 1952 for Algebra, 308 for Geometry and 600 for Statistics, adding to 2860. The overall sum of squares of the differences from the overall average is 9640. The difference of 6780 between these is also the weighted sum of the square of the differences between the subject averages and the overall average:
This gives
suggesting that most of the overall dispersion is a result of differences between topics, rather than within topics. Taking the square root
Observe that for
The limit
Pearson v. Fisher
The correlation ratio was introduced by Karl Pearson as part of analysis of variance. Ronald Fisher commented:
As a descriptive statistic the utility of the correlation ratio is extremely limited. It will be noticed that the number of degrees of freedom in the numerator of
to which Egon Pearson (Karl's son) responded by saying
Again, a long-established method such as the use of the correlation ratio [§45 The "Correlation Ratio" η] is passed over in a few words without adequate description, which is perhaps hardly fair to the student who is given no opportunity of judging its scope for himself.