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(untitled). Pearson.
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Domain-specific¶
- Correlation ratio
- The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
This sourceThe limit \eta=0 represents the special case of no dispersion among the means of the different categories, while \eta=1 refers to no dispersion within the respective categories. \eta is undefined when all data points of the complete population take the same value. ==Example== Suppose there is a distribution of test scores in three topics (categories): Algebra: 45, 70, 29, 15 and 21 (5 scores) Geometry: 40, 20, 30 and 42 (4 scores) *Statistics: 65, 95, 80, 70, 85 and 73 (6 scores). 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 squares of the differences between the subject averages and the overall average: : 5 (36-52)^2 + 4 (33-52)^2 +6 (78-52)^2 = 6780. This gives : \eta^2 = \frac{6780}{9640}=0.7033\ldots suggesting that most of the overall dispersion is a result of differences between topics, rather than within topics. Taking the square root gives : \eta = \sqrt{\frac{6780}{9640}}=0.8386\ldots. For \eta = 1 the overall sample dispersion is purely due to dispersion among the categories and not at all due to dispersion within the individual categories. For quick comprehension simply imagine all Algebra, Geometry, and Statistics scores being the same respectively, e.g. 5 times 36, 4 times 33, 6 times 78. The limit \eta = 0 refers to the case without dispersion among the categories contributing to the overall dispersion. The trivial requirement for this extreme is that all category means are the same. ==Pearson vs. 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 \eta^2 depends on the number of the arrays" Ronald Fisher (1926) "Statistical Methods for Research Workers", (excerpt).
- The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
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