The Relation of Multiserial Eta to other Measures of Correlation

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Abstract
Ordinary product-moment correlation and regression methods are frequently not immediately applicable to qualitative data, whereas multiserial r, point-multiserial r, and multiserial eta can be easily applied. The multiserial r is rejected for prediction since it tells us only what the correlation might be if certain assumptions were true and if we could measure what is not now measured. The point-multiserial r and multiserial eta are identical when the number of categories is two but differ when it is three or greater. The multiserial eta is identical with the product-moment r when categories are assigned scale values equal to their means on the continuous variable. With three or more categories, the point-multiserial r, which assumes linearity with equal step intervals, is always lower than the multiserial eta, which forces linearity by adoption of unequal step intervals based upon difference in criterion attainment. While the multiserial eta expends one degree of freedom with the weighting of each category, this is known and correctable, whereas the vague partial loss of degrees of freedom due to the ordering of categories in the point-multiserial is not correctable.

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