A study by Gupta (1960) introduces the point multi-serial correlation coefficient in addition to examining some of its properties. It is noted that the product-moment correlation coefficient can measure linear relation between qualitative characters when the number of categories of the qualitative characters is more than two, and the set scores that are assigned to these classes are well-known. That moment correlation coefficient is known as point multi-serial correlation coefficient (pms).
By definition, if Y is a discrete random variable with values y1 (i = 1, 2, …l) and probability Pi, and X are continuous random variables to the extent that when Y=y 1, then the conditional variance and mean of X are σ 2 and mi respectively. Therefore the pms can be determined by:
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P=
The main properties of p include:
M 1 = m 2 = … = m l is an adequate condition to take p to be equivalent to 0 even though it is not an essential condition.
When l = 2, p becomes invariant for linear transformation of y 1 and y 2 . Therefore, 1 and 0 can replace y 2 .
When l = 3 and y 1 + y 2 + y 3 = 0, where y 1 is greater than y 2 , which is also greater than y 3 , then p becomes unchanged if the three y values are replaced 1, 0, and -1, respectively.
Squaring the optimum value of the pms while y is equal to + β gives the multi-serial eta coefficient. Expending one degree of freedom with the Y category weighting of the multi-serial eta is known and corrected. However, the unclear and incomplete loss of the degrees of freedom resulting from categories ordering in the pms cannot be corrected. It is important to use the multi-seral eta even though one may lose all information within the ordinal or nominal scale values of Y.
References
Gupta, S. (1960). Point Biserial Correlation Coefficient and Its Generalization. Psychometrik , 25 (4).
