An article by Lev (1949) defined point bi-serial correlation as the product moment coefficient that is between variants x and y which are taking the values 0 and 1 only. By letting the observations of y to be y 1 , where i=1, …, n; y 1i be values of y which are paired with the value x= 1; y oi be values that are paired with x=0; y m, y m1 , and y m0 be the corresponding means of y, y 1 , and y 0 respectively; and n= n 1 + n 0. Therefore, the point bi-serial coefficient of correlation can be expressed by:
r =
The r distribution will readily be obtained when the y i , are distributed as:
Where, z i =
Is the y i variance about the common mean denoted by α, while p is the parameter hat represents the correlation between y i and the x i . One can easily verify that the above statistics is a maximum possible estimate of p.
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Lev then concluded that t distribution can be written as:
t =
Therefore, the distribution t is having a non-central distribution with:
δ =
=
.
These methods are useful in the calculation of test of significance and the confidence limits for p. When p = 0, t will have the distribution of students and the statistics
may be applied in testing the hypothesis, p = 0, by the means of the t tables within the degree of freedom of n – 2. Then, the non-central t distribution will determine the power function of the test.
References
Lev, J. (1949). The point biserial coefficient of correlation. Annals of Mathematical Statistics , 20 (1), 125-126.
