Triserial correlation

Assume (X, Y) has a bivariate normal distribution with correlation coefficient p and suppose the only available information on Y is the location of individual points in relation to two points of trichotomy, t1 and t2, where t1 <; t2. Let Z be 2 the discrete 3-point random variable defined by the trichotomy of Y, where Z takes on the values 0, 1, and a. Assume Y ~ N (0, 1). The correlation coefficient between X and Z, denoted p xz, is called the triserial correlation coefficient. We describe the distribution of the random variable XZ, find the maximum value of, p xz and express pxz in terms of p. It is shown that |ρxy|≤|ρE|for all choices of a; that pxy = p if and only if p = 0; and that when t = -t2, ρ Xyis maximized when a = 2. Treating the biserial t2 correlation coefficient as a special case of the triserial coefficient when t == ∞, it follows that the biserial t2 coefficient is maximized if the point of dichotomy is 0.