Abstract
Assume (X, Y) has a bivariate normal distribution with correlation coefficient ρ 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 the discrete 3-point random variable defined by the trichotomy of Y, where Z takes on the values 0, 1, αand a. Assume Y ~ N (0, 1). The correlation coefficient between X and Z, denoted ρxz , is called the triserial correlation coefficient. We describe the distribution of the random variable XZ, find the maximum value of ρxz , and express ρxz in terms of ρ. It is shown that for all choices of α that ρxz = ρ if and only if ρ = 0; and that when t1 = -t2 , ρxz is maximized when α = 2. Treating the biserial correlation coefficient as a special case of the triseria'l coefficient when t2 ==> ∞, it follows that the biserial coefficient is maximized if the point of dichotomy is 0.
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