Triserial correlation

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 ρ_xz , is called the triserial correlation coefficient. We describe the distribution of the random variable XZ, find the maximum value of ρ_xz , and express ρ_xz in terms of ρ. It is shown that for all choices of α that ρ_xz = ρ if and only if ρ = 0; and that when t₁ = -t₂ , ρ_xz 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.

Keywords:

• trichotomized information
• bivariate normal distribution
• correlation coefficient