Figure 2

Dichotomization and tetrachoric correlation estimation. Consider a sample ((x₁,y₁), (x₂,y₂),…,(x _T ,y _T )) of size T where (x,y) is distributed according to bivariate normality. Further, let x=x₁,x₂,…,x _T and y=y₁,y₂,…,y _T denote the samples from x and y, respectively. Using $\tilde{\mathit{x}}$ and $\tilde{\mathit{y}}$, as thresholds, x and y can be dichotomized resulting in the binary samples x _d and y _d . A: As an example, the density of a bivariate normal distribution (ρ=0.7) is shown (top, 3D curve) along with a sample (bottom, points in the x-y-plane) drawn from that distribution. By virtue of the two lines $x=\tilde{\mathit{x}}$ and $y=\tilde{\mathit{y}}$, the x-y-plane is divided into four quadrants, such that the counts of sample points per quadrant form a 2×2 contingency table. The (relative) frequencies in the contingency table can also be expressed in terms of x _d and y _d (e.g., n₁₁ is the number of indices where both x _d and y _d are equal to 1, and p₁₁=n₁₁T⁻¹). The probability masses corresponding to the table’s relative frequencies are equal to the respective partial volumes belonging to the four quadrants in the x-y-plane under the bivariate normal’s curve. The tetrachoric correlation coefficient r _t , for which these partial volumes resemble the relative frequencies in a given table, is an estimate of the population correlation parameter ρ belonging to the underlying distribution. B: Relationship between p₁₁ and r _t . Given x _d and y _d , r _t can be found using r _t =− cos(2π p₁₁). For details see text.