Deriving joint probabilities from marginal probabilities & polychoric correlations

Question

Given three ordered-categorical variables: $u_1, u_2, u_3$ with $K$ categories, I'm trying to derive their expected variance-covariance matrix using their marginal probabilities, thresholds, and polychoric correlation matrix.

I can derive their variances using the marginal probabilities:

Where:

$\bar{u_1} = p_1*1 + \cdots + p_k*K$

$V(u_1) = p_1*(1-\bar{u_1}) + \cdots + p_k*(K-\bar{u_1})$

But I'm stuck on how to derive $Cov(u_1, u_2)$. I've been told I'll need the joint probabilities of $u_1$ and $u_2$, but I'm not sure how to get those from the information that I have:

• Marginal probabilities
• Thresholds
• Polychoric correlation matrix

Any help greatly appreciated. Thanks!

Answer 1

Asked for help too soon sorry, I found the solution in Olsson (1979, p.447):

The joint probability $p_{ij}$ can be derived via:

$p_{ij} = \Phi_2(a_i,b_j) - \Phi_2(a_{i-1},b_j) - \Phi_2(a_i,b_{j-1}) + \Phi_2(a_{i-1},b_{j-1})$

Where $a_i,b_j$ are the thresholds and $\Phi_2$ is the bivariate normal PDF with (polychoric) correlation $p$

Olsson, U. Maximum likelihood estimation of the polychoric correlation coefficient. Psychometrika 44, 443–460 (1979). https://doi.org/10.1007/BF02296207