# Deriving joint probabilities from marginal probabilities & polychoric correlations

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!

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 $$p$$