Sampling of pairs in pairwise likelihood estimation for latent variable models with categorical observed variables

Pairwise likelihood is a limited information estimation method that has also been used for estimating the parameters of latent variable and structural equation models. Pairwise likelihood is a special case of composite likelihood methods that uses lower-order conditional or marginal log-likelihoods instead of the full log-likelihood. The composite likelihood to be maximized is a weighted sum of marginal or conditional log-likelihoods. Weighting has been proposed for increasing efficiency, but the choice of weights is not straightforward in most applications. Furthermore, the importance of leaving out higher-order scores to avoid duplicating lower-order marginal information has been pointed out. In this paper, we approach the problem of weighting from a sampling perspective. More specifically, we propose a sampling method for selecting pairs based on their contribution to the total variance from all pairs. The sampling approach does not aim to increase efficiency but to decrease the estimation time, especially in models with a large number of observed categorical variables. We demonstrate the performance of the proposed methodology using simulated examples and a real application.