The latent class model is a highly effective tool in the analysis of categorical data from social, psychological, and behavioral sciences, where populations often share hidden common characteristics. In this article, we introduce two new algorithms for estimating the parameters of a latent class model for ordered categorical data with polytomous responses. These algorithms are based on a newly defined regularized Laplacian matrix derived from the response matrix. We provide theoretical convergence rates for our algorithms by considering a sparsity parameter and demonstrate that under a mild condition on data's sparsity, our algorithms yield consistent latent class analysis. Furthermore, we introduce a metric to assess the strength of latent class analysis and develop procedures based on this metric to determine the optimal number of latent classes for real-world ordered categorical data. Extensive simulation experiments demonstrate the efficiency and accuracy of our algorithms, and we demonstrate their practical application to real-world ordered categorical data with promising results.

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