Latent class analysis for multi-layer categorical data

TL;DR

This paper introduces a novel multi-layer latent class model for complex categorical data, providing spectral methods for estimation, demonstrating theoretical consistency, and showing that multiple data layers improve analysis accuracy.

Contribution

The paper develops a new multi-layer latent class model and three spectral estimation methods, with theoretical guarantees and insights on the benefits of multiple data layers.

Findings

Increasing layers improves method performance

Debiased Gram matrix method performs best

Methods effectively learn latent classes and estimate their number

Abstract

Traditional categorical data, often collected in psychological tests and educational assessments, are typically single-layer and gathered only once.This paper considers a more general case, multi-layer categorical data with polytomous responses. To model such data, we present a novel statistical model, the multi-layer latent class model (multi-layer LCM). This model assumes that all layers share common subjects and items. To discover subjects' latent classes and other model parameters under this model, we develop three efficient spectral methods based on the sum of response matrices, the sum of Gram matrices, and the debiased sum of Gram matrices, respectively. Within the framework of multi-layer LCM, we demonstrate the estimation consistency of these methods under mild conditions regarding data sparsity. Our theoretical findings reveal two key insights: (1) increasing the number of layers can enhance the performance of the proposed methods, highlighting the advantages of considering multiple layers in latent class analysis; (2) we theoretically show that the algorithm based on the debiased sum of Gram matrices usually performs best. Additionally, we propose an approach that combines the averaged modularity metric with our methods to determine the number of latent classes. Extensive experiments are conducted to support our theoretical results and show the powerfulness of our methods in the task of learning latent classes and estimating the number of latent classes in multi-layer categorical data with polytomous responses.