Robust estimation and inference for categorical data

Empirical research in the social, health, and economic sciences is often based on categorical variables, such as questionnaire responses, self-reported health, or counting processes. Yet, just like in continuous variables, contamination might be present in such data—for instance, careless responses—which can cause severe biases in commonly employed maximum likelihood estimation.

However, robustifying estimation is challenging because categorical variables, by their nature, cannot take arbitrarily large values and may not even admit a numerical interpretation. Consequently, the extensive literature on outlier-robust M-estimation is not applicable.

As a remedy, this talk introduces a general framework for robust estimation and inference of models for categorical data, called C-estimation (“C” for categorical; Welz, 2025). In addition to offering enhanced robustness, C-estimators are shown to be asymptotically consistent, normally distributed, and fully efficient—avoiding the robustness-efficiency tradeoff inherent to M-estimation.

The presentation balances theoretical insights with an application to structural equation models (SEMs) using ordinal data, demonstrating how robustly estimated polychoric correlation matrices can improve SEM fit, enhance parameter accuracy, and detect low-quality responses. The approach is broadly compatible with existing SEM fitting methods and is implemented in the open-source R package “robcat” (https://cran.r-project.org/package=robcat).

References:

• Welz, M. (2025). Robust estimation and inference for categorical data [arXiv:2403.11954]. https://doi.org/10.48550/arXiv.2403.11954
• Welz, M., Mair, P., & Alfons, A. (2025). Robust estimation of polychoric correlation [arXiv:2407.18835]. https://doi.org/10.48550/arXiv.2407.18835