Maximum likelihood estimation for ordinal randomized response data via the EM algorithm: identifiability and efficiency

Abstract

Surveys on sensitive matters, including security behaviour, substance use, and financial misconduct, are prone to insincere answers when respondents fear disapproval or consequences. Randomized response techniques address this by letting respondents answer through a privacy-preserving device, so no individual reply is ever revealed. The approach is well developed for dichotomous and quantitative questions but has seldom been adapted to ordinal Likert-type items, among the most common measurement tools in behavioural research. This article provides that extension. We propose a scrambling mechanism for four-category ordinal responses driven by a single six-sided die and show that the design is identifiable: the population category probabilities are uniquely determined by the population distribution of scrambled responses, and this can be checked before any data are collected. Because the true category is latent, we derive an expectation-maximization (EM) algorithm with closed-form updates that converges to the unique maximiser of the observed-data likelihood, and we give the asymptotic theory in the three free coordinates the simplex actually has. The maximum-likelihood and method-of-moments estimators are asymptotically equivalent, but they differ appreciably in finite samples. Across five distributions and sample sizes up to 1,000, the maximum-likelihood estimator reduces mean squared error by up to 27% at (n=200) for polarized distributions and by construction avoids the boundary violations that force the moment estimator to be projected back onto the simplex in over half of small polarized samples. The result is a practical and reliable tool for measuring sensitive ordinal outcomes without compromising respondent trust.