A cognitive diagnosis model for latent classification of bounded continuous variables

Cognitive Diagnosis Models (CDMs) are widely used in latent-variable modeling for classification tasks that diagnose abilities or skills. Originally developed for dichotomous indicators, CDMs have been extended to polytomous and continuous responses, including bounded continuous variables (e.g. proportions or index scores on a 0-1 or 0-100 scale). We introduce a Bounded DINA (B-DINA) model, an extension of DINA for handling bounded continuous responses, using a Beta distribution with an appropriate mean-precision parameterization. We present a Bayesian estimation framework, define interpretable item parameters and compute posterior probabilities of membership in each latent-attribute profile. We explicitly address label-switching nonidentifiability and assess absolute model fit via posterior predictive <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>p</mml:mi></mml:mrow> <mml:annotation>$$ p $$</mml:annotation></mml:semantics> </mml:math> -values (PPP). Also, we have conducted a simulation study to evaluate parameter recovery for our proposed method and its performance. Further, we illustrate the model mainly with municipal data from Southeastern Brazil, where bounded indices summarize economy, education and health. Our proposed B-DINA effectively classifies municipalities and reveals relationships between observed indicators and latent attributes. As bounded continuous variables are common across the social sciences and policy analysis, our proposed B-DINA could offer a broadly applicable classification tool in the practice.