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Over the last decade, researchers have come to recognize the benefits of ideal point item response theory (IRT) models for noncognitive measurement. Although most applied studies have utilized the Generalized Graded Unfolding Model (GGUM), many others have been developed. Most notably, David Andrich and colleagues published a series of papers comparing dominance and ideal point measurement perspectives, and they proposed ideal point models for dichotomous and polytomous single-stimulus responses, known as the Hyperbolic Cosine Model (HCM) and the General Hyperbolic Cosine Model (GHCM), respectively. These models have item response functions resembling the GGUM and its more constrained forms, but they are mathematically simpler. Despite the apparent impact of Andrich’s work on ensuing investigations, the HCM and GHCM have been largely overlooked by applied researchers. This may stem from questions about the compatibility of the parameter metric with other ideal point estimation and model-data fit software or seemingly unrealistic parameter estimates sometimes produced by the original joint maximum likelihood (JML) estimation software. Given the growing list of ideal point applications and variations in sample and scale characteristics, the authors believe these HCMs warrant renewed consideration. To address this need and overcome potential JML estimation difficulties, this study developed a marginal maximum likelihood (MML) estimation algorithm for the GHCM and explored parameter estimation requirements in a Monte Carlo study manipulating sample size, scale length, and data types. The authors found a sample size of 400 was adequate for parameter estimation and, in accordance with GGUM studies, estimation was superior in polytomous conditions.

Item Parameter Estimation With the General Hyperbolic Cosine Ideal Point IRT Model