Threshold models with fixed and random effects for ordered categorical data

Sensometric assessments often give rise to ordered categorical data. These can be analysed by a threshold model, in which frequencies in the observed ordered categories are modelled by an underlying latent continuous random variable. It is customary to assume a normal or a logistic distribution. Rival methods include analysis of variance (ANOVA) and nonparametric methods based on ranks. In the present paper we will compare ANOVA techniques to the threshold model. ANOVA has the virtue of simplicity and robustness to departures from normality and homoscedasticity. For this reason, some authors favour ANOVA over other methods. However, robustness properties are good only under the global null hypothesis of exchangeability of treatments, while mean comparisons are more strongly affected when the global null does not hold. The threshold model can cope satisfactorily with this problem. In complex sampling designs (incomplete block designs with several error strata, groups of possibly unbalanced experiments), one often performs mixed linear model analyses, when the normality assumption is tenable. By analogy, the threshold model may be extended by random terms. Also, heterogeneity of variance can be allowed for. Applicability of the threshold model (with or without random terms) critically rests on the appropriateness of the distributional assumption on the latent scale. The normal and logistic distributions are most commonly used. Based on recent research on mixed models with random effects having a non-normal distribution, we employ a flexible extension of the threshold model with a nonnormal underlying latent variable to account for nonnormality. All methods are exemplified using data from different sensometric studies. While the threshold model provides great flexibility for modelling ordinal data, theory as well as simulation results show that it should be used with some caution in small samples. (C) 2003 Elsevier Science Ltd. All rights reserved.