Partial Ordered Stereotype Model: Development of a New Model for Ordinal Data

<p><strong>Ordinal variables are categorical variables whose categories have a natural ordering (e.g., Likert scale). Modelling ordinal responses requires specific methods that properly respect the discrete and natural ordering without including arbitrary assumptions, such as equally spaced categories. There is a small set of models available for ordinal data, including the proportional odds and ordered stereotype models. This thesis introduces the Partial Ordered Stereotype Model (POSM), an extension of the Ordered Stereotype Model (OSM) for ordinal response variables. The OSM, besides respecting the natural ordering of the response categories, does not assume equally spaced response categories and explicitly models the spacing by incorporating score parameters. These parameters specify the potentially unequal distances between adjacent response categories and reflect the discriminant ability of the covariates, indicating how effectively they can distinguish between ordinal response categories. However, different covariates may exhibit distinct discriminant abilities with respect to the ordinal outcome. The POSM addresses this by allowing multiple sets of score parameters within the same model, thus capturing the characteristics of each covariate in a single framework. Choosing the number of covariate groups and determining how to group them based on their discriminant ability are additional challenges for the model selection. To address this, we develop stepwise-based algorithms that simultaneously select relevant covariates and identify appropriate groupings as well as the number of groups. We assess the performance of the POSM and the proposed model selection algorithms through a comprehensive simulation study encompassing an extensive set of scenarios. We also develop an R package called 'stereord', which includes the functions to fit the POSM and to perform model selection using the designed algorithms. Finally, we demonstrate the model’s flexibility and practical value using two real-world datasets: one from happiness research, showcasing the benefits of multiple score parameter sets, and another from aquaculture, highlighting novel applications of the model.</strong></p>