The polytomous Rasch model is generalization of the dichotomous Rasch model. It is a measurement model that has potential application in any context in which the objective is to measure a trait or ability through a process in which responses to items are scored with successive integers. For example, the model is applicable to the use of Likert scales, rating scales, and to educational assessment items for which successively higher integer scores are intended to indicate increasing levels of competence or attainment.
Background and overview The polytomous Rasch model was derived by Andrich (1978), subsequent to derivations by Rasch (1961) and Andersen (1977), through the resolution of relevant terms of a general form of Rasch's model into threshold and discrimination parameters. When the model was derived, Andrich focused on the use of Likert scales in psychometrics, both for illustrative purposes and to aid in the interpretation of the model. The model is sometimes referred to as the Rating Scale Model when (i) items have the same number of thresholds and (ii) in turn, the difference between any given threshold location and the mean of the threshold locations is equal or uniform across items. This is, however, a potentially misleading name for the model because it is far more general in its application than to so-called rating scales. The model is also sometimes referred to as the Partial Credit Model, particularly when applied in educational contexts. The Partial Credit Model (Masters 1982) has an identical algebraic form but was derived from a different starting point at a later time, and is interpreted in a somewhat different manner. The Partial Credit Model also allows different thresholds for different items. Although this name for the model is often used, Andrich (2005) provides a detailed analysis of problems associated with elements of Masters' approach, which relate specifically to the type of response process that is compatible with the model, and to empirical situations in which estimates of threshold locations are disordered. These issues are discussed in the elaboration of the model that follows. The model is a general probabilistic measurement model which provides a theoretical foundation for the use of sequential integer scores, in a manner that preserves the distinctive property that defines Rasch models: specifically, total raw scores are sufficient statistics for the parameters of the models. See the main article for the Rasch model for elaboration of this property. In addition to preserving this property, the model permits a stringent empirical test of the hypothesis that response categories represent increasing levels of a latent attribute or trait, hence are ordered. The reason the model provides a basis for testing this hypothesis is that it is empirically possible that thresholds will fail to display their intended ordering. In this more general form of the Rasch model for dichotomous data, the score on a particular item is defined as the count of the number of threshold locations on the latent trait surpassed by the individual. This does not mean that a measurement process entails making such counts in a literal sense; rather, threshold locations on a latent continuum are usually inferred from a matrix of response data through an estimation process such as Conditional Maximum likelihood estimation. In general, the central feature of the measurement process is that individuals are classified into one of a set of contiguous, or adjoining, ordered categories. A response format employed in a given experimental context may achieve this in a number of ways. For example, respondents may choose a category they perceive best captures their level of endorsement of a statement (such as 'strongly agree'), judges may classify persons into categories based on well-defined criteria, or a person may categorise a physical stimulus based on perceived similarity to a set of reference stimuli. The polytomous Rasch model specialises to the model for dichotomous data when responses are classifiable into only two categories. In this special case, the item difficulty and (single) threshold are identical. The concept of a threshold is elaborated on in the following section.
The Polytomous Rasch Model First, let
X n i = x ∈ { 0 , 1 , … , m i } {\displaystyle X_{ni}=x\in \{0,1,\dots ,m_{i}\}\,}
be an integer random variable where m i {\displaystyle m_{i}} is the maximum score for item i. That is, the variable X n i {\displaystyle X_{ni}} is a random variable that can take on integer values between 0 and a maximum of m i {\displaystyle m_{i}} . In the polytomous Rasch model (Andrich 1978), the probability of the outcome X n i = x {\displaystyle X_{ni}=x} is
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