Polytomous 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.

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 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.

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.

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.

In this special case, the item difficulty and (single) threshold are identical.

is the kth threshold location of the rating scale which is in common to all the items.

Applied in a given empirical context, the model can be considered a mathematical hypothesis that the probability of a given outcome is a probabilistic function of these person and item parameters.

The threshold corresponds with the location on a latent continuum at which it is equally likely a person will be classified into adjacent categories, and therefore to obtain one of two successive scores.

In the context of assessment in educational psychology, successively higher integer scores may be awarded according to explicit criteria or descriptions which characterise increasing levels of attainment in a specific domain, such as reading comprehension.

The common and central feature is that some process must result in classification of each individual into one of a set of ordered categories that collectively comprise an assessment item.

In elaborating on features of the model, Andrich (2005) clarifies that its structure entails a simultaneous classification process, which results in a single manifest response, and involves a series of dichotomous latent responses.

Andrich (1978, 2005) shows that the polytomous Rasch model requires that these dichotomous responses conform with a latent Guttman response subspace: in which x ones are followed by m-x zeros.

, as defined earlier, is intrinsic to the structure of the polytomous Rasch model.

In the polytomous Rasch model, a score of x on a given item implies that an individual has simultaneously surpassed x thresholds below a certain region on the continuum, and failed to surpass the remaining m − x thresholds above that region.

Disordered threshold estimates indicate a failure to construct an assessment context in which classifications represented by successive scores reflect increasing levels of the latent trait.

If the locations are taken literally, classification of a person into category 1 implies that the person's location simultaneously surpasses the second threshold but fails to surpass the first threshold.

When threshold estimates are disordered, the estimates cannot therefore be taken literally; rather the disordering, in itself, inherently indicates that the classifications do not satisfy criteria that must logically be satisfied in order to justify the use of successive integer scores as a basis for measurement.

To emphasise this point, Andrich (2005) uses an example in which grades of fail, pass, credit, and distinction are awarded.

These grades, or classifications, are usually intended to represent increasing levels of attainment.

Consider also another person B, whose location is at the threshold between the regions at which a credit and distinction are most likely to be awarded.

That is, the disordering would indicate that the hypothesis implicit in the grading system - that grades represent ordered classifications of increasing performance - is not substantiated by the structure of the empirical data.