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 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

${\displaystyle X_{ni}=x\in \{0,1,\dots ,m_{i}\}\,}$

be an integer random variable where ${\displaystyle m_{i}}$ is the maximum score for item i. That is, the variable ${\displaystyle X_{ni}}$ is a random variable that can take on integer values between 0 and a maximum of ${\displaystyle m_{i}}$.

In the polytomous Rasch model (Andrich, 1978), the probability of the outcome ${\displaystyle X_{ni}=x}$ is

${\displaystyle \Pr\{X_{ni}=x,x>0\}={\frac {\exp {{\sum _{k=1}^{x}(\beta _{n}}-{\tau _{ki}})}}{1+\sum _{j=1}^{m_{i}}\exp {{\sum _{k=1}^{j}(\beta _{n}}-{\tau _{ki}})}}};}$

${\displaystyle \Pr\{X_{ni}=0\}={\frac {1}{1+\sum _{j=1}^{m_{i}}\exp {{\sum _{k=1}^{j}(\beta _{n}}-{\tau _{ki}})}}}}$

where ${\displaystyle \tau _{ki}}$ is the kth threshold location of item i on a latent continuum, ${\displaystyle \beta _{n}}$ is the location of person n on the same continuum, and ${\displaystyle m_{i}}$ is the maximum score for item i. These equations are the same as

${\displaystyle \Pr\{X_{ni}=x\}={\frac {\exp {{\sum _{k=0}^{x}(\beta _{n}}-{\tau _{ki}})}}{\sum _{j=0}^{m_{i}}\exp {{\sum _{k=0}^{j}(\beta _{n}}-{\tau _{ki}})}}}}$

where the value of ${\displaystyle \tau _{0i}}$ is chosen for computational convenience that is: ${\displaystyle \sum _{k=0}^{m_{i}}(\beta _{n}-\tau _{ki})\equiv 0}$.

The Rating Scale Model

Similarly, the Rasch "Rating Scale" model (Andrich, 1978) is

${\displaystyle \Pr\{X_{ni}=x\}={\frac {\exp {{\sum _{k=0}^{x}(\beta _{n}}-({\delta _{i}-\tau _{k}}))}}{\sum _{j=0}^{m}\exp {{\sum _{k=0}^{j}(\beta _{n}}-{(\delta _{i}-\tau _{k}}))}}}}$

where ${\displaystyle \delta _{i}}$ is the difficulty of item i and ${\displaystyle \tau _{k}}$ is the kth threshold location of the rating scale which is in common to all the items. m is the maximum score and is identical for all the items. ${\displaystyle \tau _{0}}$ is chosen for computational convenience.

Application

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 graph showing the relation between the probability of a given category as a function of person location is referred to as a Category Probability Curve (CPC). An example of the CPCs for an item with five categories, scored from 0 to 4, is shown in Figure 1.

Figure 1: Rasch category probability curves for an item with five ordered categories

A given threshold ${\displaystyle \tau _{ki}}$ partitions the continuum into regions above and below its location. 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. The first threshold of item i, ${\displaystyle \tau _{1i}}$, is the location on the continuum at which a person is equally likely to obtain a score of 0 or 1, the second threshold is the location at which a person is equally likely to obtain a score of 1 and 2, and so on. In the example shown in Figure 1, the threshold locations are −1.5, −0.5, 0.5, and 1.5 respectively.

Respondents may obtain scores in many different ways. For example, where Likert response formats are employed, Strongly Disagree may be assigned 0, Disagree a 1, Agree a 2, and Strongly Agree a 3. 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.

References

• Andersen, E.B. (1977). Sufficient statistics and latent trait models, Psychometrika, 42, 69–81.
• Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43, 561–73.
• Andrich, D. (2005). The Rasch model explained. In Sivakumar Alagumalai, David D Durtis, and Njora Hungi (Eds.) Applied Rasch Measurement: A book of exemplars. Springer-Kluwer. Chapter 3, 308–328.
• Masters, G.N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174.
• Rasch, G. (1960/1980). Probabilistic models for some intelligence and attainment tests. (Copenhagen, Danish Institute for Educational Research), expanded edition (1980) with foreword and afterword by B.D. Wright. Chicago: The University of Chicago Press.
• von Davier, M. & Rost, J. (1995). Polytomous Mixed Rasch Models. In G. H. Fischer & I. W. Molenaar (Eds.): Rasch Models - Foundations, Recent Developments and Applications. (pp. 371-379). New York: Springer. https://link.springer.com/chapter/10.1007/978-1-4612-4230-7_20
• von Davier M. (2014) Rasch Polytomous Models. In: Michalos A.C. (eds) Encyclopedia of Quality of Life and Well-Being Research. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0753-5_2412
• Wright, B.D. & Masters, G.N. (1982). Rating Scale Analysis. Chicago: MESA Press. (Available from the Institute for Objective Measurement.)

External links

• Disordered thresholds and item information
• Category Disordering and Threshold Disordering
• Andrich on disordered thresholds and 'steps'
• Directory of Rasch Software - freeware and paid
• Institute for Objective Measurement
• Rasch analysis
• Rasch Model in Stata