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The polytomous Rasch model is generalization of the
dichotomous A dichotomy is a partition of a whole (or a set) into two parts (subsets). In other words, this couple of parts must be * jointly exhaustive: everything must belong to one part or the other, and * mutually exclusive: nothing can belong simultan ...
Rasch model The Rasch model, named after Georg Rasch, is a psychometric model for analyzing categorical data, such as answers to questions on a reading assessment or questionnaire responses, as a function of the trade-off between the respondent's abilities, at ...
. It is a
measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
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
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s. For example, the model is applicable to the use of
Likert scale A Likert scale ( , commonly mispronounced as ) is a psychometric scale commonly involved in research that employs questionnaires. It is the most widely used approach to scaling responses in survey research, such that the term (or more fully the ...
s,
rating scale :''Concerning rating scales as systems of educational marks, see articles about education in different countries (named "Education in ..."), for example, Education in Ukraine.'' :''Concerning rating scales used in the practice of medicine, see arti ...
s, 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 Psychometrics is a field of study within psychology concerned with the theory and technique of measurement. Psychometrics generally refers to specialized fields within psychology and education devoted to testing, measurement, assessment, and ...
, 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 Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
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 In statistics, a statistic is ''sufficient'' with respect to a statistical model and its associated unknown parameter if "no other statistic that can be calculated from the same sample provides any additional information as to the value of the p ...
for the
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s of the models. See the main article for the
Rasch model The Rasch model, named after Georg Rasch, is a psychometric model for analyzing categorical data, such as answers to questions on a reading assessment or questionnaire responses, as a function of the trade-off between the respondent's abilities, at ...
for elaboration of this property. In addition to preserving this property, the model permits a stringent
empirical Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...
test of the
hypothesis A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can test it. Scientists generally base scientific hypotheses on previous obse ...
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 The Rasch model, named after Georg Rasch, is a psychometric model for analyzing categorical data, such as answers to questions on a reading assessment or questionnaire responses, as a function of the trade-off between the respondent's abilities, at ...
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 Continuum may refer to: * Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes Mathematics * Continuum (set theory), the real line or the corresponding cardinal number ...
are usually ''inferred'' from a matrix of response data through an estimation process such as Conditional
Maximum likelihood In statistics, maximum likelihood estimation (MLE) is a method of estimation theory, estimating the Statistical parameter, parameters of an assumed probability distribution, given some observed data. This is achieved by Mathematical optimization, ...
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_ = x \in \ \, be an integer
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
where m_i is the maximum score for item ''i''. That is, the variable X_ is a random variable that can take on integer values between 0 and a maximum of m_i . In the polytomous Rasch model (Andrich, 1978), the probability of the outcome X_ = x is : \Pr \ =\frac ; : \Pr \ =\frac where \tau_ is the ''k''th threshold location of item ''i'' on a latent continuum, \beta_n is the location of person ''n'' on the same continuum, and m_i is the maximum score for item ''i''. These equations are the same as : \Pr \ =\frac where the value of \tau_ is chosen for computational convenience that is: \sum_^ (\beta_n - \tau_) \equiv 0.


The Rating Scale Model

Similarly, the Rasch "Rating Scale" model (Andrich, 1978) is : \Pr \ =\frac where \delta_ is the difficulty of item ''i'' and \tau_ is the ''k''th 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. \tau_ 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. A given threshold \tau_ 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'', \tau_, 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 Educational psychology is the branch of psychology concerned with the scientific study of human learning. The study of learning processes, from both cognitive and behavioral perspectives, allows researchers to understand individual differences i ...
, 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.


Elaboration of the model

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. In addition, the latent dichotomous responses operate within a Guttman structure and associated response space, as is characterised to follow. Let : Y_=y\in \, k \in\ \, be a set of independent dichotomous random variables. Andrich (1978, 2005) shows that the polytomous Rasch model requires that these dichotomous responses conform with a latent Guttman response subspace: : \Omega' \ \stackrel\ \ in which ''x'' ones are followed by ''m-x'' zeros. For example, in the case of two thresholds, the permissible patterns in this response subspace are: ::0,0 \Leftrightarrow 0 ::1,0 \Leftrightarrow 1 ::1,1 \Leftrightarrow 2 where the integer score ''x'' implied by each pattern (and vice versa) is as shown. The reason this subspace is implied by the model is as follows. Let : P_ =\frac,\ k=x,\, be the probability that Y_=1 and let Q_=1-P_. This function has the structure of the
Rasch model The Rasch model, named after Georg Rasch, is a psychometric model for analyzing categorical data, such as answers to questions on a reading assessment or questionnaire responses, as a function of the trade-off between the respondent's abilities, at ...
for dichotomous data. Next, consider the following conditional probability in the case two thresholds: : \frac . It can be shown that this conditional probability is equal to : \frac which, in turn, is the probability Pr\ given by the polytomous Rasch model. From the denominator of these equations, it can be seen that the probability in this example is conditional on response patterns of \, \, or \. It is therefore evident that in general, the response subspace \Omega', as defined earlier, is ''intrinsic'' to the structure of the polytomous Rasch model. This restriction on the subspace is necessary to the justification for integer scoring of responses: i.e. such that the score is simply the count of ordered thresholds surpassed. Andrich (1978) showed that equal discrimination at each of the thresholds is also necessary to this justification. 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. In order for this to be possible, the thresholds must be in their natural order, as shown in the example of Figure 1. 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. For example, consider a situation in which there are two thresholds, and in which the estimate of the second threshold is lower on the continuum than the estimate of the first threshold. 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. In turn, this implies a response pattern {0,1}, a pattern which does not belong to the subspace of patterns that is intrinsic to the structure of the model, as described above. 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 a person A, whose location on the latent continuum is at the threshold between regions on the continuum at which a pass and credit are most likely to be awarded. 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. In the example considered by Andrich (2005, p. 25), disordered thresholds would, if taken literally, imply that the location of person A (at the pass/credit threshold) is higher than that of person B (at the credit/distinction threshold). That is, taken literally, the disordered threshold locations would imply that a person would need to demonstrate a higher level of attainment to be at the pass/credit threshold than would be needed to be at the credit/distinction threshold. Clearly, this disagrees with the intent of such a grading system. The disordering of the thresholds would, therefore, indicate that the manner in which grades are being awarded is not in agreement with the intention of the grading system. 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.


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.)


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