Rasch model Totally Explained

Everything about Rasch Model totally explained

Rasch models are used for analysing data from assessments to measure things such as abilities, attitudes, and personality traits. For example, they may be used to estimate a student's reading ability from answers to questions on a reading assessment, or the extremity of a person's attitude to capital punishment from responses on a questionnaire. Rasch models are particularly used in psychometrics, the field concerned with the theory and technique of psychological and educational measurement. In addition, they're increasingly being used in other areas, including the health profession and market research because of their general applicability.
The mathematical theory underlying Rasch models is in some respects the same as item response theory. However, Rasch models have a specific measurement property that provides a criterion for successful measurement. This formal property distinguishes Rasch models from other models used to model people's responses to items or questions. Application of the models provides diagnostic information regarding how well the criterion is met. Application of the models also provides information about how well items or questions on assessments work to measure the ability or trait.

Overview

The Rasch model for measurement

In the Rasch model, the probability of a specified response (for example right/wrong answer) is modelled as a function of person and item parameters. Specifically, in the simple Rasch model, the probability of a correct response is modelled as a logistic function of the difference between the person and item parameter. The mathematical form of the model is provided later in this article. In most contexts, the parameters of the model pertain to the level of a quantitative trait possessed by a person or item. For example, in educational tests, item parameters pertain to the difficulty of items while person parameters pertain to the ability or attainment level of people who are assessed. The higher a person's ability relative to the difficulty of an item, the higher the probability of a correct response on that item. When a person's location on the latent trait is equal to the difficulty of the item, there's by definition a 0.5 probability of a correct response in the Rasch model.
   The purpose of applying the model is to obtain measurements from categorical response data. Estimation methods are used to obtain estimates from matrices of response data based on the model.
   The Rasch model is a model in the sense that it represents the structure which data should exhibit in order to obtain measurements from the data; for example it provides a criterion for successful measurement. It is therefore a model in the sense of an ideal or standard. The perspective or paradigm underpinning the Rasch model is distinctly different from the perspective underpinning statistical modelling. Models are most often used with the intention of describing a set of data. Parameters are modified and accepted or rejected based on how well they fit the data. In contrast, when the Rasch model is employed, the objective is to obtain data which fit the model (Andrich, 2004). The rationale for this perspective is that the Rasch model embodies requirements which must be met in order to obtain measurement, in the sense that measurement is generally understood in the physical sciences.
   A useful analogy for understanding this rationale is to consider objects measured on a weighing scale. Suppose the weight of an object A is measured as being substantially greater than the weight of an object B on one occasion, then immediately afterward the weight of object B is measured as being substantially greater than the weight of object A. A property we require of measurements is that the resulting comparison between objects should be the same, or invariant, irrespective of other factors. This key requirement is embodied within the formal structure of the Rasch model. Consequently, the Rasch model isn't altered to suit data. Instead, the method of assessment should be changed so that this requirement is met, in the same way that a weighing scale should be rectified if it gives different comparisons between objects upon separate measurements of the objects.
   Data analysed using the model are usually responses to conventional items on tests, such as educational tests with right/wrong answers. However, the model is a general one, and can be applied wherever discrete data are obtained with the intention of measuring a quantitative attribute or trait.

Scaling

When all test-takers have an opportunity to attempt all items on a single test, each total score on the test maps to a unique estimate of ability and the greater the total, the greater the ability estimate. Total scores don't have a linear relationship with ability estimates. Rather, the relationship is non-linear as shown in Figure 1. The total score is shown on the vertical axis, while the corresponding person location estimate is shown on the horizontal axis. For the particular test on which the test characteristic curve (TCC) shown in Figure 1 is based, the relationship is approximately linear throughout the range of total scores from about 10 to 33. The shape of the TCC is generally somewhat ogival as in this example. However, the precise relationship between total scores and person location estimates depends on the distribution of items on the test. The TCC is steeper in ranges on the continuum in which there are a number of items, such as in the range on either side of 0 in Figures 1 and 2.
In applying the Rasch model, item locations are often scaled first, based on methods such as those described below. This part of the process of scaling is often referred to as item calibration. In educational tests, the smaller the proportion of correct responses, the higher the difficulty of an item and hence the higher the item's scale location. Once item locations are scaled, the person locations are measured on the scale. As a result, person and item locations are estimated on a single scale as shown in Figure 2.

Interpreting scale locations

For dichotomous data such as right/wrong answers, by definition, the location of an item on a scale corresponds with the person location at which there's a 0.5 probability of success. In general, the probability of a person responding correctly to a question with difficulty lower than that person's location is greater than 0.5, while the probability of responding correctly to a question with difficulty greater than the person's location is less than 0.5. When responses of a person are listed according to item difficulty, from lowest to highest, the most likely pattern is a Guttman pattern or vector; for example =1 is equal to 0.5. The black circles represent the actual or observed proportions of persons within Class Intervals for which the outcome was observed. For example, in the case of an assessment item used in the context of educational psychology, these could represent the proportions of persons who answered the item correctly. Persons are ordered by the estimates of their locations on the latent continuum and classified into Class Intervals on this basis in order to graphically inspect the accordance of observations with the model. In Figure 1, there's a close conformity of the data with the model. In addition to graphical inspection of data, a range of statistical tests of fit are used to evaluate whether departures of observations from the model can be attributed to random effects alone, as required, or whether there are systematic departures from the model.

The polytomous form of the Rasch model

The polytomous Rasch model, which is a generalisation of the dichotomous model, can be applied in contexts in which successive integer scores represent categories of increasing level or magnitude of a latent trait, such as increasing ability, motor function, endorsement of a statement, and so forth. The Polytomous response model is, for example, applicable to the use of Likert scales, grading in educational assessment, and scoring of performances by judges.

Other considerations

A criticism of the Rasch model is that it's overly restrictive or prescriptive because it doesn't permit each item to have a different discrimination. A criticism specific to the use of multiple choice items in educational assessment is that there's no provision in the model for guessing because the left asymptote always approaches a zero probability in the Rasch model. These variations are available in models such as the two and three parameter logistic models (Birnbaum, 1968). However, the specification of uniform discrimination and zero left asymptote are necessary properties of the model in order to sustain sufficiency of the simple, unweighted raw score.
   In the two-parameter logistic model (2PL-IRT; Lord & Novick, 1968) the weighted raw score is theoretically sufficient for person parameters, where the weights are given by model parameters referred to as discrimination parameters. Lord & Novick's one-parameter logistic model, 1PL, appears similar to the Rasch model in that it doesn't have discrimination parameters, but 1PL has different motivation and subtly different parameterization. The 1PL is a descriptive model which summarizes the sample as a normal distribution. The dichotomous Rasch model is a measurement model which parameterizes each member of the sample individually. There are other technical differences.
   Verhelst & Glas (1995) derive Conditional Maximum Likelihood (CML) equations for a model they refer to as the One Parameter Logistic Model (OPLM). In algebraic form it appears to be identical with the 2PL model, but OPLM contains preset discrimination indexes rather than 2PL's estimated discrimination parameters. As noted by these authors, though, the problem one faces in estimation with estimated discrimination parameters is that the discriminations are unknown, meaning that the weighted raw score "is not a mere statistic, and hence it's impossible to use CML as an estimation method" (Verhelst & Glas, 1995, p. 217). That is, sufficiency of the weighted "score" in the 2PL can't be used according to the way in which a sufficient statistic is defined. If the weights are imputed instead of being estimated, as in OPLM, conditional estimation is possible and the properties of the Rasch model are retained (Verhelst, Glas & Verstralen, 1995; Verhelst & Glas, 1995). In OPLM, the values of the discrimination index are restricted to between 1 and 15. A limitation of this approach is that in practice, values of discrimination indexes must be preset as a starting point. This means some type of estimation of discrimination is involved when the purpose is to avoid doing so.
   The Rasch model for dichotomous data inherently entails a single discrimination parameter which, as noted by Rasch (1960/1980, p. 121), constitutes an arbitrary choice of the unit in terms of which magnitudes of the latent trait are expressed or estimated. However, the Rasch model requires that the discrimination is uniform across interactions between persons and items within a specified frame of reference (for example the assessment context given conditions for assessment).

References and further reading

Alagumalai, S., Curtis, D.D. & Hungi, N. (2005). Applied Rasch Measurement: A book of exemplars. Springer-Kluwer.

Andersen, E.B. (1977). Sufficient statistics and latent trait models, Psychometrika, 42, 69-81.

Andrich, D. (1978a). A rating formulation for ordered response categories. Psychometrika, 43, 357-74.

Andrich, D. (1978b). Relationships between the Thurstone and Rasch approaches to item scaling. Applied Psychological Measurement, 2, 449-460.

Andrich, D. (1988). Rasch models for measurement. Beverly Hills: Sage Publications.

Andrich, D. (2004). Controversy and the Rasch model: a characteristic of incompatible paradigms? Medical Care, 42, 1-16.

Baker, F. The Basics of Item Response Theory. ERIC Clearinghouse on Assessment and Evaluation, University of Maryland, College Park, MD. (2001) available free with software included from IRT at Edres.org

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee’s ability. In Lord, F.M. & Novick, M.R. (Eds.), Statistical theories of mental test scores. Reading, MA: Addison-Wesley.

Bond, T.G. & Fox, C.M. (2007). Applying the Rasch Model: Fundamental measurement in the human sciences. 2nd Edn (includes Rasch software on CD-ROM). Lawrence Erlbaum.

Fischer, G.H. & Molenaar, I.W. (1995). Rasch models: foundations, recent developments and applications. New York: Springer-Verlag.

Goldstein H & Blinkhorn.S (1977). Monitoring Educational Standards: an inappropriate model. . Bull.Br.Psychol.Soc. 30 309-311

Goldstein H & Blinkhorn.S (1982). The Rasch Model Still Does Not Fit. . BERJ 82 167-170.

Hambleton RK, Jones RW. Comparison of classical test theory and item response Educational Measurement: Issues and Practice. 1993; 12(3):38-47. available in the ITEMS Series from the National Council on Measurement in Education

Harris D. Comparison of 1-, 2-, and 3-parameter IRT models. Educational Measurement: Issues and Practice;. 1989; 8: 35-41 available in the ITEMS Series from the National Council on Measurement in Education

Kuhn, T.S. (1961). The function of measurement in modern physical science. ISIS, 52, 161-193. JSTOR

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.

Rasch, G. (1961). On general laws and the meaning of measurement in psychology, pp. 321-334 in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, IV. Berkeley: University of Chicago Press, 1980.

Verhelst, N.D. and Glas, C.A.W. (1995). The one parameter logistic model. In G.H. Fischer and I.W. Molenaar (Eds.), Rasch Models: Foundations, recent developments, and applications (pp. 215-238). New York: Springer Verlag.

Verhelst, N.D., Glas, C.A.W. and Verstralen, H.H.F.M. (1995). One parameter logistic model (OPLM). Arnhem: CITO.

von Davier, M., & Carstensen, C. H. (2007). Multivariate and Mixture Distribution Rasch Models: Extensions and Applications. New York: Springer.

Wright, B.D., & Stone, M.H. (1979). Best Test Design. Chicago, IL: MESA Press.Further Information

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