Rasch model

The class of models is named after Georg Rasch, a Danish mathematician and statistician who advanced the epistemological case for the models based on their congruence with a core requirement of measurement in physics; namely the requirement of invariant comparison.[1] This is the defining feature of the class of models, as is elaborated upon in the following section. The Rasch model for dichotomous data has a close conceptual relationship to the law of comparative judgment (LCJ), a model formulated and used extensively by L. L. Thurstone,[12][13] and therefore also to the Thurstone scale.[14]

Prior to introducing the measurement model he is best known for, Rasch had applied the Poisson distribution to reading data as a measurement model, hypothesizing that in the relevant empirical context, the number of errors made by a given individual was governed by the ratio of the text difficulty to the person's reading ability. Rasch referred to this model as the multiplicative Poisson model. Rasch's model for dichotomous data – i.e. where responses are classifiable into two categories – is his most widely known and used model, and is the main focus here. This model has the form of a simple logistic function.

The brief outline above highlights certain distinctive and interrelated features of Rasch's perspective on social measurement, which are as follows:

1. He was concerned principally with the measurement of individuals, rather than with distributions among populations.
2. He was concerned with establishing a basis for meeting a priori requirements for measurement deduced from physics and, consequently, did not invoke any assumptions about the distribution of levels of a trait in a population.
3. Rasch's approach explicitly recognizes that it is a scientific hypothesis that a given trait is both quantitative and measurable, as operationalized in a particular experimental context.

Thus, congruent with the perspective articulated by Thomas Kuhn in his 1961 paper The function of measurement in modern physical science, measurement was regarded both as being founded in theory, and as being instrumental to detecting quantitative anomalies incongruent with hypotheses related to a broader theoretical framework.[15] This perspective is in contrast to that generally prevailing in the social sciences, in which data such as test scores are directly treated as measurements without requiring a theoretical foundation for measurement. Although this contrast exists, Rasch's perspective is actually complementary to the use of statistical analysis or modelling that requires interval-level measurements, because the purpose of applying a Rasch model is to obtain such measurements. Applications of Rasch models are described in a wide variety of sources.[16]

Let ${\displaystyle X_{ni}=x\in \{0,1\}}$ be a dichotomous random variable where, for example, ${\displaystyle x=1}$ denotes a correct response and ${\displaystyle x=0}$ an incorrect response to a given assessment item. In the Rasch model for dichotomous data, the probability of the outcome ${\displaystyle X_{ni}=1}$ is given by:

${\displaystyle \Pr\{X_{ni}=1\}={\frac {e^{{\beta _{n}}-{\delta _{i}}}}{1+e^{{\beta _{n}}-{\delta _{i}}}}},}$

where ${\displaystyle \beta _{n}}$ is the ability of person ${\displaystyle n}$ and ${\displaystyle \delta _{i}}$ is the difficulty of item ${\displaystyle i}$. Thus, in the case of a dichotomous attainment item, ${\displaystyle \Pr\{X_{ni}=1\}}$ is the probability of success upon interaction between the relevant person and assessment item. It is readily shown that the log odds, or logit, of correct response by a person to an item, based on the model, is equal to ${\displaystyle \beta _{n}-\delta _{i}}$. Given two examinees with different ability parameters ${\displaystyle \beta _{1}}$ and ${\displaystyle \beta _{2}}$ and an arbitrary item with difficulty ${\displaystyle \delta _{i}}$, compute the difference in logits for these two examinees by ${\displaystyle (\beta _{1}-\delta _{i})-(\beta _{2}-\delta _{i})}$. This difference becomes ${\displaystyle \beta _{1}-\beta _{2}}$. Conversely, it can be shown that the log odds of a correct response by the same person to one item, conditional on a correct response to one of two items, is equal to the difference between the item locations. For example,

${\displaystyle \operatorname {log-odds} \{X_{n1}=1\mid \ r_{n}=1\}=\delta _{2}-\delta _{1},\,}$

where ${\displaystyle r_{n}}$ is the total score of person n over the two items, which implies a correct response to one or other of the items.[1][19][20] Hence, the conditional log odds does not involve the person parameter ${\displaystyle \beta _{n}}$, which can therefore be eliminated by conditioning on the total score ${\displaystyle r_{n}=1}$. That is, by partitioning the responses according to raw scores and calculating the log odds of a correct response, an estimate ${\displaystyle \delta _{2}-\delta _{1}}$ is obtained without involvement of ${\displaystyle \beta _{n}}$. More generally, a number of item parameters can be estimated iteratively through application of a process such as Conditional Maximum Likelihood estimation (see Rasch model estimation). While more involved, the same fundamental principle applies in such estimations.

Figure 4: ICC for the Rasch model showing the comparison between observed and expected proportions correct for five class intervals of persons

The ICC of the Rasch model for dichotomous data is shown in Figure 4. The grey line maps the probability of the discrete outcome ${\displaystyle X_{ni}=1}$ (that is, correctly answering the question) for persons with different locations on the latent continuum (that is, their level of abilities). The location of an item is, by definition, that location at which the probability that ${\displaystyle X_{ni}=1}$ is equal to 0.5. In figure 4, 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. There is 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.

There are multiple polytomous extensions to the Rasch model, which generalize the dichotomous model so that it 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. These polytomous extensions are, for example, applicable to the use of Likert scales, grading in educational assessment, and scoring of performances by judges.

A criticism of the Rasch model is that it is overly restrictive or prescriptive because an assumption of the model is that all items have equal discrimination, whereas in practice, items discriminations vary, and thus no data set will ever show perfect data-model fit. A frequent misunderstanding is that the Rasch model does not permit each item to have a different discrimination, but equal discrimination is an assumption of invariant measurement, so differing item discriminations are not forbidden, but rather indicate that measurement quality does not equal a theoretical ideal. Just as in physical measurement, real world datasets will never perfectly match theoretical models, so the relevant question is whether a particular data set provides sufficient quality of measurement for the purpose at hand, not whether it perfectly matches an unattainable standard of perfection.

A criticism specific to the use of the Rasch model with response data from multiple choice items is that there is no provision in the model for guessing because the left asymptote always approaches a zero probability in the Rasch model. This implies that a person of low ability will always get an item wrong. However, low-ability individuals completing a multiple-choice exam have a substantially higher probability of choosing the correct answer by chance alone (for a k-option item, the likelihood is around 1/k).

The three-parameter logistic model relaxes both these assumptions and the two-parameter logistic model allows varying slopes.[21] 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 practice, the non-zero lower asymptote found in multiple-choice datasets is less of a threat to measurement than commonly assumed and typically does not result in substantive errors in measurement when well-developed test items are used sensibly [22]

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 is impossible to use CML as an estimation method".[23]^: 217 That is, sufficiency of the weighted "score" in the 2PL cannot 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 some of the properties of the Rasch model are retained.[24][23] 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,[1]^: 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 (i.e. the assessment context given conditions for assessment).

Application of the model provides diagnostic information regarding how well the criterion is met. Application of the model can also provide information about how well items or questions on assessments work to measure the ability or trait. For instance, knowing the proportion of persons that engage in a given behavior, the Rasch model can be used to derive the relations between difficulty of behaviors, attitudes and behaviors.[25] Prominent advocates of Rasch models include Benjamin Drake Wright, David Andrich and Erling Andersen.

• Mokken scale
• Guttman scale

• Andrich, D. (1978a). A rating formulation for ordered response categories. Psychometrika, 43, 357–74.
• Andrich, D. (1988). Rasch models for measurement. Beverly Hills: Sage Publications.
• Baker, F. (2001). The Basics of Item Response Theory. ERIC Clearinghouse on Assessment and Evaluation, University of Maryland, College Park, MD. Available free with software included from IRT at Edres.org
• 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
• Linacre, J. M. (1999). "Understanding Rasch measurement: Estimation methods for Rasch measures". Journal of Outcome Measurement. 3 (4): 382–405. PMID 10572388.
• von Davier, M., & Carstensen, C. H. (2007). Multivariate and Mixture Distribution Rasch Models: Extensions and Applications. New York: Springer.
• von Davier, M. (2016). Rasch Model. In Wim J. van der Linden (ed.): Handbook of Item Response Theory (Boca Raton: CRC Press), Routledge Handbooks.
• Wright, B.D., & Stone, M.H. (1979). Best Test Design. Chicago, IL: MESA Press.
• Wu, M. & Adams, R. (2007). Applying the Rasch model to psycho-social measurement: A practical approach. Melbourne, Australia: Educational Measurement Solutions. Available free from Educational Measurement Solutions

• Institute for Objective Measurement Online Rasch Resources
• Pearson Psychometrics Laboratory, with information about Rasch models
• Journal of Applied Measurement
• Journal of Outcome Measurement (all issues available for free downloading)
• Berkeley Evaluation & Assessment Research Center (ConstructMap software)
• Directory of Rasch Software – freeware and paid
• IRT Modeling Lab at U. Illinois Urbana Champ.
• National Council on Measurement in Education (NCME)
• Rasch Measurement Transactions
• The Standards for Educational and Psychological Testing
• The Trouble with Rasch