Rasch model

Rasch models are probabilistic measurement models which find their application primarily in psychological and attainment assessment, and are being increasingly used in other areas, including the health profession. Rasch models provide a foundation for the measurement of quantitative attributes and traits on a continuum, based on categorical data derived from interactions between persons and items. In principle, Rasch models can be applied in any experimental context in which persons interact with assessment questions or items in a manner that provides for comparisons between persons with respect to the magnitude of some attribute or trait.

Rasch measurement models


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. 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 (cf Andrich, 1978b), and therefore also to the Thurstone scale.

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. 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 the Rasch model is to obtain such measurements. Applications of the Rasch model are described in Sivakumar, Durtis & Hungi (2005).

Invariant comparison and sufficiency
The Rasch model for dichotomous data is often regarded as an item response theory (IRT) model with one item parameter. However, rather than being a particular IRT model, proponents of the model regard it as a model that possesses a property which distinguishes it from IRT models. Specifically, the defining property of Rasch models is their formal or mathematical embodiment of the principle of invariant comparison. Rasch summarised the principle of invariant comparison as follows:


 * The comparison between two stimuli should be independent of which particular individuals were instrumental for the comparison; and it should also be independent of which other stimuli within the considered class were or might also have been compared.


 * Symmetrically, a comparison between two individuals should be independent of which particular stimuli within the class considered were instrumental for the comparison; and it should also be independent of which other individuals were also compared, on the same or some other occasion (Rasch, 1961, p. 332).

Rasch models embody this principle due to the fact that their formal structure permits algebraic separation of the person and item parameters, in the sense that the person parameter can be eliminated during the process of statistical estimation of item parameters. This result is achieved through the use of conditional maximum likelihood estimation, in which the response space is partitioned according to person total scores. The consequence is that the raw score for an item or person is the sufficient statistic for the item or person parameter. That is to say, the person total score contains all information available within the specified context about the individual, and the item total score contains all information with respect to item, with regard to the relevant latent trait. The Rasch model requires a specific structure in the response data, namely a probabilistic Guttman structure.

In somewhat more familiar terms, Rasch models provide a basis and justification for obtaining person locations on a continuum from total scores on assessments. Although it is not uncommon to treat total scores directly as measurements, they are actually counts of discrete observations rather than measurements. Each observation represents the observable outcome of a comparison between a person and item. Such outcomes are directly analogous to the observation of the rotation of a balance scale in one direction or another. This observation would indicate that one or other object has a greater mass, but counts of such observations cannot be treated directly as measurements.

Rasch pointed out that the principle of invariant comparison is characteristic of measurement in physics using, by way of example, a two-way experimental frame of reference in which each instrument exerts a mechanical force upon a solid bodies to produce acceleration. Rasch (1960/1980, pp. 112-3) stated of this context: "Generally: If for any two objects we find a certain ratio of their accelerations produced by one instrument, then the same ratio will be found for any other of the instruments". It is readily shown that Newton's second law entails that such ratios are directly proportional to the ratios of the masses of the bodies.

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



\Pr \{X_{ni}=1\} =\frac{e^{{\beta_n} - {\delta_i}}}{1 + e^{{\beta_n} - {\delta_i}}}, $$

where $$ \beta_n $$ is the ability of person $$ n $$ and $$ \delta_i $$ is the difficulty of item $$ i $$. Thus, in the case of a dichotomous attainment item, $$ \Pr \{X_{ni}=1\} $$ 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 $$\beta_n - \delta_i$$. It can be shown that the log odds of a correct response by a 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,



\operatorname{log-odds} \{X_{n1}=1 \mid \ r_n=1\} = \delta_2-\delta_1,\, $$

where $$r_n$$ is the total score of person n over the two items, which implies a correct response to one or other of the items. Hence, the conditional log odds does not involve the person parameter $$\beta_n$$, which can therefore be eliminated by conditioning on the total score $$r_n=1$$. That is, by partitioning the responses according to raw scores and calculating the log odds of a correct response, an estimate $$\delta_2-\delta_1$$ is obtained without involvement of $$\beta_n$$. 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.



The form of the Rasch model for dichotomous data can be seen in Figure 1. The grey line maps person location on the latent continuum to the probability of the discrete outcome $$X_{ni}=1$$ for an item with a location of approximately 0.2 on the latent continuum. The location of an item is, by definition, that location at which the probability that $$X_{ni}=1$$ 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 educational assessment item, 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 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.

Related information and considerations
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.

A criticism of the Rasch model is that it is overly restrictive or prescriptive because it does not permit each item to have a different discrimination, as is the case in models such as the Two Parameter Logistic Model (Birnbaum, 1968). The specification of uniform discrimination is, however, a necessary property of the model in order to sustain sufficiency. Thus, the Rasch model for dichotomous data inherently entails a 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 (i.e. experimental context).