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, attitudes, or personality traits, and the item difficulty.
[1][2] For example, they may be used to estimate a student's reading ability or the extremity of a person's attitude to capital punishment from responses on a questionnaire.
In most contexts, the parameters of the model characterize the proficiency of the respondents and the difficulty of the items as locations on a continuous latent variable.
When a person's location on the latent trait is equal to the difficulty of the item, there is by definition a 0.5 probability of a correct response in the Rasch model.
Beyond data, Rasch's equations model relationships we expect to obtain in the real world.
For instance, education is intended to prepare children for the entire range of challenges they will face in life, and not just those that appear in textbooks or on tests.
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.
However, the precise relationship between total scores and person location estimates depends on the distribution of items on the test.
In applying the Rasch model, item locations are often scaled first, based on methods such as those described below.
As a result, person and item locations are estimated on a single scale as shown in Figure 2.
A single ICC is shown and explained in more detail in relation to Figure 4 in this article (see also the item response function).
It is unnecessary for responses to conform strictly to the pattern in order for data to fit the Rasch model.
Standard errors of person estimates are smaller where the slope of the ICC is steeper, which is generally through the middle range of scores on a test.
As mentioned earlier, the level of measurement error is not uniform across the range of a test, but is generally larger for more extreme scores (low and high).
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.
[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.
Specifically, the defining property of Rasch models is their formal or mathematical embodiment of the principle of invariant comparison.
This result is achieved through the use of conditional maximum likelihood estimation, in which the response space is partitioned according to person total scores.
In somewhat more familiar terms, Rasch models provide a basis and justification for obtaining person locations on a continuum from total scores on assessments.
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 solid bodies to produce acceleration.
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
(that is, correctly answering the question) for persons with different locations on the latent continuum (that is, their level of abilities).
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.
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).
[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).
A limitation of this approach is that in practice, values of discrimination indexes must be preset as a starting point.
Application of the model can also provide information about how well items or questions on assessments work to measure the ability or trait.
[25] Prominent advocates of Rasch models include Benjamin Drake Wright, David Andrich and Erling Andersen.
