Rasch model

The Rasch model is a mathematical-psychological model of probabilistic test theory (also called item response theory ) developed by the Danish statistician Georg Rasch .

overview

Psychological tests (questionnaires, performance tests ) to measure psychological characteristics can be based on various measurement models. The latter differ in how the responses to the items of a test are used to infer the level of abilities or characteristics of a person (e.g. intelligence or extraversion ) - and how the tests must be constructed accordingly. There are two main types or classes of models, the classical test theory (KTT) and the probabilistic test theory , which aims to overcome certain disadvantages of the classical test theory. The Rasch model also belongs to the latter. The advantage of probabilistic models lies in the fact that two latent variables can be deduced from the observed response behavior, which determine the response behavior: the item difficulty and the ability of the person. One effect is that the ability can then be assessed independently of the item difficulty. A similar model was found e.g. B. also used within the PISA study .

Scientific background

Item function of the Rasch model

Compared to the Guttman model, the Rasch model does not postulate a deterministic relationship between the test behavior of a test person and their personal parameters . Rather, an underlying personality trait ("latent trait") is assumed, on the expression of which the manifest solution behavior depends in a probabilistic way.

The probability of person a _v's answer to task x _i is determined by:

Model equation:

${\ displaystyle p (X_ {vi} = {1}) = {\ frac {{\ exp} {(\ theta _ {v} - \ sigma _ {i})}} {1 + {\ exp} ({ \ theta _ {v} - \ sigma _ {i}})}}}$

Likelihood:

${\ displaystyle p (X_ {vi} = x_ {vi}) = {\ frac {{\ exp} {(x_ {vi} (\ theta _ {v} - \ sigma _ {i}))}} {1 + {\ exp} ({\ theta _ {v} - \ sigma _ {i}})}}}$

where X _{vi is} a random variable which takes the value 1 if the person a _v solves the task x _i , and which takes the value 0 if the person a _v does not solve the task. θ _v is the latent ability of the person a _v , σ _i is the difficulty of the task x _i , denotes the natural exponential function . Formally, there is a logit model that converts the proportions of 0 or 1 into a continuous distribution . ${\ displaystyle \ exp}$

Parameter estimation

Confidence intervals for the estimated personal parameters θ _v

The parameters are estimated in the Rasch model using a maximum likelihood approach . There are 3 methods of estimating the person and task parameters:

A common estimate of the person and task parameters can be made, but the consistency of the statistics suffers.
Another method is the conditional maximum likelihood estimation (also known as the conditional maximum likelihood method). First, the task parameters are given under the conditional likelihood of the data, the sufficient sum statistics are estimated for the personal parameters and then the unconditional maximum likelihood estimators of the personal parameters.
The third method is the marginal maximum likelihood estimation in which assumptions are made about the distribution of the individual parameters in the population.

Compared to the KTT, an individual confidence interval can be specified in the Rasch model for each estimated personal parameter θ _v . This becomes narrow if several items provide information for the respective person ability θ _v (maximum information iff θ _v = σ _i ). It becomes broad with few items that provide information for this area (this is usually the case with extreme values).

Use

In the Rasch model, the influence of the personal ability θ _{v is} separated from the influence of the test item σ _i . This establishes a measurement in accordance with measurement theory . It is possible to compare people (or tasks) who are independent of the tasks (or people). Rasch calls this property “specific objectivity”. Furthermore, the Rasch model forms the basis for adaptive testing , since the personal parameter can be recalculated after each task and thus items can be selected that provide maximum information. A basis for change measurements will also be established. The KTT, on the other hand, requires stable personality traits and is not designed for this from a psychometric point of view.

Model test

Model test and task selection

Within the Rasch model, a model test can be carried out by estimating the task parameters σ _i in partial samples. This is possible because the estimates are independent of the incoming personal parameters (see specific objectivity). For this you can use a sample z. B. split at the median. If the estimated values obtained are plotted against each other, they should lie on a straight line through the zero point with a slope of 1. The deviation from this straight line can be used as part of the test design as a criterion for task selection (see figure). The predictions can also be checked statistically using a likelihood ratio test (Andersen, 1973). With an optimal model fit, this quotient assumes a value of 1.

literature

GH Fischer, IWMolenaar: Rasch Models. Foundations, Recent Developments, and Applications. Springer, New York 1995, ISBN 0-387-94499-0 .
EB Andersen: A goodness of fit test for the Rasch model. In: Psychometrika. Volume 38, 1973, pp. 123-140. doi: 10.1007 / BF02291180
S. Embretson, S. Reise: Item response theory for psychologists. Erlbaum, Mahwah NJ 2000, ISBN 0-585-34782-4 .
F. Gernot: Probabilistic test models in personality diagnostics. Lang, Frankfurt am Main 1993, ISBN 3-631-46030-9 .
H. Irtel: Decision-making and test-theoretical foundations of psychological diagnostics. Lang, Frankfurt am Main 1996, ISBN 3-631-49374-6 .
W. Kempf: Dynamic models for measuring social relationships. In: W. Kempf (Ed.): Probabilistic models in social psychology [Probabilistic models in social psychology]. Huber, Bern 1974, pp. 13-55.
W. Kempf: A dynamic test model and its use in the micro-evaluation of instrumental material. In: H. Spada, W. Kempf (Ed.): Structural models for thinking and learning. Huber, Bern 1977, pp. 295-318.
W. Kempf: Dynamic models for the measurement of "traits" in social behavior. In: W. Kempf, BH Repp (Ed.): Mathematical models for social psychology. Wiley, New York 1977, pp. 14-58.
W. Kempf, R. Langeheine: Item Response Models in Social Science Research . regener, Berlin 2012, ISBN 978-3-936014-29-7 .
H. Müller: Sum score and selectivity in the Rasch model. In: Psychological Rundschau. Volume 51, 2000, pp. 34-35. doi : 10.1026 // 0033-3042.51.1.34
Matthias von Davier, Claus H. Carstensen (Ed.): Multivariate and Mixture Distribution Rasch Models. Extensions and Applications. Springer, Berlin 2006, ISBN 0-387-32916-1 .
G. Rasch: Probabilistic models for some intelligence and attainment tests . Danish Institute for Educational Research, Copenhagen 1960. (The University of Chicago Press, Chicago 1980, ISBN 0-226-70553-6 )
J. Rost: What happened to the Rasch model? In: Psychological Rundschau. Volume 50, 1999, pp. 140-156. doi : 10.1026 // 0033-3042.50.3.140
J. Rost: Test theory - test construction . Huber, Göttingen 2003, ISBN 3-456-83964-2 .

Individual evidence

↑ ^a ^b M. Amelang, L. Schmidt-Atzert: Psychological diagnosis and intervention. Springer, Heidelberg 2006, p. 75.

Web links

Wikibooks: Rasch analysis with the free statistics software R - learning and teaching materials

There are a number of software packages on the market. Some executable programs are freely available, but in some cases only as demo versions with reduced functionality. Even open source software is available.

[Amelang-1] M. Amelang, L. Schmidt-Atzert: Psychological diagnosis and intervention. Springer, Heidelberg 2006, p. 75.