Rasch model

The Rasch model is a mathematical- psychological model item response theory (or item response theory ), which was developed by Danish statistician Georg Rasch.

Overview

Psychological testing (questionnaires, performance tests ) to measure psychological characteristics may be based on different measurement models. The latter differ in how (eg, intelligence or extraversion ) is concluded from the responses to the items of a test for the expression of the skills and characteristics of a person - and how the tests must be then constructed. Two model types or classes are to be distinguished above all, the Classical Test Theory (CTT ) and the probabilistic test theory, which seeks to overcome certain disadvantages of the classical test theory. The latter also includes the Rasch model. The advantage of probabilistic models is that is inferred from the observed response behavior to two latent variables that determine the response: the item difficulty and the ability of the person. An effect is that the estimate of the capacity may then be independent of the item difficulty. A similar model was, for example, within the PISA study application.

Scientific background

The Rasch model postulates compared to the Guttman model no deterministic relationship between the behavior of a test subject and the person parameter. Instead, an underlying personality trait ( " latent trait " ) is assumed by its expression depends the solution manifest behavior in a probabilistic manner.

The probability of the response of person av task at xi is determined by:

Model equation:

Likelihood:

Where Xvi is a random variable which takes the value 1 if the person av solves the problem of xi, and which takes the value 0 if the person av does not solve the task. theta.v is the latent ability of the person av, ? i is the difficulty of the task xi, denotes the natural exponential function. Formal is here a logit model, which transforms the proportions of 0 or 1 in a continuous distribution.

Parameter estimation

The parameter estimation is done in the Rasch model with a maximum likelihood approach. There are three methods for estimating the people and task parameters:

• It can be a joint estimation of the people and carried out task parameters, but it suffers from the consistency of the statistics.
• Another method is the conditional maximum likelihood estimate ( also Conditional Maximum Likelihood Method). Here, first the task parameters in the conditional likelihood of the data given estimated to achieve adequate summary statistics for the person parameters and subsequently the unconditional maximum likelihood estimator of the person parameter.
• The third method is the marginal maximum likelihood estimation to be made in the assumptions about the distribution of those parameters in the population.

Compared to KTT can be specified in the Rasch model for each estimated person parameter theta.v an individual confidence interval. This will narrow if for each persons ability theta.v several items provide information ( Maximum Information iff. Theta.v =? I ). It is wide at the little items available for this area provide (usually this is in extreme forms of case).

Benefit

In the Rasch model, a separation of the influence of the persons ability theta.v done? I the influence of the test task. Thus, a measurement according to the theory of measurement is established. Comparisons of persons (or tasks ), the duties (or individuals ) are independent, are possible. This property is called Rasch as " specific objectivity ". Furthermore, the Rasch model provides the basis for adaptive testing because the person parameters can be recalculated after each task and thereby Items can be selected according to, provide the maximum information. Also established is a basis for change measurements. In contrast, the KTT presupposes stable personality traits and is not designed from psychometric view it.

Model test

Within the Rasch model can be carried out in subsamples a model test by estimating the task parameters? I. This is possible because the estimates are independent of the incoming parameters persons (see specific objective ). For this one can split a sample, for example, at the median. Plotting the estimated values ​​obtained against each other, so they should lie on a straight line through the origin with slope 1. The deviation from this straight line can be used in the context of test construction as a criterion for job selection (see figure). The predictions can also by a likelihood ratio test ( Andersen, 1973) are statistically tested. With optimal model fit this ratio takes a value of 1.