Rasch Measurement Model

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Rasch model can be applied to measure latent traits (e.g., ability or attitude) in various disciplines such as health, education, psychology, social sciences, and economics. Latent traits are usually assessed trough the responses of a sample of subjects to a set of items. Items have 2 or more ordered response categories so that response are scored 0/1 for two response categories (dichotomous response format) or 0/1/2/... when the item is graded on more than two response categories (polytomous response format).

The Rasch model belongs to the item response theory (IRT) models. In essence, IRT methods model the probability of an individual's response to an item. The relationship between the probability of success to an item and the latent trait (e.g., the ability) is described by a function called item characteristic curve (ICC) that takes an S-shape (Figure 1).

Figure 1
Figure 1. Item characteristics curve showing the relationship between the location on the latent trait and the probability of answering the item correctly.

The psychometric Rasch model conceptualizes the measurement scale like a ruler (Figure 2). Graduations of the measurement scale are called the items. Items are located along the measurement scale according to their difficulty. Less difficult items are located on the left of the scale, most difficult items on the right. Subjects can also be located on the same measurement scale. They are located according to their ability. Subjects with a low ability are located on the left of the scale while subjects with a higher ability are located on the right. The less difficult items can be successfully achieved by the more able subjects. For instance, subject A presents a very low ability since he/she is expected to successfully achieve only the three easiest items; subject B has an intermediate ability that should enable him/her to succeed on approximately half of the items; and subject C has a highest ability and is supposed to succeed on all but the most difficult item.

Figure 2
Figure 2. Representation of the ability continuum. Thick lines indicate, from left to right, the location of items of increasing difficulty. Arrows indicate the location of subjects A, B, and C on the ability continuum.

Location of items and subjects along the measurement scale is estimated by the model from the proportion of response of each subject to each item. The scale resulting from the Rasch analysis of ordinal response of each subject to each item has the properties of an interval scale. Interval scales have known and equal intervals between two graduations. On interval scales, numbers tell us how much more of the attribute of interest is present. These scales are linear and quantitative.

The Rasch model prescribes that the probability of endorsing any response category to an item solely depends on the subject ability and the item difficulty. In other words, no other attribute of the subjects or the items the the ability being measured is theorized to account for the probability of endorsing a response. This requirement called unidimensionality is essential to provide a measure of the latent ability under investigation, unbiased by other attributes of the subject or of the items.

The specific measurement property that distinguish the Rasch model from other IRT models is the specific objectivity. This property implies that comparisons between individuals are independent of which particular items within the class considered have been used. Symmetrically, it is possible to compare items belonging to the same class - measuring the same thing -independently of which particular individuals within a class considered answered them.


Further reading

► Andrich D. Rasch models for measurement. Newbury Park, Ca: SAGE Publications, Inc; 1988.

► Bond TG, Fox CM. Applying the Rasch model: fundamental measurement in the human sciences. Mahwah: Lawrence Erlbaum Associates; 2001.

► Penta M, Arnould C, Decruynaere C, Thonnard J-L, Plaghki L. Développer et interpreter une échelle de mesure: applications du modèle de Rasch. Sprimont: Mardaga; 2005.

► Wright BD, Masters GN. Rating scale analysis. Chicago: Mesa Press, 1982.

► Wright BD, Stone MH. Best test design. Chicago: Mesa Press; 1979.


Rasch analysis (http://www.rasch-analysis.com/)

Institute for Objective Measurement (http://www.rasch.org/)

Rasch model (http://en.wikipedia.org/wiki/Rasch_model)