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Item response theory (also known as modern test theory) is a mathematical approach to quantifying latent features based on the fundamental assumption that a subject's response to an item is a function of the difference between 1) his / her skills and 2 ) the properties of. is the subject. Within this class, the Rasch model specifically defines difficulty / ability as the only parameter of interest when evaluating items. This approach was developed in the 1950s by the Danish mathematician Georg Rasch as an instrument for assessing performance in school children. In addition to its continued use in educational assessment, the Rasch model is widely used in the social sciences, which are highly dependent on patient-reported outcomes, and more recently in clinical and public health research as a tool for studying various health outcomes, including rehabilitation and violence in the community.


Purpose of the model

Basic (i.e., mass & volume) and derived (indirectly detected, e.g., density) measurements for quantifying the properties of objects are widely used in the natural sciences. However, because of the complex nature of humans, measurements that can capture the unobservable qualities of beliefs and behaviors are less straightforward (we cannot physically align parts of the human psyche, Bond & Fox 2012, p. 6). Clinicians and researchers often rely on scales, indices, and other item-based quantitative tools to infer variables to measure latent constructs. For example, questions about hopelessness can be included on a scale designed to assess depression. The Rasch model provides a mathematical framework that test developers can use to compare empirical data to assess the ability of an instrument to emulate the properties of the fundamental measurement (invariance and one-dimensionality) and thus serve as a tool for quantifying unobservable human conditions.

Applied Rasch Analysis

The Rasch model is based on the assumption that the most economical and effective predictor for a characteristic is the relationship between the difficulty of an object and a person's ability. It is based on the underlying logic that subjects have a higher probability of answering easier items correctly and a lower probability of answering more difficult items. A researcher begins instrument development by reviewing the existing literature and evaluating any prior knowledge of the latent characteristic. The instrument is then tested on a sample that meets the criteria for the target population (taking into account age, gender, health status, etc.) and the resulting data is compared with the Rasch model. Figure 1 below (Bond & Fox 2007) and the following steps outline the Rasch model's approach to assessing instrument development based on dichotomous data. Technical details can be found in the following section.

  1. Calculate the percentage for each person, this will give you a raw ordinal score. Convert the raw percentage for each person to the probability of success by incorrectly calculating the ratio of each person's percentages divided by the percentage [(p) / (1-p)]. Take the Chances of Success log to help calculate the person's abilities. This transformation solves the problem of compression at the ends of the raw values. On the plot, blue squares represent the person's skills.

  2. Repeat step 1 for the item difficulty (percent of subjects who answered the item correctly / percent of subjects who answered incorrectly). On the plot, pink circles represent the difficulty of the item.

  3. Plot these estimates against the idealized (perfect one-dimensionality) Rasch model, which is shown on the plot as a vertical line and shows the relationship between items and skills on a logit scale (also known as the log odds ratio). The average logit (probability of success) is set to 0.

  4. The size of each object provides an inverse representation of its error (smaller objects have fewer errors). It is expected that the error will increase at both ends of the series as few subjects have low or high skills and few items are rated as incorrect or correct by all subjects. Item precision will increase with instrument management, while person estimates will improve as appropriate items are added to the instrument.

  5. Assess the fit of the empirical data to the idealized perfect line of the Rasch model. Fit values ​​are read horizontally; a person who falls outside the white path (-2.0 to 2.0 for an N of 30-300) is considered poorly adjusted and therefore does not follow the expected response pattern.

  6. If article and / or person discrepancies are found, reevaluate article wording or other aspects of articles that do not match and repeat these steps. If the empirical data fit the Rasch model, the instrument total is considered sufficient statistic for future analysis.

Technical aspects

Mathematical representation
Once we have estimated the ability of the subject (Bn) and the difficulty of the item (Di), we can express the probability of success by equation 1, which states that the probability of a correct answer for subject n with item i is a logistic function of Difference between is the ability of the subject and the difficulty of the subject.

Pni (x = 1) = f (Bn-Di) = e (Bn-Di) / 1 + e (Bn-Di), where x = 1 correct & x = 0 incorrect (1)

Assessment of abilities and difficulties
* Depending on the software used, alternative estimation approaches are Joint Maximum Likelihood Estimates (JMLE), Marginal Maximum Likelihood Estimation (MMLE) or Pairwise Estimation (PAIR).

Model customization

The degree of discrepancy between the observed item performance and the expected item performance can be quantified with the help of adjustment statistics. Unweighted (outfit) and weighted (infit) mean square statistics are calculated by comparing the observed data with the model probability matrix. The residuals are assumed to follow a chi-square distribution and an acceptable fit is identified by a chi-square probability greater than 0.5. Standardized adjustment statistics are calculated based on a t-score, with acceptable values ​​ranging from -2 to +2 (expected = 0). In addition, Wald tests can be used to identify certain items that do not fit well and likelihood ratio tests can be used to assess the overall data fit.
In addition, Figure 2 (J. Sick 2010 http://jalt.org/test/sic_5.htm ) the fit can be assessed visually by creating an Item Characteristic Curve (ICC). The x-axis represents the latent feature on the logit scale and the y-axis represents the expected score for an item. The Rasch model predicts a sigmoid curve and the fit of the data can be assessed in comparison to this curve (observed data not shown in the figure). Lighter objects fall to the left of 0 and more difficult objects fall to the right. The estimated value of an item can be identified by finding its position on the x-axis when the expected value is equal to 0.5.

Model extensions

  • Rating scale model (e.g. Likert scales)

  • Partial credit model

  • Many facets

  • Information on 2- and 3-parameter IRT models


  • Concatenation: the combination of any units that have an additive (linear) relationship; the basis of basic measurements

  • Conjoint Measurement: Increasing the level of one attribute (i.e. probability of a correct answer) with increasing values ​​of two other attributes (i.e. item difficulty and person ability).

  • Construct validity: the degree to which an instrument or test measures what it is intended to measure based on its underlying theory

  • Interval scale: a measurement scale where the distance between units is the same

  • Invariance: Consistency of a measurement from one occasion to another, e.g. B. Constancy of a derived measure (i.e. density) given the variation in the underlying fundamental measures (i.e. mass and volume)

  • Latent variable: a characteristic that can be inferred by observing behavior rather than directly measuring an attribute

  • Parameter separation: Estimation of a parameter set independently of a second parameter set

  • One-dimensionality: A measurement is only valid for one attribute of an object


Textbooks & Chapters

Bond TG. and Fox CM. (2007). Applying the Rasch Model: Basic Measurement in the Human Sciences. Second ed. New York: Routledge.
This book provides an overview of the theories and principles that are central to Rash analysis, including information on model extensions (rating scales, partial credit model, many-facet models). While less time is spent applying the Rasch model, the book contains a Rasch Software (Winsteps) CD. Application of the Rasch model

Andrich D. (1998). Rasch models for measurement. Series: Quantitative Applications in the Social Sciences. London: Wise Publications.
This short book provides an overview of the general principles of the Rasch model and focuses on the simple logistic model for dichotomous data. Concepts are solidified through the use of an example throughout the text. Rasch models for measurement

Rasch, G. (1980). Probability models for some intelligence and performance tests (Extended Ed.). Chicago: University of Chicago Press.
This book forms the basis for the Rasch family of IRT models. It provides an overview of the theoretical and mathematical foundations of the model with an emphasis on the application of the approach to the educational field.

Wright, BD. & Stone MH. (1979). Best test design. Chicago: MESA Press.
This book provides an overview of the Rasch analysis for dichotomous data. Wright 1982 and Linacre 1989 give reviews of extensions of the dichotomous model.

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Methodical articles

Wright B. (1997) A History of Social Science Measurement. Available here

Belvedere SL, de Morton NA. 2010. The use of the Rasch analysis in the health care system is increasing and is used in mobility instruments for various reasons. Journal of Clinical Epidemiology. 62: 1287-1297.

Sica da Rocha N, Chachamovisch E, de Almeida Fleck MP, Tennant A. 2013. An introduction to Rasch analysis for psychiatric practice and research. Journal of Psychiatric Research. 47: 141-148.

Tennant A. Conaghan PG. 2007. The Rasch measurement model in rheumatology: what is it and why? When should it be used and what should you watch out for with a Rasch paper? Arthritis and rheumatism. 57: 8; 1358-1362.

Tesio L. Simone A. Bernardinello M. 2007. Rehabilitation and measurement of results: where does the Rasch analysis go? Eura Mediocphys. 43: 417-26.

Application item

Suglia SF, Ryan L, Wright R. 2008. Building a Scale of Exposure to Community Violence: Considering What, Who, Where, and How Often. J trauma stress. 2008 Oct; 21 (5) 479-486.

This article provides an example of how the Rasch model can be used in public health research. The authors used the continuation ratio model (an extension of the original Rasch model) to create a scale for measuring exposure to community violence (ETV) of women in an urban area. The paper presents the study population, background information on items, methodological approach, results and a discussion of the results and a comparison of the Rasch ETV model with previous variations.

Franchignoni F, Giordano A, Gianpaolo R, Rabini A, Ferriero G. 2014. Rasch validation of the activity-specific balance confidence scale and its short versions in patients with Parkinson's. J Rehabilitation Med. 46: 00-00 (before printing).

This recent article demonstrates the application of the Rasch model to a clinical population. The authors use the Rasch model to assess whether a long version of an assessment used to measure the psychological aspects of balance (i.e. fear of falling) in a Parkinson's population can be reduced to a shorter version, which takes less time. The authors carry out the analysis with WINSTEPS and give an overview of the classic vs. modern test theory.

Mair, P. and Hatzinger, R. (2007). Extended Rasch modeling: The eRm package for the application of IRT models in R. Journal of Statistical Software, 20 (9), 1-20.

This article provides an overview of Rasch modeling with the eRm package in R, including sample applications for the original (dichotomous) model and the following extensions: linear logistic test models, rating scale models, linear rating scale models, partial credit models, and linear partial credit models or Das Package enables the user to perform: person and article estimates, likelihood ratio tests for model fitting, Wald tests for article-specific fitting, statistics for residual and person fitting, and various charts to visualize fitting.




Estimation method


Profit steps

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Paired CMLE


* CMLE: Conditional estimation of the maximum probability, JMLE: Joint estimation of the maximum probability

Web pages

Wikipedia The Wikipedia page on Rasch modeling provides an easy-to-read history of the Rasch model, its functions, and its applications to dichotomous data and extensions.

RUMM Labor This website, operated by the RUMM Laboratory in Australia, provides a thorough introduction to the theory and application of Rasch Modeling, including a discussion of available software and a list of ongoing courses and workshops.

Rapid model SIG This website offers links to various articles on the subject of Rasch Modeling, which have been compiled by the Rasch Model Special Interest Group.


Rasch Measurement Special Interest Group (SIG): http://www.raschsig.org/
This special interest group is part of the American Educational Research Association. The group is concerned with the development of empirically verifiable instruments with linear measures for use in the social sciences. The annual membership fee is $ 10 and the application can be filled out online!

International Rasch Conference: http://www.rasch.co.za/conference.php
The next international Rasch conference is dedicated to the topic of the class of the Rasch models and the Rasch paradigm in the function of measurement in modern social science. The conference will take place in January 2015 at the University of Cape Town. You can find detailed information about the program on their website.

Rasch applications: http://www.statistics.com/rasch-applications-1/
The aim of this medium-level online workshop is to convey to social scientists the practical aspects of the Rasch analysis (data import, analysis, fit, interpretation) and the underlying theory that supports its application. The course lasts four weeks and uses the Winsteps software. Assignments include concept tests, a modeling project, and readings. The tuition fee is $ 629.00 and the dates can be found on the website above.

Introduction to Rasch Analysis: http://www.leeds.ac.uk/medicine/rehabmed/psychometric/Rasch%20Courses1.html
This face-to-face workshop is organized by the Psychometric Laboratory for Health Sciences at the University of Leeds in Western Australia. They currently offer courses in Australia and Western Europe; regular US-based courses organized by this or any other university are currently not offered. Introductory, intermediate and advanced courses on Rasch Analysis are offered and further information can be found on the website mentioned above.

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