Something about Bridge-Building [Test Equating] Techniques - a sensational new creation by Dr. Rasch

From the 23 May, 1957 issue of "Folkeskolen", the Danish elementary school journal.

Translated by Cecilie Kreiner, courtesy of Svend Kreiner

The headline of this article may give readers in this island kingdom associations to comfortable and queue-free transportation between the islands. However, it refers, of course, to the most sensational news at the report to the council at the Danish Institute for Educational Research, news which has already been much spoken of in the press. As a last treat, G. Rasch's account of his interesting new creation for statistical processing of psychological tests concluded the meeting covered elsewhere.

Plainly put, this new tool means that one can compare the result of one test directly with the result of a previous test, thereby building a bridge between tests meant for different grades. Whereas these tests were previously something isolated which could not be incorporated into a whole, the opposite is now the case, and one can thus produce a development curve.

An attempt at an explanation

After the council meeting, we had an interview with statistician Dr. Rasch, who is not only connected with the Danish Institute for Educational Research but also a lecturer at University of Copenhagen where, amongst other things, he teaches statistics to psychology students. The work on shaping the new tool has been going on for several years, and a full account of the work will obviously contain a terminology which will make it unintelligible outside the circle of colleagues. Dr. Rasch's explanation below is, however, largely comprehensible to the interviewer, and that should guarantee the reader's understanding.

- The work with the bridge building method commenced at a study of children who were slow readers carried out for the Ministry of social Affairs, Dr. Rasch says. During the work on this study, the problem of transferring the result from one test to another in order to create a basis for comparison arose. However, no measuring instrument existed, because a standardizing is not an actual measurement, as two children in different grades receive different scores for the same performance.

- In collaboration with head of department, Master of Psychology Carl Age Larsen, I carried out reading tests on a large number of children in the 2nd to 7th grades, in which children in the same grade were given two or more tests. Thus the tests "T 5" and "S" were compared directly in the 4th, 5th and 6th grades. When points for the illustration of the amount of errors were subsequently placed in a diagram with two figure axes so that the horizontal distances illustrated the number of errors in the "S"-test, and the vertical distances illustrated the number of errors in the "T 5"-test, the obvious result emerged at first: with the same test, the lower grades had a high percentage of errors, the higher grades had a lower percentage of errors, and the highest grades had the lowest percentage of errors. The interesting thing was, however, that the three assemblies of points which illustrated the amount of errors in the different grades immediately succeeded each other and pointed towards the 0-point. A line through them showed that approximately 12 errors in the "T 5"-test corresponded to 10 errors in the "S"-test, and that approximately 24 errors in the "T 5"-test corresponded to 20 errors in the "S"-test.

- In short: 1 error in the one item corresponds to 1.2 errors in the other item. It has turned out that the ratio remains the same when comparing two items by means of a third. An expression for the degree of difficulty of one test in relation to another has thus been found here. - Similar ratios have been found for further tests which, like these reading tests, are not systematically constructed [Rasch's Poisson model].

- Other tests, such as, for example, the "F"-test, consist of a number of individual items (texts) of continuously increasing degrees of difficulty, and here the calculations become considerable more complicated, since the individual items have to be considered first [Rasch's dichotomous model].

Two questions

- One may then ask: Is it possible to speak of the degree of difficulty of two items relative to each other? Can a person be given a number for their proficiency in solving this item? For example, can A be twice as good as B? Not just at playing football and drinking coffee, but at solving a number of items of the same kind but of different degrees of difficulty?

- Then another question immediately follows: Can item I be twice as difficult as item II?

Let us say that:

A's degree of proficiency is D,

B's degree of proficiency is d,

D must be 2 x d.

As for the items, the ratio is:

I's degree of difficulty is S,

II's degree of difficulty is s,


S must be 2 x s.

In order to answer these questions, I propose: A must solve I just as easily as B solves II.

But what is "just as easily"? This vicious question will not be avoided by applying an ordinary view to what people do. Precise physical laws may be laid down for the movements of the planets. But people? They can think of doing anything, both when solving items and in other situations. By making foolish mistakes, the proficient may accidentally solve an easy item in a wrong way. Conversely, the slow learner may chance on a correct solution to a difficult item. But in both cases there is a chance - a likelihood - great or small that the item is solved correctly. This idea, the chance that a person solves a given item correctly, can be used for giving meaning to the notorious expression "A solves I just as easily as B solves II", since quite simply it is to be understood as their chances being equally good.

Now the chances that A solves I should be determined by his proficiency D and the difficulty of the item S, and it should be the same for B with I, i.e. when both proficiency and degree of difficulty is bisected, or for that matter multiplied by another factor. The chances of solving an item thus come to depend only on the ratio between proficiency and degree of difficulty, and this turns out to be the crux of the matter.

- A detailed investigation of the individual items in the "F"-test has shown that these ideas are applicable to the "F"-test. Thereby, it follows that the result of an "F"-test is evaluated as a whole, i.e. as a measure of his proficiency in these kinds of tests. Had he been given a different "F"-test, the result of the measurement would most likely have been the same.

- A bridge has thus been built between the different "F"-tests. The result of one test can be translated into the result of another.


With this illustration, Dr. Rasch has at any rate removed the entirely abstract from the concept of bridge building techniques. This ground-breaking work will be studied and employed, not only in Denmark, but in the whole world. The significance it can have practically within school work will be accounted for in a following article.

Finn Jolander

Something about Bridge-Building [Test Equating] Techniques - a sensational new creation by Dr. Rasch … F. Jolander, Rasch Measurement Transactions, 2008, 21:4 p. 1129-30

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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