# Score equating

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**Test equating** traditionally refers to the statistical process of determining comparable scores on different forms of an exam.^{[1]} It can be accomplished using either classical test theory or item response theory.

In item response theory, *equating* is the process of equating the units and origins of two scales on which the abilities of students have been estimated from results on different tests. The process is analogous to equating degrees Fahrenheit with degrees Celsius by converting measurements from one scale to the other. The determination of comparable scores is a by-product of equating that results from equating the scales obtained from test results.

### Why is equating necessary?

Suppose that Dick and Jane both take a test to become licensed in a certain profession. Because the high stakes (you get to practice the profession if you pass the test) may create a temptation to cheat, the organization that oversees the test creates two forms. If we know that Dick scored 60% on form A and Jane score 70% on form B, do we know for sure which on has a better grasp of the material? What if form A is composed of very difficult items, while form B is relatively easy? Equating analyses are performed to address this very issue, so that scores are as fair as possible.

### Equating in Item Response Theory

In item response theory, person locations are estimated on a scale; i.e. locations are estimated in relation to a unit and origin. It is common in educational assessment to employ tests in order to assess different groups of students with the intention of establishing a common scale by equating the origins, and sometimes units, of the scales obtained from response data from the different tests. The process is referred to as equating or test equating.

In item response theory, two different kinds of equating are horizontal and vertical equating ^{[2]}. Vertical equating refers to the process of equating tests administered to groups of students with different abilities, such as students in different grades (years of schooling) ^{[3]}. Horizontal equating refers the equating of tests administered to groups with similar abilities; for example, two tests administered students in the same grade in two consecutive calendar years. Different tests are used to avoid practice effects.

In terms of item response theory, equating is just a special case of the more general process of *scaling*, applicable when more than one test is used. In practice, though, scaling is often implemented separately for different tests and then the scales subsequently equated.

A distinction is often made between two methods of equating; *common person* and *common item* equating. Common person equating involves the administration of two tests to a common group of persons. The mean and standard deviation of the scale locations of the groups on the two tests are equated using a linear transformation. Common item equating involves the use of common items embedded in two different tests. The mean item location of the common items is equated.

### Classical Approaches to Equating

In classical test theory, mean equating simply adjusts the distribution of scores so that the mean of one form is comparable to the mean of the other form. Linear equating adjusts so that the two forms have a comparable mean and standard deviation. Equipercentile equating determines the equating relationship as one where a score could have an equivalent percentile on either form.

Unlike with item response theory, equating based on classical test theory is somewhat distinct from scaling. Equating is a raw-to-raw transformation in that it estimates a raw score on Form B that is equivalent to each raw score on the base Form A. Any scaling transformation used is then applied on top of, or with, the equating.

## References

- ↑ Kolen, M.J., & Brennan, R.L. (1995). Test Equating. New York: Spring.
- ↑ Baker, F. (1983). Comparison of ability metrics obtained under two latent trait theory procedures. Applied Psychological Measurement, 7, 97-110.
- ↑ Baker, F. (1984). Ability metric transformations involved in vertical equating under item response theory. Applied Psychological Measurement, 8(3), 261-271.

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