4.7 Engelhard criteria for invariant measurement

In this work, we strive to achieve invariant measurement, a strict form of measurements that is subject to the following requirements (Engelhard Jr. 2013, 14):

  1. Item-invariant measurement of persons: The measurement of persons must be independent of the particular items used for the measuring.
  2. Non-crossing person response functions: A more able person must always have a better chance of success on an item that a less able person.
  3. Person-invariant calibration of test items: The calibration of the items must be independent of the particular persons used for calibration.
  4. Non-crossing item response functions: Any person must have a better chance of success on an easy item than on a more difficult item.
  5. Unidimensionality: Items and persons take on values on a single latent variable. Under this assumption, the relations between the items are fully explainable by the scores on the latent scale. In practice, the requirement implies that items should measure the same construct. (Hattie 1985)

Three families of IRT models support invariant measurement:

  1. Scalogram model (Guttman 1950)
  2. Rasch model (Rasch 1960; Andrich 1978; Wright and Masters 1982)
  3. Mokken scaling model (Mokken 1971; Molenaar 1997)

The Guttman and Mokken models yield an ordinal latent scale, while the Rasch model yields an interval scale (with a constant unit).