It should be noted that cA is conceptually closer to a real "guessing" parameter in the Birnbaum PRFs, and that αA represents person sensitivity to a particular subset of items.
Engelhard (in progress) describes five requirements of invariant measurement that must be met to yield useful inferences for measurement in the social, behavioral, and health sciences. These five requirements are
1. The measurement of persons must be independent of the particular items that happen to be used for the measuring: Item-invariant measurement of persons.
2. A more able person must always have a better chance of success on an item than a less able person: non-crossing person response functions.
3. The calibration of the items must be independent of the particular persons used for calibration: Person-invariant calibration of test items.
4. Any person must have a better chance of success on an easy item than on a more difficult item: non-crossing item response functions.
5. Items must be measuring a single underlying latent variable: unidimensionality.
Requirements 1 and 2 address issues related to PRFs.
The Figure illustrates the effects of crossing PRFs. Three PRFs were constructed for two situations: Rasch PRFs that do not cross (Panel A) and Birnbaum PRFs that do cross (Panel B). As shown in Panel C, non-crossing PRFs yield comparable person locations over subsets of items centered around easy items (-2 logits) to hard items (+2 logits). If PRFs do not cross, then Persons A, B, and C are ordered in the same way across item subsets. In other words, item-invariant measurement is achieved with the Rasch model.
Crossing PRFs based on the Birnbaum model (Panel D) yield person ordering that varies as a function of the difficulty of the item subsets. For example, Person A is the lowest achieving person with the lowest probability of success on the easy items, while Person A is the highest achieving person on the hard items. Easy item subsets yield persons ordered as A < B < C, while hard item subsets yield persons ordered B < C < A. In other words, the ordering of persons is not invariant over item subsets with the Birnbaum model.
This note calls attention to the idea that model-data fit can be conceptualized in terms of both IRFs and PRFs (Engelhard, in press). Typically IRFs and differential item functioning analyses are explored. Our work suggests that researchers should also begin to think more systematically about differential person functioning. It is important to recognize the items may function differently over different subgroups of persons (differential item functioning), but it is also important to recognize that persons may not function as intended in their interactions with subsets of test items (differential person functioning).
Aminah Perkins & George Engelhard, Jr.
Emory University, Division of Educational Studies
Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability, Part 5. In F.M. Lord and M.R. Novick (Eds.), Statistical theories of mental test scores. Reading, MA: Addison-Wesley Publishing Company, Inc.
Engelhard, G. (in progress). Invariant measurement: Rasch models in the social, behavioral, and health sciences. New York: Routledge.
Engelhard, G. (in press: available online). Using item response theory and model-data fit to conceptualize differential item and person functioning for students with disabilities. Educational and Psychological Measurement.
Mosier, C.I. (1940). Psychophysics and mental test theory: Fundamental postulates and elementary theorems. Psychological Review, 47, 355-366.
Mosier, C.I. (1941). Psychophysics and mental test theory. II. The constant process. Psychological Review, 48, 235-249.
Wright, B.D. (1992). IRT in the 1990s: Which Models Work Best? Rasch Measurement Transactions, 6:1, 196-200, www.rasch.org/rmt/rmt61a.htm
Wright, B.D. (1997). A history of social science measurement. Educational Measurement: Issues and Practice, Winter, 33- 45, 52.
Perkins A. & Engelhard, G. Jr. (2009) Crossing Person Response Functions, Rasch Measurement Transactions, 2009, 23:1, 1183-4
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