## PROX for Guessing and Carelessness

Test items don't guess and test items aren't careless. The greatest clarity comes from the detection of guessing and carelessness as misfit on the person level. Nevertheless, it is sometimes useful to work with an assertion that some items induce guessing and some items induce carelessness. Minimum competency testing procedures often encourage low performers to guess on items that are impossibly difficult for them. Some of the National Adult Literacy Survey test items are so easy that they provoke carelessness among the highly literate.

The dichotomous Rasch model can be extended to accommodate guessability and carelessness as item characteristics:

where ui is the upper, asymptotic "carelessness" coefficient for item i, and ci is its lower, asymptotic "guessability" coefficient, Pni, Bn and Di have their usual meanings.

For this model to be estimable, ui and ci must be specified ahead of time. They cannot be estimated simultaneously with person ability Bn and item difficulty Di. Reasonable values for ui and ci can often be deduced by inspection of the items themselves or from response summary statistics for very high and very low scoring groups of test-takers.

Having made the large assertion that carelessness and guessing can be induced by items, let us further assert that the item difficulty distribution and person ability distribution can be usefully summarized by their means and standard deviations. Then PROX (normal approximation algorithm) estimates (Cohen, 1979) of ability and difficulty will be satisfactory.

Rewriting (1)

Asserting that person abilities are normally distributed, and following the argument in RMT 8:3, p. 378,

where Si are the number of successes by the Ni persons who encountered item i. µi is the mean ability and i the standard deviation of those Ni persons. is the logistic ogive. 2.9 is the standard divisor. 2.65 is a better value (RMT 11:2 p. 569).

Rearranging, produces an estimation equation for Di, the logit difficulty of item i,

with model standard error,

For Bn, the ability of person n, consider the items in groups, such that ci=cg and ui=ug are set constant across items within group g. Then Bng is the ability estimate for person n based on group g of items:

where Rng is the raw score achieved by person n on Nng items, and µng and ng summarize the distribution of those Nng item difficulties.

Bng has a model standard error,

The {Bng} can be combined to produce an estimate for Bn (RMT 8:3 p. 376):

with standard error,

These equations can be solved iteratively, with one constraint such as Di0, to produce estimates for the measures of all elements.

When data are complete, or responses are missing at random, and ui=u, ci=c are set constant across items, then µi and i can also be treated as constant across items. Similarly, µn and n can be treated as constant across persons. Then non-iterative solutions are:

where is the mean person ability, the mean item difficulty is zero and 8.35 is the standard value. 7.0 is a better value (RMT 11:2 p. 569). Also,

To perform this by hand, see Wright & Stone (1979), Chapter 2.

John M. Linacre

Cohen, L. 1979. Approximate expressions for parameter estimates in the Rasch model. British Journal of Mathematical and Statistical Psychology 32(1) 113-120.

Wright B.D., Stone M.H. (1979) Best Test Design. Chicago: MESA Press.

PROX for Guessing and Carelessness. Linacre J. M. … Rasch Measurement Transactions, 1997, 11:2 p. 570-571.

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
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