Thomas Warm (1989) reports that "Lord (1983) found that maximum likelihood estimates of θ [person ability] are biased outward" and then he restates Lord's expression for the size of this bias:
Bias (θMLE) = - J / ( 2 * I2 )
where, for dichotomous Rasch items,
J = Σ Pθi (1-Pθi ) (1-2Pθi )
I = Σ Pθi (1-Pθi )
summed for all items, i = 1,L in the test, where Pθi is the Rasch-model probability of success of ability θ on item i.
The corrected estimate, θWLE = θMLE + (J / ( 2 * I2 ) ), which is almost always closer to the item mean than θMLE.
How effective is this bias correction? Warm uses a Monte Carlo study to demonstrate its effectiveness, but an exact algebraic investigation can be conducted.
I posited a test of 25 items, with its item difficulties uniformly spaced 0.2 logits apart. Figure 1 shows the locations (x-axis) of the items on the 25-item test. The item difficulties are centered on 0 logits.
Applying the MLE method of Wright & Douglas (1996) for estimating θ from known item difficulties, a Rasch ability estimate, M(s) is obtained for each possible raw score, s=0-25, on the test of 25 items. Since the estimates corresponding to s=0 and s=25 are infinite, they are substituted by estimates corresponding to s=0.3 and s=24.7 score-points. The MLE ability estimates are shown in Figure 1.
Figure 1. MLE and WLE for 25 dichotomous items.
Figure 2. Detail of Figure 1 showing MLE bias.
Figure 3. MLE and WLE for 12, 4-category, items.
|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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