For polytomies, see www.rasch.org/rmt/rmt122q.htm
Once item difficulties (criterion-referenced or norm-referenced) have been carefully calibrated and the measurement system constructed, we can administer some or all of the calibrated items to further examinees and measure them based on the pre-calibrated item difficulties. The approach here obtains the maximum-likelihood estimates using Newton-Raphson iteration.
1) Collect observed responses by person n to the desired subset of calibrated items.
There are L observed dichotomous responses to L of the calibrated items taken by this person, with R correct answers and W incorrect.
If R = 0, then put R = 0.5, W = L-0.5
If W = 0, then put R = L-0.5, W = 0.5
Check that R+W = L.
2) Each item, i, has a calibration U_{i} in user-scaled units. If not already in logits, convert this to logits D_{i}.
3) For person n's L observed responses on L items, compute the average item difficulty D_{mean} and the item sample variance, V:
D_{mean} = ( Σ D_{i} )/L for i=1,L
V = (Σ (D_{i} - D_{mean})² ) / (L-1) for i=1,L
4) An initial estimate of person n's ability M is the PROX estimate:
M = D_{mean} + (√(1 + V/2.9))*log_{e}(R/W)
alternatively, M = any convenient value
5) Compute expected score and variance for M:
For each item i of difficulty D_{i}, the probability of person n's success on item i = P_{i} = 1 / ( 1 + e ^{(Di - M)} )
where e = 2.7183person n's total raw score = Score = Σ( P_{i} ) for i=1,L
the model variance of person n's raw score = Variance = Σ( P_{i} (1 - P_{i}) ) for i=1,L
6) Obtain a better estimate M' of the measure M:
If, after the first iteration, the estimates overshoot (diverge, so that the changes in the estimates become bigger, not smaller),
abs(M' - M) > abs(M'' - M)
then multiply the divider by 2 and set its minimum value at 1.0:
Variance divider = max(variance*2, 1.0)Do not change an estimate by more than one logit from its value in the previous iteration.
M' = max(min(M+1,M'),M-1)
7) If abs(M' - M) > 0.01 logits, then set M'' = M and M = M' and go to (5).
8) Set M = M', and report this final ability estimate with standard error = sqrt(1/Variance). Convert measure and standard error back to scaled U units for reporting.
Note: Summary statistics for the final person measures may not match directly-estimated person distributional parameters - but, since the persons are often regarded as "incidental" parameters, no one seems too much concerned.
For explanation, see Wright B.D., Douglas G.A. 1975. Best Test and Self-Tailored Testing. Research Memorandum #19. Chicago: MESA Press.
This estimation is implemented in Mark Moulton's Excel Spreadsheet.
For an explanation of WLE, see RMT (2009), 23:1, 1188-9
Warm's bias correction is applied to each MLE estimate, M, to produce a Warm's Mean Likelihood Estimate (WLE), M_{WLE}, which is almost always closer to the mean item difficulty than M.
person n's WLE estimate = M_{WLE} = M + ( J / ( 2 * I^{2} ) )
where, for dichotomous Rasch items,
J = Σ ( P_{i} (1-P_{i} ) (1-2P_{i}) ) summed over i = 1,L
I = Σ ( P_{i} (1-P_{i} ) )
Estimating Rasch (person, ability, theta) measures with known dichotomous item difficulties: Anchored Maximum Likelihood Estimation (AMLE). Wright B.D., Douglas G.A. … Rasch Measurement Transactions, 1996, 10:2 p.499
Rasch Publications | ||||
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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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