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
Forum | Rasch Measurement Forum to discuss any Rasch-related topic |
Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement
Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.
Coming Rasch-related Events | |
---|---|
Oct. 4 - Nov. 8, 2024, Fri.-Fri. | On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com |
Jan. 17 - Feb. 21, 2025, Fri.-Fri. | On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com |
May 16 - June 20, 2025, Fri.-Fri. | On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com |
June 20 - July 18, 2025, Fri.-Fri. | On-line workshop: Rasch Measurement - Further Topics (E. Smith, Facets), www.statistics.com |
Oct. 3 - Nov. 7, 2025, Fri.-Fri. | On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com |
The URL of this page is www.rasch.org/rmt/rmt102t.htm
Website: www.rasch.org/rmt/contents.htm