## Estimating Rasch (person, ability, theta) Measures with Known Dichotomous Item Difficulties: Anchored Maximum Likelihood Estimation (AMLE)

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 Ui in user-scaled units. If not already in logits, convert this to logits Di.

• 3) For person n's L observed responses on L items, compute the average item difficulty Dmean and the item sample variance, V:

Dmean = ( Σ Di )/L for i=1,L

V = (Σ (Di - Dmean)² ) / (L-1) for i=1,L

• 4) An initial estimate of person n's ability M is the PROX estimate:

M = Dmean + (√(1 + V/2.9))*loge(R/W)

alternatively, M = any convenient value

• 5) Compute expected score and variance for M:

For each item i of difficulty Di, the probability of person n's success on item i = Pi = 1 / ( 1 + e (Di - M) )
where e = 2.7183

person n's total raw score = Score = Σ( Pi ) for i=1,L

the model variance of person n's raw score = Variance = Σ( Pi (1 - Pi) ) 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.

#### Warm's (Weighted Mean) Likelihood Estimates (WLE)

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), MWLE, which is almost always closer to the mean item difficulty than M.

person n's WLE estimate = MWLE = M + ( J / ( 2 * I2 ) )
where, for dichotomous Rasch items,
J = Σ ( Pi (1-Pi ) (1-2Pi) ) summed over i = 1,L
I = Σ ( Pi (1-Pi ) )

summed over i = 1,L

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
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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