Estimating Rasch Measures with Known Item Difficulties

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) Compute the average item difficulty for person n's L observed responses on L items:

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

5) Compute expected score and variance for M:

where e = 2.7183

6) Obtain a better estimate M' of the measure M:

If the estimates overshoot (diverge, so that the changes in the estimates become bigger, not smaller), then multiply the divider by 2 and set its minimum value at 1.0:
Variance divider = max(variance*2, 1.0)

7) If |M'-M|>.01 then set M=M', but do not change M by more than one logit, i.e., M = max(min(M+1,M'),M-1), and go back 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: A result is that 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.

Wright B.D., Douglas G.A. (1996) Estimating measures with known item difficulties. Rasch Measurement Transactions 10:2 p.499.

Estimating Rasch measures with known dichotomous item difficulties. Wright B.D., Douglas G.A. … Rasch Measurement Transactions, 1996, 10:2 p.499

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