Estimating Measures with Known Item Difficulties

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

For polytomies, see www.rasch.org/rmt/rmt122q.htm

Once item difficulties 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 responses by person n to the desired subset of calibrated items.

There are L dichotomous 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 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:

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

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