The effect of differential item functioning in anchor items on population invariance of equating

Anne Corinne HugginsUniversity of Florida

Introduction• Population invariance of equating– Test score level

• Differential item functioning– Item level

• Need to connect two facets of invariance• Certain types of DIF might cause equating

dependence across forms of a test.– Due to (1) a differential difficulty, particularly for anchor items(2) a second factor(s)(3) Item parameter drift

• IRT true score equating– A three-step process:• Specify a true score on one form• Find the corresponding IRT ability estimate• Find the IRT ability estimate’s corresponding true score

on the other form

• Place the latent ability scores on each test on the same scale

We can use the probability formula to get the relations.

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• Scaling– Mean-sigma

– Mean-mean

Here, measurement errors are ignored. If a and b were biased, A and B will be biased.

Q: How to estimate A under Rasch model?

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– The Stocking and Lord method• Minimize the difference in test characteristic curves

(TCCs) for anchor items

– The Haebara method• Minimize the difference in item characteristics curves

(ICCs)

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• Equating invariance– Compare the whole group to multiple subgroups

– Group to overall (compare one subgroup with the whole group)

Sigma Q is the unconditional SD of scores in population.

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• The factors that might influence population invariance of IRT true score equating– Differential anchor form DIF– The magnitude of DIF in anchor items– Number of DIF anchor items – Direction of DIF in anchor items– Ability differences between subpopulations

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Method

• Uniform DIF• R, BILOG-MG, SPSS• Design– Nonequivalent groups with anchor test (NEAT) design– Two forms contain 50 dichotomous items,

respectively, with 20% of common items– Ability ~ N (0,1)– 4500 participants (3000 in the references group)– 3PL model– 100 replications

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– Manipulations on DIF in anchor items• Magnitude (.3, .6, .9)• % of DIF items (20%, 49%, 60%)• Directionality of DIF• Mean ability differences• Differential anchor form DIF

– Null conditions• No DIF in anchor items and no mean ability differences• No DIF in anchor items but had mean ability differences

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Results—No DIF (RMSD(x))

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No DIF, Equal mean abilities (RSD(x))

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No DIF, Unequal mean abilities (RSD(x))

14Why didn’t DIF show the impact here?

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Conclusions Differential anchor form DIF occurs, the

magnitude of equating dependence increases.IPD occurs for different subpopulations.

When differential anchor from DIF occurs:Direction of DIF in anchor items shows the largest

effect on equating dependence.The magnitude of DIF in anchor itemsNumber of DIF anchor items Ability differences between subpopulations

Comments

• It is a good writing sample.• Errors:– P.9, since the study manipulated the direction of

DIF, the statement of magnitude of DIF should be revised properly.

– The author intended to design the sample size as the true subpopulations in the US, but the actual sample sizes did not reflect its intends.

• When DIF or item parameter drift occurs in a specific anchor item, this item should not be used as an anchor.

• We usually do not talk about DIF in anchor items because DIF detections should be done before any equating or linking.

• When all anchor items are DIF items, the results should be very similar with the ones without DIF anchor items.

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Future studies

• Sample size differences across score levels• Test length, anchor length, ratio of anchor to

test length, mixed item formats