I had a hard time of finding stylized examples of departures MI probably since it requires the conceptually slippery latent variables. So I made up some hypothetical examples inspired by some graphs in Jelte Wicherts' thesis. The graphs make the simplifying assumption that a given test score is a linear function of the single underlying latent trait. The latent trait is not observed but the slope and intercept are derived from factor analysis. If MI held in these examples the lines would be equal for the two groups.
No Comprende: Suppose a "culturally unfair" test of verbal intelligence is given to two groups: native speakers and foreigners. The confused foreigners answer at random so that test cannot distinguish the bright from the dim and measurement invariance does not hold.
Privileged: A test is given to the oppressed and the privileged. Because of test anxiety caused by the menace of the privileged the oppressed student at all levels of latent ability perform exactly 15 points less than the privileged students of the same level of ability. The fair thing to do is to give the oppressed a 15 point boost.
Differential: In this case the test have different characteristics for each set of subjects. The test is biased in favor of A (B) for those with a latent ability less (greater) than 30.
5 comments:
In the first graph, how is it possible that natives with low ability get worse results than if they had chosen answers at random?
Either the really retarded natives leave blank answers or the sample is such that part of the curve is extrapolated.
Another explanation could be that the test contains trick questions and answers that the stupid natives fall for, whereas foreigners who don't understand the language cannot be deceived in that way.
You actually see that on the SAT: there are tricky questions that above average scorers do worse on than below average scorers because the high scorers fall for the tricks and traps while the low scorers don't even notice t hem.
I would love to see an MI analysis of the SAT but I cannot find one. Given their huge sample sizes I would not be surprised if the differential pattern was picked up. Of course the magnitude of the slope-intercept differences may be minor but still something you do not want to explain in court.
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