What is invariance?
In principle, the property of invariance means that the ability of the test takers is independent of the test they take, and the difficulty of the test items is independent of the test takers. We like to believe that the attribute we are measuring is a property in an individual’s brain, something that the test taker brings to the test. Our measurement of this attribute should not depend on the sample of items we have selected to measure it. Likewise, we hope that the difficulty of the items is an inherent property of the construct, something we can qualitatively predict from theory and experience. In practice, invariance means that once measurement instruments have been calibrated to a common scale, estimates of person ability and item difficulty should not vary across tests and person samples by more than the error that is a consequence of measurement precision. To envision how this works, imagine a short test administered to a sample of test takers who are well-matched to the test. T
