Question: Lumsden, J. (1978). Tests are perfectly reliable. British Journal of Mathematical and Statistical Psychology, 31, 19-26, states that "test scaling models are self-contradictory if they assert both unidimensionality and different slopes for the item characteristic curves." Do differences in item discrimination always indicate multidimensionality?
Answer: In situations like this, it is helpful to think of parallels in physical measurement. Suppose we are measuring length with old-fashioned cloth tape-measures. These can become stretched along parts of their range. If we compared measurements of lengths with two of these stretched tape measures, we would see that, to start with, they would say the same numbers. Then the less-stretched tape measure would have higher numbers, i.e., be more discriminating. Then they might agree again. Then the other tape measure might have higher numbers. Length is unidimensional, but the "tape measure ICCs" cross, perhaps several times along their lengths. We could call "stretching", i.e., changes of length-discrimination, another dimension, in the same sense as "guessing" is another dimension. But these are not usually what is mean by "multidimensionality".
On the other hand, we might have two good cloth tape measures, but they might not always be parallel or straight. They might "snake" somewhat as we use them. Again they would sometimes agree and sometimes disagree due to crisscrossing "tape measure ICCs". Here we could agree that the problem is "multidimensionality". The tape measures are not in a straight line.
Varying Item Discrimination = Multidimensionality? Rasch Measurement Transactions, 2007, 21:2 p. 1104
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