Twice as hard...

Question: If one has, for example, three items. Item Two is one logit more difficult than Item One. Item Three is two logits more difficult than Item One. Can one say that Item Three is three times as hard as Item One, and Item Two is twice as hard?
L.C.

An answer: Thinking through parallel situations in physical measurement almost always clarifies problems like this. Suppose that, instead of items, we are investigating the heights of mountains. Mountain Two is 1 kilometer higher than Mountain One. Mountain Two is 2 kilometers higher than Mountain One. What is the ratio of the heights of the mountains? We don't know until we have chosen a zero reference point, an origin. The usual choice is "sea-level". Then, if Mountain One is 1 kilometer high, Mountain Two is twice as high, etc. So in your example, if Item One is one logit more difficult than some reference origin, then Item Two would be twice as hard, etc.

But how do we choose an origin? Even for mountains a choice must be made, because there are mountains under the sea! Where is the origin of your scale? You must choose a useful location. Typical locations are the average difficulty of all your items. Or the average ability of all persons in your sample. Or the lowest ability you plan to measure. Or the difficulty of the easiest item. But other choices could be as good, or better, for you to make best use of your measures.

Twice as hard... Linacre J.M. … Rasch Measurement Transactions, 2002, 15:4 p. 855

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