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W Difference Scores

The W scale was developed by Richard Woodcock and Marshall Dahl in consultation with Benjamin Wright. The W scale is simply a transformation of the ability/item score from a Rasch analysis that uses a logarithm with base 9 (log9) instead of the more common base e (ln). Base 9 was used because Woodcock believed it aided in interpreting the difference between personal ability and item difficulty values.

In the simplest Rasch model, the probability that person n correctly answers item i, Pni, is

Pni=exp(BnDi)1+exp(BnDi),

where Bn is person n’s ability (on a logit scale) and Di is the item’s difficulty (on the same logit scale as Bn). Equation 1 can be rearranged to isolate the relation between B and D:

In(Pni1Pni)=BnDi.

Converting B and D to the W scale simply ...

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