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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

${P}_{ni}=\frac{\mathrm{exp}\left({B}_{n}-{D}_{i}\right)}{1+\mathrm{exp}\left({B}_{n}-{D}_{i}\right)},$

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:

$\text{In}\left(\frac{{P}_{ni}}{1-{P}_{ni}}\right)={B}_{n}-{D}_{i}.$

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