## Wednesday, July 11, 2012

### Iterative and Non-Iterative PROX

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Non-Iterative PROX starts with item wrong counts, shifts the item distribution to match student right count distribution (subtracts the item mean from each value in the item distribution), and then applies expansion factors to each logit distribution. This is done in one operation.

Iterative PROX follows a similar sequence, but in small steps. Here the item difficulty logit distribution is reset (shifted) to the zero logit location each step. The average student ability mean of one iteration becomes the average item difficulty mean on the next iteration (which is then subtracted to reset the item difficulty mean to zero).

Double click any estimated ability or difficulty measure cell, after the first iteration, to see the following algorithms outlined for each cell in an Excel spreadsheet, http://www.nine-patch.com/download/CIPROX.xls. Change the constant value (2.647) to see the entire sheet recalculate.

The combined shift and expansion factor algorithm for student ability becomes revised student ability estimate (Ar) = current item difficulty mean (Dm) + current item difficulty standard deviation (DSD) x the initial student ability (Ai) logit (ln right/wrong ratio). For item difficulty it becomes revised item difficulty estimate (Dr) = current student ability mean (Am) + current student ability standard deviation (ASD) x the initial item difficulty (Di) logit (ln wrong/right ratio).

Again after adding in constants to match logistic and normal distributions, the algorithm used here for student ability location became Ar = Dm + SQRT(1 + (DSD^2/2.9)) * Ai. For item difficulty location it became Dr = (Am - SQRT(1 + (ASD^2/2.9)) * Di )* -1 on the Excel spreadsheet. These yielded the same check sums (Extreme 5 Range for Person and Item) printed on Winsteps Table 0.2 when the constant of 2.9 was adjusted to 2.647.

A plot of the average student ability and the average item difficulty measures, for each iteration, shows an orderly expansion from one iteration to the next. The spread from iterative PROX is less than for non-iterative PROX as the fifth iteration stopped the analysis to allow JMLE estimations to finish the analysis.

The black box chart shows the final locations for student ability and item difficulty means approaching 50%. The primary interest in the results of five PROX iterations is that the student ability and item difficulty means (47 & 48) match non-iterative PROX results (47 & 48) more closely than Winsteps results (40 & 47). The difference in results between PUP non-iterative PROX and Winsteps is then not found in this first, iterative PROX, stage of Winsteps. The difference must be in the second, JMLE, stage in Winsteps.