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.

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