﻿ Whexact Wilson-Hilferty standardization = Yes

# Whexact Wilson-Hilferty standardization = Yes

Some versions of Facets are restricted to WHEXACT=NO.

ZSTD INFIT is the "Standardized Weighted Mean Square" shown at the bottom of RSA p. 100.

ZSTD OUTFIT is the "Standardized Unweighted Mean Square" based on the terms on RSA p. 100.

The Wilson-Hilferty transformation converts mean-square values to their equivalent "standardized" normal deviates. See RSA p. 101 and www.rasch.org/rmt/rmt162g.htm

In Facets, the chi-square degrees of freedom (2/qi^2) are allowed to be less than 1.0.

Under certain circumstances, Wilson-Hilferty can correctly report the paradoxical finding that the mean-squares indicate overfit, but the normal deviate indicates underfit. To allow this possibility, specify WHEXACT=Y. To suppress it, specify WHEXACT=N. The final q/3 term is omitted from the transformation.

With WHEXACT= Yes, notice that the left side of this plot is non-centered vertically:

Example 1: You have obviously contradictory mean-square and standardized fit statistics. The mean-square is much less than 1, indicating considerable overfit, but the Zstd is much more than 0, indicating significant underfit. Here the Wilson-Hilferty approximation has failed. Specify WHEXACT=No

 WHEXACT=Yes WHEXACT=No

Example 2: A person takes a test of 20 dichotomous items and obtains an unweighted chi-square value of 19.5.

WHEXACT=Y
The OUTFIT mean-square is 0.975, i.e., apparently slightly overfitting. The exact normal deviate is .03, i.e., very slightly underfitting.

WHEXACT=N
The OUTFIT mean-square is 0.975, i.e., apparently slightly overfitting. The reported normal deviate is -.08, i.e., slightly overfitting.

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