Winsteps reports mean-square fit statistics. These are chi-square fit statistics divided by their degrees of freedom.
Mean-squares: the expected value of a mean-square is 1.0. When the mean-square = 1.0, then the unexplained variance in the data is the same size as the unexplained variance (randmness) predicted by the Rasch model. Mean-squares < 1.0 indicate over-fit of the data to the model = the Rasch model is explaining the data too well. Mean-squares > 1.0 indicate under-fit of the data to the model = there is too much unexplained variance in the data.
Unexpectedness: Winsteps test the hypothesis that the mean-square = 1.0 except for the predicted statistical variability. The outcome of the hypothesis test is reported as a unit-normal deviate, ZSTD. Values outside ±1.96 indicate p<.05 for the double-sided test, and the null hypothesis of "model fit" is rejected. Always investigate underfit (Mean-square > 1.0, ZSTD >0) before overfit. Underfit is a much greater threat to measurement than overfit.
Chi-squares, d.f., probabilities: instead of mean-squares and ZSTD, you may want to report these.
Chi-square = Mean-square * d.f.
d.f. are not usually integers, because of the ordinal, non-linear nature of the data.
To report probabilities, LOCAL=Prob. For instance, in Table 14:
To report probabilities, LOCAL=Prob, and d.f., IFILE= or PFILE=, and the Field Selection Dialog Box
To report Chi-squares in IFILE= or PFILE=, instead of Mean-squares: LOCAL=Prob, Chisquare = Yes
Winsteps Chi-squared values and Excel CHISQ.DIST.RT(VALUE, DF) and CHISQ.INV.RT(PROB., DF)
I checked two values in the Winsteps output, one has low d.f. and the other high d.f., and they agreed with CHISQ.INV.RT(PROB., DF) to within the rounding error.
Winsteps approximates the chi-squared function using an algorithm from Abramovitz and Stegun "Handbook of Mathematical Functions" p.299. The book was published in 1964, but is still considered authoritative. Winsteps accepts real values for the d.f., but Excel truncates real d.f. values to integers (and probably computes the gamma function). Consequently Winsteps and Excel output may differ slightly.
