Reproduced from Rating Scale Analysis (Wright BD & Masters GN, Chicago, MESA Press, 1982, p.100) with permission |
Page 100 of Rating Scale Analysis (Wright and Masters 1982), which summarizes the calculation of outfit and infit statistics, is displayed in a revised form adjacent.
The Rasch residuals are the differences between the observations and their expected values according to the Rasch model. Outfit is based on a sum of squared standardized residuals. Standardized residuals are modeled to approximate a unit normal distribution. Their sum of squares approximates a χ² distribution. Dividing this sum by its degrees of freedom yields a mean-square value, OUTFIT MEANSQ, with expectation 1.0 and range 0 to infinity. Values larger than 1.0 indicate unmodeled noise. Values are on a ratio scale, so that 1.2 indicates 20% excess noise. Values less than 1.0 indicate overfit of the data to the model, i.e., the observations are too predictable.
A Wilson-Hilferty transformation standardizes the mean-square into its OUTFIT ZSTD value. This approximates a unit-normal distribution corresponding to a t-statistic with infinite degrees of freedom. It is a test of the hypothesis "these data fit the Rasch model (exactly)."
Infit is an information-weighted form of outfit. The weighting reduces the influence of less informative, low variance, off-target responses. It is also computed in INFIT MEANSQ and INFIT ZSTD forms.
This computation is the same for persons and items with appropriate adjustment of subscripts and summations.
Note: Practical considerations in the computation of the ZSTD values.
1. the lower limit of the degrees of freedom of the mean-square statistic is set at 1.
qi is not allowed to be more than √2 for both
2. the contribution to the model variance of the OUTFIT MEANSQ by one observation is not allowed to
overwhelm the contributions of the other observations. W²ni is not allowed to be less than 0.00001
Computation of OUTFIT and INFIT Statistics. Wright BD, Masters GN. Rasch Measurement Transactions, 1990, 3:4 p.84-5
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