# Computation of OUTFIT and INFIT Statistics

 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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Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
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