# What do Infit and Outfit, Mean-square and Standardized mean?

These are all "fit" statistics. In a Rasch context they indicate how accurately or predictably data fit the model. Dichotomous fit statistics. Polytomous fit statistics.

Infit means inlier-sensitive or information-weighted fit. This is more sensitive to the pattern of responses to items targeted on the person, and vice-versa. For example, infit reports overfit for Guttman patterns, underfit for alternative curricula or idiosyncratic clinical groups. These patterns can be hard to diagnose and remedy.

Outfit means outlier-sensitive fit. This is more sensitive to responses to items with difficulty far from a person, and vice-versa. For example, outfit reports overfit for imputed responses, underfit for lucky guesses and careless mistakes. These are usually easy to diagnose and remedy.

Mean-square fit statistics show the size of the randomness, i.e., the amount of distortion of the measurement system. 1.0 is their expected values. Values less than 1.0 indicate observations are too predictable (redundancy, data overfit the model). Values greater than 1.0 indicate unpredictability (unmodeled noise, data underfit the model). Statistically, mean-squares are chi-square statistics divided by their degrees of freedom. Mean-squares are always positive. Mean-square ranges encountered in practice have been reported at Reasonable Mean-Square Fit Values.

In general, mean-squares near 1.0 indicate little distortion of the measurement system, regardless of the standardized value. Evaluate mean-squares high above 1.0 before mean-squares much below 1.0, because the average mean-square is usually forced to be near 1.0.

Outfit problems are less of a threat to measurement than Infit ones, but are easier to manage. To evaluate the impact of any misfit, replace suspect responses with missing values and examine the resultant changes to the measures.

Standardized fit statistics (Zstd in some computer output) are t-tests of the hypothesis "Do the data fit the model (perfectly)?" These are reported as z-scores, i.e., unit normal deviates. They show the improbability of the data, i.e., its significance, if the data actually did fit the model. 0.0 are their expected values. Less than 0.0 indicates too predictable. More than 0.0 indicates lack of predictability. Standardized values are positive and negative. For the relationship between mean-squares and standardized statistics, see www.rasch.org/rmt/rmt171n.htm

Standardized fit statistics are usually obtained by converting the mean-square statistics to the normally-distributed z-standardized ones by means of the Wilson-Hilferty cube root transformation.

Anchored runs:
Anchor values may not exactly accord with the current data. To the extent that they don't, fit statistics can be misleading. Anchor values that are too central for the current data tend to make the data appear to fit too well. Anchor values that are too dispersed for the current data tend to make the data appear noisy.

John M. Linacre

Mean-square ValueImplication for Measurement
> 2.0Distorts or degrades the measurement system. May be caused by only one or two observations.
1.5 - 2.0Unproductive for construction of measurement, but not degrading.
0.5 - 1.5Productive for measurement.
< 0.5Less productive for measurement, but not degrading. May produce misleadingly high reliability and separation coefficients.

Standardized ValueImplication for Measurement
≥ 3Data very unexpected if they fit the model (perfectly), so they probably do not. But, with large sample size, substantive misfit may be small.
2.0  -  2.9Data noticeably unpredictable.
-1.9  -  1.9Data have reasonable predictability.
≤ -2Data are too predictable. Other "dimensions" may be constraining the response patterns.

What do Infit and Outfit, Mean-square and Standardized mean? Linacre JM. … 16:2 p.878

What do Infit and Outfit, Mean-square and Standardized mean? Linacre JM. … Rasch Measurement Transactions, 2002, 16:2 p.878

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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
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
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