# RUMM2020 Item-Trait Chi-Square and Winsteps DIF Size

Obtaining equivalent numerical results from different software packages can be challenging. Item-trait interactions are an example. The RUMM2020 Item Fit Table shows the item-fit output for item I0104 from a RUMM2020 analysis. The Location is the Rasch item difficulty estimate with SE being its standard-error precision. The FitResid is the standardized sum of squared residuals with DF being its estimated degrees of freedom. FitResid is equivalent to the standardized OUTFIT statistic of Winsteps. Item-Trait Interaction

The ChiSq is the item-trait interaction. In this example the latent trait is stratified into four class intervals each containing a trait-group of approximately one quarter of the total person sample. Since there are 4 intervals, there are three degrees of freedom, DF, for the chi-square as indicated. The chi-square is computed from a comparison of the observed overall performance of each trait-group on the item with its expected performance. This quantifies the size of the departure of the empirical item characteristic curve from its model values, so identifying the magnitude of the item-by-trait (item-by-ability level) interaction for this item. Prob reports the statistical probability of observing the chi-square value (or worse) when the data fit the Rasch model. In this example, the chi-square has 3 degrees of freedom and so has an expected value of 3.0. Its observed value is 21.707, with a probability of that value of larger being observed by chance of only 0.000076. So we would reject the null hypothesis that the overall performance of the trait-groups fits the Rasch model. We are observing an item-trait interaction for item I0104.

This item-trait chi-square is featured in RUMM2020 documentation as an indicator of item behavior, more so than the FitResid, but there is no obviously equivalent statistic currently reported by Winsteps. This can be awkward when research teams are employing both software packages. Here is how to generate the equivalent statistic in Winsteps:

This procedure is now implemented as Winsteps Table 30.4 with \$DIF=MA3.

1. Decide on the number of trait-groups. 4 here.

2. Order the persons by measure (location). Writing the person-measure PFILE to Excel facilitates these steps.

3. Omit extreme scores. These cannot show an interaction.

4. Stratify the person-ability range into trait-groups of as equal numerical size as possible, keeping all persons with the same measure in the same group.

4. Number the trait-groups and put the trait-group number into each person label.

5. Perform a DIF analysis of item by trait-group-number.

6. Obtain the t-statistic for each item-trait DIF effect.

7. For each item, square and sum the t-statistics for the item-trait groups. This is the RUMM2020 chi-square.

8. The chi-square d.f. is the count of trait-groups less one.

In our example, the Winsteps DIF Table shows each trait-group as a Person Class. The Observations Count is the number of persons in the group. Average is their average rating. Baseline Expect is the expected value of the Observations Average. Measure is the item difficulty measure corresponding to the Baseline Expect rating on this item, Item 104. It is expected to be the same for every trait-group. The DIF Score is the difference between the Observations Average and Baseline Expect ratings. The DIF Measure is the item difficulty that would produce the Observations Average. So that DIF Size is the difference between the Baseline Measure item difficulty and the item difficulty observed for this group, the DIF Measure. S.E. is the standard error of the DIF Size. The t-statistic is a hypothesis test that the DIF Size is due to chance alone, it is the DIF Size divided by its S.E.

The Winsteps t-statistic is approximately a unit-normal deviate. Squaring and summing the four of these for item I0104 amounts to 20.05, close to the RUMM2020 ChiSq of 21.707. Thus this procedure yields approximately the same number as the RUMM2020 ChiSq. Over 72% of the Winsteps chi-square is contributed by the 4th trait-group, indicating that the item-trait interaction is primarily due to the unexpectedly poor performance by the high ability group.

These statistics are sensitive to the number of item-trait groups, so verify that an item is defective (from an item-trait perspective) by replicating this process with different numbers of item-trait groups.

RUMM2020 Item Fit Table
SeqItemTypeLocationSEFitResidDFChiSqDFProb
104I0104Poly0.2460.1372.852228.5621.70730.000076
Winsteps DIF Table
Person
Class
ObservationsBaselineDIFItem
CountAverageExpectMeasureScoreMeasureSizeS.E.tNumberName
1
2
3
4
57
55
59
60
0.53
0.62
0.68
0.47
0.40
0.55
0.62
0.70
0.25
0.25
0.25
0.25
0.12
0.07
0.05
-0.23
-0.27
-0.04
0.02
1.24
-0.52
-0.29
-0.24
0.99
0.27
0.28
0.28
0.26
-1.92
-1.06
-0.85
3.81
104
104
104
104
I0104
I0104
I0104
I0104

John M. Linacre

RUMM2020 Item-Trait Chi-Square and Winsteps DIF Size. Linacre, J.M. … Rasch Measurement Transactions, 2007, 21:1 p. 1096

Rasch Publications
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
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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