Table 30.4 Differential item functioning DIF fit summary 
In Table 30.1  the hypothesis is "this item has the same difficulty for two groups"
In Table 30.2, 30.3  the hypothesis is "this item has the same difficulty as its average difficulty for all groups"
In Table 30.4  the hypothesis is "this item has no overall DIF across all groups"
Table 30.4 summarizes the CLASS/GROUP Differential Item Functioning statistics for each item shown in Table 30.2. Table 30.2 shows a tstatistic for each CLASS/GROUP. These are summarized as chisquare statistics for each item, indicating whether the observed DIF within each item is due to chance alone. The null hypothesis is that the DIF is statistically zero across the classes. Please do not use anchoring (IAFILE=, PAFILE=, SAFILE=) when investigating this Table.
DIF class/group specification is: DIF=@GENDER

 KID SUMMARY DIF BETWEENCLASS TAP 
 CLASSES CHISQUARE D.F. PROB. UNWTD MNSQ t=ZSTD Number Name 

 2 .8772 1 .3490 .9200 .4132 6 341 
 2 2.2427 1 .1342 2.5792 1.2592 7 1432 
 2 1.9207 1 .1658 1.9963 1.0211 13 14324 

KID CLASSES/GROUPS is the count of person CLASSES or GROUPS for the item: 2 = Male and Female.
SUMMARY DIF CHISQUARE is the sum of the tstatistic values from Table 30.2, squared and normalized. See Example 1 below.
D.F. is the degrees of freedom, the count of CLASSES or GROUPS contributing to the chisquare less 1.
PROB. is the probability of the chisquare. Values less than .05 indicate statistical significance.
BETWEENCLASS are BetweenGroup Fit Statistics, testing the hypothesis: "The dispersion of the group measures accords with Rasch model expectations."
UNWTD MNSQ is the unweighted meansquare (chisquare divided by its degrees of freedom). It is the size of the misfit (expectation = 1.0, overfit <1.0, underfit >1.0).
t=ZSTD is the significance of the MEANSQUARE standardized as a unitnormal deviate (tstatistic with infinite degrees of freedom).
TAP is the item. Number is the item entry number. Name is the item label.
ItemTrait ChiSquare
The SUMMARY DIF CHISQUARE values in Table 30.4 are equivalent to the itemtrait chisquare statistics reported by RUMM with DIF=MA3 (or however many strata are chosen in RUMM). In RUMM, the trait CLASSes are obtained by ordering the persons by measure, omitting extreme scores, and dividing the ordered list as equally as possible into equal size classes, including all persons with the same measure in the same class.
The Root Mean Square Error of Approximation (RMSEA)
RMSEA is a transformation of the ItemTrait ChiSquare and so of the SUMMARY DIF CHISQUARE with DIF=MA3. Its computation is
RMSEA = √ max( [((χ²/df)  1)/(N  1)] , 0)
where
χ² = SUMMARY DIF CHISQUARE
d.f. = D.F.
N is the person sample size
Example 1. SUMMARY DIF CHISQUARE.
Let's do an analysis of Exam1.txt with Udecimals=4. Output Table 30 with DIF=@Gender
In Table 30.2, the values for item 13:
CLASS F: COUNT= 18, DIF SIZE= .7268, DIF S.E.= .6764
CLASS M: COUNT= 16, DIF SIZE= .7075, DIF S.E.= .7497
So the tstatistics are approximately:
CLASS F: t = .7268/.6764 with (181) d.f. > 1.0745 with 17 d.f.
CLASS M: t = .7075/.7497 with (161) d.f. > .9437 with 15 d.f.
Squared and normalized. The normalizing transformation is Peizer and Pratt (1968) in the Handbook of the Normal Distribution, Second Edition (Patel and Read), p. 214.
CLASS F: t**2 = 1.1546 > normalized(t**2) = 1.0843
CLASS M: t**2 = .8905 > normalized(t**2) = .8363
Chisquared = 1.0843 + .8363 = 1.9207 with (21) = 1 d.f.
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