Mantel and Mantel-Haenszel DIF statistics

Mantel-Haenszel is the industry-standard DIF statistic, but it expects complete data because it stratifies the data by raw scores. Please Google "Mantel-Haenszel". The Winsteps implementation is slightly different because it stratifies by person measure (same as raw scores for complete data), so it is robust against missing data.

 

If the data are incomplete (missing data), then there are alternative methods, such as the Rasch-Welch method. There is no industry-standard method in this situation. Most methods require deletion of persons with incomplete response strings, see epm.sagepub.com/content/69/1/18.short- Rasch-Welch uses all available data.

 


 

Differential item functioning (DIF) an be investigated using log-odds estimators, Mantel-Haenszel (1959) for dichotomies or Mantel (1963) for polytomies. The sample is divided into difference classification groups (also called reference groups and focal groups) which are shown in Table 30 and specified with DIF=. And then sliced into strata by ability measure (equivalent to raw score for complete data).

 

The usual M-H computation stratifies the sample by raw scores, so it works with case-wise deletion of cases with missing data. Winsteps stratifies cases by measure, so cases with missing data are stratified at their estimated measure. For complete data and thin-slicing, the conventional M-H computation and the Winsteps M-H computation produce the same numbers. With missing date or thick-slicing, the conventional M-H computations and the Winsteps M-H computations may differ.

 

M-H and the t-tests in Winsteps should produce the same results, because they are based on the same logit-linear theory. But, in practice, M-H will be more accurate if the data are complete and there are large numbers of subjects at every score level, so called "thin" matching. Under other circumstances, M-H may not be estimable, or must use grouped-score "thick" matching, in which case the t-test method will probably be more accurate. Similar conclusions can also be inferred from https://www.eric.ed.gov/ERICWebPortal/custom/portlets/recordDetails/detailmini.jsp?_nfpb=true&_&ERICExtSearch_SearchValue_0=ED334230&ERICExtSearch_SearchType_0=no&accno=ED334230

 

MHSLICE= controls the width of each slice, thin or thick. MHSLICE= specifies the width of the slice (in logits) of the latent variable be included in each cross-tab. The lower end of the lowest slice is always the lowest observed person measure.

 

MHSLICE = 0 bypasses Mantel-Haenszel or Mantel computation.

 

MHSLICE = .1 logits and smaller. The latent variable is stratified into thin slices. This corresponds to the slicing by raw scores with complete data

 

MHSLICE = 1 logit and larger. The latent variable is stratified into thick slices.

 

For each slice, a cross-tabulation is constructed for each pair of person classifications against each scored response level. An odds-ratio is computed from the cross-tab. Zero and infinite ratios are ignored. A homogeneity chi-square is also computed when possible.

 

Thin slices are more sensitive to small changes in item difficulty across person classifications, but more persons are ignored in inestimable cross-tabs. Thick slices are more robust because fewer persons are ignored. Use the Specification pull-down menu to set different values of MHSLICE= and then produce the corresponding Table 30.

 

In principle, when the data fit the Rasch model, the Mantel and Mantel-Haenszel estimators should concur with the Rasch DIF contrast measures. The Rasch DIF contrast weights each person equally. Mantel weights each cross-tabulation equally. Thus when the DIF estimates disagree, it indicates that the DIF in the data is non-uniform with ability level.

 

Computation:

Person classification groups are A, B, ... They are compared pairwise. Starting from the lowest person measure, each slice is MHSLICE= logits wide. There are K slices up through the highest person measure. For the target item, in the kth slice and comparing classification groups A and B, with categories renumbered from 0 to simplify the computation,

 

Person Classification Groups, g = 1, 2,  by j = 0, J categories by k = 1, K ability strata (slices)

 

Rating-Scale Category 

 

j = 0

1

2

3

j

J

Total:

Classification Group

Score = Y0

Y1

Y2

Y3

Yj

YJ

 

Reference 

g = 1

nk10

nk11

nk12

nk13

nk1j

nk1J

Nk1

Focal 

2

nk20

nk21

nk22

nk23

nk2j

nk2J

Nk2

Total:

Mk0

Mk1

Mk2

Mk3

Mkj

MkJ

Tk

 

 

Then the Mantel (1963) or Mantel-Haenszel (1959) DIF chi-square for the target item is:

 

where continuity = 0.5 for dichotomies and = 0 for polytomies.

 

The Mantel-Haenszel (1959) log-odds estimator for dichotomies and the Liu-Agresti (1996) cumulative log-odds estimator (CUMLOR) for polytomies are:

 

 

Liu, I-M, & Agresti, A. (1996).  Mantel-Haenszel-type inference for cumulative odds ratios with a stratified ordinal response.  Biometrics, 52, 1223-1234.

Mantel N. (1963) Chi-square tests with one degree of freedom: extensions of the Mantel-Haenszel procedure. J Amer Stat Assoc 58, 690-700.

Mantel, N. and Haenszel, W. (1959) Statistical aspects of the analysis of data from retrospective studies of disease. J Natl Cancer Inst 22, 719-748.

 

ETS DIF Category

with DIF Contrast and DIF Statistical Significance

C = moderate to large

|DIF| ≥  0.64 logits

prob( |DIF| ≤ 0.43 logits ) ≤ .05 (2-sided)

approximately: |DIF| > 0.43 logits + 2 * DIF S.E.

B = slight to moderate

|DIF| ≥ 0.43 logits

prob( |DIF| = 0 logits ) ≤ .05 (2-sided)

approximately: |DIF| > 2 * DIF S.E

A = negligible

-

-

C-, B- = DIF against focal group; C+, B+ = DIF against reference group

ETS (Educational Testing Service) use Delta δ units.

1 logit = 2.35 Delta δ units. 1 Delta δ unit = 0.426 logits.

Zwick, R., Thayer, D.T., Lewis, C. (1999) An Empirical Bayes Approach to Mantel-Haenszel DIF Analysis. . Journal of Educational Measurement, 36, 1, 1-28

More explanation at www.ets.org/Media/Research/pdf/RR-12-08.pdf pp. 3,4

 

Example:

----------------------------------------------------------------------------------------------------------------

| PERSON   DIF   DIF   PERSON   DIF   DIF      DIF    JOINT     Welch      Mantel-Haenszel Size ITEM           |

| CLASS  MEASURE S.E.  CLASS  MEASURE S.E.  CONTRAST  S.E.   t  d.f. Prob. Chi-squ Prob. CUMLOR Number  Name   |

|--------------------------------------------------------------------------------------------------------------|

| A        1.47   .28  P        2.75   .33     -1.28   .43 -2.96 102 .0038   7.198 .0073  -1.20      1 Response|

|--------------------------------------------------------------------------------------------------------------|

Size of Mantel-Haenszel slice = .100 logits

 

title="MH computation" 

; d.f.=1 chi=7.198 p=0.073

; log-odds = -1.20

codes=01

clfile=*

1 Better

0 Same

*

item1=1

name1=1

NI=1

pweight=$s9w2 ; weighting substitutes for entering multiple records

PAFILE=$S6W1 ; anchoring forces stratification

DIF = $4W1 ; cross-tab by Gender, F or M

&end

Response

;234567890

END LABELS    

1 FA 1  16

0 FA 1  11

1 FP 1   5

0 FP 1  20

1 MA 2  12

0 MA 2  16

1 MP 2   7

0 MP 2  19


Help for Winsteps Rasch Measurement and Rasch Analysis Software: www.winsteps.com. Author: John Michael Linacre

Facets Rasch measurement software. Buy for $149. & site licenses. Freeware student/evaluation Minifac download
Winsteps Rasch measurement software. Buy for $149. & site licenses. Freeware student/evaluation Ministep download

Rasch Books and Publications
Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, 2nd Edn, 2024 George Engelhard, Jr. & Jue Wang Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes Statistical Analyses for Language Testers (Facets), Rita Green Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind Rasch Measurement: Applications, Khine Winsteps Tutorials - free
Facets Tutorials - free
Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan
Other Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse Rasch Measurement Theory Analysis in R, Wind, Hua Applying the Rasch Model in Social Sciences Using R, Lamprianou El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Rasch Models for Measurement, David Andrich Constructing Measures, Mark Wilson Best Test Design - free, Wright & Stone
Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias Diseño de Mejores Pruebas - free, Spanish Best Test Design A Course in Rasch Measurement Theory, Andrich, Marais Rasch Models in Health, Christensen, Kreiner, Mesba Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen
As an Amazon Associate I earn from qualifying purchases. This does not change what you pay.

facebook Forum: Rasch Measurement Forum to discuss any Rasch-related topic

To receive News Emails about Winsteps and Facets by subscribing to the Winsteps.com email list,
enter your email address here:

I want to Subscribe: & click below
I want to Unsubscribe: & click below

Please set your SPAM filter to accept emails from Winsteps.com
The Winsteps.com email list is only used to email information about Winsteps, Facets and associated Rasch Measurement activities. Your email address is not shared with third-parties. Every email sent from the list includes the option to unsubscribe.

Questions, Suggestions? Want to update Winsteps or Facets? Please email Mike Linacre, author of Winsteps mike@winsteps.com


State-of-the-art : single-user and site licenses : free student/evaluation versions : download immediately : instructional PDFs : user forum : assistance by email : bugs fixed fast : free update eligibility : backwards compatible : money back if not satisfied
 
Rasch, Winsteps, Facets online Tutorials


 

 

 

Our current URL is www.winsteps.com

Winsteps® is a registered trademark