mantelHaenszel: Mantel-Haenszel DIF statistic
In difR: Collection of Methods to Detect Dichotomous Differential Item Functioning (DIF)

Description Usage Arguments Details Value Author(s) References See Also Examples

Calculates Mantel-Haenszel statistics for DIF detection.

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mantelHaenszel(data, member, match = "score", correct = TRUE, exact = FALSE,
  anchor = 1:ncol(data))

`data`	numeric: the data matrix (one row per subject, one column per item).
`member`	numeric: the vector of group membership with zero and one entries only. See Details.
`match`	specifies the type of matching criterion. Can be either `"score"` (default) to compute the test score, or any continuous or discrete variable with the same length as the number of rows of `data`. See Details.
`correct`	logical: should the continuity correction be used? (default is `TRUE`).
`exact`	logical: should an exact test be computed? (default is `FALSE`).
`anchor`	a vector of integer values specifying which items (all by default) are currently considered as anchor (DIF free) items. See Details.

This command basically computes the Mantel-Haenszel (1959) statistic in the specific framework of differential item functioning. It forms the basic command of difMH and is specifically designed for this call.

The data are passed through the data argument, with one row per subject and one column per item.

Missing values are allowed for item responses (not for group membership) but must be coded as NA values. They are discarded from sum-score computation.

The vector of group membership, specified with member argument, must hold only zeros and ones, a value of zero corresponding to the reference group and a value of one to the focal group.

The matching criterion can be either the test score or any other continuous or discrete variable to be passed in the mantelHaenszel function. This is specified by the match argument. By default, it takes the value "score" and the test score (i.e. raw score) is computed. The second option is to assign to match a vector of continuous or discrete numeric values, which acts as the matching criterion. Note that for consistency this vector should not belong to the data matrix.

By default, the continuity correction factor -0.5 is used (Holland and Thayer, 1988). One can nevertheless remove it by specifying correct=FALSE.

By default, the asymptotic Mantel-Haenszel statistic is computed. However, the exact statistics and related P-values can be obtained by specifying the logical argument exact to TRUE. See Agresti (1990, 1992) for further details about exact inference.

Option anchor sets the items which are considered as anchor items for computing Mantel-Haenszel statistics. Items other than the anchor items and the tested item are discarded. anchor must hold integer values specifying the column numbers of the corresponding anchor items. It is primarily designed to perform item purification.

In addition to the Mantel-Haenszel statistics to identify DIF items, mantelHaenszel computes the estimates of the common odds ratio α_{MH} which are used for measuring the effect size of the items (Holland and Thayer, 1985, 1988). They are returned in the resAlpha argument of the output list. Moreover, the logarithm of α_{MH}, say λ_{MH}, is asymptotically distributed and its variance is computed and returned into the varLambda argument. Note that this variance is the one proposed by Philips and Holland (1987), since it seems the most accurate expression for the variance of λ_{MH} (Penfield and Camilli, 2007).

A list with several arguments:

`resMH`	the vector of the Mantel-Haenszel DIF statistics (either asymptotic or exact).
`resAlpha`	the vector of the (asymptotic) Mantel-Haenszel estimates of the common odds ratios. Returned only if `exact` is `FALSE`.
`varLambda`	the (asymptotic) variance of the λ_{MH} statistic. Returned only if `exact` is `FALSE`.
`Pval`	the exact P-values of the MH test. Returned only if `exact` is `TRUE`.
`match`	a character string, either `"score"` or `"matching variable"` depending on the `match` argument.

Sebastien Beland
Collectif pour le Developpement et les Applications en Mesure et Evaluation (Cdame)
Universite du Quebec a Montreal
sebastien.beland.1@hotmail.com, http://www.cdame.uqam.ca/
David Magis
Department of Psychology, University of Liege
Research Group of Quantitative Psychology and Individual Differences, KU Leuven
David.Magis@uliege.be, http://ppw.kuleuven.be/okp/home/
Gilles Raiche
Collectif pour le Developpement et les Applications en Mesure et Evaluation (Cdame)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca, http://www.cdame.uqam.ca/

Agresti, A. (1990). Categorical data analysis. New York: Wiley.

Agresti, A. (1992). A survey of exact inference for contingency tables. Statistical Science, 7, 131-177. doi: 10.1214/ss/1177011454

Holland, P. W. and Thayer, D. T. (1985). An alternative definition of the ETS delta scale of item difficulty. Research Report RR-85-43. Princeton, NJ: Educational Testing Service.

Holland, P. W. and Thayer, D. T. (1988). Differential item performance and the Mantel-Haenszel procedure. In H. Wainer and H. I. Braun (Ed.), Test validity. Hillsdale, NJ: Lawrence Erlbaum Associates.

Magis, D., Beland, S., Tuerlinckx, F. and De Boeck, P. (2010). A general framework and an R package for the detection of dichotomous differential item functioning. Behavior Research Methods, 42, 847-862. doi: 10.3758/BRM.42.3.847

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

Penfield, R. D., and Camilli, G. (2007). Differential item functioning and item bias. In C. R. Rao and S. Sinharray (Eds.), Handbook of Statistics 26: Psychometrics (pp. 125-167). Amsterdam, The Netherlands: Elsevier.

Philips, A., and Holland, P. W. (1987). Estimators of the Mantel-Haenszel log odds-ratio estimate. Biometrics, 43, 425-431. doi: 10.2307/2531824

difMH, dichoDif

## Not run: 

 # Loading of the verbal data
 data(verbal)

 # With and without continuity correction
 mantelHaenszel(verbal[,1:24], verbal[,26])
 mantelHaenszel(verbal[,1:24], verbal[,26], correct = FALSE)
 
 # Exact test
 mantelHaenszel(verbal[,1:24], verbal[,26], exact = TRUE)

 # Removing item 6 from the set of anchor items
 mantelHaenszel(verbal[,1:24], verbal[,26], anchor = c(1:5,7:24))
 
## End(Not run)