Calculates Mantel-Haenszel statistics for DIF detection.
mantelHaenszel(data, member, match = "score", correct = TRUE, exact = FALSE,
anchor = 1:ncol(data))numeric: the data matrix (one row per subject, one column per item).
numeric: the vector of group membership with zero and one entries only. See Details.
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.
logical: should the continuity correction be used? (default is TRUE).
logical: should an exact test be computed? (default is FALSE).
a vector of integer values specifying which items (all by default) are currently considered as anchor (DIF free) items. See Details.
A list with several arguments:
the vector of the Mantel-Haenszel DIF statistics (either asymptotic or exact).
the vector of the (asymptotic) Mantel-Haenszel estimates of the common odds ratios. Returned only if
exact is FALSE.
the (asymptotic) variance of the \(\lambda_{MH}\) statistic. Returned only if exact is
FALSE.
the exact P-values of the MH test. Returned only if exact is TRUE.
a character string, either "score" or "matching variable" depending on the match argument.
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 \(\alpha_{MH}\) 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
\(\alpha_{MH}\), say \(\lambda_{MH}\), 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 \(\lambda_{MH}\) (Penfield and Camilli, 2007).
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. 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. 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. 10.2307/2531824
# 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))
# }
# NOT RUN {
# }
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