genLogistik: Generalized logistic regression DIF statistic

In difR: Collection of Methods to Detect Dichotomous and Polytomous Differential Item Functioning (DIF)

Generalized logistic regression DIF statistic

Description

Calculates the "generalized logistic regression" likelihood-ratio or Wald statistics for DIF detection among multiple groups.

Usage

genLogistik(data, member, match = "score", anchor = 1:ncol(data), 
 	type = "both", criterion = "LRT") 
 

Arguments

data	numeric: the data matrix (one row per subject, one column per item).
member	numeric: the vector of group membership with zero and positive integer 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.
anchor	a vector of integer values specifying which items (all by default) are currently considered as anchor (DIF free) items. See Details.
type	a character string specifying which DIF effects must be tested. Possible values are "both" (default), "udif" and "nudif". See Details.
criterion	character: the type of test statistic used to detect DIF items. Possible values are "LRT" (default) and "Wald". See Details.

Details

This command computes the generalized logistic regression statistic (Magis, Raiche, Beland and Gerard, 2011) in the specific framework of differential item functioning among (J+1) groups and J is the number of focal groups. It forms the basic command of difGenLogistic and is specifically designed for this call.

The three possible models to be fitted are:

M_0: logit (\pi_i) = \alpha + \beta X + \gamma_i + \delta_i X

M_1: logit (\pi_i) = \alpha + \beta X + \gamma_i

M_2: logit (\pi_i) = \alpha + \beta X

where \pi_i is the probability of answering correctly the item in group i (i = 0, ..., J) and X is the matching criterion. Parameters \alpha and \beta are the common intercept and the slope of the logistic curves, while \gamma_i and \delta_i are group-specific parameters. For identification reasons the parameters \gamma_0 and \delta_0 of the reference group are set to zero. The set of parameters \{\gamma_i: i = 1, ..., J\} of the focal groups (g=i) represents the uniform DIF effect across all groups, and the set of parameters \{\delta_i: i = 1, ..., n\} is used to model nonuniform DIF effect across all groups. The models are fitted with the glm function.

The matching criterion can be either the test score or any other continuous or discrete variable to be passed in the Logistik 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.

Two tests are available: the Wald test and the likelihood ratio test. With the likelihood ratio test, two nested models are fitted and compared by means of Wilks' Lambda (or likelihood ratio) statistic (Wilks, 1938). With the Wald test, the model parameters are statistically tested using an appropriate contrast matrix. Each test is set with the criterion argument, with the values "LRT" and "Wald" respectively.

The argument type determines the type of DIF effect to be tested. The three possible values of type are: type="both" which tests the hypothesis H_0: \gamma_i = \delta_i=0 for all i; type="nudif" which tests the hypothesis H_0: \delta_i = 0 for all i; and type="udif" which tests the hypothesis H_0: \gamma_i = 0 | \delta_i = 0 for all i. In other words, type="both" tests for DIF (without distinction between uniform and nonuniform effects), while type="udif" and type="nudif" test for uniform and nonuniform DIF, respectively. Whatever the tested DIF effects, this is a simultaneous test of the equality of focal group parameters to zero.

The data are passed through the data argument, with one row per subject and one column per item. Missing values are allowed but must be coded as NA values. They are discarded from the fitting of the logistic models (see glm for further details).

The vector of group membership, specified with member argument, must hold only zeros and positive integers. The value zero corresponds to the reference group, and each positive integer value corresponds to one focal group. At least two different positive integers must be supplied.

Option anchor sets the items which are considered as anchor items for computing the logistic regression DIF 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 mainly designed to perform item purification.

In addition to the results of the fitted models (model parameters, covariance matrices, test statistics), Nagelkerke's R^2 coefficients (Nagelkerke, 1991) are computed for each model and the output returns the differences in these coefficients. Such differences are used as measures of effect size by the difGenLogistic command; see Gomez-Benito, Dolores Hidalgo and Padilla (2009), Jodoin and Gierl (2001) and Zumbo and Thomas (1997).

Value

A list with nine components:

stat	the values of the generalized logistic regression DIF statistics (that is, the likelihood ratio test statistics).
R2M0	the values of Nagelkerke's R^2 coefficients for the "full" model.
R2M1	the values of Nagelkerke's R^2 coefficients for the "simpler" model.
deltaR2	the differences between Nagelkerke's R^2 coefficients of the tested models. See Details.
parM0	a matrix with one row per item and 2+J*2 columns (where J is the number of focal groups), holding successively the fitted parameters \hat{\alpha}, \hat{\beta}, \hat{\gamma}_i and \hat{\delta}_i (i = 1, ..., J) of the "full" model (M_0 if type="both" or type="nudif", M_1 if type="udif").
parM1	the same matrix as parM0 but with fitted parameters for the "simpler" model (M_1 if type="nudif", M_2 if type="both" or type="udif").
covMat	a 3-dimensional matrix of size p x p x K, where p is the number of estimated parameters and K is the number of items, holding the p x p covariance matrices of the estimated parameters (one matrix for each tested item).
criterion	the value of the criterion argument.
match	a character string, either "score" or "matching variable" depending on the match argument.

Author(s)

David Magis
Data science consultant at IQVIA Belux
Brussels, Belgium
Sebastien Beland
Faculte des sciences de l'education
Universite de Montreal (Canada)
[e-mail]
Gilles Raiche
Universite du Quebec a Montreal
[e-mail]

References

Gomez-Benito, J., Dolores Hidalgo, M. and Padilla, J.-L. (2009). Efficacy of effect size measures in logistic regression: an application for detecting DIF. Methodology, 5, 18-25. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1027/1614-2241.5.1.18")}

Jodoin, M. G. and Gierl, M. J. (2001). Evaluating Type I error and power rates using an effect size measure with logistic regression procedure for DIF detection. Applied Measurement in Education, 14, 329-349. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1207/S15324818AME1404_2")}

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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.3758/BRM.42.3.847")}

Magis, D., Raiche, G., Beland, S. and Gerard, P. (2011). A logistic regression procedure to detect differential item functioning among multiple groups. International Journal of Testing, 11, 365–386. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/15305058.2011.602810")}

Nagelkerke, N. J. D. (1991). A note on a general definition of the coefficient of determination. Biometrika, 78, 691-692. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1093/biomet/78.3.691")}

Wilks, S. S. (1938). The large-sample distribution of the likelihood ratio for testing composite hypotheses. Annals of Mathematical Statistics, 9, 60-62. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/aoms/1177732360")}

Zumbo, B. D. and Thomas, D. R. (1997). A measure of effect size for a model-based approach for studying DIF. Prince George, Canada: University of Northern British Columbia, Edgeworth Laboratory for Quantitative Behavioral Science.

See Also

difGenLogistic, genDichoDif

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Creating four groups according to gender (0 or 1) and trait anger score
 # ("Low" or "High")
 # Reference group: women with low trait anger score (<=20)
 group <- rep(0,nrow(verbal))
 group[Anger>20 & Gender==0] <- 1
 group[Anger<=20 & Gender==1] <- 2
 group[Anger>20 & Gender==1] <- 3

 # Testing both types of DIF simultaneously
 # With all items
 genLogistik(verbal[,1:24], group)
 genLogistik(verbal[,1:24], group, criterion = "Wald")

 # Removing item 6 from the set of anchor items
 genLogistik(verbal[,1:24], group, anchor = c(1:5, 7:24))
 genLogistik(verbal[,1:24], group, anchor = c(1:5, 7:24), criterion = "Wald")

 # Testing nonuniform DIF effect
 genLogistik(verbal[,1:24], group, type = "nudif")
 genLogistik(verbal[,1:24], group, type = "nudif", criterion="Wald")

 # Testing uniform DIF effect
 genLogistik(verbal[,1:24], group, type = "udif")
 genLogistik(verbal[,1:24], group, type = "udif", criterion="Wald")

 # Using trait anger score as matching criterion
 genLogistik(verbal[,1:24], group, match = verbal[,25])
 
## End(Not run)
 

difR documentation built on June 8, 2025, 1:03 p.m.