This command computes the generalized logistic regression statistic (Magis, Raiche, Beland and Gerard, 2010) 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).