The unconstrained form of the 4PL generalized logistic regression model for
probability of correct answer (i.e., \(Y_{pi} = 1\)) using IRT parameterization
is
$$P(Y_{pi} = 1|X_p, G_p) = (c_{iR} \cdot G_p + c_{iF} \cdot (1 - G_p)) +
(d_{iR} \cdot G_p + d_{iF} \cdot (1 - G_p) - c_{iR} \cdot G_p - c_{iF} \cdot
(1 - G_p)) / (1 + \exp(-(a_i + a_{i\text{DIF}} \cdot G_p) \cdot
(X_p - b_p - b_{i\text{DIF}} \cdot G_p))), $$
where \(X_p\) is the matching criterion (e.g., standardized total score) and
\(G_p\) is a group membership variable for respondent \(p\).
Parameters \(a_i\), \(b_i\), \(c_{iR}\), and \(d_{iR}\)
are discrimination, difficulty, guessing, and inattention for the reference
group for item \(i\). Terms \(a_{i\text{DIF}}\) and \(b_{i\text{DIF}}\)
then represent differences between the focal and reference groups in
discrimination and difficulty for item \(i\). Terms \(c_{iF}\), and
\(d_{iF}\) are guessing and inattention parameters for the focal group for
item \(i\). In the case that there is no assumed difference between the
reference and focal group in the guessing or inattention parameters, the terms
\(c_i\) and \(d_i\) are used.
Alternatively, intercept-slope parameterization may be applied:
$$P(Y_{pi} = 1|X_p, G_p) = (c_{iR} \cdot G_p + c_{iF} \cdot (1 - G_p)) +
(d_{iR} \cdot G_p + d_{iF} \cdot (1 - G_p) - c_{iR} \cdot G_p - c_{iF} \cdot
(1 - G_p)) / (1 + \exp(-(\beta_{i0} + \beta_{i1} \cdot X_p +
\beta_{i2} \cdot G_p + \beta_{i3} \cdot X_p \cdot G_p))), $$
where parameters \(\beta_{i0}, \beta_{i1}, \beta_{i2}, \beta_{i3}\) are
intercept, effect of the matching criterion, effect of the group membership,
and their mutual interaction, respectively.
The model argument offers several predefined models. The options are as follows:
Rasch for 1PL model with discrimination parameter fixed on value 1 for both groups,
1PL for 1PL model with discrimination parameter set the same for both groups,
2PL for logistic regression model,
3PLcg for 3PL model with fixed guessing for both groups,
3PLdg for 3PL model with fixed inattention for both groups,
3PLc (alternatively also 3PL) for 3PL regression model with guessing parameter,
3PLd for 3PL model with inattention parameter,
4PLcgdg for 4PL model with fixed guessing and inattention parameter for both groups,
4PLcgd (alternatively also 4PLd) for 4PL model with fixed guessing for both groups,
4PLcdg (alternatively also 4PLc) for 4PL model with fixed inattention for both groups,
or 4PL for 4PL model.
Three possible parameterizations can be specified in the
"parameterization" argument: "irt" returns the IRT parameters
of the reference group and differences in these parameters between the
reference and focal group. Parameters of asymptotes are printed separately
for the reference and focal groups. "is" returns intercept-slope
parameterization. Parameters of asymptotes are again printed separately for
the reference and focal groups. "logistic" returns parameters in
logistic regression parameterization as in the glm
function, and it is available only for the 2PL model.