Calculates either DIF likelihood ratio statistics or F statistics for dichotomous data based on non-linear regression model (generalized logistic regression model).
NLR(Data, group, model, constraints = NULL, type = "all", method = "nls",
match = "zscore", anchor = 1:ncol(Data), start, p.adjust.method = "none", test = "LR",
alpha = 0.05, initboot = TRUE, nrBo = 20, sandwich = FALSE)data.frame or matrix: dataset which rows represent scored examinee answers ("1"
correct, "0" incorrect) and columns correspond to the items.
numeric: binary vector of group membership. "0" for reference group, "1" for
focal group.
character: generalized logistic regression model to be fitted. See Details.
character: which parameters should be the same for both groups. Possible values
are any combinations of parameters "a", "b", "c", and "d". Default value
is NULL. See Details.
character: type of DIF to be tested. Possible values are "all" for detecting
difference in any parameter (default), "udif" for uniform DIF only (i.e., difference in
difficulty parameter "b"), "nudif" for non-uniform DIF only (i.e., difference in
discrimination parameter "a"), "both" for uniform and non-uniform DIF (i.e.,
difference in parameters "a" and "b"), or combination of parameters "a",
"b", "c", and "d". Can be specified as a single value (for all items) or as
an item-specific vector.
character: method used to estimate parameters. The options are "nls" for
non-linear least squares (default) and "likelihood" for maximum likelihood method.
character or numeric: matching criterion to be used as estimate of trait. Can be
either "zscore" (default, standardized total score), "score" (total test score),
or numeric vector of the same length as number of observations in Data.
character or numeric: specification of DIF free items. A vector of item identifiers
(integers specifying the column number) specifying which items are currently considered as anchor
(DIF free) items. Argument is ignored if match is not "zscore" or "score".
numeric: initial values for estimation of parameters. If not specified, starting
values are calculated with startNLR function. Otherwise, list with as many
elements as number of items. Each element is a named numeric vector of length 8 representing initial
values for parameter estimation. Specifically, parameters "a", "b", "c", and
"d" are initial values for discrimination, difficulty, guessing, and inattention for reference
group. Parameters "aDif", "bDif", "cDif", and "dDif" are then differences
in these parameters between reference and focal group.
character: method for multiple comparison correction. Possible values are
"holm", "hochberg", "hommel", "bonferroni", "BH", "BY",
"fdr", and "none" (default). For more details see p.adjust.
character: test to be performed for DIF detection. Can be either "LR" for
likelihood ratio test of a submodel (default), "W" for Wald test, or "F" for F-test
of a submodel.
numeric: significance level (default is 0.05).
logical: in case of convergence issues, should be starting values re-calculated based on
bootstraped samples? (default is TRUE; newly calculated initial values are applied only to
items/models with convergence issues).
numeric: the maximal number of iterations for calculation of starting values using bootstraped samples (default is 20).
logical: should be sandwich estimator used for covariance matrix of parameters when using
method = "nls"? Default is FALSE.
A list with the following arguments:
Svalthe values of test statistics.
pvalthe p-values by test.
adjusted.pvaladjusted p-values by p.adjust.method.
dfthe degress of freedom of test.
testused test.
par.m0the matrix of estimated item parameters for m0 model.
se.m0the matrix of standard errors of item parameters for m0 model.
cov.m0list of covariance matrices of item parameters for m0 model.
par.m1the matrix of estimated item parameters for m1 model.
se.m1the matrix of standard errors of item parameters for m1 model.
cov.m1list of covariance matrices of item parameters for m1 model.
conv.failnumeric: number of convergence issues.
conv.fail.whichthe indicators of the items which did not converge.
ll.m0log-likelihood of m0 model.
ll.m1log-likelihood of m1 model.
startBo0the binary matrix. Columns represents iterations of initial values re-calculations, rows represents items. The value of 0 means no convergence issue in m0 model, 1 means convergence issue in m0 model.
startBo1the binary matrix. Columns represents iterations of initial values re-calculations, rows represents items. The value of 0 means no convergence issue in m1 model, 1 means convergence issue in m1 model.
Calculation of the test statistics using DIF detection procedure based on non-linear regression (extension of logistic regression procedure; Swaminathan and Rogers, 1990; Drabinova and Martinkova, 2017).
The unconstrained form of 4PL generalized logistic regression model for probability of correct answer (i.e., \(y = 1\)) is $$P(y = 1) = (c + cDif*g) + (d + dDif*g - c - cDif*g)/(1 + exp(-(a + aDif*g)*(x - b - bDif*g))), $$ where \(x\) is by default standardized total score (also called Z-score) and \(g\) is a group membership. Parameters \(a\), \(b\), \(c\), and \(d\) are discrimination, difficulty, guessing, and inattention. Terms \(aDif\), \(bDif\), \(cDif\), and \(dDif\) then represent differences between two groups (reference and focal) in relevant parameters.
This 4PL model can be further constrained by model and constraints arguments.
The arguments model and constraints can be also combined. Both arguments can
be specified as a single value (for all items) or as an item-specific vector (where each
element correspond to one item).
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 fixed 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.
The model can be specified in more detail with constraints argument which specifies what
parameters should be fixed for both groups. For example, choice "ad" means that discrimination
(parameter "a") and inattention (parameter "d") are fixed for both groups and other parameters
("b" and "c") are not. The NA value for constraints means no constraints.
In case that model considers difference in guessing or inattention parameter, the different parameterization is used and parameters with standard errors are re-calculated by delta method.
Drabinova, A. & Martinkova, P. (2017). Detection of differential item functioning with nonlinear regression: A non-IRT approach accounting for guessing. Journal of Educational Measurement, 54(4), 498--517, 10.1111/jedm.12158.
Hladka, A. & Martinkova, P. (2020). difNLR: Generalized logistic regression models for DIF and DDF detection. The R journal, 12(1), 300--323, 10.32614/RJ-2020-014.
Swaminathan, H. & Rogers, H. J. (1990). Detecting differential item functioning using logistic regression procedures. Journal of Educational Measurement, 27(4), 361--370, 10.1111/j.1745-3984.1990.tb00754.x
# NOT RUN {
# loading data based on GMAT
data(GMAT)
Data <- GMAT[, 1:20]
group <- GMAT[, "group"]
# Testing both DIF effects using LR test (default)
# and model with fixed guessing for both groups
NLR(Data, group, model = "3PLcg")
# Using F test
NLR(Data, group, model = "3PLcg", test = "F")
# Testing both DIF effects with Benjamini-Hochberg correction
NLR(Data, group, model = "3PLcg", p.adjust.method = "BH")
# 4PL model with the same guessing and inattention
# to test uniform DIF
NLR(Data, group, model = "4PLcgdg", type = "udif")
# 2PL model to test non-uniform DIF
NLR(Data, group, model = "2PL", type = "nudif")
# 4PL model with fixed a and c parameter
# to test difference in b
NLR(Data, group, model = "4PL", constraints = "ac", type = "b")
# using maximum likelihood estimation method
NLR(Data, group, model = "3PLcg", method = "likelihood")
# }
# NOT RUN {
# }
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