NLR: DIF statistics for non-linear regression models.

In adelahladka/difNLR: DIF and DDF Detection by Non-Linear Regression Models

DIF statistics for non-linear regression models.

Description

Calculates likelihood ratio test statistics, F-test statistics, or Wald's test statistics for DIF detection among dichotomous items using non-linear regression models (generalized logistic regression models).

Usage

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)

Arguments

Data	data.frame or matrix: dataset in which rows represent scored examinee answers ("1" correct, "0" incorrect) and columns correspond to the items.
group	numeric: a binary vector of a group membership ("0" for the reference group, "1" for the focal group).
model	character: generalized logistic regression model to be fitted. See Details.
constraints	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.
type	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 any combination of parameters "a", "b", "c", and "d". Can be specified as a single value (for all items) or as an item-specific vector.
method	character: an estimation method to be applied. The options are "nls" for non-linear least squares (default), "mle" for the maximum likelihood method using the "L-BFGS-B" algorithm with constraints, "em" for the maximum likelihood estimation with the EM algorithm, "plf" for the maximum likelihood estimation with the algorithm based on parametric link function, and "irls" for the maximum likelihood estimation with the iteratively reweighted least squares algorithm (available for the "2PL" model only). See Details.
match	character or numeric: matching criterion to be used as an estimate of the trait. It can be either "zscore" (default, standardized total score), "score" (total test score), or a numeric vector of the same length as a number of observations in the Data.
anchor	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 the match is not "zscore" or "score".
start	numeric: initial values for the estimation of item parameters. If not specified, starting values are calculated with the startNLR function. Otherwise, a list with as many elements as a number of items. Each element is a named numeric vector representing initial values for estimation of item parameters. Specifically, parameters "a", "b", "c", and "d" are initial values for discrimination, difficulty, guessing, and inattention for the reference group. Parameters "aDif", "bDif", "cDif", and "dDif" are then differences in these parameters between the reference and focal groups. For the method = "irls", default initial values from the glm function are used.
p.adjust.method	character: a method for a multiple comparison correction. Possible values are "holm", "hochberg", "hommel", "bonferroni", "BH", "BY", "fdr", and "none" (default). For more details see p.adjust.
test	character: a statistical test to be performed for DIF detection. Can be either "LR" for the likelihood ratio test of a submodel (default), "W" for the Wald's test, or "F" for the F-test of a submodel.
alpha	numeric: a significance level (the default is 0.05).
initboot	logical: in the case of convergence issues, should starting values be re-calculated based on bootstrapped samples? (the default is TRUE; newly calculated initial values are applied only to items/models with convergence issues).
nrBo	numeric: the maximal number of iterations for the calculation of starting values using bootstrapped samples (the default is 20).
sandwich	logical: should the sandwich estimator be applied for computation of the covariance matrix of item parameters when using method = "nls"? (the default is FALSE).

Details

The function calculates test statistics using a DIF detection procedure based on non-linear regression models (i.e., extensions of the logistic regression procedure; Swaminathan & Rogers, 1990; Drabinova & Martinkova, 2017).

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 and constraints arguments can further constrain the 4PL model. The arguments model and constraints can also be combined. Both arguments can be specified as a single value (for all items) or as an item-specific vector (where each element corresponds 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 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.

The function uses intercept-slope parameterization for the estimation via the estimNLR function. Item parameters are then re-calculated into the IRT parameterization using the delta method.

The function offers either the non-linear least squares estimation via the nls function (Drabinova & Martinkova, 2017; Hladka & Martinkova, 2020), the maximum likelihood method with the "L-BFGS-B" algorithm with constraints via the optim function (Hladka & Martinkova, 2020), the maximum likelihood method with the EM algorithm (Hladka, Martinkova, & Brabec, 2024), the maximum likelihood method with the algorithm based on parametric link function (PLF, the default option; Hladka, Martinkova, & Brabec, 2024), or the maximum likelihood method with the iteratively reweighted least squares algorithm via the glm function.

Value

A list with the following arguments:

Sval

the values of the test statistics.

pval

the p-values by the test.

adjusted.pval

adjusted p-values by the p.adjust.method.

df

the degrees of freedom of the test.

test

used test.

par.m0

the matrix of estimated item parameters for the null model.

se.m0

the matrix of standard errors of item parameters for the null model.

cov.m0

list of covariance matrices of item parameters for the null model.

par.m1

the matrix of estimated item parameters for the alternative model.

se.m1

the matrix of standard errors of item parameters for the alternative model.

cov.m1

list of covariance matrices of item parameters for the alternative model.

conv.fail

numeric: a number of convergence issues.

conv.fail.which

the indicators of the items that did not converge.

ll.m0

log-likelihood of null model.

ll.m1

log-likelihood of alternative model.

startBo0

the binary matrix. Columns represent iterations of initial values re-calculations, rows represent items. The value of 0 means no convergence issue in the null model, 1 means convergence issue in the null model.

startBo1

the binary matrix. Columns represent iterations of initial values re-calculations, rows represent items. The value of 0 means no convergence issue in the alternative model, 1 means convergence issue in the alternative model.

Author(s)

Adela Hladka (nee Drabinova)
Institute of Computer Science of the Czech Academy of Sciences
[e-mail]

Patricia Martinkova
Institute of Computer Science of the Czech Academy of Sciences
[e-mail]

Karel Zvara
Faculty of Mathematics and Physics, Charles University

References

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, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/jedm.12158")}.

Hladka, A. (2021). Statistical models for detection of differential item functioning. Dissertation thesis. Faculty of Mathematics and Physics, Charles University.

Hladka, A. & Martinkova, P. (2020). difNLR: Generalized logistic regression models for DIF and DDF detection. The R Journal, 12(1), 300–323, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.32614/RJ-2020-014")}.

Hladka, A., Martinkova, P., & Brabec, M. (2024). New iterative algorithms for estimation of item functioning. Journal of Educational and Behavioral Statistics. Accepted.

Swaminathan, H. & Rogers, H. J. (1990). Detecting differential item functioning using logistic regression procedures. Journal of Educational Measurement, 27(4), 361–370, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/j.1745-3984.1990.tb00754.x")}

See Also

p.adjust

Examples

## Not run: 
# loading data
data(GMAT)
Data <- GMAT[, 1:20] # items
group <- GMAT[, "group"] # group membership variable

# testing both DIF effects using the LR test (default)
# and the model with fixed guessing for both groups
NLR(Data, group, model = "3PLcg")

# using the F test and Wald's test
NLR(Data, group, model = "3PLcg", test = "F")
NLR(Data, group, model = "3PLcg", test = "W")

# using the 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 parameters
# to test difference in parameter b
NLR(Data, group, model = "4PL", constraints = "ac", type = "b")

# using various estimation algorithms
NLR(Data, group, model = "3PLcg", method = "nls")
NLR(Data, group, model = "3PLcg", method = "mle")
NLR(Data, group, model = "3PLcg", method = "em")
NLR(Data, group, model = "3PLcg", method = "plf")
NLR(Data, group, model = "2PL", method = "irls")

## End(Not run)

adelahladka/difNLR documentation built on Dec. 23, 2024, 2:20 a.m.