DDF likelihood ratio statistics based on multinomial log-linear regression model.

Calculates DDF likelihood ratio statistics for nominal data based on multinomial log-linear model.

Usage

MLR(Data, group, key, type = "both", match = "zscore", anchor = 1:ncol(Data),
    p.adjust.method = "none", alpha = 0.05, parametrization)

Arguments

Data: data.frame or matrix: dataset which rows represent unscored examinee answers (nominal) and columns correspond to the items.
group: numeric: binary vector of group membership. "0" for reference group, "1" for focal group.
key: character: the answer key. Each element corresponds to the correct answer of one item.
type: character: type of DDF to be tested. Either "both" for uniform and non-uniform DDF (i.e., difference in parameters "a" and "b") (default), or "udif" for uniform DDF only (i.e., difference in difficulty parameter "b"), or "nudif" for non-uniform DDF only (i.e., difference in discrimination parameter "a"). Can be specified as a single value (for all items) or as an item-specific vector.
match: numeric or character: matching criterion to be used as an estimate of trait. Can be either "zscore" (default, standardized total score), "score" (total test score), or vector of the same length as number of observations in 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 match is not "zscore" or "score".
p.adjust.method: 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.
alpha: numeric: significance level (default is 0.05).
parametrization: deprecated. Use coef.ddfMLR for different parameterizations.

Value

A list with the following arguments:

Sval: the values of likelihood ratio test statistics.
pval: the p-values by likelihood ratio test.
adj.pval: the adjusted p-values by likelihood ratio test using p.adjust.method.
df: the degress of freedom of likelihood ratio test.
par.m0: the estimates of null model.
par.m1: the estimates of alternative model.
se.m0: standard errors of parameters in null model.
se.m1: standard errors of parameters in alternative model.
cov.m0: list of covariance matrices of item parameters for null model.
cov.m1: list of covariance matrices of item parameters for alternative model.
ll.m0: log-likelihood of m0 model.
ll.m1: log-likelihood of m1 model.
AIC.m0: AIC of m0 model.
AIC.m1: AIC of m1 model.
BIC.m0: BIC of m0 model.
BIC.m1: BIC of m1 model.

Details

$$P(y = k) = exp(b_0k + b_1k * x + b_2k * g + b_3k * x * g) / (1 + \sum exp(b_0l + b_1l * x + b_2l * g + b_3l * x * g)), $$ where $x$ is by default standardized total score (also called Z-score) and $g$ is a group membership. Probability of correct answer (specified in argument key) is $$P(y = k) = 1/(1 + \sum exp(b_0l + b_1l * x + b_2l * g + b_3l * x * g)). $$ Parameters are estimated via neural networks. For more details see multinom.

References

Agresti, A. (2010). Analysis of ordinal categorical data. Second edition. John Wiley & Sons.

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, doi:10.32614/RJ-2020-014 .

Author

Adela Hladka (nee Drabinova)
Institute of Computer Science of the Czech Academy of Sciences
Faculty of Mathematics and Physics, Charles University
hladka@cs.cas.cz

Patricia Martinkova
Institute of Computer Science of the Czech Academy of Sciences
martinkova@cs.cas.cz

Examples

if (FALSE) { # \dontrun{
# loading data
data(GMATtest, GMATkey)
Data <- GMATtest[, 1:20] # items
group <- GMATtest[, "group"] # group membership variable
key <- GMATkey # correct answers

# testing both DDF effects
MLR(Data, group, key, type = "both")

# testing uniform DDF effects
MLR(Data, group, key, type = "udif")

# testing non-uniform DDF effects
MLR(Data, group, key, type = "nudif")
} # }