DDF detection for nominal data.

Performs DDF detection procedure for nominal data based on multinomial log-linear regression model and likelihood ratio test of a submodel.

Usage

ddfMLR(Data, group, focal.name, key, type = "both", match = "zscore", anchor = NULL,
       purify = FALSE, nrIter = 10, 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. In addition, Data can hold the vector of group membership.
group: numeric or character: a dichotomous vector of the same length as nrow(Data) or a column identifier of Data.
focal.name: numeric or character: indicates the level of group which corresponds to 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: 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), "restscore" (total score without the tested item), "zrestscore" (standardized total score without the tested item), a numeric vector of the same length as a number of observations in the Data, or a numeric matrix of the same dimensions as Data (each column represents matching criterion for one item).
anchor: character or numeric: specification of DIF-free (anchor) items used to compute the matching criterion (match). Can be either NULL (default; all items are used for the calculation), or a vector of item identifiers (integers indicating column numbers or item names in `Data`) specifying which items are currently considered as anchor items. This argument is ignored if the match is not "zscore", "score", "restscore", or "zrestscore". For match = "score" or match = "zscore", the matching criterion is computed from the items specified in the anchor set. For match = "restscore" or match = "zrestscore", the same anchor items are used, except that the item currently under test is excluded from the computation.
purify: logical: should the item purification be applied? (default is FALSE). Item purification is not applied when set of anchor items in anchor is specified or when match is not "zscore", "score", "restscore", or "zrestscore".
nrIter: numeric: the maximal number of iterations in the item purification (default is 10).
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

The ddfMLR() function returns an object of class "ddfMLR". The output including values of the test statistics, p-values, and items marked as DDF is displayed by the print() method.

A list of class "ddfMLR" with the following arguments:

Sval: the values of likelihood ratio test statistics.
mlrPAR: the estimates of final model.
mlrSE: standard errors of the estimates of final model.
parM0: the estimates of null model.
parM1: the estimates of alternative model.
llM0: log-likelihood of null model.
llM1: log-likelihood of alternative model.
AIC0: AIC of null model.
AIC1: AIC of alternative model.
BIC0: BIC of null model.
BIC1: BIC of alternative model.
DDFitems: either the column identifiers of the items which were detected as DDF, or "No DDF item detected" in case no item was detected as DDF.
type: character: type of DDF that was tested.
anchor: DIF free items specified by the anchor and purify.
purification: purify value.
nrPur: number of iterations in item purification process. Returned only if purify is TRUE.
ddfPur: a binary matrix with one row per iteration of item purification and one column per item. "1" in i-th row and j-th column means that j-th item was identified as DDF in i-th iteration. Returned only if purify is TRUE.
conv.puri: logical indicating whether item purification process converged before the maximal number nrIter of iterations. Returned only if purify is TRUE.
p.adjust.method: character: method for multiple comparison correction which was applied.
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.
alpha: numeric: significance level.
Data: the data matrix.
group: the vector of group membership.
group.names: levels of grouping variable.
key: key of correct answers.
match: matching criterion.
match.name: Name of the matching criterion.

For an object of class "ddfMLR" several methods are available (e.g. methods(class = "ddfMLR")).

Details

Performs DDF detection procedure for nominal data based on multinomial log-linear regression model and likelihood ratio test of submodel. Probability of selection the $k$-th category (distractor) is $$P(y = k) = exp((a_k + a_kDif * g) * (x - b_k - b_kDif * g))) / (1 + \sum exp((a_l + a_lDif * g) * (x - b_l - b_lDif * g))), $$ where $x$ is by default standardized total score (also called Z-score) and $g$ is a group membership. Parameters $a_k$ and $b_k$ are discrimination and difficulty for the $k$-th category. Terms $a_kDif$ and $b_kDif$ then represent differences between two groups (reference and focal) in relevant parameters. Probability of correct answer (specified in argument key) is $$P(y = k) = 1/(1 + \sum exp((a_l + a_lDif * g)*(x - b_l - b_lDif * g))). $$ Parameters are estimated via neural networks. For more details see multinom.

Missing values are allowed but discarded for item estimation. They must be coded as NA for both, Data and group arguments.

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
(x <- ddfMLR(Data, group, focal.name = 1, key))

# graphical devices
plot(x, item = "Item1", group.names = c("Group 1", "Group 2"))
plot(x, item = x$DDFitems)
plot(x, item = 1)

# estimated parameters
coef(x)
coef(x, SE = TRUE)
coef(x, SE = TRUE, simplify = TRUE)

# AIC, BIC, log-likelihood
AIC(x)
BIC(x)
logLik(x)

# AIC, BIC, log-likelihood for the first item
AIC(x, item = 1)
BIC(x, item = 1)
logLik(x, item = 1)

# testing both DDF effects with Benjamini-Hochberg adjustment method
ddfMLR(Data, group, focal.name = 1, key, p.adjust.method = "BH")

# testing both DDF effects with item purification
ddfMLR(Data, group, focal.name = 1, key, purify = TRUE)

# testing uniform DDF effects
ddfMLR(Data, group, focal.name = 1, key, type = "udif")
# testing non-uniform DDF effects
ddfMLR(Data, group, focal.name = 1, key, type = "nudif")

# testing both DDF effects with different matching criteria
ddfMLR(Data, group, focal.name = 1, key, match = "score")
ddfMLR(Data, group, focal.name = 1, key, match = "restscore")
ddfMLR(Data, group, focal.name = 1, key, match = "zrestscore")
match <- rowSums(GMAT[, 1:20])
ddfMLR(Data, group, focal.name = 1, key, match = match)
match <- replicate(ncol(Data), GMAT$criterion)
ddfMLR(Data, group, focal.name = 1, key, match = match)
match <- as.data.frame(match)
ddfMLR(Data, group, focal.name = 1, key, match = match)
} # }