# Detection of Differential Item Functioning Using the Lasso Approach: Selection of Optimal $\lambda$ Value Performs DIF detection using a lasso-penalized logistic regression model for dichotomous items and selects the optimal value of the penalty parameter $\lambda$ using an information criterion. ```r lassoDIF.ABWIC(Data, group, type = "AIC", N = NULL, lambda = NULL, ...) ``` ## Arguments - `...`: Additional arguments passed to internal methods. - `Data`: A numeric data frame or matrix: either only the item responses or the item responses with a group membership column. - `group`: A numeric or character vector: either a vector of group membership or a column index/name indicating group membership in `Data`. - `type`: Character string indicating the criterion used to select the optimal $\lambda$ value. Must be one of `"AIC"`, `"BIC"`, or `"WIC"`. - `N`: Integer: total sample size. If `NULL`, it is inferred from the number of rows in `Data`. - `lambda`: Optional numeric vector of $\lambda$ values to be used in the penalization path. If `NULL`, a default sequence is used. ## Returns A list with the following components: - **DIFitems**: Indices of items flagged as exhibiting DIF. - **DIFpars**: Matrix of estimated DIF parameters for each item. - **crit.value**: Numeric vector of criterion values (e.g., AIC or BIC) across the $\lambda$ path. - **crit.type**: The criterion used to select the optimal $\lambda$ (either `"AIC"`, `"BIC"`, or `"WIC"`). - **lambda**: Vector of $\lambda$ values considered. - **opt.lambda**: The optimal $\lambda$ value selected. - **glmnet.fit**: Fitted `glmnet` model object. ## Details This function detects uniform DIF using a penalized logistic regression model based on the 2PL model. The model includes item-by-group interaction terms that are subject to lasso penalization. The optimal $\lambda$ value is selected based on either the AIC, BIC, or WIC criterion. For the selected $\lambda^*$, the function returns DIF parameters for all items, and flags items whose corresponding DIF parameters are non-zero. Note: the function's behavior is sensitive to input parameters (e.g., criterion type, sample size, $\lambda$ grid). It is strongly recommended to explore different settings and validate findings before interpreting DIF detection results. ## References Magis, D., Tuerlinckx, F., & De Boeck, P. (2015). Detection of Differential Item Functioning Using the Lasso Approach. Journal of Educational and Behavioral Statistics, 40(2), 111–135. https://doi.org/10.3102/1076998614559747 ## Author(s) David Magis Data science consultant at IQVIA Belux Brussels, Belgium Carl F. Falk Department of Psychology McGill University (Canada) carl.falk@mcgill.ca , [https://www.mcgill.ca/psychology/carl-f-falk](https://www.mcgill.ca/psychology/carl-f-falk) Sebastien Beland Faculte des sciences de l'education Universite de Montreal (Canada) sebastien.beland@umontreal.ca ## Examples ```r ## Not run: # With the Verbal data set data(verbal) Dat <-verbal[,1:20] Member <-verbal[,26] # Using AIC for selection lassoDIF.ABWIC(Dat, Member, type="AIC") # Using BIC for selection lassoDIF.ABWIC(Dat, Member, type="BIC") # With simulated data It <- 15 # number of items ItDIFa <- NULL ItDIFb <- c(1,3) NR <- 100 # number of responses for group 1 (reference) NF <- 100 # number of responses for group 2 (focal) a <- rep(1,It) # for tests: runif(It,0.2,.5) b <- rnorm(It,1,.5) Gb <- rep(2,2) # Group value for U-DIF Ga <- 0 # Group value for NU-DIF: need to be fix to 0 for U-DIF Out1 <- SimDichoDif(It,ItDIFa,ItDIFb, NR,NF,a,b,Ga,Gb) Dat<-Out1$data[,1:15] Member<-Out1$data[,16] # Using AIC for selection lassoDIF.ABWIC(Dat, Member, type="AIC") # Using BIC for selection lassoDIF.ABWIC(Dat, Member, type="BIC") # This plot shows how the estimated DIF effects for each item evolve # as the lasso penalty (lambda) increases aic.res <- lassoDIF.ABWIC(Dat, Member, type="AIC") plot_lasso_paths(aic.res$glmnet.fit) bic.res <- lassoDIF.ABWIC(Dat, Member, type="BIC") plot_lasso_paths(bic.res$glmnet.fit) ## End(Not run) ```

lassoDIF.ABWIC function