# Detection of Differential Item Functioning Using Quade-Type Association Indices for Polytomous (Ordinal) Item This function detects DIF in ordinal items using association indices based on pairwise comparisons, as proposed by Quade (1974) and extended in Woods (2009). It supports various ordinal measures of association to identify uniform DIF only. ```r difQuade(Data, group, focal.name = NULL, anchor = NULL, match = "score", type = c("ta", "e", "dxy", "dyx", "gamma"), alpha = 0.05, purify = FALSE, nrIter = 10, save.output = FALSE, output = c("out", "default")) ``` ## Arguments - `Data`: A data frame or matrix of ordinal item responses. - `group`: A vector indicating group membership. - `focal.name`: Value in `group` identifying the focal group. - `anchor`: Optional vector of anchor item indices. If `NULL`, all items are used. - `match`: Type of matching score: `"score"` (total test score) or `"restscore"` (excluding item). - `type`: Type of ordinal association index: `"ta"` (Kendall's tau-a), `"e"` (Wilson's e), `"gamma"` (Goodman & Kruskal's gamma), `"dyx"` (Somers' dyx), or `"dxy"` (Somers' dxy). - `alpha`: Significance level for DIF detection. - `purify`: Logical: should purification be applied? - `nrIter`: Number of iterations for purification. - `save.output`: Logical: should the results be saved to a text file? - `output`: Name of the output file (or `"out"` to use default). ## Details The function implements the ordinal association approach introduced by Quade (1974), where pairwise comparisons are made between respondents' item responses and total scores. Five indices are supported: * `"ta"`: Kendall's tau-a, considers all pair types including ties. * `"e"`: Wilson's e index, accounts for ties in both variables. * `"gamma"`: Goodman & Kruskal's gamma, ignores tied pairs. * `"dyx"`: Somers' dyx, conditions on the matching score. * `"dxy"`: Somers' dxy, conditions on the item response. These indices follow the methodology validated in Woods (2009), who confirmed through simulation their robustness across various ordinal DIF contexts. ## Returns An object of class `"difQuade"` with components: * `stat`, `se`, `zstat`, `p.value`: test statistics, standard errors, z-values, and p-values for each item. * `DIFitems`: Logical vector indicating flagged items. * `match`, `type`, `anchor`, `purification`: arguments used. ## References Quade, D. (1974). Nonparametric tests for the comparison of two groups of multivariate observations. **The Annals of Statistics**, 2(5), 949–960. Woods, C. M. (2009). Testing for differential item functioning with measures of partial association. **Applied Psychological Measurement**, 33(7), 538–554. ## Author(s) Sebastien Beland Faculte des sciences de l'education Universite de Montreal (Canada) sebastien.beland@umontreal.ca ## See Also `plot.difQuade`, `print.difQuade` ## Examples ```r ## Not run: # With real data # DIF detection using tau-a and purification data(SCS) Data <- SCS[, 1:10] group <- SCS$Gender # Using ta and purification res1 <- difQuade(Data, group, focal.name = 2, type = "ta", purify = TRUE) print(res1) # Here is a function thta plot the results plot(res1) # Using Goodman & Kruskal's gamma with restscore matching res2 <- difQuade(Data, group, focal.name = 2, type = "gamma", match = "restscore") print(res2) # Using Wilson's e index (recommended for tied ordinal data) res3 <- difQuade(Data, group, focal.name = 2, type = "e") print(res3) # Somers' dyx index with no purification res4 <- difQuade(Data, group, focal.name = 2, type = "dyx", purify = FALSE) print(res4) # With simulated data set.seed(1234) # original item parameters a <- rlnorm(10, -0.5) # slopes b <- runif(10, -2, 2) # difficulty d <- list() d[[1]] <- c(0, 2, .5, -.15, -1.1) d[[2]] <- c(0, 2, .25, -.45, -.75) d[[3]] <- c(0, 1, .5, -.65, -1) d[[4]] <- c(0, 2, .5, -.85, -2) d[[5]] <- c(0, 1, .25, -.05, -1) d[[6]] <- c(0, 2, .5, -.95, -1) d[[7]] <- c(0, 1, .25, -.35, -2) d[[8]] <- c(0, 2, .5, -.15, -1) d[[9]] <- c(0, 1, .25, -.25, -2) d[[10]] <- c(0, 2, .5, -.35, -1) # Uniform DIF It <- 10; NR <- 1000; NF <- 1000 ItDIFa <- NULL; Ga <- NULL ItDIFb <- c(1, 3) Gb <- rep(.5, 2) Out.Unif <- SimPolyDif(It, ItDIFa, ItDIFb, NR, NF, a, b, d, ncat = 5, Ga = Ga, Gb = Gb) Out.Unif$ipars Data <- Out.Unif$data # Using ta and purification res5 <- difQuade(Data = Data[, 1:10], group = Data$group, focal.name = "G1", type = "ta", purify = TRUE) print(res5) # Here is a function thta plot the results plot(res5) # Using Goodman & Kruskal's gamma with restscore matching res6 <- difQuade(Data = Data[, 1:10], group = Data$group, focal.name = "G1", type = "gamma", match = "restscore") print(res6) # Using Wilson's e index (recommended for tied ordinal data) res7 <- difQuade(Data = Data[, 1:10], group = Data$group, focal.name = "G1", type = "e") print(res7) # Somers' dyx index with no purification res8 <- difQuade(Data = Data[, 1:10], group = Data$group, focal.name = "G1", type = "dyx", purify = FALSE) print(res8) ## End(Not run) ```

difQuade function