Differential Response Functioning statistics

Function performs various omnibus differential item (DIF), bundle (DBF), and test (DTF) functioning procedures on an object estimated with multipleGroup(). The compensatory and non-compensatory statistics provided are described in Chalmers (2018), which generally can be interpreted as IRT generalizations of the SIBTEST and CSIBTEST statistics. For hypothesis tests, these measures require the ACOV matrix to be computed in the fitted multiple-group model (otherwise, sets of plausible draws from the posterior are explicitly required).

Usage

DRF(
  mod,
  draws = NULL,
  focal_items = 1L:extract.mirt(mod, "nitems"),
  param_set = NULL,
  den.type = "marginal",
  best_fitting = FALSE,
  CI = 0.95,
  npts = 1000,
  quadpts = NULL,
  theta_lim = c(-6, 6),
  Theta_nodes = NULL,
  plot = FALSE,
  DIF = FALSE,
  DIF.cats = FALSE,
  groups2test = "all",
  pairwise = FALSE,
  simplify = TRUE,
  p.adjust = "none",
  par.strip.text = list(cex = 0.7),
  par.settings = list(strip.background = list(col = "#9ECAE1"), strip.border = list(col =
    "black")),
  auto.key = list(space = "right", points = FALSE, lines = TRUE),
  verbose = TRUE,
  ...
)

Arguments

mod: a multipleGroup object which estimated only 2 groups
draws: a number indicating how many draws to take to form a suitable multiple imputation or bootstrap estimate of the expected test scores (100 or more). If boot = FALSE, requires an estimated parameter information matrix. Returns a list containing the bootstrap/imputation distribution and null hypothesis test for the sDRF statistics
focal_items: a character/numeric vector indicating which items to include in the DRF tests. The default uses all of the items (note that including anchors in the focal items has no effect because they are exactly equal across groups). Selecting fewer items will result in tests of 'differential bundle functioning'
param_set: an N x p matrix of parameter values drawn from the posterior (e.g., using the parametric sampling approach, bootstrap, of MCMC). If supplied, then these will be used to compute the DRF measures. Can be much more efficient to pre-compute these values if DIF, DBF, or DTF are being evaluated within the same model (especially when using the bootstrap method). See draw_parameters
den.type: character specifying how the density of the latent traits is computed. Default is 'marginal' to include the proportional information from both groups, 'focal' for just the focal group, and 'reference' for the reference group
best_fitting: logical; use the best fitting parametric distribution (Gaussian by default) that was used at the time of model estimation? This will result in much fast computations, however the results are more dependent upon the underlying modelling assumptions. Default is FALSE, which uses the empirical histogram approach
CI: range of confidence interval when using draws input
npts: number of points to use for plotting. Default is 1000
quadpts: number of quadrature nodes to use when constructing DRF statistics. Default is extracted from the input model object
theta_lim: lower and upper limits of the latent trait (theta) to be evaluated, and is used in conjunction with quadpts and npts
Theta_nodes: an optional matrix of Theta values to be evaluated in the draws for the sDRF statistics. However, these values are not averaged across, and instead give the bootstrap confidence intervals at the respective Theta nodes. Useful when following up a large sDRF or uDRF statistic, for example, to determine where the difference between the test curves are large (while still accounting for sampling variability). Returns a matrix with observed variability
plot: logical; plot the 'sDRF' functions for the evaluated sDBF or sDTF values across the integration grid or, if DIF = TRUE, the selected items as a faceted plot of individual items? If plausible parameter sets were obtained/supplied then imputed confidence intervals will be included
DIF: logical; return a list of item-level imputation properties using the DRF statistics? These can generally be used as a DIF detection method and as a graphical display for understanding DIF within each item
DIF.cats: logical; same as DIF = TRUE, however computations will be performed on each item category probability functions rather than the score functions. Only useful for understanding DIF in polytomous items
groups2test: when more than 2 groups are being investigated which two groups should be used in the effect size comparisons?
pairwise: logical; perform pairwise computations when the applying to multi-group settings
simplify: logical; attempt to simplify the output rather than returning larger lists?
p.adjust: string to be passed to the p.adjust function to adjust p-values. Adjustments are located in the adj_pvals element in the returned list. Only applicable when DIF = TRUE
par.strip.text: plotting argument passed to lattice
par.settings: plotting argument passed to lattice
auto.key: plotting argument passed to lattice
verbose: logical; include additional information in the console?
...: additional arguments to be passed to lattice

Details

The effect sizes estimates by the DRF function are $$sDRF = \int [S(C|\bm{\Psi}^{(R)},\theta) S(C|\bm{\Psi}^{(F)},\theta)] f(\theta)d\theta,$$ $$uDRF = \int |S(C|\bm{\Psi}^{(R)},\theta) S(C|\bm{\Psi}^{(F)},\theta)| f(\theta)d\theta,$$ and $$dDRF = \sqrt{\int [S(C|\bm{\Psi}^{(R)},\theta) S(C|\bm{\Psi}^{(F)},\theta)]^2 f(\theta)d\theta}$$ where $S(.)$ are the scoring equations used to evaluate the model-implied difference between the focal and reference group. The $f(\theta)$ terms can either be estimated from the posterior via an empirical histogram approach (default), or can use the best fitting prior distribution that is obtain post-convergence (default is a Guassian distribution). Note that, in comparison to Chalmers (2018), the focal group is the leftmost scoring function while the reference group is the rightmost scoring function. This is largely to keep consistent with similar effect size statistics, such as SIBTEST, DFIT, Wainer's measures of impact, etc, which in general can be seen as special-case estimators of this family.

References

Chalmers, R. P. (2018). Model-Based Measures for Detecting and Quantifying Response Bias. Psychometrika, 83(3), 696-732. doi:10.1007/s11336-018-9626-9

Author

Phil Chalmers rphilip.chalmers@gmail.com

Examples

if (FALSE) { # \dontrun{

set.seed(1234)
n <- 30
N <- 500

# only first 5 items as anchors
model <- 'F = 1-30
          CONSTRAINB = (1-5, a1), (1-5, d)'

a <- matrix(1, n)
d <- matrix(rnorm(n), n)
group <- c(rep('Group_1', N), rep('Group_2', N))

## -------------
# groups completely equal
dat1 <- simdata(a, d, N, itemtype = 'dich')
dat2 <- simdata(a, d, N, itemtype = 'dich')
dat <- rbind(dat1, dat2)
mod <- multipleGroup(dat, model, group=group, SE=TRUE,
                     invariance=c('free_means', 'free_var'))
plot(mod)
plot(mod, which.items = 6:10) #DBF
plot(mod, type = 'itemscore')
plot(mod, type = 'itemscore', which.items = 10:15)

# empirical histogram approach
DRF(mod)
DRF(mod, focal_items = 6:10) #DBF
DRF(mod, DIF=TRUE)
DRF(mod, DIF=TRUE, focal_items = 10:15)

# Best-fitting Gaussian distributions
DRF(mod, best_fitting=TRUE)
DRF(mod, focal_items = 6:10, best_fitting=TRUE) #DBF
DRF(mod, DIF=TRUE, best_fitting=TRUE)
DRF(mod, DIF=TRUE, focal_items = 10:15, best_fitting=TRUE)

DRF(mod, plot = TRUE)
DRF(mod, focal_items = 6:10, plot = TRUE) #DBF
DRF(mod, DIF=TRUE, plot = TRUE)
DRF(mod, DIF=TRUE, focal_items = 10:15, plot = TRUE)

if(interactive()) mirtCluster()
DRF(mod, draws = 500)
DRF(mod, draws = 500, best_fitting=TRUE)
DRF(mod, draws = 500, plot=TRUE)

# pre-draw parameter set to save computations
#  (more useful when using non-parametric bootstrap)
param_set <- draw_parameters(mod, draws = 500)
DRF(mod, focal_items = 6, param_set=param_set) #DIF test
DRF(mod, DIF=TRUE, param_set=param_set) #DIF test
DRF(mod, focal_items = 6:10, param_set=param_set) #DBF test
DRF(mod, param_set=param_set) #DTF test

DRF(mod, focal_items = 6:10, draws=500) #DBF test
DRF(mod, focal_items = 10:15, draws=500) #DBF test

DIFs <- DRF(mod, draws = 500, DIF=TRUE)
print(DIFs)
DRF(mod, draws = 500, DIF=TRUE, plot=TRUE)

DIFs <- DRF(mod, draws = 500, DIF=TRUE, focal_items = 6:10)
print(DIFs)
DRF(mod, draws = 500, DIF=TRUE, focal_items = 6:10, plot = TRUE)

DRF(mod, DIF=TRUE, focal_items = 6)
DRF(mod, draws=500, DIF=TRUE, focal_items = 6)

# evaluate specific values for sDRF
Theta_nodes <- matrix(seq(-6,6,length.out = 100))

sDTF <- DRF(mod, Theta_nodes=Theta_nodes)
head(sDTF)
sDTF <- DRF(mod, Theta_nodes=Theta_nodes, draws=200)
head(sDTF)

# sDIF (isolate single item)
sDIF <- DRF(mod, Theta_nodes=Theta_nodes, focal_items=6)
head(sDIF)
sDIF <- DRF(mod, Theta_nodes=Theta_nodes, focal_items = 6, draws=200)
head(sDIF)

## -------------
## random slopes and intercepts for 15 items, and latent mean difference
##    (no systematic DTF should exist, but DIF will be present)
set.seed(1234)
dat1 <- simdata(a, d, N, itemtype = 'dich', mu=.50, sigma=matrix(1.5))
dat2 <- simdata(a + c(numeric(15), rnorm(n-15, 0, .25)),
                d + c(numeric(15), rnorm(n-15, 0, .5)), N, itemtype = 'dich')
dat <- rbind(dat1, dat2)
mod1 <- multipleGroup(dat, 1, group=group)
plot(mod1)
DRF(mod1) #does not account for group differences! Need anchors

mod2 <- multipleGroup(dat, model, group=group, SE=TRUE,
                      invariance=c('free_means', 'free_var'))
plot(mod2)

# significant DIF in multiple items....
# DIF(mod2, which.par=c('a1', 'd'), items2test=16:30)
DRF(mod2)
DRF(mod2, draws=500) #non-sig DTF due to item cancellation

## -------------
## systematic differing slopes and intercepts (clear DTF)
set.seed(1234)
dat1 <- simdata(a, d, N, itemtype = 'dich', mu=.50, sigma=matrix(1.5))
dat2 <- simdata(a + c(numeric(15), rnorm(n-15, 1, .25)),
                d + c(numeric(15), rnorm(n-15, 1, .5)),
                N, itemtype = 'dich')
dat <- rbind(dat1, dat2)
mod3 <- multipleGroup(dat, model, group=group, SE=TRUE,
                      invariance=c('free_means', 'free_var'))
plot(mod3) #visable DTF happening

# DIF(mod3, c('a1', 'd'), items2test=16:30)
DRF(mod3) #unsigned bias. Signed bias (group 2 scores higher on average)
DRF(mod3, draws=500)
DRF(mod3, draws=500, plot=TRUE) #multiple DRF areas along Theta

# plot the DIF
DRF(mod3, draws=500, DIF=TRUE, plot=TRUE)

# evaluate specific values for sDRF
Theta_nodes <- matrix(seq(-6,6,length.out = 100))
sDTF <- DRF(mod3, Theta_nodes=Theta_nodes, draws=200)
head(sDTF)

# DIF
sDIF <- DRF(mod3, Theta_nodes=Theta_nodes, focal_items = 30, draws=200)
car::some(sDIF)

## ----------------------------------------------------------------
# polytomous example
# simulate data where group 2 has a different slopes/intercepts
set.seed(4321)
a1 <- a2 <- matrix(rlnorm(20,.2,.3))
a2[c(16:17, 19:20),] <- a1[c(16:17, 19:20),] + c(-.5, -.25, .25, .5)

# for the graded model, ensure that there is enough space between the intercepts,
# otherwise closer categories will not be selected often
diffs <- t(apply(matrix(runif(20*4, .3, 1), 20), 1, cumsum))
diffs <- -(diffs - rowMeans(diffs))
d1 <- d2 <- diffs + rnorm(20)
rownames(d1) <- rownames(d2) <- paste0('Item.', 1:20)
d2[16:20,] <- d1[16:20,] + matrix(c(-.5, -.5, -.5, -.5,
                                    1, 0, 0, -1,
                                    .5, .5, -.5, -.5,
                                    1, .5, 0, -1,
                                    .5, .5, .5, .5), byrow=TRUE, nrow=5)

tail(data.frame(a.group1 = a1, a.group2 = a2), 6)
list(d.group1 = d1[15:20,], d.group2 = d2[15:20,])

itemtype <- rep('graded', nrow(a1))
N <- 600
dataset1 <- simdata(a1, d1, N, itemtype)
dataset2 <- simdata(a2, d2, N, itemtype, mu = -.25, sigma = matrix(1.25))
dat <- rbind(dataset1, dataset2)
group <- c(rep('D1', N), rep('D2', N))

# item 1-10 as anchors
mod <- multipleGroup(dat, group=group, SE=TRUE,
                     invariance=c(colnames(dat)[1:10], 'free_means', 'free_var'))
coef(mod, simplify=TRUE)
plot(mod)
plot(mod, type='itemscore')

# DIF tests vis Wald method
DIF(mod, items2test=11:20,
   which.par=c('a1', paste0('d', 1:4)),
   Wald=TRUE, p.adjust='holm')

DRF(mod)
DRF(mod, DIF=TRUE, focal_items=11:20)
DRF(mod, DIF.cats=TRUE, focal_items=11:20)

## ----------------------------------------------------------------
### multidimensional DTF

set.seed(1234)
n <- 50
N <- 1000

# only first 5 items as anchors within each dimension
model <- 'F1 = 1-25
          F2 = 26-50
          COV = F1*F2
          CONSTRAINB = (1-5, a1), (1-5, 26-30, d), (26-30, a2)'

a <- matrix(c(rep(1, 25), numeric(50), rep(1, 25)), n)
d <- matrix(rnorm(n), n)
group <- c(rep('Group_1', N), rep('Group_2', N))
Cov <- matrix(c(1, .5, .5, 1.5), 2)
Mean <- c(0, 0.5)

# groups completely equal
dat1 <- simdata(a, d, N, itemtype = 'dich', sigma = cov2cor(Cov))
dat2 <- simdata(a, d, N, itemtype = 'dich', sigma = Cov, mu = Mean)
dat <- rbind(dat1, dat2)
mod <- multipleGroup(dat, model, group=group, SE=TRUE,
                     invariance=c('free_means', 'free_var'))
coef(mod, simplify=TRUE)
plot(mod, degrees = c(45,45))
DRF(mod)

# some intercepts slightly higher in Group 2
d2 <- d
d2[c(10:15, 31:35)] <- d2[c(10:15, 31:35)] + 1
dat1 <- simdata(a, d, N, itemtype = 'dich', sigma = cov2cor(Cov))
dat2 <- simdata(a, d2, N, itemtype = 'dich', sigma = Cov, mu = Mean)
dat <- rbind(dat1, dat2)
mod <- multipleGroup(dat, model, group=group, SE=TRUE,
                     invariance=c('free_means', 'free_var'))
coef(mod, simplify=TRUE)
plot(mod, degrees = c(45,45))

DRF(mod)
DRF(mod, draws = 500)

} # }