Parametric smoothed regression lines for item response probability functions

This function uses a generalized additive model (GAM) to estimate response curves for items that do not seem to fit well in a given model. Using a stable axillary model, traceline functions for poorly fitting dichotomous or polytomous items can be inspected using point estimates (or plausible values) of the latent trait. Plots of the tracelines and their associated standard errors are available to help interpret the misfit. This function may also be useful when adding new items to an existing, well established set of items, especially when the parametric form of the items under investigation are unknown.

Usage

itemGAM(
  item,
  Theta,
  formula = resp ~ s(Theta, k = 10),
  CI = 0.95,
  theta_lim = c(-3, 3),
  return.models = FALSE,
  ...
)

# S3 method for class 'itemGAM'
plot(
  x,
  y = NULL,
  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),
  ...
)

Arguments

item

a single poorly fitting item to be investigated. Can be a vector or matrix

Theta

a list or matrix of latent trait estimates typically returned from fscores

formula

an R formula to be passed to the gam function. Default fits a spline model with 10 nodes. For multidimensional models, the traits are assigned the names 'Theta1', 'Theta2', ..., 'ThetaN'

CI

a number ranging from 0 to 1 indicating the confidence interval range. Default provides the 95 percent interval

theta_lim

range of latent trait scores to be evaluated

return.models

logical; return a list of GAM models for each category? Useful when the GAMs should be inspected directly, but also when fitting multidimensional models (this is set to TRUE automatically for multidimensional models)

...

additional arguments to be passed to gam or lattice

x

an object of class 'itemGAM'

y

a NULL value ignored by the plotting function

par.strip.text

plotting argument passed to lattice

par.settings

plotting argument passed to lattice

auto.key

plotting argument passed to lattice

References

Chalmers, R., P. (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment. Journal of Statistical Software, 48(6), 1-29. doi:10.18637/jss.v048.i06

See also

itemfit

Author

Phil Chalmers rphilip.chalmers@gmail.com

Examples

if (FALSE) { # \dontrun{
set.seed(10)
N <- 1000
J <- 30

a <- matrix(1, J)
d <- matrix(rnorm(J))
Theta <- matrix(rnorm(N, 0, 1.5))
dat <- simdata(a, d, N, itemtype = '2PL', Theta=Theta)

# make a bad item
ps <- exp(Theta^2 + Theta) / (1 + exp(Theta^2 + Theta))
item1 <- sapply(ps, function(x) sample(c(0,1), size = 1, prob = c(1-x, x)))

ps2 <- exp(2 * Theta^2 + Theta + .5 * Theta^3) / (1 + exp(2 * Theta^2 + Theta + .5 * Theta^3))
item2 <- sapply(ps2, function(x) sample(c(0,1), size = 1, prob = c(1-x, x)))

# how the actual item looks in the population
plot(Theta, ps, ylim = c(0,1))
plot(Theta, ps2, ylim = c(0,1))

baditems <- cbind(item1, item2)
newdat <- cbind(dat, baditems)

badmod <- mirt(newdat, 1)
itemfit(badmod) #clearly a bad fit for the last two items
mod <- mirt(dat, 1) #fit a model that does not contain the bad items
itemfit(mod)

#### Pure non-parametric way of investigating the items
library(KernSmoothIRT)
ks <- ksIRT(newdat, rep(1, ncol(newdat)), 1)
plot(ks, item=c(1,31,32))
par(ask=FALSE)

# Using point estimates from the model
Theta <- fscores(mod)
IG0 <- itemGAM(dat[,1], Theta) #good item
IG1 <- itemGAM(baditems[,1], Theta)
IG2 <- itemGAM(baditems[,2], Theta)
plot(IG0)
plot(IG1)
plot(IG2)

# same as above, but with plausible values to obtain the standard errors
set.seed(4321)
ThetaPV <- fscores(mod, plausible.draws=10)
IG0 <- itemGAM(dat[,1], ThetaPV) #good item
IG1 <- itemGAM(baditems[,1], ThetaPV)
IG2 <- itemGAM(baditems[,2], ThetaPV)
plot(IG0)
plot(IG1)
plot(IG2)

## for polytomous test items
SAT12[SAT12 == 8] <- NA
dat <- key2binary(SAT12,
                  key = c(1,4,5,2,3,1,2,1,3,1,2,4,2,1,5,3,4,4,1,4,3,3,4,1,3,5,1,3,1,5,4,5))
dat <- dat[,-32]
mod <- mirt(dat, 1)

# Kernal smoothing is very sensitive to which category is selected as 'correct'
# 5th category as correct
ks <- ksIRT(cbind(dat, SAT12[,32]), c(rep(1, 31), 5), 1)
plot(ks, items = c(1,2,32))

# 3rd category as correct
ks <- ksIRT(cbind(dat, SAT12[,32]), c(rep(1, 31), 3), 1)
plot(ks, items = c(1,2,32))

# splines approach
Theta <- fscores(mod)
IG <- itemGAM(SAT12[,32], Theta)
plot(IG)

set.seed(1423)
ThetaPV <- fscores(mod, plausible.draws=10)
IG2 <- itemGAM(SAT12[,32], ThetaPV)
plot(IG2)

# assuming a simple increasing parametric form (like in a standard IRT model)
IG3 <- itemGAM(SAT12[,32], Theta, formula = resp ~ Theta)
plot(IG3)
IG3 <- itemGAM(SAT12[,32], ThetaPV, formula = resp ~ Theta)
plot(IG3)

### multidimensional example by returning the GAM objects
mod2 <- mirt(dat, 2)
Theta <- fscores(mod2)
IG4 <- itemGAM(SAT12[,32], Theta, formula = resp ~ s(Theta1, k=10) + s(Theta2, k=10),
   return.models=TRUE)
names(IG4)
plot(IG4[[1L]], main = 'Category 1')
plot(IG4[[2L]], main = 'Category 2')
plot(IG4[[3L]], main = 'Category 3')

} # }