Unidimensional Non- and Semiparametric Item Response Model
Conducts non- and semiparametric estimation of a unidimensional item response model for a single group allowing polytomous item responses (Rossi, Wang & Ramsay, 2002).
For dichotomous data, the function also allows group lasso penalty
(penalty_type="lasso"; Breheny & Huang, 2015; Yang & Zhou, 2015) and a ridge penalty
(penalty_type="ridge"; Rossi et al., 2002)
which is applied to the nonlinear part of the basis expansion. This approach
automatically detects deviations from a 2PL or a 1PL model (see Examples 2 and 3).
See Details for model specification.
tam.np( dat, probs_init=NULL, pweights=NULL, lambda=NULL, control=list(),
model="2PL", n_basis=0, basis_type="hermite", penalty_type="lasso",
pars_init=NULL, orthonormalize=TRUE)
## S3 method for class 'tam.np'
summary(object, file=NULL, ...)
## S3 method for class 'tam.np'
IRT.cv(object, kfold=10, ...)dat |
Matrix of integer item responses (starting from zero) |
probs_init |
Array containing initial probabilities |
pweights |
Optional vector of person weights |
lambda |
Numeric or vector of regularization parameter |
control |
List of control arguments, see |
model |
Specified target model. Can be |
n_basis |
Number of basis functions |
basis_type |
Type of basis function: |
penalty_type |
Lasso type penalty ( |
pars_init |
Optional matrix of initial item parameters |
orthonormalize |
Logical indicating whether basis functions should be orthonormalized |
object |
Object of class |
file |
Optional file name for summary output |
kfold |
Number of folds in k-fold cross-validation |
... |
Further arguments to be passed |
The basis expansion approach is applied for the logit transformation of item response functions for dichotomous data. In more detail, it this assumed that
P(X_i=1|θ)=ψ( H_0(θ) + H_1(θ)
where H_0 is the target function type and H_1 is the semiparametric
part which parameterizes model deviations. For the 2PL model (model="2PL")
it is H_0(θ)=d_i + a_i θ and for the 1PL model
(model="1PL") we set H_1(θ)=d_i + 1 \cdot θ .
The model discrepancy is specified as a basis expansion approach
H_1 ( θ )=∑_{h=1}^p β_{ih} f_h( θ)
where f_h are basis functions (possibly orthonormalized) and β_{ih} are item parameters which should be estimated. Penalty functions are posed on the β_{ih} coefficients. For the group lasso penalty, we specify the penalty J_{i,L1}=N λ √{p} √{ ∑_{h=1}^p β_{ih}^2 } while for the ridge penalty it is J_{i,L2}=N λ ∑_{h=1}^p β_{ih}^2 (N denoting the sample size).
List containing several entries
rprobs |
Item response probabilities |
theta |
Used nodes for approximation of θ distribution |
n.ik |
Expected counts |
like |
Individual likelihood |
hwt |
Individual posterior |
item |
Summary item parameter table |
pars |
Estimated parameters |
regularized |
Logical indicating which items are regularized |
ic |
List containing |
... |
Further values |
Breheny, P., & Huang, J. (2015). Group descent algorithms for nonconvex penalized linear and logistic regression models with grouped predictors. Statistics and Computing, 25(2), 173-187. doi: 10.1007/s11222-013-9424-2
Rossi, N., Wang, X., & Ramsay, J. O. (2002). Nonparametric item response function estimates with the EM algorithm. Journal of Educational and Behavioral Statistics, 27(3), 291-317. doi: 10.3102/10769986027003291
Yang, Y., & Zou, H. (2015). A fast unified algorithm for solving group-lasso penalized learning problems. Statistics and Computing, 25(6), 1129-1141. doi: 10.1007/s11222-014-9498-5
Nonparametric item response models can also be estimated with the
mirt::itemGAM function in the mirt package and the
KernSmoothIRT::ksIRT in the KernSmoothIRT package.
See tam.mml and tam.mml.2pl for parametric item response
models.
## Not run:
#############################################################################
# EXAMPLE 1: Nonparametric estimation polytomous data
#############################################################################
data(data.cqc02, package="TAM")
dat <- data.cqc02
#** nonparametric estimation
mod <- TAM::tam.np(dat)
#** extractor functions for objects of class 'tam.np'
lmod <- IRT.likelihood(mod)
pmod <- IRT.posterior(mod)
rmod <- IRT.irfprob(mod)
emod <- IRT.expectedCounts(mod)
#############################################################################
# EXAMPLE 2: Semiparametric estimation and detection of item misfit
#############################################################################
#- simulate data with two misfitting items
set.seed(998)
I <- 10
N <- 1000
a <- stats::rnorm(I, mean=1, sd=.3)
b <- stats::rnorm(I, mean=0, sd=1)
dat <- matrix(NA, nrow=N, ncol=I)
colnames(dat) <- paste0("I",1:I)
theta <- stats::rnorm(N)
for (ii in 1:I){
dat[,ii] <- 1*(stats::runif(N) < stats::plogis( a[ii]*(theta-b[ii] ) ))
}
#* first misfitting item with lower and upper asymptote
ii <- 1
l <- .3
u <- 1
b[ii] <- 1.5
dat[,ii] <- 1*(stats::runif(N) < l + (u-l)*stats::plogis( a[ii]*(theta-b[ii] ) ))
#* second misfitting item with non-monotonic item response function
ii <- 3
dat[,ii] <- (stats::runif(N) < stats::plogis( theta-b[ii]+.6*theta^2))
#- 2PL model
mod0 <- TAM::tam.mml.2pl(dat)
#- lasso penalty with lambda of .05
mod1 <- TAM::tam.np(dat, n_basis=4, lambda=.05)
#- lambda value of .03 using starting value of previous model
mod2 <- TAM::tam.np(dat, n_basis=4, lambda=.03, pars_init=mod1$pars)
cmod2 <- TAM::IRT.cv(mod2) # cross-validated deviance
#- lambda=.015
mod3 <- TAM::tam.np(dat, n_basis=4, lambda=.015, pars_init=mod2$pars)
cmod3 <- TAM::IRT.cv(mod3)
#- lambda=.007
mod4 <- TAM::tam.np(dat, n_basis=4, lambda=.007, pars_init=mod3$pars)
#- lambda=.001
mod5 <- TAM::tam.np(dat, n_basis=4, lambda=.001, pars_init=mod4$pars)
#- final estimation using solution of mod3
eps <- .0001
lambda_final <- eps+(1-eps)*mod3$regularized # lambda parameter for final estimate
mod3b <- TAM::tam.np(dat, n_basis=4, lambda=lambda_final, pars_init=mod3$pars)
summary(mod1)
summary(mod2)
summary(mod3)
summary(mod3b)
summary(mod4)
# compare models with respect to information criteria
IRT.compareModels(mod0, mod1, mod2, mod3, mod3b, mod4, mod5)
#-- compute item fit statistics RISE
# regularized solution
TAM::IRT.RISE(mod_p=mod1, mod_np=mod3)
# regularized solution, final estimation
TAM::IRT.RISE(mod_p=mod1, mod_np=mod3b, use_probs=TRUE)
TAM::IRT.RISE(mod_p=mod1, mod_np=mod3b, use_probs=FALSE)
# use TAM::IRT.RISE() function for computing the RMSD statistic
TAM::IRT.RISE(mod_p=mod1, mod_np=mod1, use_probs=FALSE)
#############################################################################
# EXAMPLE 3: Mixed 1PL/2PL model
#############################################################################
#* simulate data with 2 2PL items and 8 1PL items
set.seed(9877)
N <- 2000
I <- 10
b <- seq(-1,1,len=I)
a <- rep(1,I)
a[c(3,8)] <- c(.5, 2)
theta <- stats::rnorm(N, sd=1)
dat <- sirt::sim.raschtype(theta, b=b, fixed.a=a)
#- 1PL model
mod1 <- TAM::tam.mml(dat)
#- 2PL model
mod2 <- TAM::tam.mml.2pl(dat)
#- 2PL model with penalty on slopes
mod3 <- TAM::tam.np(dat, lambda=.04, model="1PL", n_basis=0)
summary(mod3)
#- final mixed 1PL/2PL model
lambda <- 1*mod3$regularized
mod4 <- TAM::tam.np(dat, lambda=lambda, model="1PL", n_basis=0)
summary(mod4)
IRT.compareModels(mod1, mod2, mod3, mod4)
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.