tam.np {TAM}

Unidimensional Non- and Semiparametric Item Response Model

Description

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.

Usage

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, ...)

Arguments

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 tam.mml.

model

Specified target model. Can be "2PL" or "1PL".

n_basis

Number of basis functions

basis_type

Type of basis function: "bspline" for B-splines or "hermite" for Gauss-Hermite polynomials

penalty_type

Lasso type penalty ("lasso") or ridge penalty ("ridge")

pars_init

Optional matrix of initial item parameters

orthonormalize

Logical indicating whether basis functions should be orthonormalized

object

Object of class tam.np

file

Optional file name for summary output

kfold

Number of folds in k-fold cross-validation

...

Further arguments to be passed

Details

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|\theta)=\psi( H_0(\theta) + H_1(\theta)

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(\theta)=d_i + a_i \theta and for the 1PL model (model="1PL") we set H_1(\theta)=d_i + 1 \cdot \theta . The model discrepancy is specified as a basis expansion approach

H_1 ( \theta )=\sum_{h=1}^p \beta_{ih} f_h( \theta)

where f_h are basis functions (possibly orthonormalized) and \beta_{ih} are item parameters which should be estimated. Penalty functions are posed on the \beta_{ih} coefficients. For the group lasso penalty, we specify the penalty J_{i,L1}=N \lambda \sqrt{p} \sqrt{ \sum_{h=1}^p \beta_{ih}^2 } while for the ridge penalty it is J_{i,L2}=N \lambda \sum_{h=1}^p \beta_{ih}^2 (N denoting the sample size).

Value

List containing several entries

rprobs

Item response probabilities

theta

Used nodes for approximation of \theta 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

References

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

See Also

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.

Examples

## 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)