nedelsky-methods: Functions for the Nedelsky Model

Description

Functions for simulating and estimating the Nedelsky model (Bechger et al., 2003, 2005). nedelsky.sim can be used for simulating the model, nedelsky.irf computes the item response function and can be used for example when estimating the Nedelsky model in the mirt package or using the xxirt function in the sirt package.

Usage

# simulating the Nedelsky model
nedelsky.sim(theta, b, a=NULL, tau=NULL)
# creating latent responses of the Nedelsky model
nedelsky.latresp(K)
# computing the item response function of the Nedelsky model
nedelsky.irf(Theta, K, b, a, tau, combis, thdim=1)

Arguments

theta: Unidimensional ability (theta)
b: Matrix of category difficulties
a: Vector of item discriminations
tau: Category attractivity parameters $\tau$ (see Bechger et al., 2005)
K: (Maximum) Number of distractors of the used multiple choice items
Theta: Theta vector. Note that the Nedelsky model can be only specified as models with between item dimensionality (defined in thdim).
combis: Latent response classes as produced by nedelsky.latresp.
thdim: Theta dimension at which the item loads

Details

Assume that for item $i$ there exists $K+1$ categories $0,1,...,K$. The category 0 denotes the correct alternative. The Nedelsky model assumes that a respondent eliminates all distractors which are thought to be incorrect and guesses the solution from the remaining alternatives. This means, that for item $i$, $K$ latent variables $S_{ik}$ are defined which indicate whether alternative $k$ has been correctly identified as a distractor. By definition, the correct alternative is never been judged as wrong by the respondent.

Formally, the Nedelsky model assumes a 2PL model for eliminating each of the distractors $$P(S_{ik}=1 | \theta )=invlogit[ a_i ( \theta - b_{ik} ) ] $$ where $\theta$ is the person ability and $b_{ik}$ are distractor difficulties.

The guessing process of the Nedelsky model is defined as $$P(X_i=j | \theta, S_{i1}, ..., S_{iK} )= \frac{ ( 1- S_{ij} ) \tau_{ij} }{ \sum_{k=0}^K [ ( 1- S_{ik} ) \tau_{ik} ] }$$ where $\tau_{ij}$ are attractivity parameters of alternative $j$. By definition $\tau_{i0}$ is set to 1. By default, all attractivity parameters are set to 1.

References

Bechger, T. M., Maris, G., Verstralen, H. H. F. M., & Verhelst, N. D. (2003). The Nedelsky model for multiple-choice items. CITO Research Report, 2003-5.

Bechger, T. M., Maris, G., Verstralen, H. H. F. M., & Verhelst, N. D. (2005). The Nedelsky model for multiple-choice items. In L. van der Ark, M. Croon, & Sijtsma, K. (Eds.). New developments in categorical data analysis for the social and behavioral sciences, pp. 187-206. Mahwah, Lawrence Erlbaum.

Examples

Run this code

if (FALSE) {
#############################################################################
# EXAMPLE 1: Simulated data according to the Nedelsky model
#############################################################################

#*** simulate data
set.seed(123)
I <- 20          # number of items
b <- matrix(NA,I,ncol=3)
b[,1] <- -0.5 + stats::runif( I, -.75, .75 )
b[,2] <- -1.5 + stats::runif( I, -.75, .75 )
b[,3] <- -2.5 + stats::runif( I, -.75, .75 )
K <- 3           # number of distractors
N <- 2000        # number of persons
# apply simulation function
dat <- sirt::nedelsky.sim( theta=stats::rnorm(N,sd=1.2), b=b )

#*** latent response patterns
K <- 3
combis <- sirt::nedelsky.latresp(K=3)

#*** defining the Nedelsky item response function for estimation in mirt
par <- c( 3, rep(-1,K), 1, rep(1,K+1),1)
names(par) <- c("K", paste0("b",1:K), "a", paste0("tau", 0:K),"thdim")
est <- c( FALSE, rep(TRUE,K), rep(FALSE, K+1 + 2 ) )
names(est) <- names(par)
nedelsky.icc <- function( par, Theta, ncat ){
     K <- par[1]
     b <- par[ 1:K + 1]
     a <- par[ K+2]
     tau <- par[1:(K+1) + (K+2) ]
     thdim <- par[ K+2+K+1 +1 ]
     probs <- sirt::nedelsky.irf( Theta, K=K, b=b, a=a, tau=tau, combis,
                    thdim=thdim  )$probs
     return(probs)
}
name <- "nedelsky"
# create item response function
nedelsky.itemfct <- mirt::createItem(name, par=par, est=est, P=nedelsky.icc)

#*** define model in mirt
mirtmodel <- mirt::mirt.model("
           F1=1-20
           COV=F1*F1
           # define some prior distributions
           PRIOR=(1-20,b1,norm,-1,2),(1-20,b2,norm,-1,2),
                   (1-20,b3,norm,-1,2)
        " )

itemtype <- rep("nedelsky", I )
customItems <- list("nedelsky"=nedelsky.itemfct)
# define parameters to be estimated
mod1.pars <- mirt::mirt(dat, mirtmodel, itemtype=itemtype,
                   customItems=customItems, pars="values")
# estimate model
mod1 <- mirt::mirt(dat,mirtmodel, itemtype=itemtype, customItems=customItems,
               pars=mod1.pars, verbose=TRUE  )
# model summaries
print(mod1)
summary(mod1)
mirt.wrapper.coef( mod1 )$coef
mirt.wrapper.itemplot(mod1,ask=TRUE)

#******************************************************
# fit Nedelsky model with xxirt function in sirt

# define item class for xxirt
item_nedelsky <- sirt::xxirt_createDiscItem( name="nedelsky", par=par,
                est=est, P=nedelsky.icc,
                prior=c( b1="dnorm", b2="dnorm", b3="dnorm" ),
                prior_par1=c( b1=-1, b2=-1, b3=-1),
                prior_par2=c(b1=2, b2=2, b3=2) )
customItems <- list( item_nedelsky )

#---- definition theta distribution
#** theta grid
Theta <- matrix( seq(-6,6,length=21), ncol=1 )
#** theta distribution
P_Theta1 <- function( par, Theta, G){
    mu <- par[1]
    sigma <- max( par[2], .01 )
    TP <- nrow(Theta)
    pi_Theta <- matrix( 0, nrow=TP, ncol=G)
    pi1 <- dnorm( Theta[,1], mean=mu, sd=sigma )
    pi1 <- pi1 / sum(pi1)
    pi_Theta[,1] <- pi1
    return(pi_Theta)
}
#** create distribution class
par_Theta <- c( "mu"=0, "sigma"=1 )
customTheta <- sirt::xxirt_createThetaDistribution( par=par_Theta, est=c(FALSE,TRUE),
                   P=P_Theta1 )

#-- create parameter table
itemtype <- rep( "nedelsky", I )
partable <- sirt::xxirt_createParTable( dat, itemtype=itemtype, customItems=customItems)

# estimate model
mod2 <- sirt::xxirt( dat=dat, Theta=Theta, partable=partable, customItems=customItems,
                    customTheta=customTheta)
summary(mod2)
# compare sirt::xxirt and mirt::mirt
logLik(mod2)
mod1@Fit$logLik

#############################################################################
# EXAMPLE 2: Multiple choice dataset data.si06
#############################################################################

data(data.si06)
dat <- data.si06

#*** create latent responses
combis <- sirt::nedelsky.latresp(K)
I <- ncol(dat)
#*** define item response function
K <- 3
par <- c( 3, rep(-1,K), 1, rep(1,K+1),1)
names(par) <- c("K", paste0("b",1:K), "a", paste0("tau", 0:K),"thdim")
est <- c( FALSE, rep(TRUE,K), rep(FALSE, K+1 + 2 ) )
names(est) <- names(par)
nedelsky.icc <- function( par, Theta, ncat ){
     K <- par[1]
     b <- par[ 1:K + 1]
     a <- par[ K+2]
     tau <- par[1:(K+1) + (K+2) ]
     thdim <- par[ K+2+K+1 +1 ]
     probs <- sirt::nedelsky.irf( Theta, K=K, b=b, a=a, tau=tau, combis,
                    thdim=thdim  )$probs
     return(probs)
}
name <- "nedelsky"
# create item response function
nedelsky.itemfct <- mirt::createItem(name, par=par, est=est, P=nedelsky.icc)

#*** define model in mirt
mirtmodel <- mirt::mirt.model("
           F1=1-14
           COV=F1*F1
           PRIOR=(1-14,b1,norm,-1,2),(1-14,b2,norm,-1,2),
                   (1-14,b3,norm,-1,2)
        " )

itemtype <- rep("nedelsky", I )
customItems <- list("nedelsky"=nedelsky.itemfct)
# define parameters to be estimated
mod1.pars <- mirt::mirt(dat, mirtmodel, itemtype=itemtype,
                   customItems=customItems, pars="values")

#*** estimate model
mod1 <- mirt::mirt(dat,mirtmodel, itemtype=itemtype, customItems=customItems,
               pars=mod1.pars, verbose=TRUE )
#*** summaries
print(mod1)
summary(mod1)
mirt.wrapper.coef( mod1 )$coef
mirt.wrapper.itemplot(mod1,ask=TRUE)
}

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