sirt (version 0.31-20)

smirt: Multidimensional Noncompensatory and Compensatory Item Response Model

Description

This function estimates the noncompensatory and compensatory multidimensional item response model (Reckase, 2009).

Usage

smirt(dat, Qmatrix, irtmodel="noncomp" , est.b = NULL, est.a = NULL, 
     est.c = NULL, est.d = NULL, b.init = NULL, a.init = NULL, c.init = NULL,
     d.init=NULL, Sigma.init=NULL , theta.k=seq(-6,6,len=20), theta.kDES=NULL, 
     qmcnodes=0 , mu.fixed = NULL, variance.fixed = NULL,  est.corr = FALSE, 
     max.increment = 1, numdiff.parm = 0.001, maxdevchange = 0.1, 
     globconv = 0.001, maxiter = 1000, msteps = 4, mstepconv = 0.001)

Arguments

dat

Data frame with dichotomous item responses

Qmatrix

The $Q$ matrix which specifies the loadings to be estimated

irtmodel

The item response model. Options are the noncompensatory model ("noncomp") and the compensatory model ("comp"). See Details for more explanations.

est.b

An integer matrix (if irtmodel="noncomp") or integer vector (if irtmodel="comp") for $b$ parameters to be estimated

est.a

An integer matrix for $a$ parameters to be estimated. If est.a="2PL", then all item loadings will be estimated and the variances are set to one (and therefore est.corr=TRUE).

est.c

An integer vector for $c$ parameters to be estimated

est.d

An integer vector for $d$ parameters to be estimated

b.init

Initial $b$ coefficients

a.init

Initial $a$ coefficients

c.init

Initial $c$ coefficients

d.init

Initial $d$ coefficients

Sigma.init

Initial covariance matrix $\Sigma$

theta.k

Vector of discretized trait distribution. This vector is expanded in all dimensions by using the expand.grid function. If a user specifies a design matrix theta.kDES of transformed $\bold{\theta}_p$ values (see Details and Exa

theta.kDES

An optional design matrix. This matrix will differ from the ordinary theta grid in case of nonlinear item response models.

qmcnodes

Number of integration nodes for quasi Monte Carlo integration (see Pan & Thompson, 2007). Integration points are obtained by using the function qmc.nodes. Note that when using quasi Monte Carlo nodes,

mu.fixed

Matrix with fixed entries in the mean vector. By default, all means are set to zero.

variance.fixed

Matrix with fixed entries in the covariance matrix (see Examples).

est.corr

Should only a correlation matrix instead of a covariance matrix be estimated?

max.increment

Maximum increment

numdiff.parm

Numerical differentiation parameter

maxdevchange

Convergence criterion for change in relative deviance

globconv

Global convergence criterion for parameter change

maxiter

Maximum number of iterations

msteps

Number of iterations within a M step

mstepconv

Convergence criterion within a M step

Value

• A list with following entries:
• deviance
  Deviance
• ic
  Information criteria
• item
  Data frame with item parameters
• person
  Data frame with person parameters
• EAP.rel
  EAP reliability
• mean.trait
  Means of trait
• sd.trait
  Standard deviations of trait
• Sigma
  Trait covariance matrix
• cor.trait
  Trait correlation matrix
• b
  Matrix (vector) of $b$ parameters
• se.b
  Matrix (vector) of standard errors $b$ parameters
• a
  Matrix of $a$ parameters
• se.a
  Matrix of standard errors of $a$ parameters
• c
  Vector of $c$ parameters
• se.c
  Vector of standard errors of $c$ parameters
• d
  Vector of $d$ parameters
• se.d
  Vector of standard errors of $d$ parameters
• f.yi.qk
  Individual likelihood
• f.qk.yi
  Individual posterior
• probs
  Probabilities of item response functions evaluated at theta.k
• n.ik
  Expected counts
• iter
  Number of iterations
• dat2
  Processed data set
• dat2.resp
  Data set of response indicators
• I
  Number of items
• D
  Number of dimensions
• K
  Maximum item response score
• theta.k
  Used theta integration grid
• pi.k
  Distribution function evaluated at theta.k
• irtmodel
  Used IRT model

Details

The noncompensatory item response model (irtmodel="noncomp") is defined by $$P(X_{pi}=1 | \bold{\theta}_p ) = c_i + (d_i - c_i ) \prod_l invlogit( a_{il} q_{il} \theta_{pl} - b_{il} )$$ where $i$, $p$, $l$ denote items, persons and dimensions respectively. The compensatory item response model (irtmodel="comp") is defined by $$P(X_{pi}=1 | \bold{\theta}_p ) = c_i + (d_i - c_i ) invlogit( \sum_l a_{il} q_{il} \theta_{pl} - b_{i} )$$ Using a design matrix theta.kDES the model can be made even more general in a model which is linear in item parameters $$P(X_{pi}=1 | \bold{\theta}_p ) = c_i + (d_i - c_i ) invlogit( \sum_l a_{il} q_{il} t_l ( \bold{ \theta_{p} } ) - b_{i} )$$ with known functions $t_l$ of the trait vector $\bold{\theta}_p$. Fixed values of the functions $t_l$ are specified in the $\bold{\theta}_p$ design matrix theta.kDES. The models are estimated by an EM algorithm employing marginal maximum likelihood.

References

Pan, J. & Thompson, R. (2007). Quasi-Monte Carlo estimation in generalized linear mixed models. Computational Statistics & Data Analysis, 51, 5765-5775. Reckase, M.D. (2009). Multidimensional item response theory. New York: Springer.

See Also

Other multidimensional IRT models can also be estimated with rasch.mml2 and rasch.mirtlc. See also the mirt and TAM packages for estimation of multidimensional models.

Examples

############################
## SIMULATED EXAMPLE 1: Noncompensatory and compensatory IRT models
set.seed(997)

# (1) simulate data from a two-dimensional noncompensatory
#     item response model
 
N <- 1000	# number of persons
I <- 10		# number of items
theta0 <- rnorm( N , sd= 1 )
theta1 <- theta0 + rnorm(N , sd = .7 )
theta2 <- theta0 + rnorm(N , sd = .7 )
Q <- matrix( 1 , nrow=I,ncol=2 )
Q[ 1:(I/2) , 2 ] <- 0
Q[ I,1] <- 0
b <- matrix( rnorm( I*2 ) , I , 2 )
a <- matrix( 1 , I  , 2 )

# simulate data                                     
prob <- dat <- matrix(0 , nrow=N , ncol=I )
for (ii in 1:I){
prob[,ii] <- ( plogis( theta1 - b[ii,1]  ) )^Q[ii,1]
prob[,ii] <- prob[,ii] * ( plogis( theta2 - b[ii,2]  ) )^Q[ii,2]
            }
dat[ prob > matrix(runif( N*I),N,I) ] <- 1
colnames(dat) <- paste0("I",1:I)

#***
# Model 1: Noncompensatory 1PL model
mod1 <- smirt(dat, Qmatrix=Q , maxiter=30 )
summary(mod1)

#***
# Model 2: Noncompensatory 2PL model
mod2 <- smirt(dat,Qmatrix=Q , est.a="2PL" , maxiter=30 )
summary(mod2)

#***
# Model 3: some fixed c and d parameters different from zero or one
c.init <- rep(0,I)
c.init[ c(3,7)] <- .2
d.init <- rep(1,I)
d.init[c(4,8)] <- .95
mod3 <- smirt( dat , Qmatrix=Q , c.init=c.init , d.init=d.init )
summary(mod3)

#***
# Model 4: some estimated c and d parameters (in parameter groups)
est.c <- c.init <- rep(0,I)
c.estpars <- c(3,6,7)
c.init[ c.estpars ] <- .2
est.c[c.estpars] <- 1
est.d <- rep(0,I)
d.init <- rep(1,I)
d.estpars <- c(6,9)
d.init[ d.estpars ] <- .95
est.d[ d.estpars ] <- d.estpars   # different d parameters
mod4 <- smirt(dat,Qmatrix=Q , est.c=est.c , c.init=c.init ,
    est.d=est.d , d.init=d.init  )
summary(mod4)

#***
# Model 5: Unidimensional 1PL model
Qmatrix <- matrix( 1 , nrow=I , ncol=1 )
mod5 <- smirt( dat , Qmatrix=Qmatrix )
summary(mod5)

#***
# Model 6: Unidimensional 2PL model
mod6 <- smirt( dat , Qmatrix=Qmatrix , est.a="2PL" )
summary(mod6)

#***
# Model 7: Compensatory model with between item dimensionality
# between-item dimensionality
# Note that the data is simulated under the noncompensatory condition
# Therefore Model 7 should have a worse model fit than Model 1
Q1 <- Q
Q1[ 6:10 , 1] <- 0
mod7 <- smirt(dat,Qmatrix=Q1 , irtmodel="comp" , maxiter=30)
summary(mod7)

#***
# Model 8: Compensatory model with within item dimensionality
# 		assuming zero correlation between dimensions
variance.fixed <- as.matrix( cbind( 1,2,0) )
mod8 <- smirt(dat,Qmatrix=Q , irtmodel="comp" , variance.fixed=variance.fixed ,
	maxiter=30)
summary(mod8)

#***
# Model 9: Unidimensional model with quadratic item response functions
# define theta
theta.k <- seq( - 6 , 6 , len=15 )
theta.k <- as.matrix( theta.k , ncol=1 )
# define design matrix
theta.kDES <- cbind( theta.k[,1] , theta.k[,1]^2 )
# define Q matrix
Qmatrix <- matrix( 0 , I , 2 )
Qmatrix[,1] <- 1
Qmatrix[ c(3,6,7) , 2 ] <- 1
colnames(Qmatrix) <- c("F1" , "F1sq" )
# estimate model
mod9 <- smirt(dat,Qmatrix=Qmatrix , maxiter=50 , irtmodel="comp" , 
	theta.k=theta.k , theta.kDES=theta.kDES , est.a="2PL" )           
summary(mod9)

#***
# Model 10: Two-dimensional item response model with latent interaction
#           between dimensions
theta.k <- seq( - 6 , 6 , len=15 )
theta.k <- expand.grid( theta.k , theta.k )	# expand theta to 2 dimensions
# define design matrix
theta.kDES <- cbind( theta.k , theta.k[,1]*theta.k[,2] )
# define Q matrix
Qmatrix <- matrix( 0 , I , 3 )
Qmatrix[,1] <- 1
Qmatrix[ 6:10 , c(2,3) ] <- 1
colnames(Qmatrix) <- c("F1" , "F2" , "F1iF2" )
# estimate model
mod10 <- smirt(dat,Qmatrix=Qmatrix ,irtmodel="comp" , theta.k=theta.k , 
            theta.kDES= theta.kDES , est.a="2PL" )           
summary(mod10)

#****
# Model 11: Quasi Monte Carlo integration
Qmatrix <- matrix( 1 , I , 1 )
mod11 <- smirt( dat , irtmodel="comp" , Qmatrix=Qmatrix ,
            qmcnodes=1000 )
summary(mod11)