gdm: General Diagnostic Model
In CDM: Cognitive Diagnosis Modeling
gdm R Documentation
General Diagnostic Model
Description
This function estimates the general diagnostic model
(von Davier, 2008; Xu & von Davier, 2008) which handles
multidimensional item response models with ordered discrete
or continuous latent variables for polytomous item
responses.
Usage
gdm( data, theta.k, irtmodel="2PL", group=NULL, weights=rep(1, nrow(data)),
Qmatrix=NULL, thetaDes=NULL, skillspace="loglinear",
b.constraint=NULL, a.constraint=NULL,
mean.constraint=NULL, Sigma.constraint=NULL, delta.designmatrix=NULL,
standardized.latent=FALSE, centered.latent=FALSE,
centerintercepts=FALSE, centerslopes=FALSE,
maxiter=1000, conv=1e-5, globconv=1e-5, msteps=4, convM=.0005,
decrease.increments=FALSE, use.freqpatt=FALSE, progress=TRUE,
PEM=FALSE, PEM_itermax=maxiter, ...)
## S3 method for class 'gdm'
summary(object, file=NULL, ...)
## S3 method for class 'gdm'
print(x, ...)
## S3 method for class 'gdm'
plot(x, perstype="EAP", group=1, barwidth=.1, histcol=1,
cexcor=3, pchpers=16, cexpers=.7, ... )
Arguments
data
An N \times I matrix of polytomous item
responses with categories k=0,1,...,K
theta.k
In the one-dimensional case it must be a vector.
For multidimensional models it has to be a list
of skill vectors if the theta grid differs between
dimensions. If not, a vector input can be supplied.
If an estimated skillspace (skillspace="est" should be estimated,
a vector or a matrix theta.k will be used as initial values of the estimated
\bold{θ} grid.
irtmodel
The default 2PL corresponds to the model
where item slopes on dimensions are equal for all
item categories. If item-category slopes should
be estimated, use 2PLcat. If no item slopes
should be estimated then 1PL can be selected.
Note that fixed item slopes can be specified in
the Q-matrix (argument Qmatrix).
group
An optional vector of group identifiers for
multiple group estimation.
For plot.gdm it is an integer indicating which
group should be used for plotting.
weights
An optional vector of sample weights
Qmatrix
An optional array of dimension I \times D \times K
which indicates pre-specified item loadings
on dimensions. The default for category k is the score k, i.e.
the scoring in the (generalized) partial credit model.
thetaDes
A design matrix for specifying nonlinear item response
functions (see Example 1, Models 4 and 5)
skillspace
The parametric assumption of the skillspace.
If skillspace="normal" then a univariate or
multivariate normal distribution is assumed.
The default "loglinear" corresponds to log-linear
smoothing of the skillspace distribution (Xu & von Davier, 2008).
If skillspace="full", then all probabilities of the skill space
are nonparametrically estimated. If skillspace="est", then the
\bold{θ} distribution vectors will be estimated
(see Details and Examples 4 and 5; Bartolucci, 2007).
b.constraint
In this optional matrix with C_b rows and three columns,
C_b item intercepts b_{ik} can be fixed.
1st column: item index, 2nd column: category index,
3rd column: fixed item thresholds
a.constraint
In this optional matrix with C_a rows and four columns,
C_a item intercepts a_{idk} can be fixed.
1st column: item index, 2nd column: dimension index,
3rd column: category index, 4th column: fixed item slopes
mean.constraint
A C \times 3 matrix for
constraining C means in the
normal distribution assumption (skillspace="normal").
1st column: Dimension, 2nd column: Group, 3rd column: Value
Sigma.constraint
A C \times 4 matrix for
constraining C covariances in the
normal distribution assumption (skillspace="normal").
1st column: Dimension 1, 2nd column: Dimension 2,
3rd column: Group, 4th column: Value
delta.designmatrix
The design matrix of δ parameters
for the reduced skillspace estimation (see Xu &
von Davier, 2008)
standardized.latent
A logical indicating whether in a uni- or multidimensional
model all latent variables of the first group should be normally distributed
and standardized. The default is FALSE.
centered.latent
A logical indicating whether in a uni- or multidimensional
model all latent variables of the first group should be normally
distributed and do have zero means? The default is FALSE.
centerintercepts
A logical indicating whether intercepts should be centered to have a
mean of 0 for all dimensions. This argument does not (yet) work properly
for varying numbers of item categories.
centerslopes
A logical indicating whether item slopes should be centered to have
a mean of 1 for all dimensions. This argument only works for
irtmodel="2PL". The default is FALSE.
maxiter
Maximum number of iterations
conv
Convergence criterion for item parameters and
distribution parameters
globconv
Global deviance convergence criterion
msteps
Maximum number of M steps in estimating b and
a item parameters. The default is to use 4 M steps.
convM
Convergence criterion in M step
decrease.increments
Should in the M step the increments
of a and b parameters decrease during iterations?
The default is FALSE. If there is an increase in deviance
during estimation, setting decrease.increments to TRUE
is recommended.
use.freqpatt
A logical indicating whether frequencies of unique item response patterns
should be used. In case of large data set use.freqpatt=TRUE
can speed calculations (depending on the problem).
Note that in this case, not all person parameters are calculated
as usual in the output.
progress
An optional logical indicating whether the function should print the
progress of iteration in the estimation process.
PEM
Logical indicating whether the P-EM acceleration should be
applied (Berlinet & Roland, 2012).
PEM_itermax
Number of iterations in which the P-EM method should be
applied.
object
A required object of class gdm
file
Optional file name for a file in which summary
should be sinked.
x
A required object of class gdm
perstype
Person parameter estimate type. Can be either
"EAP", "MAP" or "MLE".
barwidth
Bar width in plot.gdm
histcol
Color of histogram bars in plot.gdm
cexcor
Font size for print of correlation in plot.gdm
pchpers
Point type for scatter plot of person
parameters in plot.gdm
cexpers
Point size for scatter plot of person
parameters in plot.gdm
...
Optional parameters to be passed to or from other
methods will be ignored.
Details
Case irtmodel="1PL":
Equal item slopes of 1 are assumed in this model. Therefore,
it corresponds to a generalized multidimensional Rasch model.
logit P( X_{nj}=k | θ_n )=b_{j0} +
∑_d q_{jdk} θ_{nd}
The Q-matrix entries q_{jdk} are pre-specified by the user.
Case irtmodel="2PL":
For each item and each dimension, different item slopes a_{jd}
are estimated:
logit P( X_{nj}=k | θ_n )=b_{j0} +
∑_d a_{jd} q_{jdk} θ_{nd}
Case irtmodel="2PLcat":
For each item, each dimension and each category,
different item slopes a_{jdk}
are estimated:
logit P( X_{nj}=k | θ_n )=b_{j0} +
∑_d a_{jdk} q_{jdk} θ_{nd}
Note that this model can be generalized to include terms of
any transformation t_h of the θ_n vector (e.g. quadratic terms,
step functions or interaction) such that the model can be formulated as
logit P( X_{nj}=k | θ_n )=b_{j0} +
∑_h a_{jhk} q_{jhk} t_h( θ_{n} )
In general, the number of functions t_1, ..., t_H will be
larger than the θ dimension of D.
The estimation follows an EM algorithm as described in von Davier and
Yamamoto (2004) and von Davier (2008).
In case of skillspace="est", the \bold{θ} vectors
(the grid of the theta distribution) are estimated (Bartolucci, 2007;
Bacci, Bartolucci & Gnaldi, 2012). This model is called a multidimensional
latent class item response model.
Value
An object of class gdm. The list contains the
following entries:
item
Data frame with item parameters
person
Data frame with person parameters:
EAP denotes the mean of the individual posterior distribution,
SE.EAP the corresponding standard error,
MLE the maximum likelihood estimate at theta.k
and MAP the mode of the posterior distribution
EAP.rel
Reliability of the EAP
deviance
Deviance
ic
Information criteria, number of estimated parameters
b
Item intercepts b_{jk}
se.b
Standard error of item intercepts b_{jk}
a
Item slopes a_{jd} resp. a_{jdk}
se.a
Standard error of item slopes a_{jd} resp. a_{jdk}
itemfit.rmsea
The RMSEA item fit index (see itemfit.rmsea).
This entry comes as a list with total and group-wise item fit
statistics.
mean.rmsea
Mean of RMSEA item fit indexes.
Qmatrix
Used Q-matrix
pi.k
Trait distribution
mean.trait
Means of trait distribution
sd.trait
Standard deviations of trait distribution
skewness.trait
Skewnesses of trait distribution
correlation.trait
List of correlation matrices of trait distribution
corresponding to each group
pjk
Item response probabilities evaluated at grid theta.k
n.ik
An array of expected counts n_{cikg} of ability class c
at item i at category k in group g
G
Number of groups
D
Number of dimension of \bold{θ}
I
Number of items
N
Number of persons
delta
Parameter estimates for skillspace representation
covdelta
Covariance matrix of parameter estimates for
skillspace representation
data
Original data frame
group.stat
Group statistics (sample sizes, group labels)
p.xi.aj
Individual likelihood
posterior
Individual posterior distribution
skill.levels
Number of skill levels per dimension
K.item
Maximal category per item
theta.k
Used theta design or estimated theta trait distribution
in case of skillspace="est"
thetaDes
Used theta design for item responses
se.theta.k
Estimated standard errors of theta.k if it is
estimated
time
Info about computation time
skillspace
Used skillspace parametrization
iter
Number of iterations
converged
Logical indicating whether convergence was achieved.
object
Object of class gdm
x
Object of class gdm
perstype
Person paramter estimate type. Can be either
"EAP", "MAP" or "MLE".
group
Group which should be used for plot.gdm
barwidth
Bar width in plot.gdm
histcol
Color of histogram bars in plot.gdm
cexcor
Font size for print of correlation in plot.gdm
pchpers
Point type for scatter plot of person
parameters in plot.gdm
cexpers
Point size for scatter plot of person
parameters in plot.gdm
...
Optional parameters to be passed to or from other
methods will be ignored.
References
Bacci, S., Bartolucci, F., & Gnaldi, M. (2012).
A class of multidimensional latent class IRT models for ordinal
polytomous item responses. arXiv preprint, arXiv:1201.4667.
Bartolucci, F. (2007). A class of multidimensional IRT models for testing
unidimensionality and clustering items. Psychometrika, 72, 141-157.
Berlinet, A. F., & Roland, C. (2012).
Acceleration of the EM algorithm: P-EM versus epsilon algorithm.
Computational Statistics & Data Analysis, 56(12), 4122-4137.
von Davier, M. (2008). A general diagnostic model applied to
language testing data. British Journal
of Mathematical and Statistical Psychology, 61, 287-307.
von Davier, M., & Yamamoto, K. (2004). Partially observed mixtures of IRT models:
An extension of the generalized partial-credit model.
Applied Psychological Measurement, 28, 389-406.
Xu, X., & von Davier, M. (2008). Fitting the structured general diagnostic
model to NAEP data. ETS Research Report ETS RR-08-27. Princeton, ETS.
See Also
Cognitive diagnostic models for dichotomous data can be estimated
with din (DINA or DINO model) or gdina
(GDINA model, which contains many CDMs as special cases).
For assessment of model fit see modelfit.cor.din and
anova.gdm.
See itemfit.sx2 for item fit statistics.
For the estimation of the multidimensional
latent class item response model see the MultiLCIRT package
and sirt package (function sirt::rasch.mirtlc).
Examples
#############################################################################
# EXAMPLE 1: Fraction Dataset 1
# Unidimensional Models for dichotomous data
#############################################################################
data(data.fraction1, package="CDM")
dat <- data.fraction1$data
theta.k <- seq( -6, 6, len=15 ) # discretized ability
#***
# Model 1: Rasch model (normal distribution)
mod1 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, skillspace="normal",
centered.latent=TRUE)
summary(mod1)
plot(mod1)
#***
# Model 2: Rasch model (log-linear smoothing)
# set the item difficulty of the 8th item to zero
b.constraint <- matrix( c(8,1,0), 1, 3 )
mod2 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k,
skillspace="loglinear", b.constraint=b.constraint )
summary(mod2)
#***
# Model 3: 2PL model
mod3 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k,
skillspace="normal", standardized.latent=TRUE )
summary(mod3)
## Not run:
#***
# Model 4: include quadratic term in item response function
# using the argument decrease.increments=TRUE leads to a more
# stable estimate
thetaDes <- cbind( theta.k, theta.k^2 )
colnames(thetaDes) <- c( "F1", "F1q" )
mod4 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k,
thetaDes=thetaDes, skillspace="normal",
standardized.latent=TRUE, decrease.increments=TRUE)
summary(mod4)
#***
# Model 5: step function for ICC
# two different probabilities theta < 0 and theta > 0
thetaDes <- matrix( 1*(theta.k>0), ncol=1 )
colnames(thetaDes) <- c( "Fgrm1" )
mod5 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k,
thetaDes=thetaDes, skillspace="normal" )
summary(mod5)
#***
# Model 6: DINA model with din function
mod6 <- CDM::din( dat, q.matrix=matrix( 1, nrow=ncol(dat),ncol=1 ) )
summary(mod6)
#***
# Model 7: Estimating a version of the DINA model with gdm
theta.k <- c(-.5,.5)
mod7 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k, skillspace="loglinear" )
summary(mod7)
#############################################################################
# EXAMPLE 2: Cultural Activities - data.Students
# Unidimensional Models for polytomous data
#############################################################################
data(data.Students, package="CDM")
dat <- data.Students
dat <- dat[, grep( "act", colnames(dat) ) ]
theta.k <- seq( -4, 4, len=11 ) # discretized ability
#***
# Model 1: Partial Credit Model (PCM)
mod1 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, skillspace="normal",
centered.latent=TRUE)
summary(mod1)
plot(mod1)
#***
# Model 1b: PCM using frequency patterns
mod1b <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, skillspace="normal",
centered.latent=TRUE, use.freqpatt=TRUE)
summary(mod1b)
#***
# Model 2: PCM with two groups
mod2 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k,
group=CDM::data.Students$urban + 1, skillspace="normal",
centered.latent=TRUE)
summary(mod2)
#***
# Model 3: PCM with loglinear smoothing
b.constraint <- matrix( c(1,2,0), ncol=3 )
mod3 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k,
skillspace="loglinear", b.constraint=b.constraint )
summary(mod3)
#***
# Model 4: Model with pre-specified item weights in Q-matrix
Qmatrix <- array( 1, dim=c(5,1,2) )
Qmatrix[,1,2] <- 2 # default is score 2 for category 2
# now change the scoring of category 2:
Qmatrix[c(2,4),1,1] <- .74
Qmatrix[c(2,4),1,2] <- 2.3
# for items 2 and 4 the score for category 1 is .74 and for category 2 it is 2.3
mod4 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, Qmatrix=Qmatrix,
skillspace="normal", centered.latent=TRUE)
summary(mod4)
#***
# Model 5: Generalized partial credit model
mod5 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k,
skillspace="normal", standardized.latent=TRUE )
summary(mod5)
#***
# Model 6: Item-category slope estimation
mod6 <- CDM::gdm( dat, irtmodel="2PLcat", theta.k=theta.k, skillspace="normal",
standardized.latent=TRUE, decrease.increments=TRUE)
summary(mod6)
#***
# Models 7: items with different number of categories
dat0 <- dat
dat0[ paste(dat0[,1])==2, 1 ] <- 1 # 1st item has only two categories
dat0[ paste(dat0[,3])==2, 3 ] <- 1 # 3rd item has only two categories
# Model 7a: PCM
mod7a <- CDM::gdm( dat0, irtmodel="1PL", theta.k=theta.k, centered.latent=TRUE )
summary(mod7a)
# Model 7b: Item category slopes
mod7b <- CDM::gdm( dat0, irtmodel="2PLcat", theta.k=theta.k,
standardized.latent=TRUE, decrease.increments=TRUE )
summary(mod7b)
#############################################################################
# EXAMPLE 3: Fraction Dataset 2
# Multidimensional Models for dichotomous data
#############################################################################
data(data.fraction2, package="CDM")
dat <- data.fraction2$data
Qmatrix <- data.fraction2$q.matrix3
#***
# Model 1: One-dimensional Rasch model
theta.k <- seq( -4, 4, len=11 ) # discretized ability
mod1 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, centered.latent=TRUE)
summary(mod1)
plot(mod1)
#***
# Model 2: One-dimensional 2PL model
mod2 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k, standardized.latent=TRUE)
summary(mod2)
plot(mod2)
#***
# Model 3: 3-dimensional Rasch Model (normal distribution)
mod3 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, Qmatrix=Qmatrix,
centered.latent=TRUE, globconv=5*1E-3, conv=1E-4 )
summary(mod3)
#***
# Model 4: 3-dimensional Rasch model (loglinear smoothing)
# set some item parameters of items 4,1 and 2 to zero
b.constraint <- cbind( c(4,1,2), 1, 0 )
mod4 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, Qmatrix=Qmatrix,
b.constraint=b.constraint, skillspace="loglinear" )
summary(mod4)
#***
# Model 5: define a different theta grid for each dimension
theta.k <- list( "Dim1"=seq( -5, 5, len=11 ),
"Dim2"=seq(-5,5,len=8),
"Dim3"=seq( -3,3,len=6) )
mod5 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, Qmatrix=Qmatrix,
b.constraint=b.constraint, skillspace="loglinear")
summary(mod5)
#***
# Model 6: multdimensional 2PL model (normal distribution)
theta.k <- seq( -5, 5, len=13 )
a.constraint <- cbind( c(8,1,3), 1:3, 1, 1 ) # fix some slopes to 1
mod6 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k, Qmatrix=Qmatrix,
centered.latent=TRUE, a.constraint=a.constraint, decrease.increments=TRUE,
skillspace="normal")
summary(mod6)
#***
# Model 7: multdimensional 2PL model (loglinear distribution)
a.constraint <- cbind( c(8,1,3), 1:3, 1, 1 )
b.constraint <- cbind( c(8,1,3), 1, 0 )
mod7 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k, Qmatrix=Qmatrix,
b.constraint=b.constraint, a.constraint=a.constraint,
decrease.increments=FALSE, skillspace="loglinear")
summary(mod7)
#############################################################################
# EXAMPLE 4: Unidimensional latent class 1PL IRT model
#############################################################################
# simulate data
set.seed(754)
I <- 20 # number of items
N <- 2000 # number of persons
theta <- c( -2, 0, 1, 2 )
theta <- rep( theta, c(N/4,N/4, 3*N/8, N/8) )
b <- seq(-2,2,len=I)
library(sirt) # use function sim.raschtype from sirt package
dat <- sirt::sim.raschtype( theta=theta, b=b )
theta.k <- seq(-1, 1, len=4) # initial vector of theta
# estimate model
mod1 <- CDM::gdm( dat, theta.k=theta.k, skillspace="est", irtmodel="1PL",
centerintercepts=TRUE, maxiter=200)
summary(mod1)
## Estimated Skill Distribution
## F1 pi.k
## 1 -1.988 0.24813
## 2 -0.055 0.23313
## 3 0.940 0.40059
## 4 2.000 0.11816
#############################################################################
# EXAMPLE 5: Multidimensional latent class IRT model
#############################################################################
# We simulate a two-dimensional IRT model in which theta vectors
# are observed at a fixed discrete grid (see below).
# simulate data
set.seed(754)
I <- 13 # number of items
N <- 2400 # number of persons
# simulate Dimension 1 at 4 discrete theta points
theta <- c( -2, 0, 1, 2 )
theta <- rep( theta, c(N/4,N/4, 3*N/8, N/8) )
b <- seq(-2,2,len=I)
library(sirt) # use simulation function from sirt package
dat1 <- sirt::sim.raschtype( theta=theta, b=b )
# simulate Dimension 2 at 4 discrete theta points
theta <- c( -3, 0, 1.5, 2 )
theta <- rep( theta, c(N/4,N/4, 3*N/8, N/8) )
dat2 <- sirt::sim.raschtype( theta=theta, b=b )
colnames(dat2) <- gsub( "I", "U", colnames(dat2))
dat <- cbind( dat1, dat2 )
# define Q-matrix
Qmatrix <- matrix(0,2*I,2)
Qmatrix[ cbind( 1:(2*I), rep(1:2, each=I) ) ] <- 1
theta.k <- seq(-1, 1, len=4) # initial matrix
theta.k <- cbind( theta.k, theta.k )
colnames(theta.k) <- c("Dim1","Dim2")
# estimate model
mod2 <- CDM::gdm( dat, theta.k=theta.k, skillspace="est", irtmodel="1PL",
Qmatrix=Qmatrix, centerintercepts=TRUE)
summary(mod2)
## Estimated Skill Distribution
## theta.k.Dim1 theta.k.Dim2 pi.k
## 1 -2.022 -3.035 0.25010
## 2 0.016 0.053 0.24794
## 3 0.956 1.525 0.36401
## 4 1.958 1.919 0.13795
#############################################################################
# EXAMPLE 6: Large-scale dataset data.mg
#############################################################################
data(data.mg, package="CDM")
dat <- data.mg[, paste0("I", 1:11 ) ]
theta.k <- seq(-6,6,len=21)
#***
# Model 1: Generalized partial credit model with multiple groups
mod1 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k, group=CDM::data.mg$group,
skillspace="normal", standardized.latent=TRUE)
summary(mod1)
## End(Not run)
CDM documentation built on Aug. 25, 2022, 5:08 p.m.
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| gdm | R Documentation |
General Diagnostic Model
Description
This function estimates the general diagnostic model (von Davier, 2008; Xu & von Davier, 2008) which handles multidimensional item response models with ordered discrete or continuous latent variables for polytomous item responses.
Usage
gdm( data, theta.k, irtmodel="2PL", group=NULL, weights=rep(1, nrow(data)),
Qmatrix=NULL, thetaDes=NULL, skillspace="loglinear",
b.constraint=NULL, a.constraint=NULL,
mean.constraint=NULL, Sigma.constraint=NULL, delta.designmatrix=NULL,
standardized.latent=FALSE, centered.latent=FALSE,
centerintercepts=FALSE, centerslopes=FALSE,
maxiter=1000, conv=1e-5, globconv=1e-5, msteps=4, convM=.0005,
decrease.increments=FALSE, use.freqpatt=FALSE, progress=TRUE,
PEM=FALSE, PEM_itermax=maxiter, ...)
## S3 method for class 'gdm'
summary(object, file=NULL, ...)
## S3 method for class 'gdm'
print(x, ...)
## S3 method for class 'gdm'
plot(x, perstype="EAP", group=1, barwidth=.1, histcol=1,
cexcor=3, pchpers=16, cexpers=.7, ... )
Arguments
data |
An N \times I matrix of polytomous item responses with categories k=0,1,...,K |
theta.k |
In the one-dimensional case it must be a vector.
For multidimensional models it has to be a list
of skill vectors if the theta grid differs between
dimensions. If not, a vector input can be supplied.
If an estimated skillspace ( |
irtmodel |
The default |
group |
An optional vector of group identifiers for
multiple group estimation.
For |
weights |
An optional vector of sample weights |
Qmatrix |
An optional array of dimension I \times D \times K which indicates pre-specified item loadings on dimensions. The default for category k is the score k, i.e. the scoring in the (generalized) partial credit model. |
thetaDes |
A design matrix for specifying nonlinear item response functions (see Example 1, Models 4 and 5) |
skillspace |
The parametric assumption of the skillspace.
If |
b.constraint |
In this optional matrix with C_b rows and three columns, C_b item intercepts b_{ik} can be fixed. 1st column: item index, 2nd column: category index, 3rd column: fixed item thresholds |
a.constraint |
In this optional matrix with C_a rows and four columns,
C_a item intercepts a_{idk} can be fixed.
1st column: item index, 2nd column: dimension index,
3rd column: category index, 4th column: fixed item slopes |
mean.constraint |
A C \times 3 matrix for
constraining C means in the
normal distribution assumption ( |
Sigma.constraint |
A C \times 4 matrix for
constraining C covariances in the
normal distribution assumption ( |
delta.designmatrix |
The design matrix of δ parameters for the reduced skillspace estimation (see Xu & von Davier, 2008) |
standardized.latent |
A logical indicating whether in a uni- or multidimensional
model all latent variables of the first group should be normally distributed
and standardized. The default is |
centered.latent |
A logical indicating whether in a uni- or multidimensional
model all latent variables of the first group should be normally
distributed and do have zero means? The default is |
centerintercepts |
A logical indicating whether intercepts should be centered to have a mean of 0 for all dimensions. This argument does not (yet) work properly for varying numbers of item categories. |
centerslopes |
A logical indicating whether item slopes should be centered to have
a mean of 1 for all dimensions. This argument only works for
|
maxiter |
Maximum number of iterations |
conv |
Convergence criterion for item parameters and distribution parameters |
globconv |
Global deviance convergence criterion |
msteps |
Maximum number of M steps in estimating b and a item parameters. The default is to use 4 M steps. |
convM |
Convergence criterion in M step |
decrease.increments |
Should in the M step the increments
of a and b parameters decrease during iterations?
The default is |
use.freqpatt |
A logical indicating whether frequencies of unique item response patterns
should be used. In case of large data set |
progress |
An optional logical indicating whether the function should print the progress of iteration in the estimation process. |
PEM |
Logical indicating whether the P-EM acceleration should be applied (Berlinet & Roland, 2012). |
PEM_itermax |
Number of iterations in which the P-EM method should be applied. |
object |
A required object of class |
file |
Optional file name for a file in which |
x |
A required object of class |
perstype |
Person parameter estimate type. Can be either
|
barwidth |
Bar width in |
histcol |
Color of histogram bars in |
cexcor |
Font size for print of correlation in |
pchpers |
Point type for scatter plot of person
parameters in |
cexpers |
Point size for scatter plot of person
parameters in |
... |
Optional parameters to be passed to or from other methods will be ignored. |
Details
Case irtmodel="1PL":
Equal item slopes of 1 are assumed in this model. Therefore,
it corresponds to a generalized multidimensional Rasch model.
logit P( X_{nj}=k | θ_n )=b_{j0} + ∑_d q_{jdk} θ_{nd}
The Q-matrix entries q_{jdk} are pre-specified by the user.
Case irtmodel="2PL":
For each item and each dimension, different item slopes a_{jd}
are estimated:
logit P( X_{nj}=k | θ_n )=b_{j0} + ∑_d a_{jd} q_{jdk} θ_{nd}
Case irtmodel="2PLcat":
For each item, each dimension and each category,
different item slopes a_{jdk}
are estimated:
logit P( X_{nj}=k | θ_n )=b_{j0} + ∑_d a_{jdk} q_{jdk} θ_{nd}
Note that this model can be generalized to include terms of any transformation t_h of the θ_n vector (e.g. quadratic terms, step functions or interaction) such that the model can be formulated as
logit P( X_{nj}=k | θ_n )=b_{j0} + ∑_h a_{jhk} q_{jhk} t_h( θ_{n} )
In general, the number of functions t_1, ..., t_H will be larger than the θ dimension of D.
The estimation follows an EM algorithm as described in von Davier and Yamamoto (2004) and von Davier (2008).
In case of skillspace="est", the \bold{θ} vectors
(the grid of the theta distribution) are estimated (Bartolucci, 2007;
Bacci, Bartolucci & Gnaldi, 2012). This model is called a multidimensional
latent class item response model.
Value
An object of class gdm. The list contains the
following entries:
item |
Data frame with item parameters |
person |
Data frame with person parameters:
|
EAP.rel |
Reliability of the EAP |
deviance |
Deviance |
ic |
Information criteria, number of estimated parameters |
b |
Item intercepts b_{jk} |
se.b |
Standard error of item intercepts b_{jk} |
a |
Item slopes a_{jd} resp. a_{jdk} |
se.a |
Standard error of item slopes a_{jd} resp. a_{jdk} |
itemfit.rmsea |
The RMSEA item fit index (see |
mean.rmsea |
Mean of RMSEA item fit indexes. |
Qmatrix |
Used Q-matrix |
pi.k |
Trait distribution |
mean.trait |
Means of trait distribution |
sd.trait |
Standard deviations of trait distribution |
skewness.trait |
Skewnesses of trait distribution |
correlation.trait |
List of correlation matrices of trait distribution corresponding to each group |
pjk |
Item response probabilities evaluated at grid |
n.ik |
An array of expected counts n_{cikg} of ability class c at item i at category k in group g |
G |
Number of groups |
D |
Number of dimension of \bold{θ} |
I |
Number of items |
N |
Number of persons |
delta |
Parameter estimates for skillspace representation |
covdelta |
Covariance matrix of parameter estimates for skillspace representation |
data |
Original data frame |
group.stat |
Group statistics (sample sizes, group labels) |
p.xi.aj |
Individual likelihood |
posterior |
Individual posterior distribution |
skill.levels |
Number of skill levels per dimension |
K.item |
Maximal category per item |
theta.k |
Used theta design or estimated theta trait distribution
in case of |
thetaDes |
Used theta design for item responses |
se.theta.k |
Estimated standard errors of |
time |
Info about computation time |
skillspace |
Used skillspace parametrization |
iter |
Number of iterations |
converged |
Logical indicating whether convergence was achieved. |
object |
Object of class |
x |
Object of class |
perstype |
Person paramter estimate type. Can be either
|
group |
Group which should be used for |
barwidth |
Bar width in |
histcol |
Color of histogram bars in |
cexcor |
Font size for print of correlation in |
pchpers |
Point type for scatter plot of person
parameters in |
cexpers |
Point size for scatter plot of person
parameters in |
... |
Optional parameters to be passed to or from other methods will be ignored. |
References
Bacci, S., Bartolucci, F., & Gnaldi, M. (2012). A class of multidimensional latent class IRT models for ordinal polytomous item responses. arXiv preprint, arXiv:1201.4667.
Bartolucci, F. (2007). A class of multidimensional IRT models for testing unidimensionality and clustering items. Psychometrika, 72, 141-157.
Berlinet, A. F., & Roland, C. (2012). Acceleration of the EM algorithm: P-EM versus epsilon algorithm. Computational Statistics & Data Analysis, 56(12), 4122-4137.
von Davier, M. (2008). A general diagnostic model applied to language testing data. British Journal of Mathematical and Statistical Psychology, 61, 287-307.
von Davier, M., & Yamamoto, K. (2004). Partially observed mixtures of IRT models: An extension of the generalized partial-credit model. Applied Psychological Measurement, 28, 389-406.
Xu, X., & von Davier, M. (2008). Fitting the structured general diagnostic model to NAEP data. ETS Research Report ETS RR-08-27. Princeton, ETS.
See Also
Cognitive diagnostic models for dichotomous data can be estimated
with din (DINA or DINO model) or gdina
(GDINA model, which contains many CDMs as special cases).
For assessment of model fit see modelfit.cor.din and
anova.gdm.
See itemfit.sx2 for item fit statistics.
For the estimation of the multidimensional
latent class item response model see the MultiLCIRT package
and sirt package (function sirt::rasch.mirtlc).
Examples
#############################################################################
# EXAMPLE 1: Fraction Dataset 1
# Unidimensional Models for dichotomous data
#############################################################################
data(data.fraction1, package="CDM")
dat <- data.fraction1$data
theta.k <- seq( -6, 6, len=15 ) # discretized ability
#***
# Model 1: Rasch model (normal distribution)
mod1 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, skillspace="normal",
centered.latent=TRUE)
summary(mod1)
plot(mod1)
#***
# Model 2: Rasch model (log-linear smoothing)
# set the item difficulty of the 8th item to zero
b.constraint <- matrix( c(8,1,0), 1, 3 )
mod2 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k,
skillspace="loglinear", b.constraint=b.constraint )
summary(mod2)
#***
# Model 3: 2PL model
mod3 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k,
skillspace="normal", standardized.latent=TRUE )
summary(mod3)
## Not run:
#***
# Model 4: include quadratic term in item response function
# using the argument decrease.increments=TRUE leads to a more
# stable estimate
thetaDes <- cbind( theta.k, theta.k^2 )
colnames(thetaDes) <- c( "F1", "F1q" )
mod4 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k,
thetaDes=thetaDes, skillspace="normal",
standardized.latent=TRUE, decrease.increments=TRUE)
summary(mod4)
#***
# Model 5: step function for ICC
# two different probabilities theta < 0 and theta > 0
thetaDes <- matrix( 1*(theta.k>0), ncol=1 )
colnames(thetaDes) <- c( "Fgrm1" )
mod5 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k,
thetaDes=thetaDes, skillspace="normal" )
summary(mod5)
#***
# Model 6: DINA model with din function
mod6 <- CDM::din( dat, q.matrix=matrix( 1, nrow=ncol(dat),ncol=1 ) )
summary(mod6)
#***
# Model 7: Estimating a version of the DINA model with gdm
theta.k <- c(-.5,.5)
mod7 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k, skillspace="loglinear" )
summary(mod7)
#############################################################################
# EXAMPLE 2: Cultural Activities - data.Students
# Unidimensional Models for polytomous data
#############################################################################
data(data.Students, package="CDM")
dat <- data.Students
dat <- dat[, grep( "act", colnames(dat) ) ]
theta.k <- seq( -4, 4, len=11 ) # discretized ability
#***
# Model 1: Partial Credit Model (PCM)
mod1 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, skillspace="normal",
centered.latent=TRUE)
summary(mod1)
plot(mod1)
#***
# Model 1b: PCM using frequency patterns
mod1b <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, skillspace="normal",
centered.latent=TRUE, use.freqpatt=TRUE)
summary(mod1b)
#***
# Model 2: PCM with two groups
mod2 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k,
group=CDM::data.Students$urban + 1, skillspace="normal",
centered.latent=TRUE)
summary(mod2)
#***
# Model 3: PCM with loglinear smoothing
b.constraint <- matrix( c(1,2,0), ncol=3 )
mod3 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k,
skillspace="loglinear", b.constraint=b.constraint )
summary(mod3)
#***
# Model 4: Model with pre-specified item weights in Q-matrix
Qmatrix <- array( 1, dim=c(5,1,2) )
Qmatrix[,1,2] <- 2 # default is score 2 for category 2
# now change the scoring of category 2:
Qmatrix[c(2,4),1,1] <- .74
Qmatrix[c(2,4),1,2] <- 2.3
# for items 2 and 4 the score for category 1 is .74 and for category 2 it is 2.3
mod4 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, Qmatrix=Qmatrix,
skillspace="normal", centered.latent=TRUE)
summary(mod4)
#***
# Model 5: Generalized partial credit model
mod5 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k,
skillspace="normal", standardized.latent=TRUE )
summary(mod5)
#***
# Model 6: Item-category slope estimation
mod6 <- CDM::gdm( dat, irtmodel="2PLcat", theta.k=theta.k, skillspace="normal",
standardized.latent=TRUE, decrease.increments=TRUE)
summary(mod6)
#***
# Models 7: items with different number of categories
dat0 <- dat
dat0[ paste(dat0[,1])==2, 1 ] <- 1 # 1st item has only two categories
dat0[ paste(dat0[,3])==2, 3 ] <- 1 # 3rd item has only two categories
# Model 7a: PCM
mod7a <- CDM::gdm( dat0, irtmodel="1PL", theta.k=theta.k, centered.latent=TRUE )
summary(mod7a)
# Model 7b: Item category slopes
mod7b <- CDM::gdm( dat0, irtmodel="2PLcat", theta.k=theta.k,
standardized.latent=TRUE, decrease.increments=TRUE )
summary(mod7b)
#############################################################################
# EXAMPLE 3: Fraction Dataset 2
# Multidimensional Models for dichotomous data
#############################################################################
data(data.fraction2, package="CDM")
dat <- data.fraction2$data
Qmatrix <- data.fraction2$q.matrix3
#***
# Model 1: One-dimensional Rasch model
theta.k <- seq( -4, 4, len=11 ) # discretized ability
mod1 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, centered.latent=TRUE)
summary(mod1)
plot(mod1)
#***
# Model 2: One-dimensional 2PL model
mod2 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k, standardized.latent=TRUE)
summary(mod2)
plot(mod2)
#***
# Model 3: 3-dimensional Rasch Model (normal distribution)
mod3 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, Qmatrix=Qmatrix,
centered.latent=TRUE, globconv=5*1E-3, conv=1E-4 )
summary(mod3)
#***
# Model 4: 3-dimensional Rasch model (loglinear smoothing)
# set some item parameters of items 4,1 and 2 to zero
b.constraint <- cbind( c(4,1,2), 1, 0 )
mod4 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, Qmatrix=Qmatrix,
b.constraint=b.constraint, skillspace="loglinear" )
summary(mod4)
#***
# Model 5: define a different theta grid for each dimension
theta.k <- list( "Dim1"=seq( -5, 5, len=11 ),
"Dim2"=seq(-5,5,len=8),
"Dim3"=seq( -3,3,len=6) )
mod5 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, Qmatrix=Qmatrix,
b.constraint=b.constraint, skillspace="loglinear")
summary(mod5)
#***
# Model 6: multdimensional 2PL model (normal distribution)
theta.k <- seq( -5, 5, len=13 )
a.constraint <- cbind( c(8,1,3), 1:3, 1, 1 ) # fix some slopes to 1
mod6 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k, Qmatrix=Qmatrix,
centered.latent=TRUE, a.constraint=a.constraint, decrease.increments=TRUE,
skillspace="normal")
summary(mod6)
#***
# Model 7: multdimensional 2PL model (loglinear distribution)
a.constraint <- cbind( c(8,1,3), 1:3, 1, 1 )
b.constraint <- cbind( c(8,1,3), 1, 0 )
mod7 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k, Qmatrix=Qmatrix,
b.constraint=b.constraint, a.constraint=a.constraint,
decrease.increments=FALSE, skillspace="loglinear")
summary(mod7)
#############################################################################
# EXAMPLE 4: Unidimensional latent class 1PL IRT model
#############################################################################
# simulate data
set.seed(754)
I <- 20 # number of items
N <- 2000 # number of persons
theta <- c( -2, 0, 1, 2 )
theta <- rep( theta, c(N/4,N/4, 3*N/8, N/8) )
b <- seq(-2,2,len=I)
library(sirt) # use function sim.raschtype from sirt package
dat <- sirt::sim.raschtype( theta=theta, b=b )
theta.k <- seq(-1, 1, len=4) # initial vector of theta
# estimate model
mod1 <- CDM::gdm( dat, theta.k=theta.k, skillspace="est", irtmodel="1PL",
centerintercepts=TRUE, maxiter=200)
summary(mod1)
## Estimated Skill Distribution
## F1 pi.k
## 1 -1.988 0.24813
## 2 -0.055 0.23313
## 3 0.940 0.40059
## 4 2.000 0.11816
#############################################################################
# EXAMPLE 5: Multidimensional latent class IRT model
#############################################################################
# We simulate a two-dimensional IRT model in which theta vectors
# are observed at a fixed discrete grid (see below).
# simulate data
set.seed(754)
I <- 13 # number of items
N <- 2400 # number of persons
# simulate Dimension 1 at 4 discrete theta points
theta <- c( -2, 0, 1, 2 )
theta <- rep( theta, c(N/4,N/4, 3*N/8, N/8) )
b <- seq(-2,2,len=I)
library(sirt) # use simulation function from sirt package
dat1 <- sirt::sim.raschtype( theta=theta, b=b )
# simulate Dimension 2 at 4 discrete theta points
theta <- c( -3, 0, 1.5, 2 )
theta <- rep( theta, c(N/4,N/4, 3*N/8, N/8) )
dat2 <- sirt::sim.raschtype( theta=theta, b=b )
colnames(dat2) <- gsub( "I", "U", colnames(dat2))
dat <- cbind( dat1, dat2 )
# define Q-matrix
Qmatrix <- matrix(0,2*I,2)
Qmatrix[ cbind( 1:(2*I), rep(1:2, each=I) ) ] <- 1
theta.k <- seq(-1, 1, len=4) # initial matrix
theta.k <- cbind( theta.k, theta.k )
colnames(theta.k) <- c("Dim1","Dim2")
# estimate model
mod2 <- CDM::gdm( dat, theta.k=theta.k, skillspace="est", irtmodel="1PL",
Qmatrix=Qmatrix, centerintercepts=TRUE)
summary(mod2)
## Estimated Skill Distribution
## theta.k.Dim1 theta.k.Dim2 pi.k
## 1 -2.022 -3.035 0.25010
## 2 0.016 0.053 0.24794
## 3 0.956 1.525 0.36401
## 4 1.958 1.919 0.13795
#############################################################################
# EXAMPLE 6: Large-scale dataset data.mg
#############################################################################
data(data.mg, package="CDM")
dat <- data.mg[, paste0("I", 1:11 ) ]
theta.k <- seq(-6,6,len=21)
#***
# Model 1: Generalized partial credit model with multiple groups
mod1 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k, group=CDM::data.mg$group,
skillspace="normal", standardized.latent=TRUE)
summary(mod1)
## End(Not run)
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