gdm.Rd
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.
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 gdm summary(object, file=NULL, ...) # S3 method for gdm print(x, ...) # S3 method for gdm plot(x, perstype="EAP", group=1, barwidth=.1, histcol=1, cexcor=3, pchpers=16, cexpers=.7, ... )
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 \(\delta\) 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. |
Case irtmodel="1PL"
:
Equal item slopes of 1 are assumed in this model. Therefore,
it corresponds to a generalized multidimensional Rasch model.
$$logit P( X_{nj}=k | \theta_n )=b_{j0} +
\sum_d q_{jdk} \theta_{nd} $$
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:
$$logit P( X_{nj}=k | \theta_n )=b_{j0} +
\sum_d a_{jd} q_{jdk} \theta_{nd} $$
Case irtmodel="2PLcat"
:
For each item, each dimension and each category,
different item slopes \(a_{jdk}\)
are estimated:
$$logit P( X_{nj}=k | \theta_n )=b_{j0} +
\sum_d a_{jdk} q_{jdk} \theta_{nd} $$
Note that this model can be generalized to include terms of any transformation \(t_h\) of the \(\theta_n\) vector (e.g. quadratic terms, step functions or interaction) such that the model can be formulated as $$logit P( X_{nj}=k | \theta_n )=b_{j0} + \sum_h a_{jhk} q_{jhk} t_h( \theta_{n} ) $$ In general, the number of functions \(t_1, ..., t_H\) will be larger than the \(\theta\) 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 \(\bold{\theta}\) 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.
An object of class gdm
. The list contains the
following entries:
Data frame with item parameters
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
Reliability of the EAP
Deviance
Information criteria, number of estimated parameters
Item intercepts \(b_{jk}\)
Standard error of item intercepts \(b_{jk}\)
Item slopes \(a_{jd}\) resp. \(a_{jdk}\)
Standard error of item slopes \(a_{jd}\) resp. \(a_{jdk}\)
The RMSEA item fit index (see itemfit.rmsea
).
This entry comes as a list with total and group-wise item fit
statistics.
Mean of RMSEA item fit indexes.
Used Q-matrix
Trait distribution
Means of trait distribution
Standard deviations of trait distribution
Skewnesses of trait distribution
List of correlation matrices of trait distribution corresponding to each group
Item response probabilities evaluated at grid theta.k
An array of expected counts \(n_{cikg}\) of ability class \(c\) at item \(i\) at category \(k\) in group \(g\)
Number of groups
Number of dimension of \(\bold{\theta}\)
Number of items
Number of persons
Parameter estimates for skillspace representation
Covariance matrix of parameter estimates for skillspace representation
Original data frame
Group statistics (sample sizes, group labels)
Individual likelihood
Individual posterior distribution
Number of skill levels per dimension
Maximal category per item
Used theta design or estimated theta trait distribution
in case of skillspace="est"
Used theta design for item responses
Estimated standard errors of theta.k
if it is
estimated
Info about computation time
Used skillspace parametrization
Number of iterations
Logical indicating whether convergence was achieved.
Object of class gdm
Object of class gdm
Person paramter estimate type. Can be either
"EAP"
, "MAP"
or "MLE"
.
Group which should be used for plot.gdm
Bar width in plot.gdm
Color of histogram bars in plot.gdm
Font size for print of correlation in plot.gdm
Point type for scatter plot of person
parameters in plot.gdm
Point size for scatter plot of person
parameters in plot.gdm
Optional parameters to be passed to or from other methods will be ignored.
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.
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
).
############################################################################# # 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) if (FALSE) { #*** # 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) }