CDM (version 1.4-16)

gdm: 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 ,  maxiter=1000, 
    conv=10^(-5), globconv=10^(-5), msteps=8 , convM=.0005 ,
    decrease.increments = FALSE , use.freqpatt=FALSE , ...)

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.

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

group

An optional vector of group identifiers for multiple group estimation

weights

Sample weights

Qmatrix

An optional array $I$ 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 distributio

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 $\delta$ parameters for the reduced skillspace estimation (see Xu & von Davier, 2008)

standardized.latent

Should in a uni- or multidimensional model all latent variables of the first group be normally distributed and standardized? The default is FALSE.

centered.latent

Should in a uni- or multidimensional model all latent variables of the first group be normally distributed and have zero means? 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 8 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

use.freqpatt

Should frequencies of unique item response patterns be used. In case of large data set use.freqpatt=TRUE can speed calculations. Note that in this case, not all person parameters are calculated as usual in the output.

...

Further arguments to be passed

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 $\theta$ dimensions
• 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 dataframe
• 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
• thetaDes
  Used theta design for item responses
• time
  Info about computation time
• skillspace
  Used skillspace parametrization
• iter
  Number of iterations
• ...

Details

Case irtmodel="1PL": Equal item slopes of 1 are assumed in this model. Therefore, it corresponds to a generalized multidimensional Rasch model. $$logit P( X_{nj} = k | \theta_n ) = b_{j0} + \sum_d q_{jdk} \theta_{nd}$$ 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 $D$.

References

von Davier, M. (2008). A general diagnostic model applied to language testing data. British Journal of Mathematical and Statistical Psychology, 61, 287-307. 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.

Examples

###################################################################
# EXAMPLE 1: Fraction Dataset 1
#      Unidimensional Models for dichotomous data
###################################################################

data( data.fraction1 )
dat <- data.fraction1$data
theta.k <- seq( -6 , 6 , len=15 )   # discretized ability

#***
# Model 1: Rasch model (normal distribution)
mod1 <- gdm( dat , irtmodel="1PL" , theta.k=theta.k , skillspace="normal" ,
        centered.latent=TRUE)
summary(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 <- gdm( dat , irtmodel="1PL" , theta.k=theta.k , 
          skillspace="loglinear" , b.constraint=b.constraint  )
summary(mod2)

#***
# Model 3: 2PL model
mod3 <- gdm( dat , irtmodel="2PL" , theta.k=theta.k , 
          skillspace="normal" , standardized.latent=TRUE  )
summary(mod3)

#***
# 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 <- 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 <- gdm( dat , irtmodel="2PL" , theta.k=theta.k , 
          thetaDes = thetaDes , skillspace="normal" )
summary(mod5)

#***
# Model 6: DINA model with din function
mod6 <- 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 <- 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 )
dat <- data.Students[ , grep( "act" , colnames(data.Students) ) ]
theta.k <- seq( -4 , 4 , len=11 )   # discretized ability

#***
# Model 1: Partial Credit Model (PCM)
mod1 <- gdm( dat , irtmodel="1PL" , theta.k=theta.k , skillspace="normal" ,
           centered.latent=TRUE)
summary(mod1)

#***
# Model 1b: PCM using frequency patterns
mod1b <- 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 <- gdm( dat , irtmodel="1PL" , theta.k=theta.k , 
            group=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 <- 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
Qmatrix[c(2,4),1,1] <- .74
Qmatrix[c(2,4),1,2] <- 2.3
# score for category 1 is 1.2 and for category 2 is 1.8
mod4 <- gdm( dat , irtmodel="1PL" , theta.k=theta.k , Qmatrix=Qmatrix ,
           skillspace="normal"  , centered.latent=TRUE)
summary(mod4)

#***
# Model 5: Generalized partial credit model
mod5 <- gdm( dat , irtmodel="2PL" , theta.k=theta.k ,  
          skillspace="normal" , standardized.latent=TRUE )
summary(mod5)

#***
# Model 6: Item-category slope estimation
mod6 <- 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 <- gdm( dat0 , irtmodel="1PL" , theta.k=theta.k ,  
           centered.latent=TRUE )        
summary(mod7a)

# Model 7b: Item category slopes
mod7b <- 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 )
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 <- gdm( dat , irtmodel = "1PL" , theta.k=theta.k ,
           centered.latent=TRUE) 
summary(mod1)

#***
# Model 2: One-dimensional 2PL model
mod2 <- gdm( dat , irtmodel = "2PL" , theta.k=theta.k ,
           standardized.latent=TRUE) 
summary(mod2)

#***
# Model 3: 3-dimensional Rasch Model (normal distribution)
mod3 <- gdm( dat , irtmodel="1PL" , theta.k=theta.k , 
            Qmatrix=Qmatrix , centered.latent=TRUE , 
            globconv=5*10^(-3) , conv=10^(-4) , maxiter=10 )
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 <- gdm( dat , irtmodel="1PL" , theta.k=theta.k , 
            Qmatrix=Qmatrix , b.constraint=b.constraint , 
            skillspace="loglinear" , globconv= 5*10^(-2) , conv=10^(-3) )
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 <- gdm( dat , irtmodel="1PL" , theta.k=theta.k , 
            Qmatrix=Qmatrix , b.constraint=b.constraint , 
            skillspace="loglinear" , globconv= 5*10^(-2) , conv=10^(-3) )
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 <- gdm( dat , irtmodel="2PL" , theta.k=theta.k , 
            Qmatrix=Qmatrix , centered.latent=TRUE , 
            a.constraint=a.constraint , decrease.increments=TRUE , 
            skillspace="normal" , maxiter=400)
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 <- gdm( dat , irtmodel="2PL" , theta.k=theta.k , 
            Qmatrix=Qmatrix , b.constraint=b.constraint , 
            a.constraint=a.constraint , decrease.increments=FALSE , 
            skillspace="loglinear" , maxiter=400)
summary(mod7)