sirt (version 0.31-20)

gom.em: Discrete (Rasch) Grade of Membership Model

Description

This function estimates the grade of membership model (Erosheva, Fienberg & Joutard, 2007) by the EM algorithm assuming a discrete membership score distribution.

Usage

gom.em(dat, K=NULL, problevels=NULL, model="GOM", theta0.k=seq(-5, 5, len=15), 
    xsi0.k=exp(seq(-6, 3, len=15)), max.increment=0.3, numdiff.parm=0.001, 
    maxdevchange=10^(-5), globconv=0.001, maxiter=1000, msteps=4, mstepconv=0.001)

Arguments

dat

Data frame with dichotomous responses

K

Number of classes (only applies for model="GOM")

problevels

Probability levels for membership functions (only applies for model="GOM")

model

The type of grade of membership model. The default "GOM" is the nonparametric grade of membership model. The probabilities and membership functions specifications described in Details are called via "GOMRasch".

theta0.k

Vector of $\tilde{\theta}_k$ grid (applies only for model="GOMRasch")

xsi0.k

Vector of $\xi_p$ grid (applies only for model="GOMRasch")

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 (only applies for model="GOMRasch")
• MAP
  Maximum aposteriori estimate of the membership function
• classdesc
  Descriptives for class membership
• lambda
  Estimated response probabilities $\lambda_{ik}$
• se.lambda
  Standard error for stimated response probabilities $\lambda_{ik}$
• mu
  Mean of $(\theta_p , \xi_p)$ (only applies for model="GOMRasch")
• Sigma
  Covariance matrix of $(\theta_p , \xi_p)$ (only applies for model="GOMRasch")
• b
  Estimated item difficulties (only applies for model="GOMRasch")
• se.b
  Standard error of estimated difficulties (only applies for model="GOMRasch")
• f.yi.qk
  Individual likelihood
• f.qk.yi
  Individual posterior
• probs
  Array with response probabilities
• n.ik
  Expected counts
• iter
  Number of iterations
• I
  Number of items
• K
  Number of classes
• TP
  Number of discrete integration points for $(g_{p1},...,g_{pK})$
• theta.k
  Used grid of membership functions
• ...
  Further values

Details

The item response model of the grade of membership model (Erosheva, Fienberg & Joutard, 2007) with $K$ classes for dichotomous correct responses $X_{pi}$ of person $p$ on item $i$ is as follows (model="GOM") $$P(X_{pi}=1 | g_{p1}, \ldots , g_{pK} ) = \sum_k \lambda_{ik} g_{pk} \quad , \quad \sum_k g_{pk} = 1$$ The membership function is assumed to be discretely represented (specified by problevels). The Rasch grade of membership model poses constraints on probabilities $\lambda_{ik}$ and membership functions $g_{pk}$. The membership function of person $p$ is parametrized by a location parameter $\theta_p$ and a variability parameter $\xi_p$. Each class $k$ is represented by a location $\tilde{\theta}_k$. The membership function is defined as $$g_{pk} \propto exp[ - ( \theta_p - \tilde{\theta}_k )^2 / ( 2 \xi_p^2 )]$$ The extremal class probabilities $\lambda_{ik}$ follow the Rasch model $$\lambda_{ik} = invlogit( \tilde{\theta}_k - b_i )$$

References

Erosheva, E. A., Fienberg, S. E. & Joutard, C. (2007). Describing disability through individual-level mixture models for multivariate binary data. Annals of Applied Statistics, 1, 502-537.

See Also

For joint maximum likelihood estimation of the grade of membership model see gom.jml.

Examples

#########################################################
# EXAMPLE 1: PISA data mathematics
#########################################################
data(data.pisaMath)
dat <- data.pisaMath$data
dat <- dat[ , grep("M" , colnames(dat)) ]

#***
# Model 1: Discrete GOM with 3 classes and 5 probability levels
problevels <- seq( 0 , 1 , len=5 )
mod1 <- gom.em( dat , K=3 , problevels ,  model="GOM"  )            
summary(mod1)

#***
# Model 2: Discrete GOM with 4 classes and 5 probability levels
problevels <- seq( 0 , 1 , len=5 )
mod2 <- gom.em( dat , K=4 , problevels ,  model="GOM"  )            
summary(mod2)

#***
# Model 3: Rasch GOM
mod3 <- gom.em( dat , model="GOMRasch" , maxiter=20 )            
summary(mod3)

#***
# Model 4: 'Ordinary' Rasch model
mod4 <- rasch.mml2( dat )
summary(mod4)