sirt (version 1.5-0)

prob.guttman: Probabilistic Guttman Model

Description

This function estimates the probabilistic Guttman model which is a special case of an ordered latent trait model (Hanson, 2000; Proctor, 1970).

Usage

prob.guttman(dat, pid = NULL, guess.equal = FALSE,  slip.equal = FALSE, 
    itemlevel = NULL, conv1 = 0.001, glob.conv = 0.001, mmliter = 500)
    
## S3 method for class 'prob.guttman':
summary(object,...)

## S3 method for class 'prob.guttman':
anova(object,...)

## S3 method for class 'prob.guttman':
logLik(object,...)

## S3 method for class 'prob.guttman':
IRT.irfprob(object,...)

## S3 method for class 'prob.guttman':
IRT.likelihood(object,...)

## S3 method for class 'prob.guttman':
IRT.posterior(object,...)

Arguments

dat

An $N \times I$ data frame of dichotomous item responses

pid

Optional vector of person identifiers

guess.equal

Should the same guessing parameters for all the items estimated?

slip.equal

Should the same slipping parameters for all the items estimated?

itemlevel

A vector of item levels of the Guttman scale for each item. If there are $K$ different item levels, then the Guttman scale possesses $K$ ordered trait values.

conv1

Convergence criterion for item parameters

glob.conv

Global convergence criterion for the deviance

mmliter

Maximum number of iterations

object

Object of class prob.guttman

...

Further arguments to be passed

Value

• An object of class prob.guttman
• person
  Estimated person parameters
• item
  Estimated item parameters
• theta.k
  Ability levels
• trait
  Estimated trait distribution
• ic
  Information criteria
• deviance
  Deviance
• iter
  Number of iterations
• itemdesign
  Specified allocation of items to trait levels

References

Hanson, B. (2000). IRT parameter estimation using the EM algorithm. Technical Report. Proctor, C. H. (1970). A probabilistic formulation and statistical analysis for Guttman scaling. Psychometrika, 35, 73-78.

Examples

#############################################################################
# EXAMPLE 1: Dataset Reading
#############################################################################
data(data.read)
dat <- data.read

#***
# Model 1: estimate probabilistic Guttman model
mod1 <- prob.guttman( dat )
summary(mod1)

#***
# Model 2: probabilistic Guttman model with equal guessing and slipping parameters
mod2 <- prob.guttman( dat , guess.equal=TRUE , slip.equal=TRUE)
summary(mod2)

#***
# Model 3: Guttman model with three a priori specified item levels
itemlevel <- rep(1,12)
itemlevel[ c(2,5,8,10,12) ] <- 2
itemlevel[ c(3,4,6) ] <- 3
mod3 <- prob.guttman( dat , itemlevel=itemlevel )
summary(mod3)

#***
# Model3m: estimate Model 3 in mirt

library(mirt)
# define four ordered latent classes
Theta <- scan(nlines=1)
   0 0 0    1 0 0   1 1 0   1 1 1    
Theta <- matrix( Theta , nrow=4 , ncol=3,byrow=TRUE)

# define mirt model
I <- ncol(dat)  # I = 12
mirtmodel <- mirt::mirt.model("
        # specify factors for each item level
        C1 = 1,7,9,11
        C2 = 2,5,8,10,12 
        C3 = 3,4,6
        ")
# get initial parameter values
mod.pars <- mirt::mirt(dat, model=mirtmodel ,  pars = "values")
# redefine initial parameter values
mod.pars[ mod.pars$name == "d" ,"value" ]  <- -1
mod.pars[ mod.pars$name %in% paste0("a",1:3) & mod.pars$est    ,"value" ]  <- 2
mod.pars
# define prior for latent class analysis
lca_prior <- function(Theta,Etable){
  # number of latent Theta classes
  TP <- nrow(Theta)
  # prior in initial iteration
  if ( is.null(Etable) ){ prior <- rep( 1/TP , TP ) }    
  # process Etable (this is correct for datasets without missing data)
  if ( ! is.null(Etable) ){  
    # sum over correct and incorrect expected responses 
    prior <- ( rowSums(Etable[ , seq(1,2*I,2)]) + rowSums(Etable[,seq(2,2*I,2)]) )/I
                 }
  prior <- prior / sum(prior)  
  return(prior)
}
# estimate model in mirt
mod3m <- mirt::mirt(dat, mirtmodel , pars = mod.pars , verbose=TRUE , 
            technical = list( customTheta=Theta , customPriorFun = lca_prior) )
# correct number of estimated parameters
mod3m@nest <- as.integer(sum(mod.pars$est) + nrow(Theta)-1 ) 
# extract log-likelihood and compute AIC and BIC
mod3m@logLik
( AIC <- -2*mod3m@logLik+2*mod3m@nest )
( BIC <- -2*mod3m@logLik+log(mod3m@Data$N)*mod3m@nest )
# compare with information criteria from prob.guttman
mod3$ic
# model fit in mirt
mirt::M2(mod3m)
# extract coefficients
( cmod3m <- mirt.wrapper.coef(mod3m) )
# compare estimated distributions
round( cbind( "sirt"  = mod3$trait$prob , "mirt" = mod3m@Prior[[1]] ) , 5 )
  ##           sirt    mirt
  ##   [1,] 0.13709 0.13765
  ##   [2,] 0.30266 0.30303
  ##   [3,] 0.15239 0.15085
  ##   [4,] 0.40786 0.40846
# compare estimated item parameters
ipars <- data.frame( "guess.sirt" = mod3$item$guess , 
                     "guess.mirt" = plogis( cmod3m$coef$d ) )
ipars$slip.sirt <- mod3$item$slip
ipars$slip.mirt <- 1-plogis( rowSums(cmod3m$coef[ , c("a1","a2","a3","d") ] ) )
round( ipars , 4 )
  ##      guess.sirt guess.mirt slip.sirt slip.mirt
  ##   1      0.7810     0.7804    0.1383    0.1382
  ##   2      0.4513     0.4517    0.0373    0.0368
  ##   3      0.3203     0.3200    0.0747    0.0751
  ##   4      0.3009     0.3007    0.3082    0.3087
  ##   5      0.5776     0.5779    0.1800    0.1798
  ##   6      0.3758     0.3759    0.3047    0.3051
  ##   7      0.7262     0.7259    0.0625    0.0623
  ##   [...]

#***
# Model 4: Monotone item response function estimated in mirt

# define four ordered latent classes
Theta <- scan(nlines=1)
   0 0 0    1 0 0   1 1 0   1 1 1    
Theta <- matrix( Theta , nrow=4 , ncol=3,byrow=TRUE)

# define mirt model
I <- ncol(dat)  # I = 12
mirtmodel <- mirt::mirt.model("
        # specify factors for each item level
        C1 = 1-12  
        C2 = 1-12
        C3 = 1-12
        ")
# get initial parameter values
mod.pars <- mirt::mirt(dat, model=mirtmodel ,  pars = "values")
# redefine initial parameter values
mod.pars[ mod.pars$name == "d" ,"value" ]  <- -1
mod.pars[ mod.pars$name %in% paste0("a",1:3) & mod.pars$est    ,"value" ]  <- .6
# set lower bound to zero ton ensure monotonicity
mod.pars[ mod.pars$name %in% paste0("a",1:3)     ,"lbound" ]  <- 0
mod.pars
# estimate model in mirt
mod4 <- mirt::mirt(dat, mirtmodel , pars = mod.pars , verbose=TRUE , 
            technical = list( customTheta=Theta , customPriorFun = lca_prior) )
# correct number of estimated parameters
mod4@nest <- as.integer(sum(mod.pars$est) + nrow(Theta)-1 ) 
# extract coefficients
cmod4 <- mirt.wrapper.coef(mod4)
cmod4
# compute item response functions
cmod4c <- cmod4$coef[ , c("d" , "a1" , "a2" , "a3" ) ]
probs4 <- t( apply( cmod4c , 1 , FUN = function(ll){ 
                 plogis(cumsum(as.numeric(ll))) } ) )
matplot( 1:4 ,  t(probs4) , type="b" , pch=1:I)