sirt (version 4.1-15)

data.mcdonald: Some Datasets from McDonald's Test Theory Book

Description

Some datasets from McDonald (1999), especially related to using NOHARM for item response modeling. See Examples below.

Usage

data(data.mcdonald.act15)
data(data.mcdonald.LSAT6)
data(data.mcdonald.rape)

Format

• The format of the ACT15 data data.mcdonald.act15 is:

  num [1:15, 1:15] 0.49 0.44 0.38 0.3 0.29 0.13 0.23 0.16 0.16 0.23 ...
- attr(*, "dimnames")=List of 2
..$ : chr [1:15] "A01" "A02" "A03" "A04" ...
..$ : chr [1:15] "A01" "A02" "A03" "A04" ...

  The dataset (which is the product-moment covariance matrix) is obtained from Ch. 12 in McDonald (1999).
  

• The format of the LSAT6 data data.mcdonald.LSAT6 is:

  'data.frame': 1004 obs. of 5 variables:
$ L1: int 0 0 0 0 0 0 0 0 0 0 ...
$ L2: int 0 0 0 0 0 0 0 0 0 0 ...
$ L3: int 0 0 0 0 0 0 0 0 0 0 ...
$ L4: int 0 0 0 0 0 0 0 0 0 1 ...
$ L5: int 0 0 0 1 1 1 1 1 1 0 ...

  The dataset is obtained from Ch. 6 in McDonald (1999).
  

• The format of the rape myth scale data data.mcdonald.rape is

  List of 2
$ lambda: num [1:2, 1:19] 1.13 0.88 0.85 0.77 0.79 0.55 1.12 1.01 0.99 0.79 ...
..- attr(*, "dimnames")=List of 2
.. ..$ : chr [1:2] "male" "female"
.. ..$ : chr [1:19] "I1" "I2" "I3" "I4" ...
$ nu : num [1:2, 1:19] 2.88 1.87 3.12 2.32 2.13 1.43 3.79 2.6 3.01 2.11 ...
..- attr(*, "dimnames")=List of 2
.. ..$ : chr [1:2] "male" "female"
.. ..$ : chr [1:19] "I1" "I2" "I3" "I4" ...
  

  The dataset is obtained from Ch. 15 in McDonald (1999).

References

McDonald, R. P. (1999). Test theory: A unified treatment. Psychology Press.

Examples

if (FALSE) {
#############################################################################
# EXAMPLE 1: LSAT6 data    | Chapter 12 McDonald (1999)
#############################################################################
data(data.mcdonald.act15)

#************
# Model 1: 2-parameter normal ogive model

#++ NOHARM estimation
I <- ncol(dat)
# covariance structure
P.pattern <- matrix( 0, ncol=1, nrow=1 )
P.init <- 1+0*P.pattern
# fix all entries in the loading matrix to 1
F.pattern <- matrix( 1, nrow=I, ncol=1 )
F.init <- F.pattern
# estimate model
mod1a <- sirt::R2noharm( dat=dat, model.type="CFA", F.pattern=F.pattern,
             F.init=F.init, P.pattern=P.pattern, P.init=P.init,
             writename="LSAT6__1dim_2pno", noharm.path=noharm.path, dec="," )
summary(mod1a, logfile="LSAT6__1dim_2pno__SUMMARY")

#++ pairwise marginal maximum likelihood estimation using the probit link
mod1b <- sirt::rasch.pml3( dat, est.a=1:I, est.sigma=FALSE)

#************
# Model 2: 1-parameter normal ogive model

#++ NOHARM estimation
# covariance structure
P.pattern <- matrix( 0, ncol=1, nrow=1 )
P.init <- 1+0*P.pattern
# fix all entries in the loading matrix to 1
F.pattern <- matrix( 2, nrow=I, ncol=1 )
F.init <- 1+0*F.pattern
# estimate model
mod2a <- sirt::R2noharm( dat=dat, model.type="CFA", F.pattern=F.pattern,
                F.init=F.init, P.pattern=P.pattern, P.init=P.init,
                writename="LSAT6__1dim_1pno", noharm.path=noharm.path, dec="," )
summary(mod2a, logfile="LSAT6__1dim_1pno__SUMMARY")

# PMML estimation
mod2b <- sirt::rasch.pml3( dat, est.a=rep(1,I), est.sigma=FALSE )
summary(mod2b)

#************
# Model 3: 3-parameter normal ogive model with fixed guessing parameters

#++ NOHARM estimation
# covariance structure
P.pattern <- matrix( 0, ncol=1, nrow=1 )
P.init <- 1+0*P.pattern
# fix all entries in the loading matrix to 1
F.pattern <- matrix( 1, nrow=I, ncol=1 )
F.init <- 1+0*F.pattern
# estimate model
mod <- sirt::R2noharm( dat=dat, model.type="CFA",  guesses=rep(.2,I),
            F.pattern=F.pattern, F.init=F.init, P.pattern=P.pattern,
            P.init=P.init, writename="LSAT6__1dim_3pno",
            noharm.path=noharm.path, dec="," )
summary(mod, logfile="LSAT6__1dim_3pno__SUMMARY")

#++ logistic link function employed in smirt function
mod1d <- sirt::smirt(dat, Qmatrix=F.pattern, est.a=matrix(1:I,I,1), c.init=rep(.2,I))
summary(mod1d)

#############################################################################
# EXAMPLE 2: ACT15 data    | Chapter 6 McDonald (1999)
#############################################################################
data(data.mcdonald.act15)
pm <- data.mcdonald.act15

#************
# Model 1: 2-dimensional exploratory factor analysis
mod1 <- sirt::R2noharm( pm=pm, n=1000, model.type="EFA", dimensions=2,
             writename="ACT15__efa_2dim", noharm.path=noharm.path, dec="," )
summary(mod1)

#************
# Model 2: 2-dimensional independent clusters basis solution
P.pattern <- matrix(1,2,2)
diag(P.pattern) <- 0
P.init <- 1+0*P.pattern
F.pattern <- matrix(0,15,2)
F.pattern[ c(1:5,11:15),1] <- 1
F.pattern[ c(6:10,11:15),2] <- 1
F.init <- F.pattern

# estimate model
mod2 <- sirt::R2noharm( pm=pm, n=1000,  model.type="CFA", F.pattern=F.pattern,
            F.init=F.init, P.pattern=P.pattern,P.init=P.init,
            writename="ACT15_indep_clusters", noharm.path=noharm.path, dec="," )
summary(mod2)

#************
# Model 3: Hierarchical model

P.pattern <- matrix(0,3,3)
P.init <- P.pattern
diag(P.init) <- 1
F.pattern <- matrix(0,15,3)
F.pattern[,1] <- 1    # all items load on g factor
F.pattern[ c(1:5,11:15),2] <- 1   # Items 1-5 and 11-15 load on first nested factor
F.pattern[ c(6:10,11:15),3] <- 1  # Items 6-10 and 11-15 load on second nested factor
F.init <- F.pattern

# estimate model
mod3 <- sirt::R2noharm( pm=pm, n=1000,  model.type="CFA", F.pattern=F.pattern,
           F.init=F.init, P.pattern=P.pattern, P.init=P.init,
           writename="ACT15_hierarch_model", noharm.path=noharm.path, dec="," )
summary(mod3)

#############################################################################
# EXAMPLE 3: Rape myth scale | Chapter 15 McDonald (1999)
#############################################################################
data(data.mcdonald.rape)
lambda <- data.mcdonald.rape$lambda
nu <- data.mcdonald.rape$nu

# obtain multiplier for factor loadings (Formula 15.5)
k <- sum( lambda[1,] * lambda[2,] ) / sum( lambda[2,]^2 )
  ##   [1] 1.263243

# additive parameter (Formula 15.7)
c <- sum( lambda[2,]*(nu[1,]-nu[2,]) ) / sum( lambda[2,]^2 )
  ##   [1] 1.247697

# SD in the female group
1/k
  ##   [1] 0.7916132

# M in the female group
- c/k
  ##   [1] -0.9876932

# Burt's coefficient of factorial congruence (Formula 15.10a)
sum( lambda[1,] * lambda[2,] ) / sqrt( sum( lambda[1,]^2 ) * sum( lambda[2,]^2 ) )
  ##   [1] 0.9727831

# congruence for mean parameters
sum(  (nu[1,]-nu[2,]) * lambda[2,] ) / sqrt( sum( (nu[1,]-nu[2,])^2 ) * sum( lambda[2,]^2 ) )
  ##   [1] 0.968176
}