data.reck: Datasets from Reckase' Book Multidimensional Item Response Theory

Description

Some simulated datasets from Reckase (2009).

Usage

data(data.reck21)
data(data.reck61DAT1)
data(data.reck61DAT2)
data(data.reck73C1a)
data(data.reck73C1b)
data(data.reck75C2)
data(data.reck78ExA)
data(data.reck79ExB)

Arguments

Format

The format of the data.reck21 (Table 2.1, p. 45) is:

List of 2
$ data: num [1:2500, 1:50] 0 0 0 1 1 0 0 0 1 0 ...
..- attr(*, "dimnames")=List of 2
.. ..$ : NULL
.. ..$ : chr [1:50] "I0001" "I0002" "I0003" "I0004" ...
$ pars:'data.frame':
..$ a: num [1:50] 1.83 1.38 1.47 1.53 0.88 0.82 1.02 1.19 1.15 0.18 ...
..$ b: num [1:50] 0.91 0.81 0.06 -0.8 0.24 0.99 1.23 -0.47 2.78 -3.85 ...
..$ c: num [1:50] 0 0 0 0.25 0.21 0.29 0.26 0.19 0 0.21 ...
The format of the datasets data.reck61DAT1 and data.reck61DAT2 (Table 6.1, p. 153) is

List of 4
$ data : num [1:2500, 1:30] 1 0 0 1 1 0 0 1 1 0 ...
..- attr(*, "dimnames")=List of 2
.. ..$ : NULL
.. ..$ : chr [1:30] "A01" "A02" "A03" "A04" ...
$ pars :'data.frame':
..$ a1: num [1:30] 0.747 0.46 0.861 1.014 0.552 ...
..$ a2: num [1:30] 0.025 0.0097 0.0067 0.008 0.0204 0.0064 0.0861 ...
..$ a3: num [1:30] 0.1428 0.0692 0.404 0.047 0.1482 ...
..$ d : num [1:30] 0.183 -0.192 -0.466 -0.434 -0.443 ...
$ mu : num [1:3] -0.4 -0.7 0.1
$ sigma: num [1:3, 1:3] 1.21 0.297 1.232 0.297 0.81 ...

The dataset data.reck61DAT2 has correlated dimensions while data.reck61DAT1 has uncorrelated dimensions.
Datasets data.reck73C1a and data.reck73C1b use item parameters from Table 7.3 (p. 188). The dataset C1a has uncorrelated dimensions, while C1b has perfectly correlated dimensions. The items are sensitive to 3 dimensions. The format of the datasets is

List of 4
$ data : num [1:2500, 1:30] 1 0 1 1 1 0 1 1 1 1 ...
..- attr(*, "dimnames")=List of 2
.. ..$ : NULL
.. ..$ : chr [1:30] "A01" "A02" "A03" "A04" ...
$ pars :'data.frame': 30 obs. of 4 variables:
..$ a1: num [1:30] 0.747 0.46 0.861 1.014 0.552 ...
..$ a2: num [1:30] 0.025 0.0097 0.0067 0.008 0.0204 0.0064 ...
..$ a3: num [1:30] 0.1428 0.0692 0.404 0.047 0.1482 ...
..$ d : num [1:30] 0.183 -0.192 -0.466 -0.434 -0.443 ...
$ mu : num [1:3] 0 0 0
$ sigma: num [1:3, 1:3] 0.167 0.236 0.289 0.236 0.334 ...
The dataset data.reck75C2 is simulated using item parameters from Table 7.5 (p. 191). It contains items which are sensitive to only one dimension but individuals which have abilities in three uncorrelated dimensions. The format is

List of 4
$ data : num [1:2500, 1:30] 0 0 1 1 1 0 0 1 1 1 ...
..- attr(*, "dimnames")=List of 2
.. ..$ : NULL
.. ..$ : chr [1:30] "A01" "A02" "A03" "A04" ...
$ pars :'data.frame': 30 obs. of 4 variables:
..$ a1: num [1:30] 0.56 0.48 0.67 0.57 0.54 0.74 0.7 0.59 0.63 0.64 ...
..$ a2: num [1:30] 0.62 0.53 0.63 0.69 0.58 0.69 0.75 0.63 0.64 0.64 ...
..$ a3: num [1:30] 0.46 0.42 0.43 0.51 0.41 0.48 0.46 0.5 0.51 0.46 ...
..$ d : num [1:30] 0.1 0.06 -0.38 0.46 0.14 0.31 0.06 -1.23 0.47 1.06 ...
$ mu : num [1:3] 0 0 0
$ sigma: num [1:3, 1:3] 1 0 0 0 1 0 0 0 1
The dataset data.reck78ExA contains simulated item responses from Table 7.8 (p. 204 ff.). There are three item clusters and two ability dimensions. The format is

List of 4
$ data : num [1:2500, 1:50] 0 1 1 0 1 0 0 0 0 0 ...
..- attr(*, "dimnames")=List of 2
.. ..$ : NULL
.. ..$ : chr [1:50] "A01" "A02" "A03" "A04" ...
$ pars :'data.frame': 50 obs. of 3 variables:
..$ a1: num [1:50] 0.889 1.057 1.047 1.178 1.029 ...
..$ a2: num [1:50] 0.1399 0.0432 0.016 0.0231 0.2347 ...
..$ d : num [1:50] 0.2724 1.2335 -0.0918 -0.2372 0.8471 ...
$ mu : num [1:2] 0 0
$ sigma: num [1:2, 1:2] 1 0 0 1
The dataset data.reck79ExB contains simulated item responses from Table 7.9 (p. 207 ff.). There are three item clusters and three ability dimensions. The format is

List of 4
$ data : num [1:2500, 1:50] 1 1 0 1 0 0 0 1 1 0 ...
..- attr(*, "dimnames")=List of 2
.. ..$ : NULL
.. ..$ : chr [1:50] "A01" "A02" "A03" "A04" ...
$ pars :'data.frame': 50 obs. of 4 variables:
..$ a1: num [1:50] 0.895 1.032 1.036 1.163 1.022 ...
..$ a2: num [1:50] 0.052 0.132 0.144 0.13 0.165 ...
..$ a3: num [1:50] 0.0722 0.1923 0.0482 0.1321 0.204 ...
..$ d : num [1:50] 0.2724 1.2335 -0.0918 -0.2372 0.8471 ...
$ mu : num [1:3] 0 0 0
$ sigma: num [1:3, 1:3] 1 0 0 0 1 0 0 0 1

References

Reckase, M. (2009). Multidimensional item response theory. New York: Springer. tools:::Rd_expr_doi("10.1007/978-0-387-89976-3")

Examples

Run this code

if (FALSE) {
#############################################################################
# EXAMPLE 1: data.reck21 dataset, Table 2.1, p. 45
#############################################################################
data(data.reck21)

dat <- data.reck21$dat      # extract dataset

# items with zero guessing parameters
guess0 <- c( 1, 2, 3, 9,11,27,30,35,45,49,50 )
I <- ncol(dat)

#***
# Model 1: 3PL estimation using rasch.mml2
est.c <- est.a <- 1:I
est.c[ guess0 ] <- 0
mod1 <- sirt::rasch.mml2( dat, est.a=est.a, est.c=est.c, mmliter=300 )
summary(mod1)

#***
# Model 2: 3PL estimation using smirt
Q <- matrix(1,I,1)
mod2 <- sirt::smirt( dat, Qmatrix=Q, est.a="2PL", est.c=est.c, increment.factor=1.01)
summary(mod2)

#***
# Model 3: estimation in mirt package
library(mirt)
itemtype <- rep("3PL", I )
itemtype[ guess0 ] <- "2PL"
mod3 <- mirt::mirt(dat, 1, itemtype=itemtype, verbose=TRUE)
summary(mod3)

c3 <- unlist( coef(mod3) )[ 1:(4*I) ]
c3 <- matrix( c3, I, 4, byrow=TRUE )
# compare estimates of rasch.mml2, smirt and true parameters
round( cbind( mod1$item$c, mod2$item$c,c3[,3],data.reck21$pars$c ), 2 )
round( cbind( mod1$item$a, mod2$item$a.Dim1,c3[,1], data.reck21$pars$a ), 2 )
round( cbind( mod1$item$b, mod2$item$b.Dim1 / mod2$item$a.Dim1, - c3[,2] / c3[,1],
            data.reck21$pars$b ), 2 )

#############################################################################
# EXAMPLE 2: data.reck61 dataset, Table 6.1, p. 153
#############################################################################

data(data.reck61DAT1)
dat <- data.reck61DAT1$data

#***
# Model 1: Exploratory factor analysis

#-- Model 1a: tam.fa in TAM
library(TAM)
mod1a <- TAM::tam.fa( dat, irtmodel="efa", nfactors=3 )
# varimax rotation
varimax(mod1a$B.stand)

# Model 1b: EFA in NOHARM (Promax rotation)
mod1b <- sirt::R2noharm( dat=dat, model.type="EFA",  dimensions=3,
              writename="reck61__3dim_efa", noharm.path="c:/NOHARM",dec=",")
summary(mod1b)

# Model 1c: EFA with noharm.sirt
mod1c <- sirt::noharm.sirt( dat=dat, dimensions=3  )
summary(mod1c)
plot(mod1c)

# Model 1d: EFA with 2 dimensions in noharm.sirt
mod1d <- sirt::noharm.sirt( dat=dat, dimensions=2  )
summary(mod1d)
plot(mod1d, efa.load.min=.2)   # plot loadings of at least .20

#***
# Model 2: Confirmatory factor analysis

#-- Model 2a: tam.fa in TAM
dims <- c( rep(1,10), rep(3,10), rep(2,10)  )
Qmatrix <- matrix( 0, nrow=30, ncol=3 )
Qmatrix[ cbind( 1:30, dims) ] <- 1
mod2a <- TAM::tam.mml.2pl( dat,Q=Qmatrix,
            control=list( snodes=1000, QMC=TRUE, maxiter=200) )
summary(mod2a)

#-- Model 2b: smirt in sirt
mod2b <- sirt::smirt( dat,Qmatrix=Qmatrix, est.a="2PL", maxiter=20, qmcnodes=1000 )
summary(mod2b)

#-- Model 2c: rasch.mml2 in sirt
mod2c <- sirt::rasch.mml2( dat,Qmatrix=Qmatrix, est.a=1:30,
                mmliter=200, theta.k=seq(-5,5,len=11) )
summary(mod2c)

#-- Model 2d: mirt in mirt
cmodel <- mirt::mirt.model("
     F1=1-10
     F2=21-30
     F3=11-20
     COV=F1*F2, F1*F3, F2*F3 " )
mod2d <- mirt::mirt(dat, cmodel, verbose=TRUE)
summary(mod2d)
coef(mod2d)

#-- Model 2e: CFA in NOHARM
# specify covariance pattern
P.pattern <- matrix( 1, ncol=3, nrow=3 )
P.init <- .4*P.pattern
diag(P.pattern) <- 0
diag(P.init) <- 1
# fix all entries in the loading matrix to 1
F.pattern <- matrix( 0, nrow=30, ncol=3 )
F.pattern[1:10,1] <- 1
F.pattern[21:30,2] <- 1
F.pattern[11:20,3] <- 1
F.init <- F.pattern
# estimate model
mod2e <- sirt::R2noharm( dat=dat, model.type="CFA", P.pattern=P.pattern,
            P.init=P.init, F.pattern=F.pattern, F.init=F.init,
            writename="reck61__3dim_cfa", noharm.path="c:/NOHARM",dec=",")
summary(mod2e)

#-- Model 2f: CFA with noharm.sirt
mod2f <- sirt::noharm.sirt( dat=dat, Fval=F.init, Fpatt=F.pattern,
                 Pval=P.init, Ppatt=P.pattern )
summary(mod2f)

#############################################################################
# EXAMPLE 3: DETECT analysis data.reck78ExA and data.reck79ExB
#############################################################################

data(data.reck78ExA)
data(data.reck79ExB)

#************************
# Example A
dat <- data.reck78ExA$data
#- estimate person score
score <- stats::qnorm( ( rowMeans( dat )+.5 )  / ( ncol(dat) + 1 ) )
#- extract item cluster
itemcluster <- substring( colnames(dat), 1, 1 )
#- confirmatory DETECT Item cluster
detectA <- sirt::conf.detect( data=dat, score=score, itemcluster=itemcluster )
  ##          unweighted weighted
  ##   DETECT      0.571    0.571
  ##   ASSI        0.523    0.523
  ##   RATIO       0.757    0.757

#- exploratory DETECT analysis
detect_explA <- sirt::expl.detect(data=dat, score, nclusters=10, N.est=nrow(dat)/2  )
  ##  Optimal Cluster Size is  5  (Maximum of DETECT Index)
  ##     N.Cluster N.items N.est N.val         size.cluster DETECT.est ASSI.est
  ##   1         2      50  1250  1250                31-19      0.531    0.404
  ##   2         3      50  1250  1250             10-19-21      0.554    0.407
  ##   3         4      50  1250  1250           10-19-14-7      0.630    0.509
  ##   4         5      50  1250  1250         10-19-3-7-11      0.653    0.546
  ##   5         6      50  1250  1250       10-12-7-3-7-11      0.593    0.458
  ##   6         7      50  1250  1250      10-12-7-3-7-9-2      0.604    0.474
  ##   7         8      50  1250  1250    10-12-7-3-3-9-4-2      0.608    0.481
  ##   8         9      50  1250  1250  10-12-7-3-3-5-4-2-4      0.617    0.494
  ##   9        10      50  1250  1250 10-5-7-7-3-3-5-4-2-4      0.592    0.460

# cluster membership
cluster_membership <- detect_explA$itemcluster$cluster3
# Cluster 1:
colnames(dat)[ cluster_membership==1 ]
  ##   [1] "A01" "A02" "A03" "A04" "A05" "A06" "A07" "A08" "A09" "A10"
# Cluster 2:
colnames(dat)[ cluster_membership==2 ]
  ##    [1] "B11" "B12" "B13" "B14" "B15" "B16" "B17" "B18" "B19" "B20" "B21" "B22"
  ##   [13] "B23" "B25" "B26" "B27" "B28" "B29" "B30"
# Cluster 3:
colnames(dat)[ cluster_membership==3 ]
  ##    [1] "B24" "C31" "C32" "C33" "C34" "C35" "C36" "C37" "C38" "C39" "C40" "C41"
  ##   [13] "C42" "C43" "C44" "C45" "C46" "C47" "C48" "C49" "C50"

#************************
# Example B
dat <- data.reck79ExB$data
#- estimate person score
score <- stats::qnorm( ( rowMeans( dat )+.5 )  / ( ncol(dat) + 1 ) )
#- extract item cluster
itemcluster <- substring( colnames(dat), 1, 1 )
#- confirmatory DETECT Item cluster
detectB <- sirt::conf.detect( data=dat, score=score, itemcluster=itemcluster )
  ##          unweighted weighted
  ##   DETECT      0.715    0.715
  ##   ASSI        0.624    0.624
  ##   RATIO       0.855    0.855

#- exploratory DETECT analysis
detect_explB <- sirt::expl.detect(data=dat, score, nclusters=10, N.est=nrow(dat)/2  )
  ##   Optimal Cluster Size is  4  (Maximum of DETECT Index)
  ##
  ##     N.Cluster N.items N.est N.val         size.cluster DETECT.est ASSI.est
  ##   1         2      50  1250  1250                30-20      0.665    0.546
  ##   2         3      50  1250  1250             10-20-20      0.686    0.585
  ##   3         4      50  1250  1250           10-20-8-12      0.728    0.644
  ##   4         5      50  1250  1250         10-6-14-8-12      0.654    0.553
  ##   5         6      50  1250  1250       10-6-14-3-12-5      0.659    0.561
  ##   6         7      50  1250  1250      10-6-14-3-7-5-5      0.664    0.576
  ##   7         8      50  1250  1250     10-6-7-7-3-7-5-5      0.616    0.518
  ##   8         9      50  1250  1250   10-6-7-7-3-5-5-5-2      0.612    0.512
  ##   9        10      50  1250  1250 10-6-7-7-3-5-3-5-2-2      0.613    0.512
}

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