conf.detect: Confirmatory DETECT and polyDETECT Analysis

Description

This function computes the DETECT statistics for dichotomous item responses and the polyDETECT statistic for polytomous item responses under a confirmatory specification of item clusters (Stout, Habing, Douglas & Kim, 1996; Zhang & Stout, 1999a, 1999b; Zhang, 2007; Bonifay, Reise, Scheines, & Meijer, 2015).

Item responses in a multi-matrix design are allowed (Zhang, 2013).

Usage

conf.detect(data, score, itemcluster, bwscale = 1.1, progress = TRUE,  thetagrid = seq(-3, 3, len = 200))
"summary"(object, digits = 3, file = NULL, ...)

Arguments

data

An $N \times I$ data frame of dichotomous or polytomous responses. Missing responses are allowed.

score

An ability estimate, e.g. the WLE, sum score or mean score

itemcluster

Item cluster for each item. The order of entries must correspond to the columns in data.

bwscale

Bandwidth factor for calculation of conditional covariance (see ccov.np)

progress

Display progress?

thetagrid

A vector which contains theta values where conditional covariances are evaluated.

object

Object of class conf.detect

digits

Number of digits for rounding in summary

file

Optional file name to be sinked for summary

...

Further arguments to be passed

Value

A list with following entries: A list with following entries:

Details

The result of DETECT are the indices DETECT, ASSI and RATIO (see Zhang 2007 for details) calculated for the options unweighted and weighted. The option unweighted means that all conditional covariances of item pairs are equally weighted, weighted means that these covariances are weighted by the sample size of item pairs. In case of multi matrix item designs, both types of indices can differ.

The classification scheme of these indices are as follows (Jang & Roussos, 2007; Zhang, 2007):

Strong multidimensionality

DETECT > 1.00

Moderate multidimensionality

.40 < DETECT < 1.00

Weak multidimensionality

.20 < DETECT < .40

Essential unidimensionality

DETECT < .20

Maximum value under simple structure	ASSI=1
RATIO=1	Essential deviation from unidimensionality
ASSI > .25	RATIO > .36
Essential unidimensionality	ASSI < .25
RATIO < .36	Maximum value under simple structure

Note that the expected value of a conditional covariance for an item pair is negative when a unidimensional model holds. In consequence, the DETECT index can become negative for unidimensional data (see Example 3). This can be also seen in the statistic MCOV100 in the value detect.

References

Bonifay, W. E., Reise, S. P., Scheines, R., & Meijer, R. R. (2015). When are multidimensional data unidimensional enough for structural equation modeling? An evaluation of the DETECT multidimensionality index. Structural Equation Modeling, 22, 504-516. Jang, E. E., & Roussos, L. (2007). An investigation into the dimensionality of TOEFL using conditional covariance-based nonparametric approach. Journal of Educational Measurement, 44, 1-21.

Stout, W., Habing, B., Douglas, J., & Kim, H. R. (1996). Conditional covariance-based nonparametric multidimensionality assessment. Applied Psychological Measurement, 20, 331-354.

Zhang, J., & Stout, W. (1999a). Conditional covariance structure of generalized compensatory multidimensional items. Psychometrika, 64, 129-152.

Zhang, J., & Stout, W. (1999b). The theoretical DETECT index of dimensionality and its application to approximate simple structure. Psychometrika, 64, 213-249.

Zhang, J. (2007). Conditional covariance theory and DETECT for polytomous items. Psychometrika, 72, 69-91.

Zhang, J. (2013). A procedure for dimensionality analyses of response data from various test designs. Psychometrika, 78, 37-58.

Examples

Run this code

#############################################################################
# EXAMPLE 1: TIMSS mathematics data set (dichotomous data)
#############################################################################
data(data.timss)

# extract data
dat <- data.timss$data
dat <- dat[ , substring( colnames(dat),1,1) == "M" ]
# extract item informations
iteminfo <- data.timss$item
# estimate Rasch model
mod1 <- rasch.mml2( dat )
# estimate WLEs
wle1 <- wle.rasch( dat , b = mod1$item$b )$theta

# DETECT for content domains
detect1 <- conf.detect( data = dat , score = wle1 ,
                    itemcluster = iteminfo$Content.Domain )
  ##          unweighted weighted
  ##   DETECT      0.316    0.316
  ##   ASSI        0.273    0.273
  ##   RATIO       0.355    0.355

## Not run: 
# # DETECT cognitive domains
# detect2 <- conf.detect( data = dat , score = wle1 ,
#                     itemcluster = iteminfo$Cognitive.Domain )
#   ##          unweighted weighted
#   ##   DETECT      0.251    0.251
#   ##   ASSI        0.227    0.227
#   ##   RATIO       0.282    0.282
# 
# # DETECT for item format 
# detect3 <- conf.detect( data = dat , score = wle1 ,
#                     itemcluster = iteminfo$Format )
#   ##          unweighted weighted
#   ##   DETECT      0.056    0.056
#   ##   ASSI        0.060    0.060
#   ##   RATIO       0.062    0.062
# 
# # DETECT for item blocks
# detect4 <- conf.detect( data = dat , score = wle1 ,
#                     itemcluster = iteminfo$Block )
#   ##          unweighted weighted
#   ##   DETECT      0.301    0.301
#   ##   ASSI        0.193    0.193
#   ##   RATIO       0.339    0.339 ## End(Not run)

# Exploratory DETECT: Application of a cluster analysis employing the Ward method
detect5 <- expl.detect( data = dat , score = wle1  , 
                nclusters = 10 , N.est = nrow(dat)  )
# Plot cluster solution
pl <- graphics::plot( detect5$clusterfit , main = "Cluster solution" )
stats::rect.hclust(detect5$clusterfit, k=4, border="red")

## Not run: 
# #############################################################################
# # EXAMPLE 2: Big 5 data set (polytomous data)
# #############################################################################
# 
# # attach Big5 Dataset
# data(data.big5)
# 
# # select 6 items of each dimension
# dat <- data.big5
# dat <- dat[, 1:30]
# 
# # estimate person score by simply using a transformed sum score
# score <- stats::qnorm( ( rowMeans( dat )+.5 )  / ( 30 + 1 ) )
# 
# # extract item cluster (Big 5 dimensions)
# itemcluster <- substring( colnames(dat) , 1 , 1 )
# 
# # DETECT Item cluster
# detect1 <- conf.detect( data = dat , score = score , itemcluster = itemcluster )
#   ##        unweighted weighted
#   ## DETECT      1.256    1.256
#   ## ASSI        0.384    0.384
#   ## RATIO       0.597    0.597
# 
# # Exploratory DETECT
# detect5 <- expl.detect( data = dat , score = score  , 
#                      nclusters = 9 , N.est = nrow(dat)  )
#   ## DETECT (unweighted)
#   ## Optimal Cluster Size is  6  (Maximum of DETECT Index)
#   ##   N.Cluster N.items N.est N.val      size.cluster DETECT.est ASSI.est RATIO.est
#   ## 1         2      30   500     0              6-24      1.073    0.246     0.510
#   ## 2         3      30   500     0           6-10-14      1.578    0.457     0.750
#   ## 3         4      30   500     0         6-10-11-3      1.532    0.444     0.729
#   ## 4         5      30   500     0        6-8-11-2-3      1.591    0.462     0.757
#   ## 5         6      30   500     0       6-8-6-2-5-3      1.610    0.499     0.766
#   ## 6         7      30   500     0     6-3-6-2-5-5-3      1.557    0.476     0.740
#   ## 7         8      30   500     0   6-3-3-2-3-5-5-3      1.540    0.462     0.732
#   ## 8         9      30   500     0 6-3-3-2-3-5-3-3-2      1.522    0.444     0.724
# 
# # Plot Cluster solution
# pl <- graphics::plot( detect5$clusterfit , main = "Cluster solution" )
# stats::rect.hclust(detect5$clusterfit, k=6, border="red")
# 
# #############################################################################
# # EXAMPLE 3: DETECT index for unidimensional data
# #############################################################################
# 
# set.seed(976)
# N <- 1000
# I <- 20
# b <- base::sample( seq( -2 , 2 , len=I) )
# dat <- sim.raschtype( rnorm(N) , b = b )
# 
# # estimate Rasch model and corresponding WLEs
# mod1 <- TAM::tam.mml( dat )
# wmod1 <- TAM::tam.wle(mod1)$theta
# # define item cluster
#  
# itemcluster <- c( rep(1,5) , rep(2,I-5) )
# 
# # compute DETECT statistic 
# detect1 <- conf.detect( data = dat, score = wmod1, itemcluster = itemcluster )
#   ##            unweighted weighted
#   ##  DETECT        -0.184   -0.184
#   ##  ASSI          -0.147   -0.147
#   ##  RATIO         -0.226   -0.226
#   ##  MADCOV100      0.816    0.816
#   ##  MCOV100       -0.786   -0.786
# ## End(Not run)

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