sirt: conf.detect – R documentation

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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).

An exploratory DETECT analysis can be conducted using the expl.detect function.

Usage

conf.detect(data, score, itemcluster, bwscale=1.1, progress=TRUE,
        thetagrid=seq(-3, 3, len=200), smooth=TRUE, use_sum_score=FALSE, bias_corr=TRUE)

## S3 method for class 'conf.detect'
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?
`smooth`	Logical indicating whether smoothing should be applied for conditional covariance estimation
`thetagrid`	A vector which contains theta values where conditional covariances are evaluated.
`use_sum_score`	Logical indicating whether sum score should be used. With this option, the bias corrected conditional covariance of Zhang and Stout (1999) is used.
`bias_corr`	Logical indicating whether bias correction (Zhang & Stout, 1999) should be utilized if `use_sum_score=TRUE`.
`object`	Object of class `conf.detect`
`digits`	Number of digits for rounding in `summary`
`file`	Optional file name to be sunk for `summary`
`...`	Further arguments to be passed

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

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.

Value

A list with following entries:

`detect`	Data frame with statistics DETECT, ASSI, RATIO, MADCOV100 and MCOV100
`ccovtable`	Individual contributions to conditional covariance
`ccov.matrix`	Evaluated conditional covariance

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(4), 504-516. doi: 10.1080/10705511.2014.938596

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), 1-21. doi: 10.1111/j.1745-3984.2007.00024.x

Stout, W., Habing, B., Douglas, J., & Kim, H. R. (1996). Conditional covariance-based nonparametric multidimensionality assessment. Applied Psychological Measurement, 20(4), 331-354. doi: 10.1177/014662169602000403

Zhang, J. (2007). Conditional covariance theory and DETECT for polytomous items. Psychometrika, 72(1), 69-91. doi: 10.1007/s11336-004-1257-7

Zhang, J. (2013). A procedure for dimensionality analyses of response data from various test designs. Psychometrika, 78(1), 37-58. doi: 10.1007/s11336-012-9287-z

Zhang, J., & Stout, W. (1999a). Conditional covariance structure of generalized compensatory multidimensional items. Psychometrika, 64(2), 129-152. doi: 10.1007/BF02294532

Zhang, J., & Stout, W. (1999b). The theoretical DETECT index of dimensionality and its application to approximate simple structure. Psychometrika, 64(2), 213-249. doi: 10.1007/BF02294536

Examples

#############################################################################
# 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 <- sirt::rasch.mml2( dat )
# estimate WLEs
wle1 <- sirt::wle.rasch( dat, b=mod1$item$b )$theta

# DETECT for content domains
detect1 <- sirt::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 <- sirt::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 <- sirt::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 <- sirt::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 <- sirt::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 <- sirt::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 <- sirt::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 <- sample( seq( -2, 2, len=I) )
dat <- sirt::sim.raschtype( stats::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 <- sirt::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)

sirt

Supplementary Item Response Theory Models

v3.10-118

GPL (>= 2)

Authors

Alexander Robitzsch [aut,cre] (<https://orcid.org/0000-0002-8226-3132>)

Initial release

2021-09-22 17:45:34