CDM (version 8.2-6)

sequential.items: Constructing a Dataset with Sequential Pseudo Items for Ordered Item Responses

Description

This function constructs dichotomous pseudo items from polytomous ordered items (Tutz, 1997). Using this method, developed test models for dichotomous data can be applied for polytomous item responses after transforming them into dichotomous data. See Details for the construction.

Ma and de la Torre (2016) proposed a sequential GDINA model. Interestingly, the proposed model can be fitted with the gdina function in this CDM package while item responses has to be transformed with the sequential.items function for obtaining dichotomous pseudoitems. The Q-matrix for the sequential model of Ma and de la Torre (2016) can be used in the GDINA model for the dichotomous pseudoitems. This approach is implemented for automatic use in gdina.

Usage

sequential.items(data)

Value

A list with following entries

dat.expand

A data frame with dichotomous pseudo items

iteminfo

A data frame containing some item information

maxK

Vector with maximum number of categories per item

Arguments

data

A data frame with item responses

Details

Assume that item \(j\) possesses \(K \geq 3\) categories. We label these categories as \(k=0,1,\ldots,K-1\). The original item responses \(X_{nj}\) for person \(n\) at item \(j\) is then transformed into \(K-1\) pseudo items \(Y_{j1}, \ldots, Y_{j,K-1}\).

The first pseudo item response \(Y_{nj1}\) is defined as 1 iff \(X_{nj} \geq 1\). The second item responses \(Y_{nj2}\) is 1 iff \(X_{nj} \geq 2\), it is 0 iff \(X_{nj}=1\) and it is missing (NA in the dataset) iff \(X_{nj}=0\). The construction proceeds in the same manner for other categories (see Tutz, 1997). The pseudo items can be recognized as 'hurdles' a participant has to master to get a score of \(k\) for the original item.

The pseudo items are treated as conditionally independent which implies that IRT models or CDMs which assume local independence can be employed for estimation.

For deriving item response probabilities of the original items from response probabilities of the pseudo items see Tutz (1997, p. 141ff.).

References

Ma, W., & de la Torre, J. (2016). A sequential cognitive diagnosis model for polytomous responses. British Journal of Mathematical and Statistical Psychology, 69(3), 253-275.

Tutz, G. (1997). Sequential models for ordered responses. In W. van der Linden & R. K. Hambleton. Handbook of modern item response theory (pp. 139-152). New York: Springer.

Examples

#############################################################################
# EXAMPLE 1: Constructing sequential pseudo items for data.mg
#############################################################################

data(data.mg, package="CDM")
dat <- data.mg
items <- colnames(dat)[ which( substring( colnames(dat),1,1)=="I" ) ]
##    [1] "I1"  "I2"  "I3"  "I4"  "I5"  "I6"  "I7"  "I8"  "I9"  "I10" "I11"
data <- dat[,items]

# construct sequential dichotomous pseudo items
res <- CDM::sequential.items(data)

# item information table
res$iteminfo
  ##      item itemindex category pseudoitem
  ##   1    I1         1        1         I1
  ##   2    I2         2        1         I2
  ##   3    I3         3        1         I3
  ##   4    I4         4        1    I4_Cat1
  ##   5    I4         4        2    I4_Cat2
  ##   6    I5         5        1    I5_Cat1
  ##   7    I5         5        2    I5_Cat2
  ##     [...]

# extract dataset with pseudo items
dat.expand <- res$dat.expand
colnames(dat.expand)
  ##    [1] "I1"       "I2"       "I3"       "I4_Cat1"  "I4_Cat2"  "I5_Cat1"
  ##    [7] "I5_Cat2"  "I6_Cat1"  "I6_Cat2"  "I7_Cat1"  "I7_Cat2"  "I7_Cat3"
  ##   [13] "I8"       "I9"       "I10"      "I11_Cat1" "I11_Cat2" "I11_Cat3"

# compare original items and pseudoitems

#**** Item I1
stats::xtabs( ~ paste(data$I1) + paste(dat.expand$I1) )
  ##                 paste(dat.expand$I1)
  ##   paste(data$I1)     0     1    NA
  ##               0   4339     0     0
  ##               1      0 33326     0
  ##               NA     0     0   578

#**** Item I7

stats::xtabs( ~ paste(data$I7) + paste(dat.expand$I7_Cat1) )
  ##                 paste(dat.expand$I7_Cat1)
  ##   paste(data$I7)     0     1    NA
  ##               0   3825     0     0
  ##               1      0 14241     0
  ##               2      0 14341     0
  ##               3      0  5169     0
  ##               NA     0     0   667

stats::xtabs( ~ paste(data$I7) + paste(dat.expand$I7_Cat2) )
  ##                 paste(dat.expand$I7_Cat2)
  ##   paste(data$I7)     0     1    NA
  ##               0      0     0  3825
  ##               1  14241     0     0
  ##               2      0 14341     0
  ##               3      0  5169     0
  ##               NA     0     0   667

stats::xtabs( ~ paste(data$I7) + paste(dat.expand$I7_Cat3) )
  ##                 paste(dat.expand$I7_Cat3)
  ##   paste(data$I7)     0     1    NA
  ##               0      0     0  3825
  ##               1      0     0 14241
  ##               2  14341     0     0
  ##               3      0  5169     0
  ##               NA     0     0   667

if (FALSE) {
#*** Model 1: Rasch model for sequentially created pseudo items
mod <- CDM::gdm( dat.expand, irtmodel="1PL", theta.k=seq(-5,5,len=21),
             skillspace="normal", decrease.increments=TRUE)
}