mcdina: Multiple Choice DINA Model

Description

The function mcdina implements the multiple choice DINA model (de la Torre, 2009; see also Ozaki, 2015; Chen & Zhou, 2017) for multiple groups. Note that the dataset must contain integer values $1,\ldots, K_j$ for each item. The multiple choice DINA model assumes that each item category possesses different diagnostic capacity. Using this modeling approach, different distractors of a multiple choice item can be of different diagnostic value. The Q-matrix can also contain integer values which allows the definition of polytomous attributes.

Usage

mcdina(dat, q.matrix, group=NULL, itempars="gr", weights=NULL,
    skillclasses=NULL, zeroprob.skillclasses=NULL,
    reduced.skillspace=TRUE, conv.crit=1e-04,
    dev.crit=0.1, maxit=1000, progress=TRUE)
# S3 method for mcdina
summary(object, digits=4, file=NULL,  ...)
# S3 method for mcdina
print(x, ...)

Value

A list with following entries

item: Data frame with item parameters
posterior: Individual posterior distribution
likelihood: Individual likelihood
ic: List with information criteria
q.matrix: Used Q-matrix
pik: Array of item-category probabilities
delta: Array of item parameters
se.delta: Array of standard errors of item parameters
itemstat: Data frame containing item definitions
n.ik: Array of expected counts
deviance: Deviance
attribute.patt: Probabilities of latent classes
attribute.patt.splitted: Splitted attribute pattern
skill.patt: Marginal skill probabilities
MLE.class: Classified skills for each student (MLE)
MAP.class: Classified skills for each student (MAP)
EAP.class: Classified skills for each student (EAP)
dat: Used dataset
skillclasses: Used skill classes
group: Used group identifiers
lc: Data frame containing definitions of each item category
lr: Data frame containing the relation of each latent class and each item category
iter: Number of iterations
itempars: Used specification of item parameter estimation type
converged: Logical indicating whether convergence was achieved.

Arguments

dat: A required $N \times J$ data matrix containing integer responses (1, 2, $\ldots$, $K$) of $N$ respondents to $J$ test items.
q.matrix: A required matrix specifying which item category is intended to measure which skill. The Q-matrix has $K+2$ columns for a model with $K$ skills. In the first column should be the item index, in the second column the category integer and the rest of the columns contains the 'ordinary' Q-matrix specification. See data.cdm01$q.matrix for the layout of such a Q-matrix.
group: An optional vector of group identifiers for multiple group estimation.
itempars: A character or a character vector of length $J$ indicating whether item parameters should separately estimated within each group. The default is "gr", for group-invariant item parameters choose "jo".
weights: An optional vector of sample weights.
skillclasses: An optional matrix for determining the skill space. The argument can be used if a user wants less than the prespecified number of $2^K$ skill classes.
zeroprob.skillclasses: An optional vector of integers which indicates which skill classes should have zero probability. Default is NULL (no skill classes with zero probability).
reduced.skillspace: An optional logical indicating whether the skill space should be reduced to cover only bivariate associations among skills (see Xu & von Davier, 2008).
conv.crit: Convergence criterion for change in item parameter values
dev.crit: Convergence criterion for change in deviance values
maxit: Maximum number of iterations.
progress: An optional logical indicating whether the function should print the progress of iteration in the estimation process.
object: Object of class mcdina.
digits: Number of digits to display in summary.mcdina
file: Optional file name for a file in which summary should be sinked.
x: Object of class mcdina
...: Further arguments to be passed.

Details

The multiple choice DINA model defines for each item category $jc$ the necessary skills to master this attribute. Therefore, the vector of skills $\bold{\alpha}$ is transformed into item-specific latent responses $\eta_{j}$ which are functions of $\bold{\alpha}$ and Q-matrix entries $q_{jc}$ (just like in the DINA model). If there are $K_j$ item categories for item $j$, then there exist at most $K_j$ values of the latent response $\eta_j$.

The multiple choice DINA model estimates the item response function as $$ P( X_{nj}=k | \eta_{nj}=l )=p_{jkl} $$ with the constraint $\sum_k p_{jkl}=1 $.

References

Chen, J., & Zhou, H. (2017) Test designs and modeling under the general nominal diagnosis model framework. PLoS ONE 12(6), e0180016.

de la Torre, J. (2009). A cognitive diagnosis model for cognitively based multiple-choice options. Applied Psychological Measurement, 33, 163-183.

Ozaki, K. (2015). DINA models for multiple-choice items with few parameters: Considering incorrect answers. Applied Psychological Measurement, 39(6), 431-447.