Multiple Choice DINA Model

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.

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, ...)

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 $.

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.

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.

Xu, X., & von Davier, M. (2008). Fitting the structured general diagnostic model to NAEP data. ETS Research Report ETS RR-08-27. Princeton, ETS.

Note

If dat and q.matrix correspond to the 'ordinary format' which is used in gdina, then the function mcdina will detect it and convert it into the necessary format (see Example 2).