  Longitudinal item response data are common in social science, educational
science, and psychology, among other disciplines. Studying the time-varying
relationships between items is crucial for educational assessment or designing
marketing strategies from survey questions. Although dynamic network models
have been widely developed, we cannot apply them directly to item response data
because there are multiple systems of nodes with various types of local
interactions among items, resulting in multiplex network structures. We propose
a new model to study these temporal interactions among items by embedding the
functional parameters within the exponential random graph model framework.
Inference on such models is difficult because the likelihood functions contain
intractable normalizing constants. Furthermore, the number of functional
parameters grows exponentially as the number of items increases. Variable
selection for such models is not trivial because standard shrinkage approaches
do not consider temporal trends in functional parameters. To overcome these
challenges, we develop a novel Bayes approach by combining an auxiliary
variable MCMC algorithm and a recently-developed functional shrinkage method.
We apply our algorithm to survey and review data sets, illustrating that the
proposed approach can avoid the evaluation of intractable normalizing constants
as well as the detection of significant temporal interactions among items.
Through a simulation study under different scenarios, we examine the
performance of our algorithm. Our method is, to our knowledge, the first
attempt to select functional variables for models with intractable normalizing
constants.
