  This paper introduced a clustered estimator of the Network Information
Criterion (NICc) to approximate leave-one-cluster-out cross-validated deviance,
which can be used as an alternative to cluster-based cross-validation when
modeling clustered data. Stone proved that Akaike Information Criterion (AIC)
is an asymptotic equivalence to leave-one-observation-out cross-validation if
the parametric model is true. Ripley pointed out that the Network Information
Criterion (NIC) derived in Stone's proof, is a better approximation to
leave-one-observation-out cross-validation when the model is not true. For
clustered data, we derived a clustered estimator of NIC, referred to as NICc,
by substituting the Fisher information matrix in NIC with its estimator that
adjusts for clustering. This adjustment imposes a larger penalty in NICc than
the unclustered estimator of NIC when modeling clustered data, thereby
preventing overfitting more effectively. In a simulation study and an empirical
example, we used linear and logistic regression to model clustered data with
Gaussian or binomial response, respectively. We showed that NICc is a better
approximation to leave-one-cluster-out deviance and prevents overfitting more
effectively than AIC and Bayesian Information Criterion (BIC). NICc leads to
more accurate model selection, as determined by cluster-based cross-validation,
compared to AIC and BIC.
