  The use of Bayesian information criterion (BIC) in the model selection
procedure is under the assumption that the observations are independent and
identically distributed (i.i.d.). However, in practice, we do not always have
i.i.d. samples. For example, clustered observations tend to be more similar
within the same group, and longitudinal data is collected by measuring the same
subject repeatedly. In these scenarios, the assumption in BIC is not satisfied.
The concept of effective sample size is brought up and improved BIC is defined
by replacing the sample size in the original BIC expression with the effective
sample size, which will give us a better theoretical foundation in the
circumstance that mixed-effects models involve. Numerical experiment results
are also given by comparing the performance of our new BIC with other widely
used BICs.
