I am using the Bayesian information criterion (BIC) for model selection. I have $n$ groups of observations, and each group has $m$ univariate or scalar responses. I use BIC to compare two models.
The first model (Model 1) assumes the $m$ responses in each of the $n$ groups are independent. This means that we have $n\times m$ i.i.d. observations under Model 1.
The second model (Model 2) assumes the $m$ responses in each of the $n$ groups are correlated. The maximum likelihood estimators are obtained by considering such correlation under Model 2.
My question is that when I want to use BIC for model selection, I must specify the sample size. For Model 1, the sample size is $n\times m$. But for Model 2, it seems that the sample size should be $n$, because we only have $n$ i.i.d. (multivariate) observations. Is it true? Or are there better ways for model selection?