BIC and AIC(c) and group data

Question

I would like to compare two models using the BIC and AICc. Doing so seems fairly straightforward if both models are fit to only one dataset. However, I have data from 10 participants, and there is no reason to assume that they are all well-described by the same set of parameters. Is it appropriate to sum the log-likelihoods of a model across all participants and multiply the number of parameters by the number of participants to obtain a "group BIC/AICc" for that model, or is there another way of obtaining the BIC/AICc in such a situation?

Answer 1

I did not see the rationale of using your "group BIC/AIC". Your proposed group BIC/AIC is comparing a different pair of models. Let $f(\mathbf{x}_i \mid \alpha)$ and $g(\mathbf{y}_i \mid \beta)$ be two competing models. For simplicity, suppose we have two groups, ande denote $D_i$ the group dummy variable. $D_i=1$ indicates that $i$ belongs to group $1$, and $D_i=0$ refers to the case that $i$ belongs to group $2$. Correspondingly, let $\alpha_1$ and $\alpha_2$ be the two sets of parameters specifying the distribution of group $1$ and $2$, respectively. $\beta_1$ and $\beta_2$ are defined similarly. Define two new densities/models: $$\tilde{f}(\mathbf{x}_i, D_i \mid \alpha_1, \alpha_2)=f(\mathbf{x}_i\mid \alpha_1)^{D_i}f(\mathbf{x}_i\mid \alpha_2)^{1-D_i},$$ $$\tilde{g}(\mathbf{y}_i, D_i \mid \beta_1, \beta_2)=g(\mathbf{y}_i\mid \beta_1)^{D_i}g(\mathbf{y}_i\mid \beta_2)^{1-D_i}.$$ Note that $$\log\tilde{f}(\mathbf{x}_i, D_i \mid \alpha_1, \alpha_2)=D_i \log f(\mathbf{x}_i\mid \alpha_1)+(1-D_i)\log f(\mathbf{x}_i\mid \alpha_2),$$ $$\log\tilde{g}(\mathbf{y}_i, D_i \mid \beta_1, \beta_2)=D_i \log g(\mathbf{y}_i\mid \beta_1)+(1-D_i)\log g(\mathbf{y}_i\mid \beta_2).$$ If you fit the above two models to the two groups of data, the ordinary AIC/BIC would be your AIC/BIC. I am not sure if this connection might help you.