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In some areas, it is common to fit a model separately to multiple clusters in a data set, for instance fitting a cognitive model separately to data from each participant in an experiment.

Model comparison is a bit more complicated in this scenario, since rather than having a single deviance, AIC, BIC, or Bayes Factor per model, we have one score per model per participant.

A way that works for Bayes Factors

Stephan et al (2009; NeuroImage) discuss this issue with an emphasis on Bayesian analysis of MRI data, and identify two approaches.

In the Fixed Effects approach, we assume that all subjects’ data are generated by the same model, and so the Bayes Factor for the group is just the product of individual participants' Bayes Factors:

$$ BF_{\text{Group}} = \prod_i BF_i $$

In the Random Effects approach, introduced in that paper, we assume that there is a true distribution of models in the population that is, some participants' data is generated by model 1, some by model 2, etc., the distribution of these models is described by a multinomial distribution with model probability parameters $r$, and the posterior distribution over these model probability parameters is described by a Dirichlet distribution with concentration parameters $\alpha$.

$$ \begin{align} \text{Data}_i &\sim \text{Model}_i\\ \text{Model}_i &\sim \text{Multinomial}(r)\\ r &\sim \text{Dirichlet}(\alpha) \end{align} $$

Estimating this model allows us to infer useful quantities such as how likely it is that a specific model generated the data of a randomly chosen subject, and what is the probability that model M is the most prevalent in the population. In practice, these parameters are estimated using either model Bayes Factors, marginal likelihood from variational methods, or BIC scores.

Doing the same for AIC

My question is whether similar methods exist for evaluating collections of AIC scores obtained by fitting various models to various participants?

It seems pretty reasonable to calculate

$$ \begin{align} AIC_{\text{Group}} &= \sum_i^n AIC_i\\ &= \sum_i^n 2k_i + (\sum_i^n -2\text{ln}(\hat L_i)) \end{align} $$

since this is the same as calculating AIC for a single model with $nk$ parameters and log-likelihood of $\sum_i^n \text{ln}(\hat L_i)$.

I've also seen some papers just running t-tests on the individual AIC scores:

t.test(aic.score ~ model, paired=T, data=aic.scores)

Is there any more principled solution to this problem?


Notes

  • An R implementation of the Stephan et al (2009) random effects procedure can be found here.
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Summing up the AIC is the same as "stacking up" your individual models, akin to having an interaction term in linear regression. For example, if the model for subject 1 is \begin{equation} y_1 = \alpha_11 + \beta_1 x_1 + \epsilon_1, \quad \epsilon_1 \sim \mathcal{N}(0, \sigma_1^2I) \end{equation} and the model for subject 2 is \begin{equation} y_2 = \alpha_21 + \beta_2 x_2 + \epsilon_2, \quad \epsilon_2 \sim \mathcal{N}(0, \sigma_2^2I) \end{equation} You can fit a joint model \begin{equation} \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} = \begin{pmatrix} \alpha_11 \\ \alpha_21 \end{pmatrix} + \begin{pmatrix} \beta_1x_1 \\ \beta_2x_2 \end{pmatrix} + \begin{pmatrix} \epsilon_1 \\ \epsilon_2 \end{pmatrix}, \quad \begin{pmatrix} \epsilon_1 \\ \epsilon_2 \end{pmatrix} \sim \mathcal{N}\left (0, \begin{pmatrix} \sigma_1^2I & 0 \\ 0 & \sigma_2^2I \end{pmatrix} \right) \end{equation} and your AIC would be the sum of your two sub-models, since the log likelihood and the number of parameters of the joint model are simply the sum of the submodels' log likelihood and number of parameters.

Of course, in practice, we usually assume the models share something. In the above example, we usually assume $\sigma_1^2=\sigma_2^2$. Moreover, if there are many subjects, we may assume $\alpha_i$ come from some distribution, and have a random effects model.

I suppose the models you use are more complicated than linear regression, but the principle is the same. Summing the AIC up is basically equivalent to a "fixed effects" overall model, where the sub-models don't share anything. When the sub models are grossly different, I suppose the approach by Stephan et al (2009) is akin to having a "model of models". Importantly, their approach supposes you can specify a prior $\text{Dirichlet}(\alpha)$ for the different models. The AIC approach is not Bayesian, so I am not sure you can easily adapt their procedure for AIC.

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    $\begingroup$ Great, thanks. To be clear, you're saying you don't think there's any real alternative to summing up the individual AIC values? $\endgroup$ – Eoin Apr 16 '20 at 13:26
  • $\begingroup$ That's as far as I know. Perhaps others can enlighten me on this also. $\endgroup$ – Tim Mak Apr 17 '20 at 3:54

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