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I have a technical question concerning calculating AICc for two possible models.

For the data set I am working with there are 12 subjects and 10 phases of the experiment. Two different models, a Gamma distribution and an Inverse Gaussian distribution were fit to the data of each individual subject for each phase. Using a function I've found useful in exploratory unimodal distribution fitting (downloaded it off of Matlab's File Exchange) I got AICc values for each subject for each phase.

Thus, within each phase of the experiment there are 12 AICc values for each model, one per subject. My question is as follows:

Given the AICc values are already calculated for each individual subject, is it okay to sum the AICc values across subjects for each phase to yield to AICc values, one for the gamma distribution and one for the Inverse Gaussian and then calculate the difference or weights based off those two sums?

The alternative is to get the Log-likelihood for each animal, sum across animals, and then calculate AICc for each model within each phase. However, it seems to me that this is equivalent to my first solution...

Additionally, to determine the best model for the entire dataset (i.e. across phases) does the summing seem appropriate?

Does anyone have any insight into this situation (getting the Log-likelihoods is doable, I'd need to edit the function, but the AICc are the default output of the function).

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So, the AIC is $2k - 2ln(L)$ where $k$ is the number of parameters in the model. If I understand what you mean by AIC for a subject, you've got $2k - ln(L_i)$ for each of $n$ subjects, and if you add them up you'll have $n2k - 2\sum_{i=1}^n ln(L_i)$, so you've taken the complexity penalty n times instead of just once. Subtract $2k(n-1)$ and that should be equivalent to getting the log likelihoods then converting.

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    $\begingroup$ Doesn't this depend on whether or not the same $ k $ parameters were used to fit the $ n $ subjects? $\endgroup$ Commented Feb 16, 2014 at 0:20
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    $\begingroup$ Yes! I assumed the question was asking about $n$ observations fit with the same model. AIC is designed to tell you about how well one model fits one data set. But reading the question again, it looks like each subject has its own model, that is, its own set of estimated parameters for each of the two "models" (gamma and inverse gaussian) that the OP is trying to decide between. $\endgroup$ Commented Feb 17, 2014 at 14:17
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I encountered a very similar problem and want to share my thoughts, despite it's already 9 years after OP posted the original question.

If I understood correctly, OP's question can be formulated as such:

Is gamma distribution or inverse gaussian distribution a better way to model the 12 participants across 10 phases?

The best way to think about this, if it fits your assumptions, is actually to model all your data together and include the relevant experimental phase and/or individual ID as covariates. Then you'll only have 2 AICc values to compare.

To put my thoughts in context of the other reply in this post, you have modelled $10\times n \times k$ parameters, where $n=12$. The AIC is therefore $(10n)2k - 2 \sum^{n}_{i=1} ln(L_i)$, so equivalent to summing the AIC.

However, I would suggest to review the need of modeling parameters on a participant-level. If deemed necessary, consider using mixed-effect models and choose an appropriate class of information criterion, which may not be straightforward in itself.

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