Model selection penalises against over-parametrisation, i.e., including too many unnecessary parameters in the model, which can sometimes mislead your results. The AIC score is one approach for selecting between models. It is calculated separately for each model using the following formula,
\(AIC = 2K - 2ln(L)\),
where \(K\) is the number of free parameters and \(L\) is the likelihood. As a rule of thumb, the model with a lower AIC score is preferred if the difference in AIC values is >2. A model with many additional parameters would need to improve the likelihood substantially to be preferred over one with fewer parameters.
R’s AIC function can calculate the AIC scores for you.
Part of the output includes the df values, which refers to
“degrees of freedom” or the number of free parameters in each model. In our case, the number of parameters
associated with each substitution model is the total number of branches
in the unrooted tree (2 * num_taxa - 3 = 31) plus the number of
parameters in each model.
The JC model assumes transition rates and base
frequencies are both equal. We can’t actually estimate the substitution
rate without reference to time, so the substitution rate \(\mu\) is fixed to 1. Thus, since the JC
model doesn’t have any free parameters, df = 31.
The GTR model relaxes these assumptions. Each
possible transition (e.g., A to T or vice versa) has a separate rate.
One of the transitions is fixed to 1 and the other 5 rates are estimated
relative to this one, adding 5 parameters to the model. Since the
proportion of each base (A,T,G,C) must sum to 1, we only need to
estimate 3 additional parameters for the base frequencies. Thus, the
total number of free parameters for the GTG is df = 39 (31
+ 5 + 3).
As you should be able to see, the AIC score for the GTR model is substantially lower, despite the addition of 8 parameters.