# Using DIC for model selection: (in)valid comparisons

Imagine I have the following 7 glmm models where $b_1$ through $b_3$ are fixed effects.

$M_1 = y \sim b_1 \times b_2 \times b_3$

$M_2 = y \sim b_2 \times b_3$

$M_3 = y \sim b_1 \times b_3$

$M_4 = y \sim b_1 \times b_2$

$M_5 = y \sim b_3$

$M_6 = y \sim b_2$

$M_7 = y \sim b_1$

DIC can be used to asses the relative fit of models. However, I would like to know what comparisons would be valid to make because I am unsure and struggling to know what route to take. Imagine the following scenario, where the DIC for each model is shown in red. I am not sure what would and would not be valid comparisons/correct interpretations of using DIC compare the models. Would it be valid to compare, for example, the following:

A) $M_1$ with $M_2$ - I think this is a valid comparison, and shows whether dropping the parameter $b_1$ improves the fit of the model. If $\Delta$ DIC > 2 then model fit is improved by dropping the term.

B) $M_1$ with $M_5$ - I suspect that this is not a valid comparison. It tests whether dropping two parameters at the same time improves model fit. I think I should first try dropping both $b_1$ and $b_2$ individually, then proceeding to testing $M_2$ and $M_3$ against $M_5$, retaining the most complex model where dropping terms does not improve fit.

C) $M_5$ with $M_6$ - I suspect that this is not a valid comparison. It shows the effect of substituting the parameters, which may mean different data is included.

Essentially it boils down to this. Can one only use DIC to compare models with sequentially dropped effects, or can I drop multiple effects simultaneously and/or compare models with different fixed effects?

Then how would I draw conclusions from this? Would it be best to present the model $M_1$ as the best fitting model, because it is the most complex model of all of my models where $\Delta$ DIC is <2 when dropping parameter $b_3$... or, that models $M_5$ and $M_6$ (and $M_1$) are the best fitting models when using parameters $b_3$, $b_2$, (and $b_1$) respectively.

• so then I could look at all of my models, and just pick the one with the lowest DIC as being the best fit - in my example it would be model $m_5$ – rg255 May 31 '16 at 13:11