# Calculating DeltaAIC

I've been advised by a colleague that when performing backwards elimination for model selection using AIC as my criterion, I should remove terms individually - starting from the most complex interaction, and working down to the main effects, comparing the AIC for the reduced model and comparing it to the full model (the original model containing all terms).

For example, if I had 3 main effects, and their interactions in my model I could write the full model $y_1$ and remove the three way interaction to make model $y_2$:

$y_1 = (a \times b \times c) + (a \times b) + (a \times c) + (b \times c) + a + b + c$

$y_2 = (a \times b) + (a \times c) + (b \times c) + a + b + c$

I find that removing the term reduces the AIC by >-2, so I should remove the three way interaction. I then reduce $y_2$ further by removing the first two way interaction ($a \times b$).

$y_3 = (a \times c) + (b \times c) + a + b + c$

Following the advice given to me by my colleague I would then compare AIC of $y_3$ to that of $y_1$ and reject it if the AIC reduces by >-2. However, it strikes me that if the three way interaction, which is missing from both $y_2$ and $y_3$, is extremely uninformative such that $\Delta$AIC >>-2 then I will remove the $a \times b$ interaction unless it is highly informative such that it overcomes the effect of removing the three way interaction.

This brings me to my question: in backwards elimination, should one compare AIC of the reduced model to:

a) the full model (e.g. $AIC_{y1} - AIC_{y3}$)

b) the model upon which the reduced model is based (e.g. $AIC_{y2} - AIC_{y3}$)

c) something else...