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I am going through a model selection process with a mixed-model with 3 variables: A, B, and C. B and C are orthogonal factors. B or C may interact with A, so my full model would be:

fixed:

Y ~ A + B + C + A*B + A*C

random:

~1|D

When I run my analysis, I get four models sharing the lowest AICc.

Y ~ B + A*C
Y ~ A + B + A*C
Y ~ B + C + A*C
Y ~ A + B + C + A*C

I am very confused about what is going on. Obviously there is an interaction between A and C, but how can they all have the same AICc value? Where do I go from here in model selection?

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So apparently all these formulas are equivalent in R. No matter which model I run, the data summary includes parameters for A, B, C, and A*C. I guess if you have an interaction between A and C, then by default, A and C must be in the model. You learn something new everyday!

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    $\begingroup$ you can add an interaction term in R's lm() without the main effects through writing a:b $\endgroup$ – Chris Novak Apr 19 '15 at 20:22

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