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Crossed effects and model selection

Richard Feldman 
Hello all,

I am using AIC to select the best model from a model set. I am confused 
about what models to include when there are cross-level interactions.

Here is the model:

model <- glmer(Y ~ V1 * V2 + (V1|SITE), data=Data), which expands to

model <- glmer(Y ~ V1 + V2 + V1:V2 + (V1|SITE), data=Data)

V1 is a level-1 predictor and V2 is a level-2 (i.e. SITE) level predictor.

I am wondering whether the following model can be used in the set of models:

model.int <- glmer(Y ~ V1:V2 + (V1|SITE), data=Data), which actually tests:

model.int <- glmer(Y ~ V2 + V1:V2 + (V1|SITE), data=Data)


This model implies that the slope of V1 is modeled without an intercept:

The level-1 model is:

Yij = b0j + b1j*V1 + eij

The level-1 intercept and slope at level-2 are then:

B0j = p00 + p01*V2 + r0j

B1j = p10 + p11*V2 + r1j

Substituting the above into the full equation leads to:

Yij = p00 + p01*V2 + r0j + p10*V1 + p11*V1*V2 + r1j*V1 + eij

But since model.int doesn't contain a "main effect" V1, p10 must be zero 
and the slope has a zero intercept:

B1j = 0 + p11*V2 + r1j,

I would interpret this as meaning there is no ?average? effect of V1 on 
the response variable, Y, and that the effect of V1 on Y can only be 
interpreted based on the V2 value of the site. Is the above possible or 
must the slope-model retain its intercept parameter?

Thank you in advance!

Richard