[R-sig-ME] What is the consequence of neglecting to consider variability of a fixed effect across levels of a random effect

[Mike Lawrence]

In the scenario where I have one random effect and one fixed effect, I
typically attempt to evaluate the evidence for an effect of the fixed
effect variable by:

    restricted_model = lmer(
        formula = dv ~ (1|rand) + 1
    )
    unrestricted_model = lmer(
        formula = dv ~ (1|rand) + fixed
    )
    bits = (AIC(restricted_model)-AIC(unrestricted_model))*log2(exp(1))
    #"bits" of evidence: sign signifies whether the effect (+) or null (-) is
    #    supported, absolute magnitude signifies strength of evidence.
    print(bits)

However, this comparison ignores the possibility that the fixed effect
may vary across levels of the random effect and that these deviations
may correlate with the random effects' intercept deviations. I'm not
explicitly interested in the existence of either form of deviation,
but I wonder if anyone knows whether ignoring their possibility has
consequences for the evaluation of the general effect of the fixed
effect. Specifically, compared to the above, how would the following
be expected to affect the evidence evaluation?

    restricted_model = lmer(
        formula = dv ~ (fixed|rand) + 1
    )
    unrestricted_model = lmer(
        formula = dv ~ (fixed|rand) + fixed
    )
    bits = (AIC(restricted_model)-AIC(unrestricted_model))*log2(exp(1))
    print(bits)



Cheers,

Mike

--
Mike Lawrence
Graduate Student
Department of Psychology
Dalhousie University

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http://tr.im/mikes_public_calendar

~ Certainty is folly... I think. ~