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I've carried out random effects models on my dependent variables. Some of these give an $F$ statistic that is not significant, meaning my model is not significant. What exactly does this mean? Does it mean that I should disregard the whole thing, even though individual variables in the model are significant?

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The multiple comparisons procedures ASSUME that the omnibus F test gives a significant resutls, and so you are trying to discern which while protecting your overall Type I error rate. Your individual comparisons are not reliable. It means that the effects you posit in your model cannot be differentiated from a null model (i.e., no parameters)

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