I have rather a simple question but for some reason have not been able to find a clear answer. My apologies if I have missed a previous answer in my searching.

Here is my question: What does it mean if you have a highly significant regression coefficient for a predictor, but overall the model is not significant due to the other non-significant predictors in the model? Do you just state that the model was non-significant and conclude you cannot predict the dependent variable, ignoring the significant relationship between your dependent measure and one of your predictors? This seems to be ignoring an interesting relationship in the data, but equally I realise that one can’t throw in every possible predictor under the sun and pick one that happens to be significant.

More specifically to my problem, I have a regression model containing 3 predictor variables (A, B and C) and all corresponding two-way and three-way interactions between them. The overall model is not significant as shown by the F test, but our three-way interaction term (which we a-priori predicted would be significant) is significant at p = .025. Can we interpret this or do we have to ignore the whole thing? And does the answer to this also therefore apply to the significance of the ‘corrected model’ in ANOVA too?

Sorry for the simplicity of the question, and thank you for any help you can give!