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From http://blog.minitab.com/blog/adventures-in-statistics/what-is-the-f-test-of-overall-significance-in-regression-analysis:

Typically, if you don't have any significant P values for the individual coefficients in your model, the overall F-test won't be significant either. However, in a few cases, the tests could yield different results. For example, a significant overall F-test could determine that the coefficients are jointly not all equal to zero while the tests for individual coefficients could determine that all of them are individually equal to zero.

Could you please provide a graphical and/or real life example?

Luis
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Suppose you measure the length of people's left leg and of their right leg. Then you construct a logistic regression predicting the probability that they have bumped their head on an overhead beam in an old house in the last year. You might get a significant overall result because tall people are more likely to bump their head but in the regression neither leg will appear significant by itself as it does not add anything to the model over and above the other one since the length of the two legs are highly correlated.

mdewey
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    Ha! The scenario is appealing enough but I think it represents a slightly different scenario in which multicollinearity is present side both leg lengths are expected to be equal, assuming we haven't measured s population off amputees or those with leg-length discrepancies. – prince_of_pears Dec 04 '16 at 16:17
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    @prince_of_pears I think that mdewey has it right. The presence of a significant ANOVA result suggests that overall there is something about the predictors that can accomplish prediction. The tests on individual predictors, however, ignores variance in Y that is predictable based on more than one predictor, so they can all be nonsignificant if there is mostly shared variance predicted. The USJudgeRatings data set in R comes close to illustrating this, as there are ton of strong bivariate predictors for predicting RTEN, but few are significant in a multiple regression. – J Taylor Dec 04 '16 at 17:59
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    @prince_of_pears To finish the thought, multicollinearity need not be absolute in order to remove enough unique variance in Y that is predicted by the individual X values to render all the Xs nonsignificant. – J Taylor Dec 04 '16 at 18:01
  • I agree with you, I was just saying that the example need not be so extreme. The effects could even be nominally in different directions and still arrive at the same conditions. – prince_of_pears Dec 04 '16 at 18:51