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If the linear regression's f-test is insignificant but its coefficients are significant in t-test, can I use this regression and its coefficients?

In academic journals, I find people use linear regression without reporting f-test values but use its coefficients as long as they are significant.

Is it ok to use the linear regression's coefficients ignoring its f-test values?

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    $\begingroup$ The possible causes of such a situation are extensively discussed at stats.stackexchange.com/questions/24720. But if the F-test tells you the fit is insignificant, then you have no justification to look any further: doing so could validly be characterized as "p-hacking" or "data snooping." $\endgroup$ – whuber Jun 29 '17 at 14:52
  • $\begingroup$ So your short answer to the question is "No, you cannot use the coefficients although they are significant if the regression f-test is insignificant". $\endgroup$ – Eric Jun 29 '17 at 15:00
  • $\begingroup$ If you need a short answer: No. $\endgroup$ – Mark White Jun 29 '17 at 15:01
  • $\begingroup$ While this sequential testing procedure is used sometimes, there are situations where it seems quite rubbish. E.g. when confirming the result of a two-sample comparison by running a linear model with potential confounders. It all depends on the goal of your analysis. $\endgroup$ – Michael M Jun 29 '17 at 15:02
  • $\begingroup$ Then if the f-test is insignificant but coefficients are significant, I understand I cannot use these coefficients. But if I still need to report this regression, is it normal to show regression with insignificant f-test result but with significant coefficients although I do not interpret them at all? $\endgroup$ – Eric Jun 29 '17 at 15:15
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F test is the first thing to do in regression after identifying the features and if it fails then it means that none of the predictors/features for the given degrees of freedom is significant, so stop there. Now, I get this question a lot that if we have any single significant predictor present in the model then why bother an $F$ test. The answer here is that suppose we have $100$ predictors and we do not do an $F$ test and just rely on $t$ test to see if any predictor is significant using p-value $\le 0.05$ and then proceed with those which are significant, then there is a serious problem with this approach which is, that by chance we will have $5$ out of $100$ predictors which won't be significant but will have low p value due to chance alone. This is the reason an $F$ test is important and the first criteria to fit a model.

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