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I understand that if there's an indication of lack of fit for a linear regression model (say, as indicated by a lack of fit F -test), then the assumptions underlying an F-test for significance of regression are violated, and that test should not be performed.

My question is, does this also apply to individual t -tests on each regressor, to test if a regressor is significant? In other words, if my model has lack of fit, will my individual t-tests for regressor significance tell me anything useful?

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  • $\begingroup$ If your model does an awful job of predicting, do you really care what it says is driving its predictions? Put another way, if someone says he can guess if a coin will land heads or tails, but they only get it right 50% of the time, do you care what he says he's considering about the humidity in the room where he's flipping the coin? $\endgroup$
    – Dave
    Aug 10, 2019 at 17:25
  • $\begingroup$ I don't agree with your analogy. Lack of fit and poor prediction are not the same thing. The canonical example: fit a straight line to a confined range of quadratic data, get very good predictions within the range, but there's lack of fit. Whether the model predicts "well" is based on many factors, such as intended purpose, that don't need to be considered to answer this question. $\endgroup$
    – Julian Drago
    Aug 10, 2019 at 17:38
  • $\begingroup$ That said, my motivation here is that even with lack of fit, there could still be a case made to not include regressors that don't contribute to the model. My specific question is whether individual t-tests can still be used in the case of lack of fit. $\endgroup$
    – Julian Drago
    Aug 10, 2019 at 17:39

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