Imagine a regression with 3 predictors and one dependent variable with a sample size over 400. Only one predictor has a positive significant effect, p value equals to .02 aprox. R-squared is low (less than .03) and the R-squared IC 95% includes negative values: I have read that this means my model has worse fit than a horizontal line. Therefore, how to interpret this result? If my model is poor, how can I explain the significant effect of one predictor?
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To check for the significance of predictors, i.e. if at least one of your predictors is useful to predict the response, you should look at the F-statistic. In a multiple regression case, you should test whether all your coefficients are zero.
$ H_o: \beta_1 = \beta_2 = ... = \beta_p = 0$
$H_a:$ at least one beta is non-zero.
You can test this with the F-statistic because it adjusts for the number of possible predictors we have. This is not the case if you look at the individual p values. In the latter case, it is possible that one predictor has a p-value below 0.05 by chance, especially if the number of predictors is large. This is because, loosely speaking, by definition around 5 out of 100 predictors will have a p-value below 0.05.
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$\begingroup$ Thanks for your answer. In my case I have few predicotrs (probable relevant predicto was not included), so do you think that R-squared IC 95% implies a poor model and therefore, the significant effect of one predictor was by chance? $\endgroup$ Apr 27, 2021 at 0:18
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$\begingroup$ Yes, I think that's a reasonable explanation. $\endgroup$ Apr 27, 2021 at 16:51