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Why are variables statistically insignificant?
If the p-value for a variable is less than your significance level, your sample data provide enough evidence to reject the null hypothesis for the entire population. The regression output example below shows that the South and North predictor variables are statistically significant because their p-values equal 0.000.
Should I include covariates?
Train the model with the covariate and without using the training data. Whichever model does a better job predicting in the test data should be used. Adding covariates reduces the bias in your predictions, but increases the variance. Out of sample fit is the judge of this tradeoff.
How do you report statistically insignificant results?
A more appropriate way to report non-significant results is to report the observed differences (the effect size) along with the p-value and then carefully highlight which results were predicted to be different.
Are there reasons to keep or drop covariates?
There are reasons to keep covariates and reasons to drop covariates. Statistical significance should not be a key factor, in the vast majority of cases. Covariates may be of such substantive importance that they have to be there. The effect size of a covariate may be high, even if it is not significant.
Is there a reason to remove insignificant predictors?
The long answer is “yes”. There are few reasons to remove insignificant predictors and many reasons not to. As far as interpreting them you do so ignoring the -value just as you might interpret other predictors: with confidence intervals for effects over interesting ranges of the predictor.
Are there any non-significant covariates that should be taken out?
There is nothing in the answer to that question that suggests that non-significant covariates be taken out, though, so right now I am inclined to believe that they should stay in.
Why is the effect size of a covariate important?
Covariates may be of such substantive importance that they have to be there. The effect size of a covariate may be high, even if it is not significant. The covariate may affect other aspects of the model. The covariate may be a part of how your hypothesis was worded.