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In my regression, I have a predictor (measured in proportions) with very low variance (.002). It had no significant effect on the outcome due to a very high standard error of the regression coefficient.

When reporting on the effect in a paper, is it right to say there was no effect of my predictor in my study? Even though I know the lack of effect is most likely due to the low variance, and I add that thought in the paper? It feels like I might mislead the non-attentive readers, or I am reporting useless stuff.

Maybe it is best practice to discard this predictor from the model, and note in the paper that it was impossible to test this effect due to the data limitations?

And as extra I wanted to ask if it is possible to get a lower variance when taking the log of the values? I observed this with my variable.

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    $\begingroup$ Whether .002 is "low" depends on the scaling of the regressor. If you measure something in meters, its variance may be 2000, while if you recode the exact same data to kilometers, the variance will suddenly be .0002. There is really no good answer to your question without knowing the context and how your regressor could have an effect. $\endgroup$ – Stephan Kolassa Jan 5 '13 at 15:15
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    $\begingroup$ @StephanKolassa is right about the scaling issue, but if the variance really is very low in sensible terms, then I think you should delete it from the model because it can't make a difference. The ultimate would be a variance of 0 (that is, all subjects have the same value) and then the coefficient is really meaningless. $\endgroup$ – Peter Flom Jan 5 '13 at 15:18
  • $\begingroup$ You are right! I edited, stating the used scale is a proportion. $\endgroup$ – Marloes Jan 5 '13 at 15:21
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    $\begingroup$ The decision to exclude the variable should have been made before looking at the statistical significance of the variable. Otherwise, biased standard errors of all the coefficients will result. $\endgroup$ – Frank Harrell Jan 5 '13 at 18:31
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    $\begingroup$ In general, excluding variables because of observed $\hat{\beta}$ or $P$-values leads to multiple biases in the model. Including the variable does not lead to a bias. If on the other hand you excluded a variable without unblinding yourself to $Y$, the standard errors will not be biased low. $\endgroup$ – Frank Harrell Jan 5 '13 at 21:59
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@StephanKolassa is right about the scaling issue, but if the variance really is very low in sensible terms, then I think you should delete it from the model because it can't make a difference. The ultimate would be a variance of 0 (that is, all subjects have the same value) and then the coefficient is really meaningless.

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