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I am performing longitudinal mixed modelling. What does it mean when a regressor (predictor) alone is not significant for the model (P>0.05) but when I add another predictor, which turns out to be non-significant, the first one is suddenly significant?: e.g.

In the following model age is not significant

Test_performance ~ age + sex, random=(~1 | subjects)

But when I add another variable (e.g. brain size), age is suddenly significant, event though brain size is not.

How is this possible and what does it mean?

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This is usually due to multicollinearity. The whole point of the null is that it is the true model. You can't add a variable or your significance tests are meaningless. You should have performed a correction if you were going to continually add variables until you found significance. Most likely the two predictors are highly correlated.

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  • $\begingroup$ One question: Do you mean it is a wrongful approach to create several models with an increasing amount of variables? $\endgroup$ Commented Dec 8, 2016 at 14:34
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    $\begingroup$ The fact that the variables can be correlated means that the inclusion of one lessens the importance of one. This is a variable selection issue. Systematic procedures for comparing models such as step-wise selection are often used. Then selection criteria such as Akaike's information criterion are used to pick the final model. $\endgroup$ Commented Dec 8, 2016 at 14:39
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    $\begingroup$ I think user25459 is saying exactly that @CoffeSurfer. Including too many variables can hurt the accuracy of prediction. $\endgroup$ Commented Dec 8, 2016 at 14:42
  • $\begingroup$ This is a FAQ. Although collinearity is a possible explanation, it's not the only one. $\endgroup$
    – whuber
    Commented Dec 8, 2016 at 14:59
  • $\begingroup$ it should be noted that step-wise procedures can be highly biased. You would be better off reading a bit about variable selection procedures before continuing $\endgroup$
    – bdeonovic
    Commented Dec 8, 2016 at 18:20

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