# Two variables significant independently, but only one in a model?

I want to see how temperature and precipitation correlate with number of fires. If i do correlations independently, both are statistically significant, r(temp)=0.5, p<0.05, r(precip)=-0.3, p<0.05.

If I do multiple linear regression, the overall model is significant as well as the temperature, but precipitation is not. Does it mean that temperature is a good predictor and precipitation is not a good predictor by itself, but it is a good predictor when temperature is also considered? VIF if < 2.

• It may rain less when it is hot leading to some confounding. Your sample may be small making inference unstable so that some small change turns a p=0.049 that could be published as truth into a uninteresting p=0.051. Or any of many other possible reasons. Commented Nov 18, 2015 at 19:17
• The data were calculated for a year on a weekly basis, so I have 52 observations (for each week number of fires and avg. temperature and precipitation). The correlation between precipitation and temperature is r=-0.3. Commented Nov 18, 2015 at 19:21

When doing a regression with multiple regressors, you consider that each of your independent variable has a direct impact on the dependend variable, controlling for other variables.

When assessing the significance of a model, you have two choices:

1. Univariate significance test
2. Multivariate significance test

Here, you use the univariate test and conclude that one is not a good predictor (t-test). Also, since you only have two variables, your multivariate test amounts to testing "Are all my coefficients zero ?" through a F-test (I suppose, since it's a linear model) and the answer of your test is "No, there is at least one coefficient that is not zero."

I generally advise to keep variables in the model even when they are not significant (well, if the p-value is 0.99, I safely remove it) and when you have enough degrees of freedom because they can reduce estimation bias in estimating the coefficient. Does your coefficient in front of temperature vary greatly when controlling for precipitations ? Are the coefficients statistically different from each other with/without precipitation ?

The point underlying the previous paragraph is this: what is better ? Having a biased estimator that is precisely estimated, or an unbiased estimator that is less precisely estimated ? I favor the second.

To sum up: if adding new regressors solves a bias problem, keep them even though they are not significant. If adding a new regressor does not add anything new (Log-likelihood ratio test), then discard it. My gut tells me that it solves a bias problem, given that the correlation between those two variables is sizeable.

• Thank you for your answer, even though it goes well over my comprehension level. I will have to do a lot more googling and reading. Commented Nov 19, 2015 at 12:25

I would interpret this as meaning that precipitation does not have a significant effect when controlling for temperature. Considering the complexity of climate data, though, you would want to make sure you know your dataset really well, and that you are controlling for other relevant factors, before you make strong assertions.