# Interpreting significance when regression coefficients are combined

Suppose we have the following regression, with Sun=0 for partial sun and Sun=1 for full sun:

Plant Height = a0 + a1*Bacteria + a2*Sun + a3*Bacteria*Sun


And here is the sample output from R:

According to this post, for plants with full sun (Sun=1), the effects of Bacteria is: -0.08346 + 0.03368 = -0.04978, meaning that for two plants with full sun, the plant with one unit more bacteria will be 0.04978cm shorter.

So my question is, is this difference significant?

Since we get the above conclusion by combining the coefficients of Bacteria and Bacteria*Sun, but as in the output, the first coefficient has a significant p-value=0.00214 while the other coefficient for the interaction term has a large p-value=0.34887.

So when we add these coefficients, how should we interpret its significance?

What you should do is conduct a partial F-test. The null hypothesis for a 'full' F-test tests whether all regression coefficients are equal to zero. In your case, this null is $H_{0}: a_{1} = a_{2} = a_{3} = 0$. R's lm should print this for you.

A partial F-test is when you wish to test whether a subset of the regression coefficients is equal to zero. In your case, you wish to test $H_{0}: a_{1} = a_{3} = 0$. Instructions to implement a partial F-test in R and a brief discussion of regression hypothesis testing can be found here.

Instead of trying to "guess" based on the results of the two students t tests for these coefficients you should test both of them jointly. Doing so helps mitigate your probability of committing type 2 errors (wikipedia).

• In R, you can use the code anova(m1,m2), where m1 is the original model, and m2 is the model without $a_1,a_3$. – Drew75 Mar 6 '14 at 7:11
• I think the null is $H_0: a_1+a_3=0$. – jaradniemi Mar 6 '14 at 21:06

The easiest way to answer this question is to create a new variable PartialSun=1-Sun and rerun the regression with PartialSun in place of Sun. Then, the coefficient for Bacteria will be directly interpretable as the expected effect of bacteria for full sun plants. The output from R, will provide the pvalue.