How do I find if intercept is significantly different from zero? (using R)

Question

This is probably simple question, but after searching for a bit I didn't find (or alternatively understand) an answer. (I'm sure a question like this has been posted again and again, but it is quite hard to find the exact thing when your knowledge of the subject is very limited.)

I'm trying to figure how I can find if intercept is significantly different from 0 (or any number).

So I have some made-up numbers:

Coefficients:    
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.005         0.001   -2     0.08 .  
mkt          1.2           0.03    50     <2e-16 ***

I do remember that the Pr(>|t|) refers to the significance, but significance of what?

Answer 1

The column Pr(>|t|) refers to a test that the corresponding coefficient is zero. In your case it is greater than 0.05 and so there is insufficient evidence that it is non-zero. Having said all that your output does look rather unusual as a unit change in your other predictor leads to a massive change in the outcome. I think you should investigate the output of plot(model) to see if there is anything unusual to be seen there.

Answer 2

Mathematically, you (and R) are testing:

The null hypothesis: $H_{0}: \beta_{0} = 0$

vs.

The alternative hypothesis: $H_{0}: \beta_{0} \neq 0$

When we say the word significant what we really mean is significantly different from zero.

1) Something that is said to be significant, is significantly different from zero.
2) Something that is said to be not significant, is not significantly different from zero.

---

In 1) there is enough evidence in the data to reject the null hypothesis. (You reject the statement that says $H_{0}: \beta_{0} = 0$).

In 2) there is not enough evidence in the data to reject the null hypothesis. Hence, you say that you fail to reject the null hypothesis.

NOTE: We never say that we "accept" the alternative hypothesis, your conclusion should be limited to reject or fail to reject the null hypothesis.

Finally, more details of your specific problem can be found in the section "2.2 Inferences Concerning $\beta_{0}$" of the book Applied Linear Statistical Models, 5th Ed, by Kutner, Nachtsheim, Neter and Li.