Hypothesis Tests in Regression

The MINITAB output of the regression command contains several useful hypothesis tests. Recall we have the model y = β0 + β1 x. Here is part of the output of the regression command for the Hubble data:

Predictor Coef SE Coef T P
Constant -40.78 83.44 -0.49 0.630
Distance 454.16 75.24 6.04 0.000

There are two hypothesis tests here:

Constant:

H0: β0 = 0 (intercept is zero)
Ha: β0 ≠ 0 (intercept is not zero)

If we fail to reject H0, we conclude that the constant is not statistically significantly different from 0 (at the sample size of the data set!).

If we reject H0, we conclude that the constant is statistically significantly different from 0.

Consequences:
We are fitting the model y=β0+β1x
If H0 is true then β0=0
so the model becomes y=β1x

this is called a no-intercept model.

One consequence of this model is that if x=0 then y=β1·0 = 0, so the point (0,0) is always on this line.

Note: The decision whether an intercept should be fit or not is best made based on the background of the data and whether if x=0 then y=0 makes sense.

Example Say we have data with x = Number of Hurricanes in a year and y = $ Amount of Damage done by the Hurricanes. Now if x=0 (there were no hurricanes) obviously y=0 (no damage), so a no-intercept model is appropriate (even if the corresponding hypothesis test says otherwise!)

Predictor:

H0: β1 = 0 (slope is zero, predictor has no relationship with response)
Ha: β1 ≠ 0 (slope is zero, predictor has some relationship with response)

Consequences:
our model is y = β0 + β1 x. If H0 is true then β1=0
so the model becomes y=β0

But there is no more predictor x here! So if we fail to reject H0 it means that the predictor has no statistically significant relationship with the response (at least not for the sample size of the dataset).
If we do reject H0 we conclude that there is a statistically significant relationship between predictor and the response y.
Note: in a simple regression model such as we have here this test is the same as the test for Pearson's correlation coefficient.

Hubble data:

Constant:

1) α=0.05
2) H0: β0 = 0 (intercept is zero)
3) Ha: β0 ≠ 0 (intercept is not zero)
4) p=0.630
5) p>α, so we fail to reject the null hypothesis, the constant is stat. consistent with 0 (at the sample size of the dataset)

Distance:

1) α=0.05
2) H0: β1 = 0 (no relationship between distance and velocity)
3) Ha: β1 ≠ 0 (some relationship between distance and velocity)
4) p=0.000
5) p<α, so we reject the null hypothesis, there is a statistically significant relationship between velocity and distance.

For more on these hypothesis tests see page 524 of the textbook.