Does the response depend on X?

In a normal linear model, the response has a distribution whose mean, µy, depends linearly on the explanatory variable,

Y  ~  normaly , σ)

If the slope parameter, β1, is zero, then the response has a normal distribution that does not depend on X.

Y   ~   normal0 ,  σ)

If the slope is zero, there is no association between Y and X.

In experimental data where lurking variables have been avoided, we can further say that X does not affect Y.

Hypothesis test

This can be tested formally with a hypothesis test for whether β1 is zero. The methodology is similar to that for tests about a population mean or proportion and will be described in the rest of this section.

It is important to remember that a single data set can provide evidence about whether β1 = 0, but it usually does not allow a definite conclusion to be reached.

Model for the effect of a drug in reducing heart rate

We consider linear models for the effect of a drug (mg) in lowering the heart rate of adults. The response is the reduction in heart rate (before minus after receiving the drug) in beats per minute.

Adjust the model slope to β1 = 3.0 (drag the slider)
If this model is appropriate, heart rate will decrease on average by 3 beats per minute for every extra mg of the drug that is administered.
Adjust the model slope to β1 = 0.0
If this model is appropriate, the distribution of heart rate reduction is the same for all doses of the drug.

Click the button at the top right to rotate to a display of density against y. Drag the slider for X to verify that the response distribution does not depend on the dose of drug — the drug has no effect on reducing heart rate when β1 = 0.

Testing whether β1 is zero therefore tests whether the drug has any effect.

(Note that the models all predict a positive reduction in heart rate, even with a zero dose of the drug. The mean response with a zero dose, β1, is a placebo effect — a measure of the psychological effect of knowing that a drug has been administered.)


The diagram below shows the same range of models, but allows us to see typical data from the models.

The slider again allows the model's slope to be altered. Change the slope to zero.

Click Take sample a few times to see typical experimental data from the model. (There are 4 measurements at each of 4 levels of the drug.)

The least squares line usually has non-zero slope, so a single data set cannot immediately tell you whether β1 is zero.