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 price on sales of a New Zealand wine

We consider linear models for how the price of a popular New Zealand cabernet sauvignon red wine affects its sales in a supermarket chain, measured as a proportion of total red wine sales in a week. The relationship between price and sales will be nonlinear at high prices, but is expected to be reasonably linear within a price range of $12 to $20 per bottle.

Adjust the model slope to β1 = -0.03 (drag the slider)
If this model is appropriate, the proportion of total sales will decrease on average by 0.03 (3 percent) for every extra dollar in price.
Adjust the model slope to β1 = 0.0
If this model is appropriate, the distribution of sales is the same for prices between $12 and $20 per bottle.

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 price — the price has no effect on sales when β1 = 0.

Testing whether β1 is zero therefore tests whether price has any effect on sales.


The diagram below shows the same range of models, but allows us to see typical data from the models. These are data that might be observed if each of the 4 prices were tried in the supermarket chain for 4 separate weeks, randomised over a 16-week period.

The slider again allows the model's slope to be altered. Change the slope to zero (so that price has no effect on sales).

Click Take sample a few times to see typical experimental data from the model.

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