## 35.5 Regression for understanding

The regression equation can be used to *understand* the relationship
between the two variables.
Consider again the red deer regression equation:

\[\begin{equation} \hat{y} = 4.398 - 0.181 x. \tag{35.1} \end{equation}\] What does it tell us about the relationship between \(x\) and \(y\)?

### 35.5.1 The meaning of \(b_0\)

\(b_0\) is the *predicted* value of \(y\) when \(x=0\).
Equation (35.1) predicts a molar weight of \(4.398\) for
a deer zero years of age,
which is likely to be nonsense:
it is *extrapolating*
beyond the data
(the youngest deer in the sample is aged 4.4 years).

*relationship*between the two variables.

### 35.5.2 The meaning of \(b_1\)

\(b_1\) tells us how much the value of \(y\) changes
(on average)
when the value of \(x\) *increase* by one.
For the red-deer data,
\(b_1\) tells us how much the
molar weight changes
(on average)
when age increases by one year.

Each extra year of age is associated with a change of
\(-0.181\) grams in molar weight;
that is,
a *decrease* in molar weight by a mean of \(0.181\).
The molars of some individual deer will lose more weight than this in some years,
and some will lose less weight than this in some years…
the value is a *mean* weight loss per year.

To demonstrate,
for \(x=10\), \(y\) is predicted to be \(\hat{y}= 2.588\).
For deer one year older than this (i.e. \(x=11\)) we predict
\(y\) to be \(b_1 = -0.181\) higher
(or,
equivalently, 0.181 *lower*).
That is,
we would predict
\(\hat{y}= 2.588 - 0.181 = 2.407\).
(This is the same prediction made by using
\(x=11\) in Eq. (35.1).)

If the value of \(b_1\) is *positive*,
then the predicted values of \(y\) *increase* as the values of \(x\) *increase*.

*negative*, then the predicted values of \(y\)

*decrease*as the values of \(x\)

*increase*.

This interpretation of \(b_1\) explains the relationship: Each extra year of age reduces the weight of the molars by 0.181 grams, on average, in male red deer. The units of the slope are the units of the response variable divided by the units of the explanatory variable (so in the deer example, the slope is \(-0.181\) grams per year).

Observe what happens if the slope is *zero*.
Since \(b_1\) is the change in \(y\) (on average) when
\(x\) increase by one,
\(b_1 = 0\) means that the
value of \(y\) changes by *zero* if the value of \(x\) changes by one.
In other words,
if the value of \(x\) changes,
the predicted value of \(y\) doesn’t change.
This is equivalent to saying that
there is *no relationship* between the variables.
(We would also find \(r=0\).)

*no linear relationship*between \(x\) and \(y\). In this case, the correlation coefficient is also zero.