Setting their causal assumptions to the side (which aren't well explained in this part of the book), the correlation between radio spending and sales is more than double that of newspaper spending based on the scatters on the raw data from ISLR, the R equivalent to ISLP (James et al., 2013).
Looking at the strong associations of all indicators except for newspaper spending, this leads one to believe that the effect of newspaper spending was suppressed after other stronger variables entered the model. Suppression is when an independent variable has a weak correlation with the outcome variable, is correlated with another independent variable, and increases the $R^2$ (variance explained). It can also occur when a regression produces a negative coefficient despite the fact that all correlations between the predictor and outcome variables are non-negative, which I show as the case in the regression fitting below (Friedman & Wall, 2005). In terms of what the regression is doing, we can reconsider a simple case of two predictors algebraically as such (lecture notes here):
$$
\beta_1 = \frac{r_{y_1} - r_{y_2} r_{12}}{1 - r^2_{12}}
$$
Where the $\beta_1$ in some part functions as a measure of the partial associations:
One can think of the above equation for the standardized regression
coefficient for the first predictor, $\beta_1$, as a correction to the
correlation between $x_1$ and $y$ (i.e., $r_{y_1}$), where the second
term in the numerator, $r_{y_2} r_{12}$, is subtracted. Assuming for a
moment that the correlation $r_{y_1}$ is positive, the value of
$\beta_1$ will tend to increase if either $r_{y_1}$ or $r_{12}$ are
negative, because subtraction of a negative quantity is the same as
adding that quantity. In such a case, $\beta_1$ will be larger than
$r_{y_1}$, which is not the expected direction. As a second example,
assuming again that $r_{y_1}$ is positive, but, this time, both
$r_{y_2}$ and $r_{12}$ are both positive. If their product, $r_{y_2}
r_{12}$, happens to be larger than $r_{y_1}$, then $\beta_1$ will be
negative, in essence changing an initial positive association with the
dependent variable into a negative (partial) association. Note also
that the correlation between $x_1$ and $x_2$ can make a difference in
the estimate of $\beta_1$, because of its appearance in the
denominator of the equation. Larger values of $r^2_{12}$ will tend to
increase the value of $\beta_1$.
Therefore, based on the correlations between the predictors with each other and the outcome, the regression will "figure out" which one is more predictive of the outcome and suppress the other, as it to a degree obscures the primary effects. This seems to be the case when you fit this in R:
Call:
lm(formula = Sales ~ TV + Newspaper + Radio, data = Advertising)
Residuals:
Min 1Q Median 3Q Max
-8.8277 -0.8908 0.2418 1.1893 2.8292
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.938889 0.311908 9.422 <2e-16 ***
TV 0.045765 0.001395 32.809 <2e-16 ***
Newspaper -0.001037 0.005871 -0.177 0.86
Radio 0.188530 0.008611 21.893 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.686 on 196 degrees of freedom
Multiple R-squared: 0.8972, Adjusted R-squared: 0.8956
F-statistic: 570.3 on 3 and 196 DF, p-value: < 2.2e-16
We can see that $\beta_{\text{Newspaper}} = -0.001$ despite that it's correlation is $r = .23$. This doesn't happen with TV spending because it is about as strong at predicting sales as the radio spending. Much of this is due to the other strong sales indicators "absorbing" the explained variance of the outcome and leaving less for the newspaper spending to determine, even if the overall $R^2$ may improve with inclusion of the suppressed effect. Another example of sign-switching from other effects can be found here.
Does that actually mean that newspaper spending has no effect on sales? No. Mathematically we can say less is predicted by newspaper spending after accounting for the other spending indicators. Causally, we can't infer this from purely observational data, of which any number of contaminating factors could be influencing the outcome.
References
- Friedman, L., & Wall, M. (2005). Graphical views of suppression and multicollinearity in multiple linear regression. The American Statistician, 59(2), 127–136. https://doi.org/10.1198/000313005X41337
- James, G. M., Witten, D., Hastie, T. J., & Tibshirani, R. (2013). An introduction to statistical learning: With applications in R (Corrected at 6th printing 2015). Springer Springer Science+Business Media.
