In linear regression, is the $R^2$ value enough to assess whether the relationship between the independent and dependent variable is linear?

Question

In linear regression, is the $R^2$ value enough to assess whether the relationship between the independent and dependent variable is linear? It gives the amount of variability in the dependent variable explained by the independent variable. I know that you can plot residuals versus the x value or residuals versus the y value and see if there is a pattern (if there is a pattern then the relationship is not linear). But doesn't the correlation coefficient give enough information about linearity?

Answer 1

If you look at Anscombe's quartet you can see examples of linear with noise, linear with outliers and non-linear sets of data with the same $r^2$, means and variances.

This image is from the Wikipedia article

Answer 2

Usually not. The model

$$y_i = \beta + \varepsilon_i,$$

$\varepsilon \sim \text{iid}$, $\mathbb{E}[\varepsilon]=0$ for the relation between $(y_i)$ and $(x_i)$ is perfectly linear, yet has an $r^2$ of zero.

For other examples of what $r^2$ does not say about linearity, see the illustrations in my reply at R-squared: useful or dangerous?.

Linearity is generally assessed by goodness of fit testing; for instance, by including additional terms in a follow-on regression and testing whether they are both significant and important in the application. One person's nonlinearity is just another person's randomness, so there's no omnibus method. Nevertheless, usually $r^2$ is just too crude.

Answer 3

In addition to the above answers, a commonly used (in econometrics) test for general regression nonlinearity is Ramsey's RESET test. Suppose you ran your main regression and obtained residuals $\hat\epsilon_i$ and fitted values $\hat y_i$ in it. Then RESET test is the test of the overall significance in an auxiliary regression of $\hat\epsilon_i$ on powers of $\hat y_i$. From regression geometry, we already know that $\hat\epsilon_i$ are orthogonal to the zeroth and the first power of $\hat y_i$, so it makes sense to run it as $\hat\epsilon_i \sim \hat y_i^2 + \hat y_i^3 + \ldots$, in R-llike pseudocode. The test is implemented in R as resettest in lmtest package, and in Stata, as estat ovtest after regress.