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

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?

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 • +1 To be contrary or irksome, one might argue that in some sense the last three are all "equally" non-linear, but the contrast with the first one (which is a classic linear scatterplot) speaks volumes. – whuber Oct 4 '11 at 23:09

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 Is $R^2$ 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.

• So is looking at a scatterplot fine (or looking at a residual plot)? – question Oct 4 '11 at 22:09
• @question Yes, a plot of residual vs fit can tell you a lot. – whuber Oct 4 '11 at 22:11
• If there is a pattern in the residuals vs fit plot.....then there is nonlinearity? – question Oct 4 '11 at 22:14
• @question If it (a) is a pattern in which the mean residual varies and (b) it would be unacceptable to treat that pattern as random, then--practically by definition--there is nonlinearity. Some patterns do not indicate lack of linearity but suggest other phenomena such as heteroscedasticity, outliers, or high-leverage points, so we shouldn't assume all deviations from randomness are evidence of nonlinearity. – whuber Oct 4 '11 at 22:16

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-like pseudocode. The test is implemented in R as resettest in lmtest package, and in Stata, as estat ovtest after regress.