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?
Solved – In linear regression, is the $R^2$ value enough to assess whether the relationship between the independent and dependent variable is linear
diagnosticr-squaredregression
Related Solutions
Correlation refers to linear dependence. However, you can have non-linear dependencies. Here is the standard plot from the Wikipedia page on correlation and linear dependence:
The bivariate distributions in the bottom row all have zero correlation, but clear patterns. Thus, although standard (OLS) regression methods enforce a zero correlation between the residuals and the predicted values, there can still be a detectable pattern that indicates the functional form is mis-specified. Consider the following plot, taken from this CV question: How do I interpret this fitted vs residuals plot? As I argue in my answer there, it provides evidence of mis-specified functional form.
Best Answer
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