Can we use Correlation coeffiecients in non linear case

Question

So I am having a data of several variables (one is a dependent variable and the others are independent variables). I am not sure if the relation between each of these independent variables and the dependent variable, whether this relation is linear or not.

Is it possible (more precisely is it significant) to use Partial Correlation Coefficients (PCC) to check the correlation between each of the independent vairables and the dependent variable? For me I am not sure that we can use PCC in non linear cases?

Answer 1

Did you try some plots? Show us. And, partial correlations correspond to correlations of residuals, see https://en.wikipedia.org/wiki/Partial_correlation. So, the corresponding plots would be plots of residuals, which very well could be helpful.

More details. With variables $x,y,z$ we want to "partial out" $z$. Then estimate the regressions $$ x=\alpha_0+\alpha_1 z + \epsilon_x \\ y=\beta_0 +\beta_1 z + \epsilon_y $$ with residuals $\hat{x}=x-\hat{\alpha_0} -\hat{\alpha_1} z$ and $ \hat{y}=y-\hat{\beta_0} -\hat{\beta_1} z$. Then the partial correlation is $$ \text{cor}(\hat{x}, \hat{y} ) $$ and you can make a plot of this residuals, which will show any nonlinearities. All of this still makes sense if the linear regressions above is replaced by nonlinear regressions.

For an example see the video: https://www.youtube.com/watch?v=gLenrMgrlc8