Linear and non-linear regression analysis I'm currently reading Maths and Stats for Web Analytics and Conversion Optimisation by Himanshu Sharma and noticed the following regarding regression analysis:
"If there is no or weak linear relationship between two variables or in other words the correlation between the two variables is zero or weak then this relationship is not good enough to predict anything. Therefore there is no point in running regression analysis."
This strikes me as ignoring non-linear regression analysis. I could understand if the last sentence was "Therefore there is no point in running linear regression analysis" but the author excludes all forms of regression.
My question is, even if the R is low, if you chart the data and see a curved scatter plot, should you be looking to run non-linear regression analysis as opposed to scrapping analysis entirely?
It is implied the R calc is Pearson's.
 A: The statement is at best misleading and at worst wrong and you don't need to go to nonlinear regression to prove it wrong. Here is the statement again:

If there is no or weak linear relationship between two variables or in
  other words the correlation between the two variables is zero or weak
  then this relationship is not good enough to predict anything.
  Therefore there is no point in running regression analysis.

This ignores:


*

*Moderation effects

*Mediation

*Quadratic relationships (which are easily examined within linear regression)

*The fact that finding a small effect is often interesting and scientifically important.

A: First of: you are right, the statement by the authors is incorrect/incomplete. 
To see why, let's go through the quote bit by bit:

"If there is no or weak linear relationship between two variables or in other words the correlation between the two variables is zero or weak ... "

The key word here is linear. The correlation between two variables measures the strength of a linear relationship. So far, so good.

" ... then this relationship is not good enough to predict anything. Therefore there is no point in running regression analysis."

As you suspected, this is where the author implicitly assume that the only relationship we can predict is a linear relationship. This is certainly not true. 
Why is this not true? In OLS (ordinary least squares, normal linear regression), we assume that the conditional expectation $E(Y|X)$ has a linear functional form, i.e. $E(Y|X) = X\beta$. This functional form is the easiest to deal with and is surprisingly accurate in many situations. However, the conditional expectation $E(Y|X)$ can have any other functional form: $E(Y|X) = f(Y, X)$. As long as we know the functional form of $E(Y|X)$, there are ways to estimate regression coefficients with nonlinear regression techniques. If we know that $f(Y,X)$ is a highly nonlinear function, then the correlation coefficient between $X$ and $Y$ can be zero. So, despite the correlation coefficient to be zero, we can still predict useful values with nonlinear regression models. 
I guess, since nonlinear regression techniques are beyond the scope of the book, the author decided to ignore their existence in general. He should've been a bit more precise in his statements
