quadratic relationship

Question

Detection of linear relationship is possible with correlation coefficient. If absolute value of correlation coefficient is 1, then the relationship is linear. Is there any way for detecting quadratic relationship? is there any way for detecting relationship between more than 2 variables?

Answer 1

First fit the line $y=ax+b$ by least squares; then look for a correlation between $(x-\bar x)^2$ and the residuals, where $\bar x$ is the average of the $x$-values.

And remember, how big a correlation is statistically significant depends on the sample size, so use a t-test or an F-test for that.

And remember that fitting a polynomial by least squares is an instance of linear regression: Why is polynomial regression considered a kind of linear regression?

Later addendum in response to a comment:

First fit a hyperplane $y = a_1 x_1 + \cdots + a_k x_k$ by least squares. Then do an anova with the residuals from that fit as the response variable, and as predictors the variables $x_i - \bar x_i$, where $\bar x_i$ is the average of the values of $\bar x_i$ in all cases. Do an F-test with the sum of square sdue to all of the predictors in that anova as the numerator sum of squares, and the sum of square of the residuals from the anova as the denominator sum of squares.