# statistical tests for null hypothesis - what if model is non linear?

I am reading the "An Introduction to Statistical Learning" (Gareth James & alii, Springer) as a primer to machine learning.

I am reading the part in linear regressors, and learnt there are different tests for measuring correlations and significance of correlations between predictors (also named variables)- under the assumption that the model may be linear.

What about if the relationship between variables is (or assumed to be) non-linear ?

I also read that anyway many linear models concepts underpins a lot of statistical models.

Can you explain which could be a good practice when one has to model a context that he/she does not know yet?

• Shall we test linearity because it is simpler , and if not found, we'll proceed with other tests ?

• What about though, for testing correlations between variables in non-linear models ?

• What do you mean by the model is non-linear? Do you mean if the relationship between your variables in non-linear? Depends on what kind of non-linearity we are talking about, can you be specific? Dec 10, 2021 at 7:42
• thanks! edited. I don't know what kind of non-linearity. My question is about using T-test, R^2, P-values, Pearson... all tests for correlations between variables under the assumption that correlation is linear. But are these tests relevant if the correlation is thought to be non-linear ? I am looking for answer that let me appreciate how to typically approach a problem whose context is unknown. Dec 10, 2021 at 14:19
• Pearson correlation still cariies information even if the input-output mapping is non linear. Eg assume $y = x^2$, one can use pearson correlation between $y$ and $x^2$ (since this is now a linear relation). But even Pearson correlation between $y$ and $x$ might still convey information Dec 10, 2021 at 19:14