# Linear regression correlation coefficient significance vs. importance?

I ran a simple linear regression in Excel between variables x and y.

Pearson’s r is 0.3, and the p-value for the x coefficient is 0.5 - far above my alpha of 5%. Therefore, I conclude the correlation is “insignificant.”

Here’s my confusion - the correlation coefficient for my model is 0.3 so my sample data are in need correlated in that sense. It’s a weak correlation, but a correlation at that. We deem is statistically insignificant because the p-value is too high, indicating we failed to reject the null hypothesis which I assume is “the correlation between population x and population y is 0.”

Doesn’t the 0.3 correlation have some use? I am looking at financial data, so the correlation shows that my sample data are not strongly correlated. My question is this: since the correlation is insignificant, does this mean the 0.3 correlation must be completely disregarded? Is it of no use? If it is of some use, how?

• If your $p$ value is $0.5$ then it means theres a 50% chance you would see the data you obtained if the null hypothesis were true. The 0.3 correlation could simply be due to chance, and not due to any "real" correlation. Commented May 18, 2021 at 2:50
• A sample correlation coefficient has a confidence interval in the same way a regression coefficient does. We can fail to reject the null hypothesis that $\rho = 0$ if the $p$-value for $r$ is larger than our $\alpha$ (in this case, $0.05$). It's like seeing a positive $\hat{\beta}$ that has a confidence interval containing $0$ - there's some evidence of a positive relationship between the variables, but that doesn't mean that it's a statistically significant one based on your $\alpha$. Commented May 18, 2021 at 3:49
• But is the p value for the regression coefficient the same as the p value for the correlation? Commented May 18, 2021 at 3:50

My question is this: since the correlation is insignificant, does this mean the 0.3 correlation must be completely disregarded?

The problem with p.value is that it dependent on the sample size $$n$$, i.e., the true correlation may be nonzero, however, if you have a noisy data and small $$n$$, you may fail to reject the null hypothesis. On the other hand, if your sample size is large enough even with a noisy data, the fact that the p.value is $$0.5$$, is a good indication that there is no linear correlation between you variables. A good reality check may be done by performing nonparametric Bootstrap. I.e., sample $$N$$ times $$n$$ pairs of data with replacement, and inspect how the estimated $$r$$ changes. If your $$r$$ falls frequently on both sides of the sign (i.e., sometimes positive and sometimes negative) then you may conclude that your point estimator of $$0.3$$ is merely due to chance and have no practical use.