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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?

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    $\begingroup$ 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. $\endgroup$
    – Jake Brown
    Commented May 18, 2021 at 2:50
  • $\begingroup$ 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 $\alpha$. $\endgroup$
    – Amaan M
    Commented May 18, 2021 at 3:49
  • $\begingroup$ But is the p value for the regression coefficient the same as the p value for the correlation? $\endgroup$ Commented May 18, 2021 at 3:50

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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.

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