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I have a question about interpreting the Pearson Correlation Coefficient results. As you can see in the figure below, the r-value is close to 0 (no correlation) but the p-value is 0.98 (no confidence/no significance). So, it is correct to say "no correlation with high significance" means "either positive or negative correlation"?

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  • $\begingroup$ A high $p$-value means weak evidence, so "high significance" would be misleading. $\endgroup$
    – Frans Rodenburg
    May 26, 2021 at 11:58
  • $\begingroup$ OK that means there is weak evidence that there is no correlation. Is that true? That means there might be some correlations (positive or negative). $\endgroup$
    – mahmood
    May 26, 2021 at 12:02
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    $\begingroup$ The null-hypothesis is that there is no linear correlation. Your estimate says that it is $0.01$ and your $p$-value says that you have extremely weak evidence against the null. $\endgroup$
    – Frans Rodenburg
    May 26, 2021 at 12:03
  • $\begingroup$ @FransRodenburg: May I know what are they grey curves? I expect that in the middle where the curves are close to each other, the correlation is higher (better evidences), but that figure doesn't show that. $\endgroup$
    – mahmood
    May 26, 2021 at 13:42

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If you did what I think you did, that is estimated a (Pearson) correlation coefficient and performed a null hypothesis test, then the results are telling you that the correlation coefficient is equal to $0.01$ and that the p-value is equal to $0.98$.

The p-value is referring to the null hypothesis (which you are trying to reject), which is that the correlation coefficient is equal to $0$, the alternative being that the correlation coefficient is not equal to $0$ (for a two-sided test). Since you did not reject your null hypothesis (assuming an $\alpha<0.98$, usually $0.05$), because your p-value is equal to $0.98$, then you keep your null hypothesis of no correlation (the coefficient being equal to $0$), despite the estimated coefficient of $0.01$.

Note: your data does not really appear to be linear in the first place, so a Pearson correlation coefficient is probably not appropriate.

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  • $\begingroup$ Sorry I got confused. As a I read the texts, small p-values (<0.05) shows the significance. Therefore, R=0.8&P=0.01 means strong positive correlation, R=0.01&P=0.01 means strong no correlation and R=-0.8&P=0.01 means strong negative correlation. That means R=0.01&P=0.8 means weak no correlation. Isn't that right? $\endgroup$
    – mahmood
    May 26, 2021 at 12:01
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    $\begingroup$ @mahmood That is incorrect. A small p-value could indicate strong evidence of a weak (but nonzero) correlation. Try this in R: set.seed(2021); x <- rnorm(10000); y <- x + rnorm(10000, 0, 50); cor.test(x, y). I get a small p-value $<0.01$ but a small correlation coefficient of $0.026$. What happens is that the large sample size of $10,000$ gives the test the sensitivity to detect that the correlation is not quite zero. $\endgroup$
    – Dave
    May 26, 2021 at 12:13
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I'm going to advise you to dispense with the specifics of the math for a moment. Look at your plot. Do those data look linear? At all?

If your answer to that question is "no," then you can stop trying to interpret the results of your linear regression. The results you've gotten are consistent with the regression being uninterpretable - but you don't need the mathematical results to see that this is an inappropriate model that you should abandon. All you need is to glance at the graph.

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