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I am having trouble isolating these two concepts.

If the sample correlation, using Pearson’s r, is 0 and statistically significant then we can conclude the data are not linearly related - there is no correlation.

If the sample correlation, whatever it may be, is statistically insignificant then we fail to reject the null hypothesis that the population correlation is 0. So essentially we are saying the population correlation is 0.

How are these two observation different?

user3138766
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    You should always specify X in "statistically significantly different from X." Often X is zero, but that's not the only option. Thus statement #1 is hard to parse under the default interpretation. Your second statement is not how hypothesis testing works. Perhaps this example will give you some intuition and let you see the problem with statement #2 and perhaps #1. – dimitriy May 18 '21 at 00:57
  • My mistake on the last part. I ran a regression between two variables and got a correlation value. The p value associated with the regression coefficient is 0.54. Does this simply mean the correlation my regression spit out cannot be relied on? How would you describe it? – user3138766 May 18 '21 at 01:08
  • Regression doesn't directly estimate a correlation, it estimates coefficients. In the simplest one variable case, the slope coefficient is related to the correlation. I would suggest that you edit your question to add the new info and the regression output. – dimitriy May 18 '21 at 01:22
  • I rewrote the question as a different question. I am not too good with this site. Would you please flag this question for deletion? – user3138766 May 18 '21 at 01:27

1 Answers1

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I see a two errors.

First, any use of "statistically significant" refers to a null hypothesis. Therefore, when the first line refers to a correlation that is statistically significant yet equal to zero, the null hypothesis must not be that the correlation is zero. However, then rejecting a null hypothesis that the correlation equals some other value does not give evidence of zero correlation. For instance, if the null hypothesis is that the correlation is $0.5$, rejecting could mean that correlation is $0.4$ or $-0.8$. What could be meant here is that an equivalence test was performed and gave statistically significant evidence that the correlation is, in some sense, "close" to zero. Then it would be reasonable to conclude that there is not a practically significant linear relationship between the two variables.

Second, failure to reject a null hypothesis is not the same as confirming the null hypothesis. Therefore, just because you wind up with a correlation that is not statistically significant (that is, for a null hypothesis of zero) does not mean that the variables lack a correlation.

With this in mind, the two statements seem completely compatible. In the first case, a significant result from an equivalence test shows the correlation to be close enough to zero that we conclude there to be no practically important linear relationship. In the second case, we have no strong evidence of a linear relationship, so we cannot conclude that there exists a linear relationship.

Dave
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