is there a test that compares the observed correlation to a certain value ? for instance : is my correlation equal to 0.8 (0.8 not being any population's correlation, it's just a value that is fixed) i'd like to know the statistic of the test (if there is any)
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1$\begingroup$ Does this answer your question? $\endgroup$– DayneCommented Nov 28, 2020 at 12:20
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$\begingroup$ unfortunately it doesn't $\endgroup$– math geekCommented Nov 28, 2020 at 12:39
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4$\begingroup$ Can you be more specific, how it doesn't? The answer there gives the (asymptotic) distribution of the a function of sample correlation coefficient for some assumed value of population correlation. This can be used to get the critical value for your hypothesis. $\endgroup$– DayneCommented Nov 28, 2020 at 12:43
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1$\begingroup$ Does this answer your question? Distribution of sample correlation $\endgroup$– mdeweyCommented Nov 28, 2020 at 13:29
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$\begingroup$ Yes, there's a standard test: compare the Fisher Z-transform of the correlation to the Z-transform of the value. See blogs.sas.com/content/iml/2017/09/20/… for one online resource. A fuller account can be found in standard textbooks. $\endgroup$– whuber ♦Commented Nov 29, 2020 at 22:57
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1 Answer
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Here is a path to an answer, where I first cite efforts in a regression setting discussed over six pages, for example, in this educational source: Non-Zero Null Tests for Simple Linear Regression.
So, how does this analysis of a regression case constitute an answer for a correlation test in general? Consider a special case of transformed values for Y and a single variable X, where the data was centered around the respective sample means and scale by their respective standard deviations.
Now, I claim we have an instance where the standardized regression coefficient are so-called beta coefficients. The latter, for the case of one explanatory variable, equates to the Pearson correlation coefficient. Supporting reference, for example, to quote:
Standardized β coefficients: This mini-lesson is to introduce the concept of standardized regression coefficients in R. A standardized regression coefficient is simply the β estimate from a regression on standardized variables. A standardized variable is a variable that has a mean of 0 and a standard deviation of 1. One reason for standardizing variables is that you can interpret the β estimates as partial correlation coefficients.
where with one variable X, it is the Pearson correlation coefficient.
