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What is the maximum number of pairs allowed in a Spearman correlation to calculate the p-value in order to check the significance of the correlation coefficient?

As per the articles I've found online and their formulae that I tried, I'm getting a p-value of above 1, I want to know where I'm going wrong.

My data sets' sizes vary from 30 data points to 150 data points. I have the Spearman correlation coefficient but I want to test for significance.

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    $\begingroup$ Use this in R: cor.test(x, y, method = "spearman") $\endgroup$ Feb 20, 2023 at 17:37
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    $\begingroup$ Better yet, run ?cor.test and read how the function does hypothesis testing for Spearman correlation. $\endgroup$
    – Dave
    Feb 20, 2023 at 17:40
  • $\begingroup$ You don't give enough detail for people to figure out where you're going wrong. $\endgroup$
    – Glen_b
    Feb 21, 2023 at 4:45
  • $\begingroup$ Sample size and precision for correlations is covered at hbiostat.org/bbr/corr with R code. It's best to assume that all correlations are non-zero and realize that testing against zero isn't very meaningful. Think about using confidence intervals instead. $\endgroup$ Feb 12 at 14:22

2 Answers 2

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You definitely shouldn't be getting a p value greater than 1.

You can, of course, do the calculation for Spearman's rho by hand, and then look up in a table the p value for that rho and your given sample size.

However, it's easy enough to do with available software.

For example, you can run the following R code here, without installing software: rdrr.io/snippets/ .

A = c(1,2,3,4,5,6,7,8,9,10)
B = c(2,4,3,5,7,4,4,8,9,11)

cor.test(~ A + B, method="spearman")

In any case, you could use this approach to double-check your calculations.

You could also use free software like Jamovi, which is gui-based, fairly intuitive, and will allow you to import your data, from, say, a .csv file.

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There is no maximum sample size allowable for calculating Spearman's rank correlation coefficient.

To test $\text{H}_{0}\text{: }\rho_{\text{S}} = 0$ against $\text{H}_{\text{A}}\text{: }\rho_{\text{S}} \ne 0$, you can use a $t$ test statistic (let's choose $\alpha = 0.01$):

$$t = \frac{r_{\text{S}}-\rho_{\text{S0}}}{\sigma_{r_{\text{S}}}} \approx \frac{r_{\text{S}}}{s_{r_{\text{S}}}}$$

Where $s_{r_{\text{S}}} = \sqrt{\frac{1-r^2_{\text{S}}}{n-2}}$.

For example, of you calculate $r_{\text{S}}=0.6$ in a sample of 20 paired observations, then $s_{r_{\text{S}}}=\sqrt{\frac{1 - .6^2}{20-2}} = 0.1885618$, and so $t = 3.181981$. Therefore $p = Pr(|T_{\nu=18}|\ge |3.181981|) = 0.00258146\times 2 = 0.00516292$.

Because $0.00516292 < 0.01$ (i.e. $p < \alpha$), we reject $\text{H}_{0}$, and can conclude that we found evidence that $\rho_{\text{S}}$ is not equal to 0 significant at the $\alpha = 0.01$ level.

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