# One-sided hypothesis test for correlation

In the textbooks I have access to (and that discuss hypothesis testing for correlation), I only met examples, where the null-hypothesis was $\rho=0$, and the alternative hypothesis was $\rho\ne 0$. My question is about using a one-sided alternative hypothesis $\rho>0$. Is this meaningful?

This question has been asked before, but it has not been answered. There was a comment next to the linked question, that said that the null-hypothesis should be $\rho\le 0$ in case we would like a one-sided alternative hypothesis, but I have problems with this comment. As I understand, the t-distribution that is used for testing the correlation coefficient is only valid when $\rho=0$, so we have no choice, but using this as the null-hypothesis.

So, to summarize: can we test $H_0:\rho=0$ against $H_1:\rho>0$ using $R\sqrt{\dfrac{n-2}{1-R^2}}$ and the t-distribution with degree of freedom $n-2$?

• I don't agree with the premise of this question. "Correlation" is a symmetric measure of association, at least in terms of a Pearson or Spearman correlation -- the most common uses of the term. Commented Jul 27, 2016 at 12:07
• Can you please explain a bit this comment? Correlation is indeed symmetric measure of association, but it can be positive or negative. Commented Jul 27, 2016 at 12:16
• See additional comments (posted just now) under the linked question. Commented Jul 27, 2016 at 12:38
• See Justification of one-tailed hypothesis testing for how to think about the distribution of the test statistic under a null hypothesis that isn't of the simple form $\theta=0$ (or another exactly specified value). Commented Jul 27, 2016 at 13:01
• @DJohnson: Sorry, I pressed enter and I cannot edit my previous post. So here is the question: A company claims, that travelling distance to work is independent of salary. To test this, 20 employees are asked about salary and travel distance. For this sample, r=-0.35 was found. Perform a one-tailed test at the 5% significance level to test whether the travel distance and salary are independent. Commented Jul 27, 2016 at 14:02

Yes. Instead of using a two-sided critical value from a t-distribution with $n-2$ degrees of freedom (e.g., $\pm 2.09$ for $n=22$ and $\alpha=.05$, two-sided), you would use just the upper critical value (e.g., $+1.72$ for $n=22$ and $\alpha=.05$, one-sided).
• Thanks. Can you also please help me with what the conclusion would be if the data supports rejecting the null-hypothesis? Is it: "there is reason to believe that the variables are not correlated" (so rejecting $\rho=0$ ) or "there is reason to believe, that the variables are positively correlated" (so accepting $\rho>0$)? Since $H_0$ and $H_1$ are not negations of each other, these are different conclusions. As I understand, the conclusion should be the first one, but I would like to be sure. Commented Jul 27, 2016 at 12:31
• @FerencBeleznay The null hypothesis is $H_0: \rho \le 0$, so if the data support rejecting the null, you are rejecting the null hypothesis that the true correlation is 0 or negative (which in turn suggests that the true correlation is positive). Commented Jul 27, 2016 at 18:17
• No, for the one-sided test, the null hypothesis is $\rho \le 0$. Obviously, if I can reject $\rho = 0$, then I can also reject $\rho = -0.5$ or $\rho = -1$, so we still use $\rho = 0$ as the null. Commented Jul 28, 2016 at 6:47
• @FerencBeleznay: Wolfgang's point is explained in more detail at Justification of one-tailed hypothesis testing. (Though I feel you're quite entitled to decide between $\rho=0$ & $\rho \leq 0$ as the null depending on the situation.) Commented Jul 28, 2016 at 8:50