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 $n-2$?

  • $\begingroup$ 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. $\endgroup$
    – user78229
    Commented Jul 27, 2016 at 12:07
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    $\begingroup$ Can you please explain a bit this comment? Correlation is indeed symmetric measure of association, but it can be positive or negative. $\endgroup$ Commented Jul 27, 2016 at 12:16
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    $\begingroup$ See additional comments (posted just now) under the linked question. $\endgroup$
    – amoeba
    Commented Jul 27, 2016 at 12:38
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    $\begingroup$ 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). $\endgroup$ Commented Jul 27, 2016 at 13:01
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    $\begingroup$ @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. $\endgroup$ Commented Jul 27, 2016 at 14:02

1 Answer 1


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).

  • $\begingroup$ 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. $\endgroup$ Commented Jul 27, 2016 at 12:31
  • $\begingroup$ @FerencBeleznay: See Why do statisticians say a non-significant result means “you can't reject the null” as opposed to accepting the null hypothesis?. In this case though, you're accepting the alternative, which is what you've stipulated it to be: see Is it possible to accept the alternative hypothesis?. $\endgroup$ Commented Jul 27, 2016 at 13:03
  • $\begingroup$ @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). $\endgroup$
    – Wolfgang
    Commented Jul 27, 2016 at 18:17
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    $\begingroup$ 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. $\endgroup$
    – Wolfgang
    Commented Jul 28, 2016 at 6:47
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    $\begingroup$ @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.) $\endgroup$ Commented Jul 28, 2016 at 8:50

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