Hypothesis testing for the correlation coefficient (at least one of them) is based on a t-distribution of the test statistic $t=\frac{r}{\sqrt{(1-r^2)/(N-2)}}$ where $r$ is the sample correlation coefficient and $N$ is the number of cases.

Can someone tell me what assumptions are made in order for this $t$ to have a $t$-distribution ? I assume that bivariate normality is one of them ?

  • $\begingroup$ I highly suspect it does not exactly follow a t-distribution, even though it may be approximated well by one. The least squares regression coefficient does exactly follow a t-distribution when the errors are normally distributed. Tests of the OLS parameter are consistent with tests of correlation coefficient, meaning they converge to the same answer as n gets big. Another assumption to consider is that the null must be true; when the null is not true, statistics normally distributed as T will have non central T distributions. $\endgroup$ – AdamO Dec 27 '17 at 18:43
  • $\begingroup$ @AdamO: I see what you mean, but I am not sure it answers my question: As you say the $\hat{\beta}_1$ in a regression $y=\beta_0+\beta_1 x +\epsilon$ is linked to the correlation coefficient. Moreover, as you say, (if all necessary conditions are met) $t=\hat{\beta}_1/s_{\hat{\beta}_1}$ is t-distributed. So your reasoning may hold if this $t$ is exactly equal to the formula in my question. Is that the case ? $\endgroup$ – user83346 Dec 29 '17 at 4:37

The assumptions that are required are as follows;

  • independent observations;
  • the population correlation, ρ = 0;
  • normality: the 2 variables involved are bivariately normally distributed in the population. However, this is not needed for a reasonable sample size -say, N ≥ 20 or so.

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