Test if a correlation is bigger than a specified value?

Question

I have calculated the correlation $r$ between $X$ and $Y$. I'd like to test $H_0: r \leq \rho$, $H_1: \rho < r$.

For the case when $r = 0$, I can apply the transform $t = \rho \frac{\sqrt{n-2}}{\sqrt{1 - \rho^2}}$, which follows the $t_{n-2}$ distribution. I can then perform a one-tailed test and see if $\int_t^{\infty} p_{t_{n-2}} < \alpha$, rejecting the null hypothesis if it is.

How can I generalize this case to the described scenario when $r \not= 0$?

Edit: how would I determine the sample size required to perform this test with specified $\alpha$ and $\beta$ (1 - power)?

Answer 1

Using the Z transform its pretty easy to determine the sample size for this test. Suppose we want to test $H_0: r \leq \rho_1$, and we specify that for a correlation of $\rho_2$ ($\rho_1 < \rho_2$) our test must have probability of type I error $\alpha$ and probability of type II error $\beta$.

We consider the problem in terms of Z transformed variables. (for correlation, a Fisher transform is commonly used) Let $c$ denote the threshold value on $N(\rho_1', \frac{1}{\sqrt{n-3}})$, such that for $c < r'$ we reject the null hypothesis. Where $\rho_1'$ is the transform of $\rho_1$.

Consider the assumptions of the null hypothesis. Since we specified the probability of making a type I error as less than $\alpha$, we must have $c \geq \rho_1' + z_{1-\alpha} \frac{1}{\sqrt{n-3}}$.

Now consider the assumptions of the alternative hypothesis. For $r \geq \rho_2'$, we must have probability of type II error as less than $\beta$. Therefore, $c \leq \rho_2' -z_{1 - \beta} \frac{1}{\sqrt{n-3}}$.

It follows that $$\rho_1' + z_{1-\alpha} \frac{1}{\sqrt{n-3}} \leq \rho_2' -z_{1 - \beta} \frac{1}{\sqrt{n-3}}.$$ This can be rearranged to $n \geq 3 + (\frac{z_{1-\alpha} + z_{1-\beta}}{\rho_2' - \rho_1'})^2$.

Comments and corrections welcome.

Answer 2

The standard approach: Calculate a confidence interval for the true correlation $\rho$ and check if it excludes the hypothesized value $\rho_o$. In your case, if the lower 95 percent limit is higher than $\rho_o$, reject H0 in favour of H1 at the approximate 5% level.

The confidence interval is usually calculated via the t distribution.

Example

We are interested in the working hypothesis $\rho_o > 0.3$ at the significance level of 0.05. We crunch a parametric two-sided, equal-tailed 90% confidence interval in Python and R:

Python:

from scipy.stats import pearsonr
from sklearn import datasets

X, _ = datasets.load_iris(as_frame=True, return_X_y=True)
cor = pearsonr(X["petal length (cm)"], X["petal width (cm)"])
cor.confidence_interval(confidence_level=0.9)
# ConfidenceInterval(low=0.9515705602254544, high=0.9715644582981584)

R

cor.test(~ Petal.Length + Petal.Width, data = iris, conf.level = 0.9)
# 90 percent confidence interval:
#   0.9515706 0.9715645

Since even the lower limit 0.952 is above the hypothesized value $\rho_o=0.3$, we claim (at the approximate level 0.05) that our working hypothesis is true.