Question

The hypothesis of \(H_0: \rho = \rho_0\) (a constant) can be tested by making use of the transformation:

• A. \(Z = \dfrac{1}{2}\log\left(\dfrac{1+\rho}{1-\rho}\right)\)
• B. \(Z = \log_{10}\left(\dfrac{1-\rho}{1+\rho}\right)\)
• C. \(Z = \log_{10}\left(\dfrac{1+\rho}{1-\rho}\right)\)
• D. \(Z = \dfrac{1}{2}\log\left(\dfrac{1-\rho}{1+\rho}\right)\)

Hint:

Think of R. A. Fisher's variance-stabilising transformation for the correlation coefficient.

Answer 1

The Correct Option is A

Step 1: The sample correlation coefficient \(r\) has a skewed sampling distribution whose shape depends on the unknown \(\rho\), so testing \(H_0: \rho = \rho_0\) directly from \(r\) is awkward for any \(\rho_0 \ne 0\).

Step 2: R. A. Fisher proposed the variance-stabilising transformation \(Z = \dfrac{1}{2}\log_e\left(\dfrac{1+\rho}{1-\rho}\right)\), applied both to the sample value \(r\) and to the hypothesised \(\rho_0\).

Step 3: Under \(H_0\), the statistic \(Z = \dfrac{1}{2}\log_e\left(\dfrac{1+r}{1-r}\right)\) is approximately normal with mean \(\dfrac{1}{2}\log_e\left(\dfrac{1+\rho_0}{1-\rho_0}\right)\) and variance \(\dfrac{1}{n-3}\), independent of the actual value of \(\rho_0\), which makes a simple normal test possible.

Step 4: Matching with the given options, this is exactly option (A): \(Z=\dfrac12\log\left(\dfrac{1+\rho}{1-\rho}\right)\).