Step 1: Concept Overview:
The goal is to derive a general formula for higher-order partial correlation coefficients, given that all zero-order correlations are the same. This involves repeated application of the partial correlation formula to identify a pattern.
Step 2: Core Formula:
The recursive formula for partial correlation is:
\[ r_{12.3...k} = \frac{r_{12.3...(k-1)} - r_{1k.3...(k-1)} r_{2k.3...(k-1)}}{\sqrt{(1 - r^2_{1k.3...(k-1)})(1 - r^2_{2k.3...(k-1)})}} \]
We'll use this formula iteratively, starting with the first order, to discover a recurring pattern.
Step 3: Detailed Solution:
Assume all zero-order correlations are identical: \(r_{ij} = \rho\).
First-order partial correlation (e.g., \(r_{12.3}\)):
\[ r_{12.3} = \frac{r_{12} - r_{13}r_{23}}{\sqrt{(1 - r^2_{13})(1 - r^2_{23})}} = \frac{\rho - \rho . \rho}{\sqrt{(1 - \rho^2)(1 - \rho^2)}} = \frac{\rho(1-\rho)}{1-\rho^2} = \frac{\rho(1-\rho)}{(1-\rho)(1+\rho)} = \frac{\rho}{1+\rho} \]
Thus, all first-order partial correlations equal \(\rho_1 = \frac{\rho}{1+\rho}\).
Second-order partial correlation (e.g., \(r_{12.34}\)):
Applying the recursive formula using first-order partials:
\[ r_{12.34} = \frac{r_{12.3} - r_{14.3}r_{24.3}}{\sqrt{(1 - r^2_{14.3})(1 - r^2_{24.3})}} = \frac{\rho_1 - \rho_1 . \rho_1}{1 - \rho_1^2} = \frac{\rho_1(1-\rho_1)}{(1-\rho_1)(1+\rho_1)} = \frac{\rho_1}{1+\rho_1} \]
Substituting the value of \(\rho_1\):
\[ r_{12.34} = \frac{\frac{\rho}{1+\rho}}{1 + \frac{\rho}{1+\rho}} = \frac{\frac{\rho}{1+\rho}}{\frac{1+\rho+\rho}{1+\rho}} = \frac{\rho}{1+2\rho} \]
Therefore, all second-order partial correlations equal \(\rho_2 = \frac{\rho}{1+2\rho}\).
Third-order partial correlation (e.g., \(r_{12.345}\)):
Following the recursive pattern:
\[ r_{12.345} = \frac{\rho_2}{1+\rho_2} \]
Substituting the value of \(\rho_2\):
\[ r_{12.345} = \frac{\frac{\rho}{1+2\rho}}{1 + \frac{\rho}{1+2\rho}} = \frac{\frac{\rho}{1+2\rho}}{\frac{1+2\rho+\rho}{1+2\rho}} = \frac{\rho}{1+3\rho} \]
The general formula for the \(k\)-th order partial correlation coefficient is \( \frac{\rho}{1+k\rho} \).
Step 4: Conclusion:
Each third-order partial correlation coefficient is \( \frac{\rho}{1+3\rho} \).