Question

What would be the sum of determination coefficients if all causes are accounted for?

• A. 1
• B. 0
• C. - 0.50
• D. 0.50

Hint:

Path Analysis Rule (Sewall Wright): The sum of coefficients of determination for all causes (including residual) MUST EQUAL EXACTLY 1.0 (100%).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Sewall Wright's Path Analysis: when all independent direct, indirect, and residual environmental causes contributing to total phenotypic variance are completely accounted for, the sum of all path coefficients of determination equals 1.0.
Key Formula or Approach:
\[ \sum d = \sum p_i^2 + 2 \sum p_i p_j r_{ij} = 1.00 \quad [\text{Complete Variance Determination}] \]

Step 2: Detailed Explanation:
In population genetics and Sewall Wright's method of Path Coefficients:
- The coefficient of determination ($d$) measures the fraction of the total variance in a dependent variable (phenotype) that is directly or indirectly caused by an independent causal factor.
- By the mathematical law of total probability and variance decomposition, if all possible causal factors (including residual unaccounted error $e$) are included:
\[ \sum \text{Determination Coefficients} = 1.00 \]
- Total phenotypic variance is completely ($100\%$) explained.

Step 3: Final Answer:
Thus, the sum of determination coefficients is 1, corresponding to option (A).