Question

If n is the number of treatments and r is the number of replications, what is the error degrees of freedom in RBD ?

• A. r-1
• B. n-1
• C. n*r-1
• D. (n-1)*(r-1)

Hint:

To remember this quickly:
\[ \text{Error df} = \text{Treatment df} \times \text{Replication df} = (n-1)(r-1) \]
This basic property helps verify the correctness of the error term in RBD calculations.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A Randomized Block Design (RBD) is an experimental design where experimental units are grouped into homogeneous blocks to control a single source of variation.
Key Formula or Approach:
The degrees of freedom (df) partition in the Analysis of Variance (ANOVA) table for an RBD with \( n \) treatments and \( r \) replications (blocks) is given by:
- Total df = \( nr - 1 \)
- Block df = \( r - 1 \)
- Treatment df = \( n - 1 \)
- Error df = Total df \( - \) Block df \( - \) Treatment df

Step 2: Detailed Explanation:
Let us perform the algebraic simplification to find the error degrees of freedom:
\[ \text{Error df} = (nr - 1) - (r - 1) - (n - 1) \]
Expanding the expression:
\[ \text{Error df} = nr - 1 - r + 1 - n + 1 \]
Grouping the terms:
\[ \text{Error df} = nr - n - r + 1 \]
Factoring the terms by grouping:
\[ \text{Error df} = n(r - 1) - 1(r - 1) \]
\[ \text{Error df} = (n - 1)(r - 1) \]
Thus, the algebraic expression for the error degrees of freedom is \( (n-1)(r-1) \).

Step 3: Final Answer:
The error degrees of freedom in RBD is (n-1)*(r-1).