Randomized Complete Block Design

1. Experimental
Designs using
RCBD

2. Single-Factor Experiment in Randomized
Complete Block Design (RCBD)

3. DESCRIPTION OF THE DESIGN
✓ RCBD is used when difference in characteristics is seen between
observation units
✓ In this design, we group similar experimental units into blocks.
✓ The experimental units within a block should be as uniform as possible. In
effect, blocking can reduce the experimental error and minimized or
eliminate confounding effect.
✓ Each treatment should be assigned randomly to the experimental units
within the block.
✓ The blocks will serve as replicates of the treatments, since every treatment
is applied to each block.

4. RANDOMIZATION PROCEDURE
• Divide the experimental units into b equal blocks
• Subdivide the first block into t experimental units, where t is the number of
treatments.
• Number the experimental units from 1 to t. Then in a box put the treatment
number 1 up to t.
• Pick a treatment number from the box and it will be assigned to
experimental unit 1.
• Pick another treatment and assign it to unit 2. Continue until you have
assigned all the treatments to the experimental units in block 1.
• Repeat the steps above for each remaining blocks. Each treatment must
appear once in every block.

5. POSSIBLE LAYOUT
Block 1
T1 T2
T3 T4
Block 2
T1 T2
T3 T4
Block 3
T1 T2
T3 T4

6. MODEL
• The equation that is used to predict the value of y given the values of the
factor/s of the experiment can be written as
𝑦𝑖𝑗 = 𝜇 + 𝛼𝑖 + 𝛽𝑗 + 𝜀𝑖𝑗
where
𝑦𝑖𝑗 is the observed value of the experimental unit receiving ith treatment in the
jth block
𝜇 represents the overall mean
𝛼𝑖 is the effect of the ith treatment
𝛽𝑗 is the effect of the jth block
𝜀𝑖𝑗 is the random error

7. MODEL ASSUMPTIONS
• The observations in each treatment-block combinations are independent
• The observations in each treatment-block combinations are normally
distributed with equal variances
• There is no interaction between blocks and treatments, that is, the pattern
of the treatment effect across the different blocks must be parallel

8. EXAMPLE
A study was conducted to compare four fish pellets. Five ponds are used in this study.
Each pond was divided to four cages. In effect, each pond receives the four types of
pellets. Fifty (50) fishes are stocked in each cage. The total weight, in kg, in each cage
will be recorded.
Factor: Fish pellets
Treatment: Four different fish pellets
Experimental unit: Cage with 50 fishes
Randomization: RCBD
Blocks: 5 Ponds
Experimental Design: Single-Factor Experiment in Randomized Complete Block Design
(RCBD)

9. LAYOUT
Pond 1
P1 P2
P3 P4
Pond 2
P1 P2
P3 P4
Pond 3
P1 P2
P3 P4
Pond 4
P1 P2
P3 P4
Pond 5
P1 P2
P3 P4

10. THE DATA

11. Two-Factor Factorial Experiment in
Randomized Complete Block Design (RCBD)

12. DESCRIPTION OF THE DESIGN
✓ Recall that factorial experiment is an experiment in which the treatments
consist all possible combinations of the selected levels in two or more
factors.
✓ Also recall that in an RCBD, a set of experimental units is grouped (blocked)
in a way that variability among the units of the same block or the
experimental error is minimized.
✓ In a Two-Factor Factorial Experiment in Randomized Complete Block Design
(RCBD), blocking will be done considering 2 factors and the treatments
consist all possible combinations of the selected levels in 2 factors.

13. THE MODEL
𝑦𝑖𝑗𝑘 = 𝜇 + 𝛼𝑖 + 𝛽𝑗 + (𝛼𝛽)𝑖𝑗+𝜌𝑘 + 𝜀𝑖𝑗𝑘
where
𝜇 is the overall mean
𝛼𝑖 is the effect of the ith level for Factor A
𝛽𝑗 is the effect of jth level for Factor B
(𝛼𝛽)𝑖𝑗 is the effect due to the interaction of the ith level of Factor A and the jth level of
Factor B
𝜌𝑘 is the effect of the kth block
𝜀𝑖𝑗𝑘 is the random error associated with the kth block in the treatment with ith level of
Factor A and jth level of Factor B

14. EXAMPLE
The yield for 3 varieties of wheat was compared in an experiment. Another
factor considered was the pre-treatment of seeds before planting with 2 levels.
Three (3) fields were used in the experiment and each field was divided into 6
plots of equal sizes. In each plot, 100 seeds were planted.
Factors: A. Variety of wheat (3 levels); B. Pre-treatment (2 levels)
Experimental Unit: Plots with 100 seeds
Blocks: 3 fields
Treatment Combinations:
Var 1 w/o pre-treatment (V1P1) Var 1 w/ pre-treatment (V1P2)
Var 2 w/o pre-treatment (V2P1) Var 2 w/ pre-treatment (V2P2)
Var 3 w/o pre-treatment (V3P1) Var 3 w/ pre-treatment (V3P2)

15. LAYOUT
Field 1
V1P2 V2P2 V3P1
V3P2 V1P1 V2P1
Field 2
V2P2 V1P1 V1P2
V3P1 V2P1 V3P2
Field 3
V2P1 V3P1 V2P2
V1P2 V3P2 V1P1

16. THE DATA