Summary
Randomized Complete Block Design (RCBD) is widely used to reduce experimental error by grouping similar experimental units into blocks and randomly assigning all treatments within each block.
RCBD is especially suitable when field variability or known gradients such as soil fertility exist.
Its effectiveness depends on assumptions like homogeneity within blocks, independence of observations, normal distribution and equal variances.
The design allows precise comparison of multiple treatments under limited experimental conditions.
In the given wheat yield example, RCBD helps compare ten genotypes across four replications.
The Least Significant Difference (LSD) test is used to classify varieties based on significant mean differences.
Agri Analyze simplifies RCBD analysis by enabling easy data upload, statistical testing and report generation.
1.Introduction
Experimental design is a systematic approach in scientific research, essential for investigating relationships among variables. It ensures valid and interpretable results through randomization, replication and control.
Randomization distributes extraneous variables evenly, reducing bias.
Replication increases reliability and precision by accounting for variability within experimental units.
Control ensures that observed differences are due to the independent variable.
Designs range from simple completely randomized designs to complex ones like randomized complete block designs, factorial designs and Latin squares. These designs help isolate variable effects and understand their interactions.
Effective experimental design is crucial for drawing valid conclusions and advancing scientific knowledge across various fields.
2. Randomized Complete Block Design (RCBD)
Randomized Complete Block Design (RCBD) is a fundamental experimental design used extensively in scientific research to control for variability within experimental units.
In RCBD, each block contains all treatments, with random assignment within blocks, controlling for variability and ensuring comprehensive treatment comparison.
Hence, it is called "Randomized Complete Block Design."
This design reduces experimental error and enhances the precision of treatment comparisons by accounting for block-to-block variability.
RCBD is particularly useful in experiments with known or suspected gradients in conditions, such as soil fertility in agricultural studies.
It is essential for drawing valid inferences about treatment effects while minimizing the influence of extraneous factors.
3. When RCBD is used?
The RCBD is employed in agricultural research under specific conditions to achieve reliable and precise results.
Here are scenarios when RCBD is used:
- Heterogeneous Experimental Units: When there is significant variability within the experimental field, such as differences in soil fertility, moisture, or topography,
RCBD helps control this variability by grouping similar units into blocks.
- Known Gradients : When there are known gradients in the experimental area (e.g., fertility gradients across a field),
RCBD is used to ensure that each treatment is tested across all levels of the gradient, reducing the impact of these gradients on treatment comparisons.
- Multiple Treatments: When comparing multiple treatments (e.g., different crop varieties, fertilizers, or pest control methods),
RCBD ensures that each treatment is equally represented within each block, allowing for accurate comparison.
- Limited Experimental Units: In cases where the number of experimental units is limited, RCBD maximizes the use of available units by reducing experimental error, thus enhancing the precision of the results.
- Small-Scale Trials: For small-scale trials where variability within the experimental area can significantly impact results, RCBD provides a robust method to control for this variability.
4. Assumptions of RCBD
The RCBD operates under several key assumptions to ensure valid and reliable results:
- Homogeneity within Blocks: The experimental units within each block are assumed to be homogeneous, meaning they are similar in terms of characteristics that could affect the response variable (e.g., soil fertility, moisture levels).
- Independence of Errors: Observations from different experimental units are assumed to be independent of each other. The response of one unit does not influence the response of another.
- Additivity of Effects: The effects of blocks and treatments are additive, meaning there are no interactions between blocks and treatments.
- Random Assignment: Treatments are randomly assigned to experimental units within each block to ensure unbiased estimates of treatment effects.
- Normality: The response variable for each treatment is assumed to be normally distributed within each block.
- Equal Variance: The variance of the response variable is assumed to be the same for all treatments within each block.
- No Missing Data: It is assumed that there are no missing data points. Each treatment is represented in every block.
5.Analysis of Variance (ANOVA) for RCBD
In a Randomized Complete Block Design (RCBD), the Analysis of Variance (ANOVA) model is used
to analyze the data and test the significance of treatment effects. The ANOVA model for RCBD
can be expressed as follows:
Yij = μ + τi + βj + εij
Where,
- Yij is the observation for the jth unit in the ith treatment group
- μ is the overall mean of the response variable
- βj is the effect of the jth replication (j = 1, 2, …, r)
- τi is the effect of the ith treatment (i = 1, 2, …, t)
- εij is the random error associated with the observation
6. Randomization steps in RCBD
Randomization in a Randomized Complete Block Design (RCBD) is a crucial step to ensure unbiased allocation of treatments to experimental units within each block.
Here are the detailed steps for randomization in RCBD:
- Identify the Treatments: List all the treatments to be tested in the experiment. Let's assume there are t treatments
- Define the Blocks: Identify and define the blocks based on homogeneous characteristics. Each block will contain all the treatments. Let's assume there are r blocks (replications).
- Assign Treatments Randomly within Each Block: To randomly assign treatments within each block in an RCBD, list all treatments and use a randomization method such as random number tables, computer software, or drawing lots. Document the random allocation for each block to ensure clear and unbiased treatment distribution across the experimental units.
- Record the Assignment: Document the random allocation of treatments for each block to ensure the layout plan is clear and can be followed accurately during the experiment.
- Repeat for All Blocks: Repeat the randomization process for each block until all treatments have been randomly assigned to plots within every block.
- Verify Randomization: Ensure that each treatment appears once in every block and that the allocation is indeed random. This can be done by checking the documentation or using software outputs.
- Create a Layout Plan:Develop a visual representation or map of the experimental layout showing the randomized assignment of treatments within each block.
ANOVA Outline of RCBD
Here’s a brief outline of the ANOVA procedure for a RCBD:
| Source of Variation |
d.f. |
Sum of Squares |
Mean Sum of Squares (SS/df) |
Cal. F |
| Replication |
r - 1 |
$$
\sum_{j=1}^{r} Y_{.j}^{2}
-
\frac{\left(\sum_{i=1}^{t}\sum_{j=1}^{r} Y_{ij}\right)^2}{rt}
$$
|
MSR |
$$
\frac{MSR}{MSE}
$$
|
| Treatment |
t - 1 |
$$
\sum_{i=1}^{t} Y_{i.}^{2}
-
\frac{\left(\sum_{i=1}^{t}\sum_{j=1}^{r} Y_{ij}\right)^2}{rt}
$$
|
MST |
$$
\frac{MST}{MSE}
$$
|
| Error |
(r-1)(t-1) |
By subtraction |
MSE |
— |
| Total |
rt - 1 |
$$
\sum_{i=1}^{t}\sum_{j=1}^{r} Y_{ij}^{2}
-
\frac{\left(\sum_{i=1}^{t}\sum_{j=1}^{r} Y_{ij}\right)^2}{rt}
$$
|
— |
— |
Standard Error of Mean (SEm)
$$ SEm = \sqrt{\frac{MSE}{r}} $$
Coefficient of Variation (CV%)
$$ CV\% = \frac{\sqrt{MSE}}{\bar{Y}_{..}} \times 100 $$
Where,
$$
\bar{Y}_{..} \text{ is the general mean}
$$
Critical Difference at 5% Level of Significance
$$ CD_{0.05} = t_{(0.05,\,ne)} \times \sqrt{2} \times SEm $$
Example of Randomized Complete Block Design (RCBD)
Ten wheat varieties were evaluated in a yield trial against a local check using a
Randomized Complete Block Design (RCBD) with
four replications at the Vijapur farm.
The observed yield data (q/ha) are presented below.
| Genotypes |
Replications |
Total of Genotype |
Mean of Genotype |
| R-1 |
R-2 |
R-3 |
R-4 |
| Haura |
148 |
132 |
148 |
132 |
560 |
140 |
| HY-12 |
155 |
156 |
157 |
160 |
628 |
157 |
| HY-65-4 |
112 |
136 |
126 |
150 |
524 |
131 |
| HY-11-6 |
112 |
114 |
100 |
118 |
444 |
111 |
| HY-12-5-3 |
124 |
125 |
126 |
125 |
500 |
125 |
| HY-5-7-2 |
92 |
94 |
98 |
96 |
380 |
95 |
| HY-11-8 |
116 |
124 |
130 |
134 |
504 |
126 |
| Kalyan Sona |
115 |
121 |
122 |
126 |
484 |
121 |
| Sonalika |
131 |
131 |
132 |
130 |
524 |
131 |
| GW-24 |
145 |
149 |
150 |
156 |
600 |
150 |
| Replication Total |
1250 |
1282 |
1289 |
1327 |
5148 |
— |
Null Hypothesis for Varieties and Replications
H0 (Varieties):
There are no significant differences among the varieties under study.
H0 (Replications):
There are no significant differences among the replications (blocks) under study.
Correction Factor (CF)
$$
CF = \frac{\left(\sum_{i=1}^{t}\sum_{j=1}^{r} Y_{ij}\right)^2}{rt}
= \frac{5148^2}{40}
= 662547.60
$$
General Mean (GM)
$$
GM = \frac{GT}{rt}
= \frac{5148}{40}
= 128.70
$$
Total Sum of Squares (Total SS)
$$
\text{Total SS}
= \sum_{i=1}^{t}\sum_{j=1}^{r} Y_{ij}^2
- \frac{\left(\sum_{i=1}^{t}\sum_{j=1}^{r} Y_{ij}\right)^2}{rt}
$$
$$
= 148^2 + 132^2 + \cdots + 156^2 - 662547.60
$$
$$
= 13242.40
$$
Treatment Sum of Squares (Treatment SS)
$$
\text{Treatment SS}
= \frac{\sum_{i=1}^{t} Y_{i.}^2}{r}
- \frac{\left(\sum_{i=1}^{t}\sum_{j=1}^{r} Y_{ij}\right)^2}{rt}
$$
$$
= \frac{560^2 + 628^2 + \cdots + 600^2}{4}
- 662547.60
$$
$$
= 11688.40
$$
Replication Sum of Squares (Replication SS)
$$
\text{Replication SS}
= \frac{\sum_{j=1}^{r} Y_{.j}^2}{t}
- \frac{\left(\sum_{i=1}^{t}\sum_{j=1}^{r} Y_{ij}\right)^2}{rt}
$$
$$
= \frac{1250^2 + 1282^2 + 1289^2 + 1327^2}{10}
- 662547.60
$$
$$
= 299.80
$$
Error Sum of Squares (Error SS)
$$
\text{Error SS}
= \text{Total SS} - \text{Replication SS} - \text{Treatment SS}
$$
$$
= 13242.40 - 299.80 - 11688.40
$$
$$
= 1254.20
$$
Mean Squares for All Sources
Replication Mean Square:
$$
MS_{Rep}
= \frac{\text{Replication SS}}{\text{Replication DF}}
= \frac{299.80}{3}
= 99.93
$$
Treatment Mean Square:
$$
MS_{Treat}
= \frac{\text{Treatment SS}}{\text{Treatment DF}}
= \frac{11688.40}{9}
= 1298.70
$$
Error Mean Square:
$$
MS_{Error}
= \frac{\text{Error SS}}{\text{Error DF}}
= \frac{1254.20}{27}
= 46.45
$$
Calculated F for Replication and Treatment
Cal F of Replication
$$
= \frac{99.93}{46.45}
= 2.15
$$
Cal F of Treatment
$$
= \frac{1298.70}{46.45}
= 27.95
$$
Conducting LSD Test for Multiple Mean Comparison
Arrange varieties means in descending order, find difference (d) between two consecutive means and follow procedure given below:
| Varieties |
Mean |
| HY-12 |
157 |
| GW-24 |
150 |
| Haura |
140 |
| HY-65-4 |
131 |
| Sonalika |
131 |
| HY-11-8 |
126 |
| HY-12-5-3 |
125 |
| Kalyan sona |
121 |
| HY-11-6 |
111 |
| HY-5-7-2 |
95 |
Find difference between two consecutive means
d= 157-150 =7. If d >= CD, then both means are significantly different.
There is no need to find d between HY-12 and GW-24. If d < CD, then both means are not significantly different OR we can say both are at par.
Here, both varieties are significantly at par.
But d = 157 (HY-12) – 140 (Haura) =17, Here d >= CD, then both means are significantly different.
d = 150 (GW-24) – 140 (Haura) =10, Here d >= CD, then both means are significantly different.
d = 140 (Haura) – 131 (HY-65-4) = 9, Here d < CD, then both means are not significantly different.
Same for Sonalika, these varieties are significantly at par.
Follow same procedure to exhaust all means.
Final LSD test for varieties given as under:
| Varieties |
Mean |
Group |
| HY-12 |
157 |
a |
| GW-24 |
150 |
a |
| Haura |
140 |
b |
| HY-65-4 |
131 |
bc |
| Sonalika |
131 |
bc |
| HY-11-8 |
126 |
cd |
| HY-12-5-3 |
125 |
cd |
| Kalyan sona |
121 |
d |
| HY-11-6 |
111 |
e |
| HY-5-7-2 |
95 |
f |
Steps to Perform RCBD Analysis in Agri Analyze
Step 1: Create a CSV file with columns for
Genotype and Yield (Gain).
| Genotypes |
Replications |
Yield |
| Haura |
R1 |
148 |
| HY-12 |
R1 |
155 |
| HY-65-4 |
R1 |
112 |
| HY-11-6 |
R1 |
112 |
| HY-12-5-3 |
R1 |
124 |
| HY-5-7-2 |
R1 |
92 |
| HY-11-8 |
R1 |
116 |
| Kalyan sona |
R1 |
115 |
| Sonalika |
R1 |
131 |
| GW-24 |
R1 |
145 |
| Haura |
R2 |
132 |
| HY-12 |
R2 |
156 |
| HY-65-4 |
R2 |
136 |
| HY-11-6 |
R2 |
114 |
| HY-12-5-3 |
R2 |
125 |
| HY-5-7-2 |
R2 |
94 |
| HY-11-8 |
R2 |
124 |
| Kalyan sona |
R2 |
121 |
| Sonalika |
R2 |
131 |
| GW-24 |
R2 |
149 |
Step 2: Click on ANALYTICAL TOOL ->DESIGN OF EXPERIMENT ->RCBD ANALYSIS ->ONE FACTOR RCBD ANALYSIS
Step 3: Open link https://www.agrianalyze.com/OneFactorRCBD.aspx (For first time users free registration is mandatory)
Step 4: Link Here to download sample file Sample File Download
Step 5: Select treatment, replication and dependent variable (e.g., Gain).
Step 6: Select a test for multiple comparisons,
such as the Least Significant Difference (LSD) test, to determine significant differences among groups.
Same as for Duncan’s New Multiple Range Test (DNMRT), Tukey’s HSD Test.
Step 7: Click submit, pay a nominal fee, and download the output report with detailed interpretation.
Output Report:
Link of the output report
Video Tutorial:
Link of the Youtube Tutorial
Quiz :
RCBD Quiz
Reference
Gomez, K. A., & Gomez, A. A. (1984). Statistical Procedures for Agricultural Research. John wiley & sons. 25-30.
The blog is written by:
Darshan Kothiya, PhD Scholar, Department of Agricultural Statistics, BACA, AAU, Anand