Randomized Complete Block Design (RCBD)

Definition

A randomized complete block design is an experimental design for comparing t treatments in b blocks. The blocks consist of t homogeneous experimental units. Treatments are randomly assigned to experimental units within a block, with each treatment appearing exactly once in every block. The design is called complete because we observe the complete set of treatments within every block.

Model

The model for an observation in a randomized complete block design can be written in the form

Data

Data for a randomized complete block design

Hypothesis Testing

Treatments

\(H_{0}:\mu_{1}=\mu_{2}=...=\mu_{t}\) (treatment has no effect on response)

\(H_{1}:\) Not all \(\mu_{i}\)’s are equal
Block

\(H_{0}:\beta_{1}=\beta_{2}=...=\beta_{b}\) (block has no effect on response)

\(H_{1}:\) Not all \(\beta_{j}\)’s are equal

Analysis of variance table for a randomized complete block design:

with

The notation is presented here.

Advantages and Disadvantages

Advantages

The design is useful for comparing t treatment means in the presence of a single extraneous source of variability.
The statistical analysis is simple.
The design is easy to construct.
It can be used to accommodate any number of treatments in any number of blocks.

Disadvantages

Because the experimental units within a block must be homogeneous, the design is best suited for a relatively small number of treatments.
This design controls for only one extraneous source of variability (due to blocks). Additional extraneous sources of variability tend to increase the error term, making it more difficult to detect treatment differences.
The effect of each treatment on the response must be approximately the same from block to block.

Example

A researcher conducted an experiment to compare the effects of three different insecticides on a variety of string beans. To obtain a sufficient amount of data, it was necessary to use four different plots of land. Since the plots had somewhat different soil fertility, drainage characteristics, and sheltering from winds, the researcher decided to conduct a randomized complete block design with the plots serving as the blocks. Each plot was subdivided into three rows. A suitable distance was maintained between rows within a plot so that the insecticides could be confined to a particular row. Each row was planted with 100 seeds and then maintained under the insecticide assigned to the row. The insecticides were randomly assigned to the rows within a plot so that each insecticide appeared in one row within all four plots. The response yij of interest was the number of seedlings that emerged per row. The data and means are given below.

Write an appropriate statistical model for this experimental situation.
Run an analysis of variance to compare the effectiveness of the three insecticides. Use alpha 0.05. Summarize your results in an AOV table.

Answer:

We recognize this experimental design as a randomized complete block design with b = 4 blocks (plots) and t = 3 treatments (insecticides) per block. The appropriate statistical model is:
Hypothesis Testing and AOV Table

I.Hypotheses:

\(H_{0}:\mu_{1}=\mu_{2}=\mu_{3}\)

\(H_{1}:\) at least one \(\mu_{i}\) is different from the rest makes

II.Significance Level: 5%

III.Test Statistics

Complete ANOVA Table:

Complete ANOVA Table: (Using R)

#Data
library(readxl)
example1<-read_excel("D:/MATERI KULIAH S2 IPB/ASPRAK 1/RCBD.xlsx")
example1

## # A tibble: 12 x 3
##    block  treatments    respon
##    <chr>  <chr>          <dbl>
##  1 plot_1 insecticide_1     56
##  2 plot_2 insecticide_1     48
##  3 plot_3 insecticide_1     66
##  4 plot_4 insecticide_1     62
##  5 plot_1 insecticide_2     83
##  6 plot_2 insecticide_2     78
##  7 plot_3 insecticide_2     94
##  8 plot_4 insecticide_2     93
##  9 plot_1 insecticide_3     80
## 10 plot_2 insecticide_3     72
## 11 plot_3 insecticide_3     83
## 12 plot_4 insecticide_3     85

#anova table

anovatable<-aov(respon~treatments+block, data=example1)
summary(anovatable)

##             Df Sum Sq Mean Sq F value   Pr(>F)    
## treatments   2   1832   916.0  211.38 2.74e-06 ***
## block        3    438   146.0   33.69 0.000377 ***
## Residuals    6     26     4.3                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

IV.Rejection Region:

Reject H0 if \(F_{test}>F_{2;6;0.05}=5.14\) .

V.Conclusion

Because F_test=211.38>F_table=5.14, H0 is rejected. Therefore, there is a difference in the mean number of seedlings among the three insecticides.

Excercise

A horticulturist is designing a study to investigate the effectiveness of five methods for the irrigation of blueberry shrubs. The methods are surface, trickle, center pivot, lateral move, and subirrigation. There are 10 blueberry farms available for the study representing a wide variety of types of soil, terrains, and wind gradients. The horticulturist wants to use each of the five methods of irrigation on all 10 farms to moderate the effect of the many extraneous sources of variation that may impact the blueberry yields. Each farm is divided into five and the response variable will be the weight of the harvested fruit from each of the plots of blueberry shrubs.

Run an analysis of variance to compare the effectiveness of five methods for the irrigation of blueberry shrubs. Use alpha 0.05. Summarize your results in an AOV table.