Randomized Complete Block Design (RCBD) - R Program

Randomized Complete Block Design is a standard design in which experimental units are grouped in to blocks or replicates. This is completely different from the randomized complete design. In this case each replicate is randomized separately and each treatment has the same probability of being assign to a given experimental unit within a replicate.



Written by: Pawani Liyanaarachchi ,BSc (Hons) in Applied science (statistics), University of Sri Jayewardenepura



In this tutorial, we will study followings.

  • Introduction of the Randomized Complete Block Design
  • Statistical Model for the Randomized Complete Block Design
  • Hypothesis Testing for Randomized Complete Block Design
  • Anova table
  • Example problem for Randomized Complete Block Design


Introduction of the Randomized Complete Block Design

  • The similar experimental units are grouped into blocks or replicates so that the observed differences are largely due to true differences between treatments.
  • Units within the blocks are homogeneous and between blocks are heterogeneous.
  • Treatments are assigned at random to the object in the blocks in each block.
randomized complete Block Design

Statistical Model for the Randomized Complete Block Design

Mean Model

Yij = µij + εij { i=1,2.....,a , j=1,2,.....,b


  • a : number of treatments
  • b : number of blocks
  • µij : mean of the ith treatment of jth block
  • εij : random error
  • Here εij ~ NID (0,σ2)

Effect Model

Yij = µ + τi + βj + εij { i=1,2,....a , j=1,2,.....b

  • a : number of treatments
  • b: number of blocks
  • µ : overall mean
  • Here εij ~ NID(0,σ2).
  • τi : effect of the ith treatment
  • βj : effect of the jth block
  • εij : random error
  • Further Σ1i=1τi = 0 and Σ1i=1βj = 0


Hypothesis Testing for Randomized Complete Block Design

The interested hypothesis is for the mean model :

H0 : µ1 = µ2 = ...... µk vs H1 : µi ≠ µj



The interested hypothesis is for the effects model :

H0 : τ1 = τ2 = ...... τk = τi ≠ 0 for at least one i





Notations

  • Yij : observation in ith treatment and jth block
  • Yi. : total of all observations taken under treatment i → yi. = Σbj=1yij
  • Y.j : total of all observations taken under block j → y.j = Σai=1yij
  • y.. : grand total of all observations
  • N = ab is the total number of observations


Anova table for Randomized Complete Block Design

Anova table for randomized Complete block design

ANOVA - Computing Formulas

Eqauations for anova table

Equation for anova table

Equation for anova table

Example problem for Randomized Complete Block Design

A chemical engineer wants to test the effectiveness of four chemical agents on the strength of a particular type of cloth. Because there might be variability from one bolt to another, the chemist decides to use a randomized block design, with the bolt of cloth considered as blocks. Four bolts are selected and applied all chemicals in random order to each bolt . Estimate the result.


Data table for randomized complete block design

According to the above theoretical model and also using R code (in R studio platform) we can solve this problem.



R Program for Randomized Complete Block Design (RCBD) - R Program

Download R Program for Randomized Complete Block Design (RCBD)

  1. ## setup framework for blocks and treatments
  2. Chemical_type<-c(rep("1",4),rep("2",4),rep("3",4),rep("4",4))
  3. Bolt<-c(rep(c("O1","O2","O3","O4"),4))
  4. ##Assign data to variable "strength"
  5. strength<-c(73,68,71,67,73,67,72,70,75,68,73,68,73,71,75,69)
  6. y<-data.frame(Chemical_type,Bolt,strength)
  7. matrix(y$Bolt,4,byrow = T)
  8. matrix(y$strength,4,4,byrow = T)
  9. ## Apply a linear model
  10. yfit<-lm(strength~Bolt+Chemical_type,y)
  11. ## Compute the analysis of variance table
  12. anova(yfit)


Results of Randomized Complete Block Design (RCBD) - R Program

result of randomized complete block design example




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