RANDOMIZED COMPLETE BLOCK DESIGN (RCBD) BY:

SITI AISYAH NAWAWI

Description of the Design

• RCBD is an experimental design for comparing a treatment in b blocks.

• The blocks consist of a homogeneous experimental unit.

• Treatments are randomly assigned to experimental units within a block, with each treatment appearing exactly once in every block.

Cont..

• So, complete mean that each block contain the all the treatments.

• Completely randomize block design mean that each block have all treatment and the treatments are randomize with the all block.

When is the design useful?

• The experimental unit is not homogeneous but can sort experimental unit into homogeneous group that we call block.

• An extraneous source of validity (nuisance factor) is present.

• The treatments are assigning at random the experimental unit within each block.

• Nuisance factor is a design factor that probably has an effect on the response, but we are not interested in that effect.

Cont..

• When the nuisance factor is known and controllable, blocking can be used to systematically eliminate its effect on the statistical comparisons among treatments.

Advantage and disadvantage

Advantages • Generally more precise than the

CRD. • No restriction on the number of

treatments or replicates. • Some treatments may be replicated

more times than others. • Missing plots are easily estimated.

Cont..

Disadvantages • Error df is smaller than that for the CRD

(problem with a small number of treatments).

• If there is a large variation between experimental units within a block, a large error term may result (this may be due to too many treatments).

• If there are missing data, a RCBD experiment may be less efficient than a CRD

Designing a simple RCBD experiment

For example, an agricultural scientists wants to study the effect of 4 different fertilizers (A,B,C,D) on corn productivity. He has three fields (1,2,3) ranging in size from 4-6 ha. Since this is a large experiment, 1 ha is devoted to each fertilizer type in each field. But, the fields have different crop histories, herbicide use, etc. Field identity is an extraneous variable (block)

!➢ Treatment : Types of fertilizer (A,B,C,D) ➢ Block : Fields (1,2,3) ➢ Experimental unit : Corn ➢ Dependent variable : Production of corn

• Randomization for block 1 First, find 4 three digit random number from random number table. Rank the random number from smallest to largest.

Random Number Ranking (experimental

Treatment

625 2 A

939 4 B

493 1 C

713 3 D

• Randomization for block 2 Find the next 4 three digit random number from random number table. Rank the random number from smallest to largest.

Random Number Ranking (experimental

Treatment

496 2 A

906 4 B

440 1 C

690 3 D

• Randomization for block 3 Find the next 4 three digit random number from random number table. Rank the random number from smallest to largest.

Random Number Ranking (experimental

Treatment

253 2 A

081 1 B

901 4 C

521 3 D

• The following table shows the plan of experiment with the treatments have been allocated to experimental units according to RCBD

!! experimental unit number

!

!Treatment

Block (Field)

1 2 3

A 2 1 2

B 4 4 1

C 1 2 4

D 3 3 3

Linear Model and the ANOVA

ANOVA table

!!!

!! *a= number of treatment * b= number of block

Hypothesis testing

• Testing the equality of treatment mean H0 : µ1= µ2=…= µa

H0 : At least one µi≠µj

!α = 0.05 Test Statistics : F0 (F calculated)

Critical value : Fα,(a-1),(a-1)(b-1)

Decision : Reject H0 if F calculated > F table

Conclusion :

Multiple comparison

Multiple comparison: Least Significant Difference(LSD) test

LSD compares treatment means to see whether the difference of the observed means of treatment pairs exceeds the LSD numerically. LSD is calculated by !!!where is the value of Student’s t (2-tail)with error df at 100 % level of significance, n is the no. of replication of the treatment. For unequal replications, n1 and n2 LSD=

bMSEt ba

2)1)(1(,2/ −−α

t 2/α α

)11(21

)1)(1(,2/ bbt MSEba +×−−α

Multiple comparison: Tukey’s test

Compares treatment means to see whether the difference of the observed means of treatment pairs exceeds the Tukey’s numerically. Tukey’s is calculated by !Where f is df error .

bMSEfaT q ),(

αα =

Example

An agricultural scientists wants to study the effect of 4 different fertilizers (A,B,C,D) on corn productivity. He has six fields (1,2,3,4,5,6) ranging in size from 4-6 ha. Since this is a large experiment, 1 ha is devoted to each fertilizer type in each field. But, the fields have different crop histories, herbicide use, etc. Field identity is an extraneous variable (block)

Cont..

1) State treatment, block and experimental unit. Treatment : Types of fertilizer (A,B,C,D) Block : Fields (1,2,3,4,5,6) Experimental unit : Corn Dependent variable : Production of corn(KG)

Cont..

Types of fertilizer

Batch of Resin (Block) Treatment Total

Average

1 2 3 4 5 6

A 90.3 89.2 98.2 93.9 87.4 97.9 556.9 92.82

B 92.5 89.5 90.6 94.7 87.0 95.8 550.1 91.68

C 85.5 90.8 89.6 86.2 88.0 93.4 533.5 88.92

D 82.5 89.5 85.6 87.4 78.9 90.7 514.6 85.77

B l o c k Totals

350.8 359.0 364.0 362.2 341.3 377.8 2155.1

Cont..

2)Write down the linear statistical model for this experiment and explain the model terms?

Cont..

2)Calculate the analysis of variance manually and construct the table?

Source of variation

Sum of Squares

Degrees of Freedom

Mean Square

F

Fertilizer 178.17 3 59.39 8.11

Block 192.25 5 38.45  

Error 109.89 15 7.33  

Total 480.31 23    

Cont..

3)Test the hypothesis H0: All fertilizers give the same mean corn production (types of fertilizer do not

affects the mean corn production) H1: At least two fertilizers give different mean corn production (fertilizer affects

the mean corn production)

!α = 0.05

!Test Statistics : 8.11 Critical value : F0.05,3,15 =3.29

Decision : Reject H0 if F calculated > F table

!Conclusion: There is significant difference among the fertilizer on mean yield

!

Model comparison

• Model comparison