Randomized Complete

Block Design

NORBERTO E. MILLADepartment of Statistics

Visayas State Universitybertmilla@vsu.edu.ph

Outline

1. Basic features of RCBD experiments

2. Analysis of variance for RCBD experiments

3. Some examples

Randomized Complete Block Design

one of the most widely used experimental designs.

especially suited for field experiments where the number of treatments is not large and there exists a conspicuous factor based on which homogenous sets of experimental units can beidentified

Randomized Complete Block Design

primary distinguishing feature of the RCBD is the presence of blocks of equal size, each of which contains all the treatments

experimental units are grouped into blocks such that variability within each block is minimized and variability among blocks is maximized

Randomized Complete Block Design

reduce the experimental error by eliminating the contribution of known sources of variation among the experimental units

blocking is most effective when the experimental area has a predictable pattern of variability.

blocks should perpendicular to the direction of the gradient

Randomized Complete Block Design

reduce the experimental error by eliminating the contribution of known sources of variation among the experimental units

blocking is most effective when the experimental area has a predictable pattern of variability.

blocks should perpendicular to the direction of the gradient

Randomized Complete Block Design

Consider a field experiment involving 6 treatments (A thru F) and 3 replications

gradient

A B C

C C E

B E F

D A B

E F D

F D A

REP 1 REP 2 REP 3

ANOVA for RCBD

There are three sources of variability in a RCBD : treatment, replication (or block) and experimental error

𝑌𝑖𝑗 = 𝜇 + 𝜏𝑖 + 𝛽𝑗 + 𝜀𝑖𝑗Where: 𝑌𝑖𝑗=Response

µ=Overall mean𝜏𝑖=Treatment effect𝛽𝑗=Block effect

𝜀𝑖𝑗=Random variation

ANOVA for RCBD

Variation due to treatment

Variation due to blocking factor

Random variation (Experimental error)

Source of Variation SS df MS F

Treatment SSTr t-1 MSTr MSTr/MSE

Block SSB b-1 MSB MSB/MSE

Error SSE (t-1)(b-1) MSE

Total SST n-1

An example

Consider an an experiment, wherein eightprovenances of Gmelina arborea werecompared with respect to the girth atbreast-height (gbh) of the trees attainedafter 6 years of planting

An example

Treatment Rep 1 Rep 2 Rep 3

1 30.85 38.01 35.1

2 30.24 28.43 35.93

3 30.94 31.64 34.95

4 29.89 29.12 36.75

5 21.52 24.07 20.76

6 25.38 32.14 32.19

7 22.89 19.66 26.92

8 9.44 24.95 37.99

An example

ANOVA for RCBD in Stata

Menu: Statistics>Linear models and related>ANOVA/MANOVA

>Analysis of variance and covariance

ANOVA for RCBD in Stata

Command: anova gbh treatment rep

Interpretations:

1. There is significant block effect. Blocking is effective.

2. There is significant difference in the mean gbh among the six treatments.

Total 678.41892 23 29.496475

Residual 140.98418 14 10.070299

rep 110.97981 2 55.489907 5.51 0.0172

treatment 426.45492 7 60.922131 6.05 0.0021

Model 537.43473 9 59.71497 5.93 0.0017

Source Partial SS df MS F Prob>F

ANOVA for RCBD (post hoc)

Command: pwcompare treatment,mcompare(tukey) groups

the 5% level.

label are not significantly different at

Note: Margins sharing a letter in the group

8 30.79333 1.832148 ABC

7 23.15667 1.832148 BC

6 29.90333 1.832148 ABC

5 22.11667 1.832148 C

4 31.92 1.832148 AB

3 32.51 1.832148 A

2 31.53333 1.832148 AB

1 34.65333 1.832148 A

treatment

Margin Std. Err. Groups

Tukey