The Randomized Complete Block Design (RCBD), Exercises of Design

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The Randomized Complete Block

Design (RCBD)

Trudi Grant Department of Horticulture and Crop Science OARDC, The Ohio State University 2010

• The objective of this tutorial is to give a brief introduction to the design of a randomized complete block design (RCBD) and the basics of how to analyze the RCBD using SAS.

The field or space is divided into uniform units to account for any variation so that observed differences are largely due to true differences between treatments.

Treatments are then assigned at random to the subjects in the blocks-once in each block

The defining feature of the Randomized Complete Block Design is that each block sees each treatment exactly once

Advantages of the RCBD

Generally more precise than the completely randomized design (CRD).

No restriction on the number of treatments or replicates.

Some treatments may be replicated more times than others.

Missing plots are easily estimated.

The Layout of the Experiment

• Choose the number of blocks (minimum 2)
  • e.g. 4
• Choose treatments (assign numbers or letters for each) - e.g. 6 trt – A,B, C, D, E, F

1

F

E

D

C

B

A

2 3 4 Blocks

Treatments

The number of blocks is the number of replications

Any treatment can be adjacent to any other treatment, but not to the same treatment within the block

Treatments are assigned at random within blocks of adjacent subjects, each treatment once per block.

Image credit: Francis Lab, The Ohio State University

1. Column A – list of blocks
   2. Column B Enter =rand() to generate a random number
      3. Copy and paste command in remaining cells

4. Select cells
   5. Select data then select sort 6. Then sort by column with random numbers

Output in excel showing randomized blocks in first column. This is repeated for each block to randomize the treatments

The SAS System 14:30 Monday, August 4, 2008 3 Obs block trt 12 22 BC 34 22 AD 56 22 EF 78 11 BC 109 11 EA 1112 11 FD 1314 33 DA 1516 33 CF 1718 33 BE 1920 44 AF 2122 44 BC 2324 44 DE

SAS output showing randomized blocks and treatments

1

F

E

D

C

A

2 3 4

B

E

B

C

A

D

D

F

A

E

F

E

D

C C

A B

F

B Experimental design showing randomized blocks and treatments

Image credit: Francis Lab, The Ohio State University

Model for RCBD

• Yij - any observation for which i is the treatment factor j is the blocking factor
• μ - the mean
• T i - the effect for being in treatment i
• B j is the effect for being in block j

ANOVA table

Source Degrees of Freedom squares (SS)^ Sums of Mean squares F

Blocks b-1 Block SS BMS=BSS/b-1 BMS/ RMS

Treatment t-1 Treatment SS TMS=TSS/t-1 TMS/ RMS

Residual (t-1)(b-1) Residual SS RMS=RSS/ (t-1)(b-1) Total tb-1 SS Total t=number of treatments, b=number of blocks GM = grand mean, BM = block mean and TM= treatment mean BSS = Sum (BM-GM)^2 TSS = Sum (TM-GM)^2 RSS = Sum (V-BM-TM+GM)^2

SAS Editor

The program steps are determined by the experimental design, how you collected your samples and how you want your data presented.

SAS code for Analysis of RCBD

Sample SAS GLM statements:

PROC GLM; CLASS BLOCKS TREATS; MODEL WC = BLOCKS TREATS; RUN;