Schematic layouts of some CRD experiments.
1) CRD: Treatment levels are t=5 where each treatment occurs in 3 cells (experimental units : n=3) and each cell is sampled once.  CRD with experimental error.

A

A

D

B

C

B

D

A

E

E

D

C

B

E

C


Model

Yij = m + ti  +  eij 

Source

Expected Mean Squares

   Treatments

s2   +  n s2t     

   Error

s2

 

2) CRD: Treatment levels are t=5 where each treatment occurs in 3 cells (experimental units : n=3) and each cell is sampled twice (sampling units : d=2).  CRD with both experimental error and sampling error. 

 

A       A

A       A

D       D

B       B

C       C

B       B

D       D

A       A

E       E

E       E

D       D

C       C

B       B

E       E

C       C

 

Model

Yijk = m + ti  + dij  +  eijk

Source

Expected Mean Squares

  Treatment

 s2   +  ns2d  +  nds2t     

  Experimental Error

 s2   +  ns2d

  Sampling Error

 s2 

 

3) CRD?: Treatment levels=5; each treatment occurs in one cell (experimental unit) and each cell is sampled three times.  CRD with no experimental error, this experiment has sampling error only (Pseudo-replication). 

 

C         C         C

A         A         A

D         D         D

E         E         E

B         B         B

 

Model

Yijk = m + ti   +  eijk  

Source

Expected Mean Squares

  Treatments

s2   +  ns2d  +  nds2t     

  Sampling Error

s2 

 

Schematic layouts of some RBD experiments. 

 4) RBD: Three blocks (b=3) and five treatment levels (t=5) where each block has 5 cells (experimental units) and each treatment occurs once in each block and each is sampled once.  RBD with experimental error only 

 

Block 1

Block 2

Block 3

C

B

D

A

C

B

D

A

E

E

D

A

B

E

C


Model

Yij = m + ti  + bj  +  eij  

Source

Expected Mean Squares

  Treatment

 s2   +  nbs2t     

  Block

 s2   +  nts2b

  Error

 s2 

 
5) RBD: Three blocks (b=3) and five treatment levels (t=5) where each block has 5 cells, and each treatment occurs once in each block and each cell is sampled twice (d=2).   In this case the residual error term measures only within cell variation. 

Block 1

Block 2

Block 3

C      C

B      B

D      D

A      A

C      C

B      B

D      D

A      A

E      E

E      E

D      D

A      A

B      B

E      E

C      C

 

Model

Yijk = m + ti  +  bj   + tbij  +  eijk

Source

Expected Mean Squares

Treatment

 s2   +  ns2tb  +  nds2t     

   Block

 s2   +  ns2tb  +  nds2b     

   Block*Treatment

 s2   +  ns2tb

   Error

 s2 

6) RBD: Blocks=3, Treatment levels=5; each block has 10 cells and each treatment occurs twice in each block.  Each cell (treatment by block and within block replicate combination) is sampled once. In this case the residual error term measures between cell variation.

Block 1

Block 2

Block 3

D

C

A

E

C

A

A

E

B

A

D

B

A

E

E

C

A

E

B

C

C

D

D

B

D

B

B

D

E

C

Model

Yijk = m + ti  +  bj   + tbij  +  eijk

Source

Expected Mean Squares

   Treatment

 s2   +  ns2tb  +  nds2t     

   Block

 s2   +  ns2tb  +  nds2b     

   Block*Treatment

 s2   +  ns2tb

   Error

 s2 

Layouts of some other experiments.
7) CRD Split-plot : Main plot treatment levels are t1=5, sub-plot treatment levels t2=2 where each main plot treatment occurs in d=3 cells (experimental units) and each cell is subdivided into two parts (d=2) for one application of each level of the sub-plot treatment.

E2  |  E1

B2  |  B1

D1  |  D2

A2  |  A1

C1  |  C2

B1  |  B2

D1  |  D2

B1  |  B2

C2  |  C1

E2  |  E1

D2  |  D1

A1  |  A2

A1  |  A2

E2  |  E1

C1  |  C2

Model

Yijk = m + t1i   +  dij  + t2k  + t1t2ik +  eijk 

Source

Expected Mean Squares

  Treatment

 s2   +  nds2d   +  ns2t1t2  +  nd t2s2t1  

  Error a

 s2   + t2s2d

  SubPlot Treatment

 s2   +  ns2t1t2 +  nt1s2t2     

  Interaction

 s2   +  ns2t1t2     

  Error b

 s2 

 8) RBD: Blocks=3, Treatment Levels=5; each block has 10 cells and each treatment occurs twice in each block.  Each cell (treatment by block combination) is sampled twice. 


Block

 

Block

 

Block

 

D     D

C     C

A     A

E     E

C     C

A      A

A     A

E     E

B     B

A     A

D     D

B      B

A     A

E     E

E     E

C     C

A     A

E      E

B     B

C     C

C     C

D     D

D     D

B      B

D     D

B     B

B     B

D     D

E     E

C      C

Model

Yijk = m + ti  +  bj   + tbij   + dij   +  eijl

Source

Expected Mean Squares

   Treatment

s2   +  nst2  +  nsstb2 +  nsbst2     

   Block

s2   +  nsg2  +  nsstb2 +  nstsb2     

   Treatment*Block

s2   +  nsg2  +  nsstb2 

   Rep(Treatment*Block)

s2   +  nsg2 

   Error

s2 


 9) Latin Square Design (LSD): Three rows (r=3) and three columns (c=3) for three treatment levels (t=3) where each treatment occurs in each row and column.  Experimental units are row by column combination cells.  There could also be several separate squares (blocks) and several sampling units within cells for Latin Square Designs.

C

B

A

A

C

B

B

A

C

Model

Yijk = m + ri  +  bj  + tk  +  eijk  

Source

Expected Mean Squares

   Row

 s2   +  nrs2r

   Column

 s2   +  nrs2b

   Treatment

 s2   +  nr s2t     

   Error

 s2 

 



Modified: August 16, 2004
James P. Geaghan