Latin Square Designs

Latin Square Designs

• The experimenter is concerned with a single factor having p levels. However, variability from two other sources can be controlled in the experiment. 
• If we can control the effect of these other two variables by grouping experimental units into blocks having the same number of treatment levels as the factor of interest, then a latin square design may be appropriate.  
• Consider a square with p rows and p columns corresponding to the p levels of each blocking variable. If we assign the p treatments to the rows and columns so that each treatment appears exactly once in each row and in each column, then we have a p × p latin square design. 
• It is called a “latin” square because we assign “latin” letters A, B, C, . . . to the treatments. Examples of a 4 × 4 and a 6 × 6 latin square designs are

  A	C	B	D
  D	A	C	B
  B	D	A	C
  C	B	D	A

  A	B	C	D	E	F
  B	C	D	E	F	A
  C	D	E	F	A	B
  D	E	F	A	B	C
  E	F	A	B	C	D
  F	A	B	C	D	E

  
  
• By blocking in two directions, the MSE will (in general) be reduced. This makes detection of significant results for the factor of interest more likely. 
• The experimental units should be arranged so that differences among row and columns represent anticipated/potential sources of variability
• In industrial experiments, one blocking variable is often based on units of time. The other blocking variable may represent an effect such as machines or operators.Column

  1	A	C	B	D
  2	D	A	C	B
  3	B	D	A	C
  4	C	B	D	A

  
    
         Operator

  1		2	3	4	5
  M	A	B	C	D	E
  Tu	B	C	D	E	A
  W	C	D	E	A	B
  Th	D	E	A	B	C
  F	E	A	B	C	D

  
  
      
• In agricultural experiments, the experimental units are subplots of land. We would then have the subplots laid out so that soil fertility, moisture, and other sources of variation in two directions are controlled.
• In greenhouse experiments, the subplots are often laid out in a continuous line. In this case, the rows may be blocks of p adjacent subplots and the columns specify the order within each row block. 
  
  

  Rep.  1

  Rep.  2

  Rep.  3

  Rep.  4

  Rep.  5

  Rep. 6

  Rep. 7

  A

  D

  G

  B

  E

  C

  F

  G

  F

  B

  C

  A

  E

  D

  B

  C

  D

  G

  F

  A

  E

  E

  G

  A

  D

  C

  F

  B

  C

  B

  F

  E

  D

  G

  A

  F

  E

  C

  A

  B

  D

  G

  D

  A

  E

  F

  G

  B

  C

  
   After converting Rep. 1 to Rep. 7 into row blocks, we get
  
  

  A	D	G	B	E	C	F
  G	F	B	C	A	E	D
  B	C	D	G	F	A	E
  E	G	A	D	C	F	B
  C	B	F	E	D	G	A
  F	E	C	A	B	D	G
  D	A	E	F	G	B	C

  
  
• Like the RCBD, the latin square design is another design with restricted randomization. Randomization occurs with the initial selection of the latin square design from the set of all possible latin square designs of dimension p and then randomly assigning the treatments to the letters A, B, C, . . .. The following notation will be used:

  p = the number of treatment levels, row blocks and column blocks.
  y_ijk = the observation for the j^th treatment within the i th row and k th column.
  N =  p² = total number of observations
  y··· = the sum of all p² observations
  R_i = the sum for row block i
  C_k = the sum for column block k
  T_j = the sum for treatment j
  ȳ·· = the grand mean of all observations = y···/p²
  ȳ_i = the i^th row block mean 

  ȳ_...k = the k^th column block mean

  ȳ_..j· = the j^th treatment mean

  
  
• There are three subscripts (i, j, k), but we only need to sum over two subscripts. 
• The standard statistical model associated with a latin square design is: 


y_ijk = µ + α_i + τ_j + β_k + ε_{ijk                          _____________} (18)

where µ is the baseline mean, α_i is the block effect associated with row i , β_k is the block effect associated with column k , τ_j is the j th treatment effect, and ε_ijk is a random error which is assumed to be IIDN(0, σ²
). 

To get estimates for the parameters in (18), we need to impose three constraints. If we assume \[\sum\limits_{i=1}^{p}{{{\alpha }_{i}}}=0\] , \[\sum\limits_{j=1}^{p}{{{\τ }_{j}}}=0\] , \[\sum\limits_{k=1}^{p}{{{\beta }_{k}}}=0\]  then the least squares estimates are 

µ̂ =                                            α̂_i =                                    β̂_k =                           τ̂_j = 

Substitution of the estimates into (18) yields

y_ijk  = µ̂ + α̂_i + τ̂_j + β̂_k  + ε_ijk 

 = ȳ··· + (ȳi·· − ȳ···) + (ȳ·j· − ȳ···) + (ȳ··k − ȳ···) + ε_ijk (19)

where eijk is the ijkth residual from a latin square design and 

eijk =