A Field Guide to Experimental 
Designs

The Latin Square design

The Latin square design is used where the researcher desires to control the variation in an experiment that is related to rows and columns in the field.

Field marks:

  • Treatments are assigned at random within rows and columns, with each treatment once per row and once per column.
  • There are equal numbers of rows, columns, and treatments.
  • Useful where the experimenter desires to control variation in two different directions

Sample layout:
Different colors represent different treatments. There are 4 treatments (A-D) assigned to 4 rows (I-IV) and 4 columns (1-4).

Latin square sample layout

Row I     A   B   C   D
Row II    C   D   A   B
Row III   D   C   B   A
Row IV    B   A   D   C

Column    1   2   3   4

 

ANOVA table format:

Source of
variation
Degrees of
freedoma
Sums of
squares (SSQ)
Mean
square (MS)
F
Rows (R) r-1 SSQR SSQR/(r-1) MSR/MSE
Columns (C) r-1 SSQC SSQC/(r-1) MSC/MSE
Treatments (Tr) r-1 SSQTr SSQTr/(r-1) MSTr/MSE
Error (E) (r-1)(r-2) SSQE SSQE/((r-1)(r-2))  
Total (Tot) r2-1 SSQTot    
awhere r=number of treatments, rows, and columns.

Sample ANOVA table:

Source of
variation
Degrees of
freedom
Sums of
squares (SSQ)
Mean
square (MS)
F
Rows 3 40.77 13.59 5.91a
Columns 3 125.39 41.80 18.16a
Treatments 3 160.57 53.52 23.26a
Error 6 13.81 2.30  
Total 15 340.54    
aF test with 3,6 degrees of freedom at P=0.05 is 4.76

Sample SAS GLM statements:

PROC GLM;
  CLASS ROWS COLUMNS TREATS;
  MODEL WHATEVER = ROWS COLUMNS TREATS;
RUN;

Compare with:

 

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Wednesday, August 16, 2000