1. Latin Square Designs
KNNL – Sections

2. Description
Experiment with r treatments, and 2 blocking factors: rows (r levels) and columns (r levels)
Advantages:
Reduces more experimental error than with 1 blocking factor
Small-scale studies can isolate important treatment effects
Repeated Measures designs can remove order effects
Disadvantages
Each blocking factor must have r levels
Assumes no interactions among factors
With small r, very few Error degrees of freedom; many with big r
Randomization more complex than Completely Randomized Design and Randomized Block Design (but not too complex)

3. Randomization in Latin Square
Determine r , the number of treatments, row blocks, and column blocks
Select a Standard Latin Square (Table B.14, p. 1344)
Use Capital Letters to represent treatments (A,B,C,…) and randomly assign treatments to labels
Randomly assign Row Block levels to Square Rows
Randomly assign Column Block levels to Square Columns
4x4 Latin Squares (all treatments appear in each row/col):

4. Latin Square Model

5. Analysis of Variance

6. Post-Hoc Comparison of Treatment Means & Relative Efficiency

7. Comments and Extensions
Treatments can be Factorial Treatment Structures with Main Effects and Interactions
Row, Column, and Treatment Effects can be Fixed or Random, without changing F-test for treatments
Can have more than one replicate per cell to increase error degrees of freedom
Can use multiple squares with respect to row or column blocking factors, each square must be r x r. This builds up error degrees of freedom (power)
Can model carryover effects when rows or columns represent order of treatments