Latin Square Designs

1. Latin Square Designs KNNL – Sections 28.3-28.7

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