Latin Square Designs KNNL – Sections 28.3-28.7

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)

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):

Latin Square Model

Analysis of Variance

Post-Hoc Comparison of Treatment Means & Relative Efficiency

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