Latin Square Designs
Introduction
A Latin square design blocks on two nuisance factors simultaneously: rows and columns of a \(t \times t\) grid, with \(t\) treatments arranged so each treatment appears once in each row and once in each column. It is a highly efficient design when two blocking factors are present.
Prerequisites
Completely randomised design; blocking.
Theory
Model: \(y_{ijk} = \mu + \rho_i + \gamma_j + \tau_k + \varepsilon_{ijk}\), with row, column, and treatment effects. Requires \(t^2\) experimental units (one per cell).
Benefits: \(t\) treatments replicated \(t\) times with only \(t^2\) units; controls for row and column nuisance variation.
Assumptions
No interactions between treatment, rows, and columns; homogeneous errors; design is square (\(t = t\)).
R Implementation
library(agricolae)
set.seed(2026)
t <- 4
trts <- LETTERS[1:t]
ls_design <- design.lsd(trts, seed = 2026)
book <- ls_design$book
print(book)
# Simulate outcomes
book$y <- rnorm(t^2, mean = c(10, 12, 14, 11)[as.integer(book$trts)], sd = 1.5)
fit <- aov(y ~ row + col + trts, data = book)
summary(fit)Output & Results
Latin square layout and ANOVA partitioning variation into row, column, treatment, and residual.
Interpretation
“The Latin square ANOVA partitioned the 16 observations’ variation into row, column, and treatment effects; the treatment F-test (F(3, 6) = 7.2, p = 0.02) was significant with row blocking accounting for 20 % of total variation.”
Practical Tips
- Latin squares have low residual df with small \(t\); consider replication or a larger design for small effects.
- Analysis assumes no row-treatment or column-treatment interaction; confirm this from the context.
- For \(t > 7\), Latin squares become unwieldy; consider incomplete block designs.
- Graeco-Latin squares extend to a third blocking factor.
- Randomly permute rows, columns, and treatment labels for a valid Latin square realisation.