The Cochran and Cox Method
Cochran and Cox's Experimental Designs (1957) is the foundational reference for designed agricultural experiments. The book lays out the algebra and the practical layout for the entire family of randomised, blocked and split designs that still dominates field trial analysis. This page covers the parts of Cochran and Cox most relevant to a split-plot analysis.
Where Cochran and Cox sit in design history
Cochran and Cox systematised the agricultural design literature begun by Fisher and Yates in the 1920s and 1930s. They introduced explicit ANOVA partitions for split-plot, split-split-plot, strip-plot, lattice (balanced and quasi), confounded factorials, and incomplete block designs. The book is still cited in modern agronomy because the partitions and the sketch-by-hand layouts have not changed in 60 years; only the software has.
Worked split-plot example
The classical Cochran and Cox split-plot example places main-plot factor A (commonly irrigation) on whole plots and sub-plot factor B (commonly variety or fertiliser) on sub plots within each whole plot. The randomization is hierarchical, the ANOVA has two error terms (error a for main plots, error b for sub plots), and the F tests use the appropriate error each time. The exact algebra appears in Chapter 7 of the 1957 edition.
Pooling errors and reading the F ratios
Cochran and Cox give explicit guidance on when to pool error terms across an experiment and when to keep them separate. Pooling reduces df and increases power but is only valid when the variances are statistically homogeneous. Modern practice tends to be more conservative and avoids pooling unless a homogeneity-of-variance test (Bartlett, Levene) clearly accepts equality. The F tests in a split-plot read like a standard ANOVA table but with two rows of error: error a for the main effect of A, error b for the main effect of B and the A by B interaction.