Design of Experiments and Analysis of Variance
Domain 7: Design of Experiments and Analysis of Variance
This domain tests the principles of experimental design and the arithmetic of one-way and randomised block ANOVA, ending in the F test for equality of treatment means.
Core rules
- Principles: randomisation removes allocation bias; replication supplies an error estimate and raises precision; local control (blocking) removes known nuisance variation.
- Designs: completely randomised design (CRD) for homogeneous units; randomised block design (RBD) with one blocking factor; Latin square design (LSD) with two blocking factors, needing k² units for k treatments.
- One-way model: x_ij = μ + τ_i + e_ij with e_ij independent N(0, σ²) - normality, equal variances, independence and additivity are the assumptions.
- Partition: TSS = Σ(x_ij − x̄..)²; SST = r Σ(x̄_i − x̄..)²; SSE = TSS − SST.
- Degrees of freedom: N − 1 total, k − 1 treatment, N − k error.
- F = MST/MSE = [SST/(k − 1)]/[SSE/(N − k)]; reject H₀: all τ_i = 0 when F exceeds the table value at (k − 1, N − k).
- In an RBD the block sum of squares r Σ(x̄.j − x̄..)² is removed first and the error df becomes (k − 1)(r − 1).
Worked example
Three treatments with four replications each; totals 36, 40, 44; Σx² = 1214; N = 12; grand total 120. Correction factor = 120²/12 = 1200, so TSS = 1214 − 1200 = 14. SST = 4[(9 − 10)² + (10 − 10)² + (11 − 10)²] = 8, hence SSE = 14 − 8 = 6. MST = 8/2 = 4, MSE = 6/9 = 0.667, F = 6.00. Since F₀.₀₅(2, 9) = 4.26, the treatment means differ significantly.
Traps
- Omitting the correction factor G²/N, which inflates every sum of squares.
- Using N − 1 instead of N − k for the error df.
- Running the F test as a two-tailed test.
- Ignoring the block SS in an RBD, which inflates MSE and can hide a real treatment effect.
- Taking several readings from one experimental unit and counting them as independent replications.