Factorial design 2026-10-07
A factorial design includes every combination of specified factor levels. Balanced replication permits separate estimation of marginal effects and interaction terms. The factor allocation, not the number of subsamples, determines the appropriate error term for each effect.
Both factors are assigned at orchard level. Therefore the experimental units are the twelve orchards; the trees are observational units within them. The six combinations form a balanced factorial design, replicated twice. The orchard ANOVA stratum has statistical degrees of freedom. Spray uses , pruning uses , and their interaction term uses , leaving six for error.
Dividing each treatment sum of squares in ANOVA by its statistical degrees of freedom and using as the denominator gives all missing entries:
Orchard sourceDegrees of freedomMean squareVariance ratio, one significant figure
Spray19984
Pruning25602
Spray by pruning22020.8
Residual6240Not applicable
The unrounded F-test statistics are , and . The within-orchard tree mean square in ANOVA, 180, is not the treatment error denominator: using it would confuse subsampling with independent replication. The tree ANOVA stratum has statistical degrees of freedom; is the uncorrected total, and the corrected total is 359.