A projector-rank partition theorem for exact degrees of freedom in experimental design

Abstract

In many experimental designs -- split-plots, blocked or nested layouts, fractional factorials, and studies with missing or unequal replication -- standard ANOVA procedures no longer tell us exactly how many independent pieces of information each effect truly contributes. We provide a general degrees of freedom $(\mathrm{df})$ partition theorem that resolves this ambiguity. For $N$ observations, we show that the total information in the data (i.e., $N-1$ $\mathrm{df}$) can be split exactly across experimental effects and randomization strata by projecting the data onto each stratum and counting the $\mathrm{df}$ each effect contributes there. This yields integer $\mathrm{df}$ -- not approximations -- for any mix of fixed and random effects, blocking structures, fractionation, or imbalance. This result yields closed-form $\mathrm{df}$ tables for unbalanced split-plot, row-column, lattice, and crossed-nested designs. We introduce practical diagnostics -- the $\mathrm{df}$-retention ratio $\rho$, df deficiency $\delta$, and variance-inflation index $\alpha$ -- that measure exactly how many $\mathrm{df}$ an effect retains under blocking or fractionation and the resulting loss of precision, thereby extending Box-Hunter's resolution idea to multi-stratum and incomplete designs. Classical results emerge as corollaries: Cochran's one-stratum identity; Yates's split-plot $\mathrm{df}$; resolution-$R$ identified when an effect retains no $\mathrm{df}$. Empirical studies on split-plot and nested designs, a blocked fractional-factorial design-selection experiment, and timing benchmarks show that our approach delivers calibrated error rates, recovers information to raise power by up to 60% without additional runs, and is orders of magnitude faster than bootstrap-based $\mathrm{df}$ approximations.