2025/06/02 by G, Nagananda K
#FOS: Mathematics #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.2506.01619
In many experimental designs\textemdash split-plots, blocked or nested layouts, fractional factorials, and studies with missing or unequal replication\textemdash standard ANOVA procedures no longer tell us exactly how many independent pieces of information each effect truly carries. We provide a general degrees of freedom (df) partition theorem that resolves this ambiguity. For N observations, we show that the total information in the data (\ie, N-1 df) can be split exactly across experimental effects and randomization strata by projecting the data onto each stratum and counting the df each effect contributes there. This yields integer df\textemdash not approximations\textemdash for any mix of fixed and random effects, blocking structures, fractionation, or imbalance. This result yields closed-form df tables for unbalanced split-plot, row-column, lattice, and crossed-nested designs. We introduce practical diagnostics\textemdash the df-retention ratio ρ, df deficiency δ, and variance-inflation index α\textemdash that measure exactly how many 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 df; resolution-R identified when an effect retains no 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 df approximations.