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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2506.01619 |
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| _version_ | 1866912976890494976 |
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| author | G, Nagananda K |
| author_facet | G, Nagananda K |
| contents | 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 $ρ$, df deficiency $δ$, and variance-inflation index $α$ -- 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01619 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A projector-rank partition theorem for exact degrees of freedom in experimental design G, Nagananda K Statistics Theory 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 $ρ$, df deficiency $δ$, and variance-inflation index $α$ -- 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. |
| title | A projector-rank partition theorem for exact degrees of freedom in experimental design |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2506.01619 |