Published In
Journal of Statistical Planning and Inference
Document Type
Pre-Print
Publication Date
12-1-2026
Subjects
Experimental design, ANOVA, Degrees of freedom, Projection matrices, lias analysis
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 (df) partition theorem that resolves this ambiguity. For N observations, we show that the total information in the data (i.e., 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—not approximations—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 crossednested designs. We introduce practical diagnostics—the df-retention ratio ρ, df deficiency δ, and variance-inflation index α—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. The framework, therefore, offers both a unifying theory and immediately useful tools for planning and analyzing complex experiments.
Rights
© Copyright the author(s) 2026
Locate the Document
DOI
10.1016/j.jspi.2026.106409
Persistent Identifier
https://archives.pdx.edu/ds/psu/44584
Citation Details
Publshed as: K.G., N. (2026). A projector-rank partition theorem for exact degrees of freedom in experimental design. Journal of Statistical Planning and Inference, 245, 106409.
Description
This is the author’s version of a work that was accepted for publication. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published as: (2026). A projector-rank partition theorem for exact degrees of freedom in experimental design. Journal of Statistical Planning and Inference, 245.