The precision of fractional factorial designs under randomization
Abstract
A regular fraction of a two-level factorial design runs fewer treatment combinations (cells), with more units in each. With the number of units fixed, more units per cell do not by themselves reduce the variance. Averaged over the fractions of one defining contrast subgroup, a fraction's variance is smaller than the full design's by the unit-to-unit variation of the aliased effects, a gain the usual variance estimator misses. Assuming the aliased effects are zero, we propose a full-design estimator that weights the fractions' contrasts by inverse variance. Its estimated variance never exceeds the saturated estimator's. In large samples, it is proportional to the harmonic mean of the fractions' average cell variances, and a fraction's, on average, to the arithmetic mean. We test this assumption, but choosing between the two estimators by the test can lose coverage. The results extend to unequal allocations, and the large-sample results to covariate adjustment.
Related
- D-Optimal Saturated Factorial Designs with Three Factors
- Data-Adaptive Rerandomization for 2K Factorial Designs
- Incorporating therapist capacity, availability and turnover in trials of therapist-delivered interventions: Non-orthogonal factorial designs
- Re-randomization-probability weighting for population-adjusted comparisons of maintenance outcomes across trial designs
- New Tools for Finite-Time Fuzzy Control of Fractional-Order Systems: Stability Lemmas, Composite Disturbance Observers, and Singularity-Free Designs
BibTeX
@misc{chen2026precision,
title = {{The precision of fractional factorial designs under randomization}},
author = {Xinyuan Chen},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01949},
eprint = {2611.01949},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01949},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The precision of fractional factorial designs under randomization
AU - Xinyuan Chen
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01949
UR - https://arxiv.org/abs/2611.01949
AB - A regular fraction of a two-level factorial design runs fewer treatment combinations (cells), with more units in each. With the number of units fixed, more units per cell do not by themselves reduce the variance. Averaged over the fractions of one defining contrast subgroup, a fraction's variance is smaller than the full design's by the unit-to-unit variation of the aliased effects, a gain the usual variance estimator misses. Assuming the aliased effects are zero, we propose a full-design estimator that weights the fractions' contrasts by inverse variance. Its estimated variance never exceeds the saturated estimator's. In large samples, it is proportional to the harmonic mean of the fractions' average cell variances, and a fraction's, on average, to the arithmetic mean. We test this assumption, but choosing between the two estimators by the test can lose coverage. The results extend to unequal allocations, and the large-sample results to covariate adjustment.
ER -