Factorial and Design-of-Experiments (DOE)

When several factors might affect a response, testing them one factor at a time (OFAT) is both wasteful and blind to interactions. Factorial designs vary factors together, so you estimate every main effect and interaction from the same runs, with better precision per run. This file covers the family of DOE designs and the concepts (resolution, aliasing) needed to read them. Generate them with scripts/doe_designs.py.

Table of contents

• Why factorial beats OFAT
• Full factorial (2^k)
• Fractional factorial (2^(k-p))
• Resolution and aliasing
• Screening designs (Plackett-Burman)
• Response-surface designs
• Space-filling designs
• Choosing a design

Why factorial beats OFAT

Vary one factor while holding others fixed and you (1) spend runs inefficiently and (2) can never see interactions — cases where the effect of A depends on the level of B, which are the rule, not the exception, in real systems. A factorial varies all factors simultaneously across runs; each effect is estimated using all the data, so a 2^k factorial is more precise than k separate OFAT studies of the same size.

Full factorial

A 2^k design runs every combination of k factors at two levels (low/high, coded −1/+1). It estimates all k main effects and all 2^k − k − 1 interactions.

• Runs = 2^k: 8 for 3 factors, 16 for 4, 32 for 5. Practical to ~5 factors.
• Use when you have a handful of factors and want a full picture including interactions.
• two_level_factorial({"temp": (20,60), "conc": (1,10), "pH": (6,8)}) → 8 runs.
• For factors with more than two levels, use full_factorial with explicit level lists (runs = product of level counts — grows fast).

Add center points (all factors at their midpoint) to a two-level design to get a cheap check for curvature: if the center response departs from the factorial average, a linear model is inadequate and you need a response-surface design.

Fractional factorial

When k is large, 2^k is too many runs — but most high-order interactions are negligible (the sparsity-of-effects principle). A 2^(k−p) fractional factorial runs a carefully chosen fraction (1/2, 1/4, ...) of the full design, trading the ability to estimate some interactions for far fewer runs.

• fractional_factorial(factors, generator="a b c abc") builds a half-fraction of 4 factors in 8 runs. The generator string (Yates notation) assigns each factor to a column; a multi-letter token aliases that factor with an interaction.
• The price is aliasing: some effects become indistinguishable. You must know which.

Resolution and aliasing

Aliasing (confounding) means two effects are estimated by the same contrast — the data cannot separate them. Which effects are aliased is summarized by the design's resolution:

Always state the resolution and inspect the alias structure before interpreting a fractional design. Concluding "factor C has no effect" is unsafe if C is aliased with a real interaction (it could cancel out). When in doubt, choose a higher-resolution generator (more runs) or add runs to de-alias (fold-over / augment the design).

Screening designs

When the goal is to find the vital few factors out of many (5, 10, 20+), use a screening design that estimates main effects only, as cheaply as possible: - Plackett-Burman (plackett_burman): runs = the next multiple of 4 above k (e.g. 12 runs for up to 11 factors). Resolution III — two-factor interactions are heavily confounded with main effects. Perfect for triage: run it, keep the few factors with large effects, then study those with a full or higher-resolution factorial. - Resolution III fractional factorials serve the same purpose.

Screen first, optimize later — don't try to learn interactions and find the optimum in one cheap design.

Response-surface designs

Two-level designs fit only a flat (linear + interaction) model; they cannot locate an interior optimum or describe curvature. To fit a quadratic and optimize, use a response-surface methodology (RSM) design over continuous factors:

• Central composite design (CCD) (central_composite): a 2^k factorial + center points + axial ("star") points. The axial points add the levels needed to estimate quadratic terms. With face="circumscribed" (default) the axial points sit outside the factorial box (so actual factor levels exceed your stated low/high); use face="inscribed" or "faced" to keep everything within the original range.
• Box-Behnken (box_behnken, needs ≥3 factors): a quadratic design that avoids the extreme all-low/all-high corners — useful when those corners are unsafe, expensive, or infeasible. More economical than a CCD for 3–5 factors.

Workflow: screen → factorial (find important factors & rough region) → response surface (model curvature, locate optimum), often moving the experimental region between steps (path of steepest ascent).

Space-filling designs

For computer experiments / simulations (deterministic or expensive models) where classical replication and blocking don't apply, you want even coverage of a high-dimensional input space: - Latin hypercube (latin_hypercube): each factor's range is divided into n_samples equal bins, sampled once each, arranged to spread points apart (criterion="maximin"). Gives good coverage with relatively few points and is the standard input design for surrogate/emulator modeling and sensitivity analysis.

Choosing a design

In all cases, randomize run order (the scripts do by default) so factors aren't confounded with time-related drift, and add center points to two-level designs as a curvature check.