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 the appropriate full design separates main effects and interactions; reduced designs trade some of that information for fewer runs. 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. With one observation per combination, the model containing every effect plus an intercept is saturated: it has zero residual degrees of freedom. Replicate independently or pre-specify a defensible reduced model before claiming standard errors or significance.

• 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 often confounded with main effects. Use 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.

The current pydoe==1.5.0 implementation supports run counts in the families 2^a, 12*2^a, and 20*2^a. It can reject a requested factor count whose next multiple of four is unsupported (25 factors require 28 runs and raise ValueError). Inspect the actual matrix before budgeting; choose a supported design or a suitable fractional factorial. Reduced subsets may have better aliasing properties than the full saturated design, so inspect actual aliases. See the current factorial API.

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. Compare actual run counts and the same center-point replication: Box-Behnken is not always smaller (for five factors, its 40 non-center runs exceed a half-fraction CCD with 26). The bundled CCD wrapper uses the full factorial and does not build that half-fraction. See NIST response-surface comparisons.

The wrappers default to a single center run (center=(0, 1) for CCD; center=1 for Box-Behnken). That supplies no pure-error replication. Pre-plan independent replicate runs, inspect quadratic-model rank, and budget lack-of-fit assessment. CCD center counts refer to factorial and axial construction blocks; the wrapper returns a globally shuffled table without block IDs. Preserve or construct those IDs before using a physically blocked experiment.

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 space coverage may be the main objective, 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 common input design for surrogate/emulator modeling. A generic LHS alone is not a Saltelli/Sobol sensitivity estimator design. Stochastic simulations may still need replicated random seeds; experimental batches may still need blocking.

Choosing a design

For physical experiments, randomize run order within the allowed blocks to avoid systematic alignment with drift. DOE wrappers default to a global shuffle; Latin hypercube defaults to randomize=False and already has stochastic sample placement. Add replicated center points to two-level continuous designs for an aggregate curvature check; they cannot separate all pure quadratic coefficients.