Choosing a design

The right design depends on what stage of experimentation you are in, how many factors you have, what randomization restrictions exist, and how many runs you can afford.

Source: ../diagrams/design_selection.dot

The map starts with structural constraints because they determine the experimental unit. A mixture experiment, a split-plot experiment, and a blocked experiment are not interchangeable with a completely randomized design even when they have a similar run count. Once those constraints are respected, choose the design family that matches the statistical goal.

By experimental goal

Goal	Typical situation	Design family
Screen many factors	6+ factors, few runs, want the vital few	Plackett-Burman, definitive screening, fractional factorial
Estimate main effects and interactions	2–5 factors, moderate budget	Full factorial
Compare treatments	One factor, homogeneous units	Completely randomized design
Compare treatments with a nuisance source	Batches, days, operators	Randomized complete block design
Find an optimum	Few factors, curvature expected	Response surface (CCD, Box-Behnken)
Work under constraints	Irregular region, fixed run count	Optimal design (D, A, G, I)
Hard-to-change factors	Temperature changes are expensive	Split-plot
Formulation work	Components sum to a total	Mixture

Screening

With many factors and a small budget, start by finding which factors matter.

from industrialstats.designs.base import Factor
from industrialstats.designs.screening import PlackettBurmanDesign

factors = [Factor(f"X{i}", [-1, 1], "continuous") for i in range(1, 8)]
design = PlackettBurmanDesign(factors, seed=42)
print(design.generate_design())

Plackett-Burman designs estimate main effects in very few runs, but their interaction aliasing is complex. Treat the result as a shortlist, not as a final model.

DefinitiveScreeningDesign is also available for three-level quantitative screening when curvature information matters. Its current conference-matrix construction is validated for the main-effect orthogonality, main-effect versus two-factor-interaction orthogonality, pure-quadratic estimability, foldover symmetry, and seeded randomization properties documented by the package. Mixed continuous/two-level categorical DSDs remain outside the implemented scope.

Factorials and resolution

A regular fractional factorial trades run count for aliasing. Its resolution summarises the cost:

Resolution	Meaning
III	Main effects aliased with two-factor interactions
IV	Main effects clear of two-factor interactions; those interactions aliased with each other
V	Main effects and two-factor interactions clear of each other

Prefer resolution V when you intend to interpret interactions, and resolution IV when you mainly need clean main effects.

from industrialstats.designs.base import Factor
from industrialstats.designs.fractional_factorial import FractionalFactorialDesign

factors = [Factor(name, [-1, 1], "continuous") for name in "ABCDE"]

design = FractionalFactorialDesign(factors, fraction="1/2", randomize=False)
design.generate_design()
print(design.resolution_analysis())

That half fraction of five factors is 16 runs at resolution V, so every main effect and two-factor interaction is estimable. Taking a quarter fraction instead would halve the runs to eight but drop it to resolution III, aliasing main effects with two-factor interactions — acceptable for screening, but not if you intend to interpret interactions.

If a screening run leaves ambiguity, foldover augments the design to break the aliases that matter.

Response surface designs

Once the important factors are known and you expect curvature, move to a response surface design.

Source: ../diagrams/response_surface_workflow.dot

Response-surface methodology is sequential rather than a single design choice. A local first-order model can guide steepest ascent or descent. Once curvature becomes important, move to a second-order design, fit a quadratic surface, diagnose model adequacy, interpret the local geometry with canonical or ridge analysis, optimize one or several responses, and confirm the predicted operating point experimentally.

• Central composite (CCD) augments a factorial with axial and centre points. It estimates a full quadratic model and can be built from an existing factorial you have already run.
• Box-Behnken needs no extreme corner points, which helps when running all factors at their high settings simultaneously is impractical or unsafe.

from industrialstats.designs.base import Factor
from industrialstats.designs.response_surface import ResponseSurfaceDesign

factors = [
    Factor("temperature", [180, 220], factor_type="continuous"),
    Factor("pressure", [10, 20], factor_type="continuous"),
]

design = ResponseSurfaceDesign(factors, design_type="CCD", center_points=4)

Centre points matter: they give a pure-error estimate and a lack-of-fit test. Include several.

Blocking

When a nuisance source such as batch, day, or operator varies during the experiment, block on it rather than hoping randomization absorbs it.

from industrialstats.designs.rcbd import RandomizedCompleteBlockDesign

design = RandomizedCompleteBlockDesign(
    treatments=["A", "B", "C"],
    blocks=["Day1", "Day2", "Day3"],
    blocking_factor="Day",
    seed=42,
)

For regular two-level factorials, treatment-defined block generators are a different mechanism from RCBD nuisance blocking. The package exposes their defining contrasts and confounded effects explicitly rather than assigning blocks by row position.

Optimal designs

When the design region is constrained, the run budget is fixed, or the model is non-standard, a coordinate-exchange search over a candidate set is more appropriate than a catalogue design.

Source: ../diagrams/optimal_design_workflow.dot

An optimal design is defined jointly by the candidate region, the target model, the run budget, and the criterion. industrialstats builds the corresponding model matrix, rejects singular information matrices, searches candidate replacements by coordinate exchange, compares multiple starting designs, and retains the best criterion value. The resulting design should then be checked with efficiency and prediction-variance diagnostics rather than treated as optimal independently of the model assumptions that created it.

The available criteria answer different statistical questions:

• D-optimality maximizes log det(X.T @ X) and targets joint coefficient precision.
• A-optimality minimizes the trace of the inverse information matrix and targets average parameter variance.
• G-optimality minimizes the worst prediction variance over the candidate set.
• I-optimality minimizes the mean prediction variance over the candidate set.

Choose G or I when prediction across the region is the primary goal, and D or A when coefficient estimation is the priority. Changing the model terms, candidate set, criterion, or run budget defines a different optimization problem.

Mixture designs

Mixture experiments are structurally different from ordinary factorial experiments because component proportions are constrained by

[ \sum_{i=1}^{q} x_i = 1, \qquad x_i \ge 0. ]

Source: ../diagrams/mixture_geometry.dot

For three components, the feasible region is a triangle: vertices are pure components, edges are binary blends, and interior points are ternary blends. MixtureDesign currently generates simplex-lattice points, filters them through optional feasibility constraints, can randomize the retained run order with a seed, and can plot the three-component simplex.

from industrialstats.designs.advanced import MixtureDesign
from industrialstats.designs.base import Factor

components = [
    Factor("A", [], "continuous"),
    Factor("B", [], "continuous"),
    Factor("C", [], "continuous"),
]

design = MixtureDesign(components, order=2, randomize=True, seed=42)
mixtures = design.generate_design()

Do not analyse the component proportions as independent factorial factors. If one component changes, at least one other component must change to preserve the simplex constraint. The current package provides mixture-design construction and visualization, but not a package-native Scheffé mixture-model fitting/inference layer; any downstream model must respect the compositional constraint explicitly.

Randomization and reproducibility

Every generator that exposes a seed should be run with that seed recorded alongside the experiment. Reproducibility depends on the exact design family and its randomization contract rather than on reordering a generated table later.

Where randomization must be restricted — as in split-plot designs, where whole plots are hard to change — use the design family that encodes that restriction rather than reordering a fully randomized design by hand. The analysis has to match the randomization structure to give correct standard errors.