Experimental Design

Overview

The design of a study — how units are assigned to conditions, what is held constant, what is varied, and in what structure — determines what questions the data can answer. No analysis can rescue a confounded or pseudoreplicated design after the fact. This skill is about the decisions made before data collection: picking a design that isolates the effect of interest, randomizing to license causal claims, blocking to remove known nuisance variation, and structuring multi-factor experiments so effects are estimable rather than tangled together.

The three ideas behind almost every good design (Fisher's principles):

• Randomization — assign treatments at random so that confounders, known and unknown, are balanced in expectation. This is what turns a comparison into a causal claim.
• Replication — independent repetition at the right level, so you can estimate variability and your effects aren't artifacts of a single unit. The most common fatal error is pseudoreplication: counting repeated measurements on the same unit as independent replicates.
• Blocking / local control — group similar units (by batch, day, site, litter) and randomize within blocks, removing that nuisance variation from the error term instead of letting it inflate noise.

This skill helps you choose among design types, generate the actual randomization or DOE layout (with reproducible scripts), and avoid the structural mistakes that make data uninterpretable.

When to Use This Skill

• Planning any comparative experiment or trial and deciding how to assign units
• Randomizing subjects/samples to arms (simple, blocked, stratified, or cluster)
• Removing nuisance variation by blocking or stratification
• Designing multi-factor experiments: full or fractional factorial, screening designs
• Optimizing a response over continuous factors (response-surface designs)
• Within-subject / repeated-measures, crossover, split-plot, or Latin-square designs
• Cluster- or group-randomized designs (sites, clinics, classrooms, litters)
• Deciding the number and level of replicates and avoiding pseudoreplication
• Sequential, group-sequential, or adaptive designs with interim analyses
• Laying out plates/batches and randomizing run order to defeat drift

Installation

uv venv --python 3.13 .venv-design
uv pip install --python .venv-design/bin/python "numpy==2.5.3" "pandas==3.0.6" "pydoe==1.5.0"

pyDOE3 was archived in May 2026; active development returned to pydoe. The bundled wrappers target pydoe==1.5.0, imported as lowercase pydoe, and return designs in real factor units. All seven DOE wrappers and both script demos were executed with Python 3.13, NumPy 2.5.3, pandas 3.0.6, and SciPy 1.18.1. On Windows, the environment interpreter is .venv-design/Scripts/python.exe. Archive the environment versions and exported schedules: a seed alone does not promise identical output across package or script upgrades.

Choosing a design

Start from the question and the structure of your units, not from a favorite design.

What are you trying to learn?
│
├─ Compare a few predefined conditions (A vs B vs C)?
│   ├─ Units independent, possibly with a known nuisance factor (day, batch, site)?
│   │     → Completely randomized (no nuisance) or RANDOMIZED BLOCK design.
│   ├─ Each unit can receive every condition in sequence (washout possible)?
│   │     → CROSSOVER / repeated-measures design (watch carry-over and correlation).
│   └─ You can only randomize groups, not individuals (schools, clinics)?
│         → CLUSTER-randomized design (account for clustering; see pseudoreplication).
│
├─ Screen MANY factors (5+) to find the few that matter?
│     → FRACTIONAL FACTORIAL or PLACKETT-BURMAN screening design.
│
├─ Quantify main effects AND interactions among a handful of factors?
│     → FULL 2^k FACTORIAL design.
│
├─ Find the settings that OPTIMIZE a response (curvature matters)?
│     → RESPONSE-SURFACE design: central composite or Box-Behnken.
│
└─ Explore a simulation/computer model over a continuous space?
      → SPACE-FILLING design: Latin hypercube.

Detailed guidance per branch:

• Randomization, blocking, stratification, controls → references/randomization_and_blocking.md
• Factorial, fractional-factorial, screening, response-surface, DOE concepts (aliasing, resolution) → references/factorial_and_doe.md
• Crossover, repeated-measures, split-plot, Latin-square, cluster, nested designs → references/design_types.md
• Sequential, group-sequential, and adaptive designs (interim analyses) → references/sequential_and_adaptive.md

Generating the design

Two scripts produce ready-to-use, reproducible layouts. Run them from the skill's scripts/ directory or add it to sys.path. Seeds support reproducible layouts within the recorded software environment. The default seed is for demonstrations; choose and securely record a study-specific seed for a real allocation. These helpers do not implement an enrollment system or conceal future assignments from recruiters.

Randomization / allocation schedules — scripts/randomization.py

from randomization import (
    simple_randomization, block_randomization,
    stratified_block_randomization, cluster_randomization,
    assign_factorial_runs, arm_balance,
)

# Permuted blocks balance at completed-block boundaries.
# A partial final block or interim prefix need not have the requested ratio.
sched = block_randomization(n=60, arms=["treatment", "control"], seed=42)

# Balance a prognostic variable across arms by randomizing within each stratum
sched = stratified_block_randomization({"siteA": 30, "siteB": 30},
                                       arms=["drug", "placebo"], ratio=(2, 1), seed=42)

# Randomize whole clusters, not individuals (the cluster is the unit)
sched = cluster_randomization(["clinic1", "clinic2", "clinic3", "clinic4"], seed=42)

arm_balance(sched)            # sanity-check the counts per arm
sched.to_csv("allocation_schedule.csv", index=False)

Choosing among them: simple is fine for large n but can produce imbalance with small n; block enforces the ratio within each complete block; stratified block does this independently within each stratum; cluster is mandatory when the intervention is delivered at a group level. See references/randomization_and_blocking.md.

For a sequence of stratum labels, unit_id is the original one-based input position, including when strata are interleaved. Join the exported schedule to your subject IDs using that position and verify every ID exactly once. A dict input creates grouped planning slots rather than assigning an existing roster. Missing stratum labels and duplicate cluster IDs are rejected.

DOE matrices — scripts/doe_designs.py

from doe_designs import (
    full_factorial, two_level_factorial, fractional_factorial,
    plackett_burman, central_composite, box_behnken, latin_hypercube,
)

# Factors as real-world (low, high) ranges -> design comes back in real units
factors = {"temp_C": (20, 60), "conc_mM": (1, 10), "pH": (6, 8)}

# Full 2^3: all main effects + all interactions (8 runs), run order randomized
design = two_level_factorial(factors, seed=42)

# Screen 7 factors cheaply (main effects only)
many = {f"factor_{i}": (0, 1) for i in range(7)}
design = plackett_burman(many, seed=42)

# Optimize over 2 factors with curvature (response-surface)
design = central_composite(
    {"temp_C": (20, 60), "conc_mM": (1, 10)},
    center=(2, 2), face="inscribed", seed=42,
)

design.to_csv("experimental_runs.csv", index=False)

Before running a central composite design, inspect each factor's actual minimum and maximum. The default face="circumscribed" places axial points beyond the supplied low/high factorial settings; those arguments are not hard operating limits. If the stated ranges are physical limits, choose face="inscribed" or face="faced", then recheck all combinations. Do not clip out-of-range rows: clipping changes the design geometry and its statistical properties. See the NIST CCD comparison.

The CCD example reserves four center runs; execute them independently to estimate pure error. The wrapper default has only one. Confirm all settings are feasible, the intended model matrix has full rank, and independent replication supplies residual degrees of freedom. Center points check aggregate curvature; they do not identify each quadratic term without axial or other suitable runs.

DOE run order is randomized by default (Latin hypercube defaults to no added run_order) so factors are not systematically aligned with time/drift (machine warm-up, reagent aging). See references/factorial_and_doe.md for picking generators, reading the alias structure, and choosing resolution. These wrappers shuffle globally: for split plots, plates, or a CCD run in separate batches, retain block IDs and randomize only within the permitted structure.

The mistakes that ruin studies

These are structural — they can't be fixed in analysis, only in design.

1. Pseudoreplication. Treating repeated measurements of one unit as independent replicates: 3 mice with 100 cells each is n = 3 (mice), not n = 300 (cells), for any treatment applied to the mouse. The replicate must be at the level the treatment is randomized. This single error invalidates a large share of published experiments. Randomize and replicate at the right level; analyze with the nesting respected (mixed model). See references/design_types.md.
2. Confounding by a nuisance variable. Running all treatment samples on Monday and all controls on Tuesday confounds treatment with day. Randomize across, or block on, every nuisance factor you can name (batch, day, plate, technician, instrument, position).
3. No or broken randomization. Convenience assignment (first-come → treatment) lets confounders sneak in. Use a seeded schedule and follow it.
4. No proper control. Without a concurrent control (and, where relevant, a vehicle/sham and blinding), you can't separate the treatment effect from time, placebo, or handling effects.
5. Batch effects mistaken for biology. In omics especially, process samples in a randomized/blocked order across batches; never let batch align with the condition.
6. Edge/position effects on plates. Evaporation and thermal gradients make plate edges differ. Randomize or block sample positions; don't put all controls in column 1.
7. Aliasing ignored in fractional designs. A low-resolution fractional factorial confounds main effects with interactions; know your alias structure before concluding a factor "has no effect."
8. Optimizing without curvature. A two-level factorial alone cannot estimate pure quadratic terms; you'll miss an interior optimum. Use a response-surface design.

Workflow

1. State the question, the unit, and the response. What is randomized? What is measured? At what level is a true independent replicate? This determines everything.
2. List nuisance factors (batch, day, site, operator, position) — plan to block, stratify, or randomize across each.
3. Pick the design using the decision tree and reference files.
4. Decide replication at the correct level (and get n from the statistical-power skill for the chosen design).
5. Generate the layout with randomization.py / doe_designs.py, seeded.
6. Randomize run/processing order and plate/batch positions within the design restrictions; verify roster IDs, completed-block balance, range limits, and rank.
7. Document the design, software versions, seed, and schedule. Pre-specify the analysis; restrict access to the seed and future assignments during enrollment.
8. Match the analysis to the design — blocks, strata, clusters, and nesting must appear in the model (hand off to statistical-analysis / statsmodels).

Resources

Scripts

• scripts/randomization.py — seeded allocation schedules: simple_randomization, block_randomization, stratified_block_randomization, cluster_randomization, assign_factorial_runs, arm_balance.
• scripts/doe_designs.py — DOE matrices in real units: full_factorial, two_level_factorial, fractional_factorial, plackett_burman, central_composite, box_behnken, latin_hypercube.

References

• references/randomization_and_blocking.md — randomization methods, blocking, stratification, controls, blinding, batch/plate layout.
• references/factorial_and_doe.md — factorial and fractional designs, resolution and aliasing, screening, and response-surface methodology.
• references/design_types.md — completely randomized, randomized block, crossover, repeated-measures, split-plot, Latin-square, cluster, and nested designs; the pseudoreplication problem in depth.
• references/sequential_and_adaptive.md — group-sequential designs, alpha spending, interim stopping, and adaptive sample-size re-estimation.

Related skills

• statistical-power — required sample size / power for the design you've chosen.
• statistical-analysis — running and reporting the analysis after collection.
• statsmodels / pymc — fitting the models the design implies.

Key references

• Fisher, R. A. (1935). The Design of Experiments.
• Montgomery, D. C. (2019). Design and Analysis of Experiments (10th ed.).
• Hurlbert, S. H. (1984). Pseudoreplication and the design of ecological field experiments. Ecological Monographs, 54(2), 187–211.
• Lazic, S. E. (2016). Experimental Design for Laboratory Biologists.