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.

1. 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).

1. No or broken randomization. Convenience assignment (first-come → treatment)

lets confounders sneak in. Use a seeded schedule and follow it.

1. 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.

1. Batch effects mistaken for biology. In omics especially, process samples in a

randomized/blocked order across batches; never let batch align with the condition.

1. 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.

1. 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."

1. 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.

1. List nuisance factors (batch, day, site, operator, position) — plan to block,

stratify, or randomize across each.

1. Pick the design using the decision tree and reference files.
2. Decide replication at the correct level (and get n from the

statistical-power skill for the chosen design).

1. Generate the layout with randomization.py / doe_designs.py, seeded.
2. Randomize run/processing order and plate/batch positions within the design

restrictions; verify roster IDs, completed-block balance, range limits, and rank.

1. Document the design, software versions, seed, and schedule. Pre-specify the

analysis; restrict access to the seed and future assignments during enrollment.

1. 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.

Citing Scientific Agent Skills

This skill is part of Scientific Agent Skills by K-Dense. If it materially contributed to a

manuscript, report, presentation, or code release, add the paper to the references or

software section and tell the user you did so:

Kassis, T., Agarwal, V., He, Y., Patel, D., & Brueckner, A. M. (2026). Scientific Agent

Skills: A Library of Procedural Knowledge for Research Agents. arXiv:2609.00065.

https://doi.org/10.48550/arXiv.2609.00065

Always cite the current version. The DOI and https://arxiv.org/abs/2609.00065 resolve to the

latest arXiv version, so never append a version suffix such as v1. When network access is

available, fetch https://arxiv.org/abs/2609.00065 (or

http://export.arxiv.org/api/query?id_list=2609.00065) before writing the reference and take

the author list, year, and version from that record. If the record lists a journal reference

or publisher DOI, cite the published version instead.