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eduKateSG · Why Mathematics? · Fractional Factorial Designs: Alias Structures, Resolution and Efficient Screening
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 8–14
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 15–21
Check assumptions
See where design and calibration matter.Route 4 · Chapters 22–28
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 29–42
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when a full experiment has too many factor combinations to run? Fractional factorial designs choose a structured subset of runs. Generators, defining relations and resolution reveal exactly which effects are aliased, turning a small screening experiment into an honest map of what can and cannot be learned.
Did You Know? A four-factor two-level full factorial needs 16 runs, but a half-fraction needs 8. The saving is real because some effects become mathematically indistinguishable, not because the missing information somehow disappears.
Full chapter index
Chapters 1–7 · Understand the question
Chapters 8–14 · Reproduce the example
Chapters 15–21 · Check assumptions
Chapters 22–28 · Explore applications
Chapters 29–42 · Study and verify
- Final decision
- A spreadsheet workflow
- A coding workflow
- Applications and careers
- A student investigation
- Common misconceptions
- Connection to recent mathematics articles
- Guidance for students and parents
- Practice questions
- Frequently asked questions
- A reporting checklist
- The design matrix is the evidence contract
- A final fraction audit
- Useful next reading
Engineers, teachers and AI teams may have many possible factors but limited time. A screening design identifies a small set worth deeper study before optimisation.
Four factors at two levels create 2^4=16 combinations. The full design estimates all main effects and interactions when replication and noise permit.
A 2^(4−1) half-fraction uses eight combinations. It keeps orthogonal structure while deliberately confounding selected effects.
Let D=ABC using coded levels -1 and +1. For every chosen run, the D column is the product of A, B and C.
Multiplying both sides by D gives I=ABCD. This shortest word has length four, so the design is resolution IV.
Multiply I=ABCD by an effect. A=BCD, while AB=CD. The algebra lists every effect that produces the same design column.
| Full design | Fraction | Generator | Runs |
|---|---|---|---|
| 2^4 | 2^(4-1) | D=ABC | 8 |
Main effects are not aliased with two-factor interactions, but two-factor interactions are paired with other two-factor interactions.
The illustrative design has AB=CD, AC=BD and AD=BC. An observed AB signal could be CD or a mixture; the data alone cannot decide.
Main effects can be aliased with two-factor interactions. This may be acceptable for rough screening only when interaction sparsity is plausible.
| Effect | Aliased with |
|---|---|
| A | BCD |
| AB | CD |
| AC | BD |
| AD | BC |
Main effects and two-factor interactions are clear of each other, though two-factor interactions can alias with three-factor interactions. More runs are usually required.
NIST defines resolution from the shortest word and summarises the aliasing patterns of resolution III, IV and V designs. Read the NIST handbook.
Screening assumes only a few effects are active. This is a scientific working assumption, not a theorem, and should guide confirmation.
Lower-order effects are often more plausible than high-order interactions. A strong higher-order effect can still break that expectation.
An interaction is often more plausible when at least one parent main effect is active. Heredity helps interpretation but cannot resolve aliases from data alone.
The fraction chooses combinations; randomisation protects them from time, machine warm-up or learner fatigue.
Adding centre runs for quantitative factors can detect overall curvature, though a two-level fraction cannot identify separate quadratic terms.
Replicating runs estimates pure error and improves precision. Repeating the same biased order does not protect against drift.
A foldover reverses signs for selected factors and can break important aliases. Choose the follow-up based on the specific ambiguous effects.
Screen first, add confirmation runs, then move to response-surface optimisation. One small fraction should not carry every conclusion.
Designs with the same resolution can have different counts of short defining words. Minimum-aberration choices reduce severe aliasing.
If runs span days or batches, block the design using high-order columns. The block effect then aliases with chosen interactions and must be documented.
A failed run can destroy orthogonality and complicate the alias structure. Re-run when possible or analyse the actual rank and variance inflation.
A learning experiment screens feedback timing, example type, retrieval schedule and group size. Eight combinations may reveal promising factors, but aliased interactions need follow-up.
Training experiments can screen learning rate, augmentation, batch size and regularisation. Random seeds and compute batches act as replication or blocks.
Inspect design-matrix correlations, alias chains, residuals, run order, effect plots and leverage. Never report an effect name without its alias set.
Build the eight-run matrix, verify D=ABC, multiply columns to confirm aliases and add a foldover to separate AB from CD.
Fractionation cannot estimate every effect, protect against changing processes or validate an untested region. Efficiency is purchased with assumptions.
Report generators, defining relation, resolution, complete relevant alias structure, randomisation, blocks, centre points and confirmation runs.
Use the highest feasible resolution that protects the effects you care about. Let scientific priorities determine which aliases are acceptable.
Create columns for run, coded factors, generator checks, block, order, response and every estimable contrast.
Construct the design matrix, verify column products and rank, print alias chains and unit-test the planned randomisation.
Process engineers, agronomists, education researchers and AI experimenters use fractions to screen many controllable choices efficiently.
Create a 2^(4−1) fraction with D=ABC, simulate an AB effect and show why it is indistinguishable from CD before foldover.
A small p-value does not identify one member of an alias chain, resolution is not data quality, and sparsity is an assumption rather than a guarantee.
The factorial-experiment guide gives the full-design foundation. The multiple-testing guide explains screening claims.
Students should multiply coded columns by hand for one alias. Parents can ask whether an education experiment confirmed signals in new combinations.
- Count full and fractional runs.
- Derive I=ABCD.
- Find the AB alias.
- Add a foldover.
- Inspect a missing run.
### What is aliasing? Two or more effects share the same design column.
### Is resolution IV always enough? No.
### Why centre points? To detect overall curvature.
### Must screening be followed up? Strongly recommended.
Report factors and levels, fraction, generator, defining relation, resolution, alias sets, blocks, randomisation, replication, missing runs, model and confirmation plan.
Every estimable claim is encoded in its columns before a response is measured.
Confirm the intended effects are not fatally aliased, all runs match the generator and conclusions name unresolved alternatives.
Continue through the Mathematics Learning Hub. Recalculate the central example, identify the assumptions and explain which claim the result can and cannot support.
Extended practice laboratory
Practice case 1: List factors
Setup: Start from the article’s fractional factorial mechanism for case 1. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 2: Choose two levels
Setup: Start from the article’s fractional factorial mechanism for case 2. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 3: Count full runs
Setup: Start from the article’s fractional factorial mechanism for case 3. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 4: Choose fraction
Setup: Start from the article’s fractional factorial mechanism for case 4. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 5: Set generator
Setup: Start from the article’s fractional factorial mechanism for case 5. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 6: Write defining relation
Setup: Start from the article’s fractional factorial mechanism for case 6. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 7: Derive A alias
Setup: Start from the article’s fractional factorial mechanism for case 7. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 8: Derive AB alias
Setup: Start from the article’s fractional factorial mechanism for case 8. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 9: Identify resolution
Setup: Start from the article’s fractional factorial mechanism for case 9. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 10: Randomise runs
Setup: Start from the article’s fractional factorial mechanism for case 10. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 11: Replicate centre
Setup: Start from the article’s fractional factorial mechanism for case 11. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 12: Estimate main effects
Setup: Start from the article’s fractional factorial mechanism for case 12. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 13: Inspect interaction alias
Setup: Start from the article’s fractional factorial mechanism for case 13. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 14: Use foldover
Setup: Start from the article’s fractional factorial mechanism for case 14. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 15: Add confirmation runs
Setup: Start from the article’s fractional factorial mechanism for case 15. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 16: Check sparsity assumption
Setup: Start from the article’s fractional factorial mechanism for case 16. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 17: Check hierarchy
Setup: Start from the article’s fractional factorial mechanism for case 17. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 18: Add blocks
Setup: Start from the article’s fractional factorial mechanism for case 18. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 19: Check run order
Setup: Start from the article’s fractional factorial mechanism for case 19. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 20: Fit coded model
Setup: Start from the article’s fractional factorial mechanism for case 20. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 21: Plot effects
Setup: Start from the article’s fractional factorial mechanism for case 21. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 22: Simulate hidden interaction
Setup: Start from the article’s fractional factorial mechanism for case 22. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 23: Audit design matrix
Setup: Start from the article’s fractional factorial mechanism for case 23. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 24: Name limits
Setup: Start from the article’s fractional factorial mechanism for case 24. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 25: Write conclusion
Setup: Start from the article’s fractional factorial mechanism for case 25. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.