Seven factors at two levels require 128 runs in full combination. Selecting any 16 rows because resources are limited can destroy balance and effect separation. A fractional factorial is not random omission; it is a structural economy that deliberately chooses which effects will be seen together.
This unit asks one question.
When run count is reduced, which effects become indistinguishable in the data?
Select half of 2³ structurally
Make all four A and B combinations and set C=AB; this gives a four-run 2^(3−1) half fraction. Multiply this generator by C to obtain the defining relation I=ABC.
In the design table, the C and AB columns are identical. If response moves with that column, the data cannot distinguish a C main effect from an AB interaction. The alias chains are:
- I = ABC
- A = BC
- B = AC
- C = AB
The number labelled “estimate of A” is actually the combined A and BC contrast. Interpret it mainly as A only under an assumption that BC is near zero.
Alias is stronger than high correlation
With complete aliasing, two model columns are identical or sign-reversed, so separate coefficients cannot be estimated from the same data. In more general nonregular or optimal designs, columns may not be identical but estimates can still be correlated and imprecise.
One software coefficient does not mean every candidate effect was estimated. Inspect the alias matrix, estimability, and correlation of estimates before running the design.
Its economy relies on assumptions of effect sparsity and small high-order interactions. If an aliased interaction is large, it is wrongly attributed to a main effect. Separate important interaction candidates through factor assignment, higher resolution, foldover, or follow-up experiments.
Resolution summarizes the most severe alias order
Resolution is the shortest word length, excluding identity, in the defining relation.
- Resolution III: main effects alias with two-factor interactions.
- Resolution IV: main effects are separate from two-factor interactions, but two-factor interactions alias with one another.
- Resolution V: main effects and two-factor interactions are separate; two-factor interactions alias with three-factor interactions.
Higher resolution generally requires more runs. A higher number is not always best: the needed structure depends on whether the aim is rapid screening of many factors or estimation of a particular interaction.
A full factorial is sometimes called resolution infinity in the effect-alias sense. Blocking can confound high-order interactions with block effects, connecting the cost of blocking in U18 to the same alias language.
Choosing a fraction contains hidden assumptions
Fractional factorial practicality often relies on useful, but non-lawlike, principles:
- sparsity: only a few of many candidate effects are truly large
- hierarchy: if a high-order interaction matters, related lower-order terms deserve consideration
- heredity: interactions tend to appear with related main effects
Even a small main effect can accompany a large crossover interaction, so do not treat heredity as guaranteed. Place mechanistically important combinations clearly in the design.
In-Silico Lab: verify identical columns directly
- Inspect all four runs to confirm C and AB columns match row by row.
- Derive each alias chain by multiplying the defining relation by A, B, or C.
- If the A contrast is 8, decide whether A=8 or A+BC=8.
- Consider which columns separate after expanding to the eight-run full factorial.
Read the alias for 4-run half split directly
2³ If you select C=AB as the generator in half of the complete design, check which columns are completely equal.
If this is your first time: What should I press?
- 1. Read the question firstIn the Lab title, check the one thing you will compare this time.
- 2. Change just one conditionInitially, change only one of the inputs: n, effect, or spread.
- 3. new composite specimen pressureNew synthetic data is created. The same conditions may vary depending on the sample.
- 4. Pictures and calculation results CompareWrite in one sentence what moves and what stays the same before and after the change.
If it gets stuckresetGo back to see the default results and change just one condition. This Lab is not a correct answer tester but a pattern observation tool.
Synthetic observations of the same settings
| run | A | B | C=AB | ABC |
|---|---|---|---|---|
| 1 | -1 | -1 | 1 | 1 |
| 2 | -1 | 1 | -1 | 1 |
| 3 | 1 | -1 | -1 | 1 |
| 4 | 1 | 1 | 1 | 1 |
calculation result
I = ABCA = BCB = ACC = ABEven if the estimated value of A is large, it is not possible to separate whether the cause is A or BC with just these four runs. The assumption of small higher-order interactions is the price of savings.
educational synthetic modelbjs-factorial-sequence-v1. Actual research judgments must separately reflect experimental units, missingness, distribution, multiplicity, pre-planning, and domain criteria.
Read JMP Design Evaluation before execution
Specify what other effects are incorporated into each effect estimate.
Check resolution and estimateability before execution.
A non-zero correlation is a sign of weakened effect separation.
The Alias Table is a gate before design selection, not an appendix after interpretation. If an expected active interaction aliases with a main effect, change the fraction or factor assignment. The key is to verify that the candidate model is estimable, not merely to follow a design-generation menu.
A fractional factorial is the first wave of a sequential experiment
After using a screening fraction to find promising factors, do not declare the final optimum from the same data. Depending on the situation, the next step may be:
- foldover by adding the opposite fraction
- selected added runs to break aliases
- a full factorial on the important factors only
- center points to inspect curvature
- response-surface design to extend the range
Do not place every decision on a first-wave null/active classification; resolve ambiguous aliases with follow-up experiments.
Example result statement
We used a five-factor
2^(5−1)Resolution V half fraction. All main effects and two-factor interactions in the pre-specified candidate model were clear of one another, while each two-factor term was aliased with three-factor or higher terms. A and AB appeared promising, but because interpretation depends on the sparsity assumption, we planned independent confirmation and selected runs to verify the structure.
For Resolution III screening, write honestly “A+BC alias contrast,” not simply “A effect.”
Takeaways
- A fraction is a structured part of a full design.
- A generator yields a defining relation and alias chains.
- Aliased effects cannot be estimated separately from the same data.
- Resolution summarizes the most severe order of effect confounding.
- Sparsity, hierarchy, and heredity are useful assumptions, not guarantees.
- Follow-up experiments must resolve aliases from fractional results.
This sequence stops at the requested automatic boundary, U20. Candidate next units—screening follow-up experiments, center points, and curvature—must not be authored until a separate restart.
Official supplementary resources
- NIST/SEMATECH · Constructing a 2^(3−1) design
- NIST/SEMATECH · Confounding and aliasing
- NIST/SEMATECH · Design resolution
The design and Lab in this article are educational synthetic material.