01 · Trade combinations for identifiable questions
Trade combinations for identifiable questions
A fractional factorial design runs a deliberately selected fraction of a full factorial. It is useful when several candidate drivers make a complete experiment expensive. The saving comes with a cost: some effects have identical columns and cannot be distinguished using that fraction. This is aliasing or confounding. A balanced design can still be unable to tell two mechanisms apart.
Start by identifying the effects needed for the decision. A main effect screening study can tolerate some aliases if higher order interactions are plausibly negligible. A study where several two factor interactions are likely needs more protection. Choose a design based on its alias structure, operational feasibility, replication, and available follow-up budget, not just the smallest run count.
02 · Build a defining relation
Build a defining relation
For four two level factors, a full design requires sixteen combinations. An eight run half fraction can be generated by independently varying A, B, and C and setting D = ABC. Multiplying by D gives the defining relation I = ABCD, because each coded factor squared equals one. I denotes the all positive identity column.
Multiply any effect by each word in the defining relation to obtain its aliases. Here A = BCD, B = ACD, C = ABD, and D = ABC. Two factor aliases include AB = CD, AC = BD, and AD = BC. The shortest nonidentity word has length four, so this is resolution IV. Main effects are not aliased with two factor interactions, but two factor interactions are paired with each other. Resolution is a compact warning, not a substitute for the full alias table.
03 · Worked warehouse screening example
Worked warehouse screening example
Let A be batch size, B route policy, C scanner prompt, and D packing lane setting. Choose the fraction D = ABC. In standard A, B, C order the eight D values are −,+,+,−,+,−,−,+. Fictional cycle times are 22.75, 21.75, 22.25, 21.25, 20.25, 19.25, 16.75, and 15.75 minutes.
These teaching values were generated from y = 20−2A−B−0.5C−1.5AB+0.75CD. On the chosen fraction, CD equals AB for every row. Thus the observable AB column has combined coefficient −1.5+0.75 = −0.75 and apparent effect −1.5 minutes. The main effect estimates are A = −4, B = −2, C = −1, and D = 0. The data alone cannot reveal that the AB and CD coefficients were actually different and partly cancelled.
If a team reports AB = −1.5 as a uniquely established interaction, it has overinterpreted the design. The defensible result is the estimated alias combination AB + CD under this sign convention. Mechanistic knowledge may make one member more plausible, but it does not add a missing statistical contrast. The eight unreplicated rows also have no residual degrees of freedom if all independent design columns are fitted. These exact teaching numbers contain no independently estimated process noise.
04 · Add information to separate the aliases
Add information to separate the aliases
Run the complementary fraction D = −ABC to break this defining relation when combined with the first fraction. In the complementary fraction CD = −AB, so the observable interaction coefficient becomes bAB−bCD = −2.25, giving effect −4.5 minutes. Combining the two coefficient estimates, (−0.75−2.25)/2 = −1.5 estimates AB, and (−0.75+2.25)/2 = 0.75 estimates CD.
This augmentation doubles the combinations to the complete four factor design. Alternative targeted augmentations can separate a specific ambiguous interaction with fewer runs when the intended model permits. Do not assume every foldover breaks every alias. Reversing all signs in an even word defining relation such as ABCD leaves that word unchanged, so a naive mirror image of this particular half fraction does not produce the complementary fraction. Check the actual generators and combined design rank.
05 · Understand resolution and assumptions
Understand resolution and assumptions
Resolution III designs alias main effects with two factor interactions. They can be efficient preliminary screens but are risky where interactions are credible. Resolution IV protects main effects from two factor interactions while leaving two factor effects mutually aliased. Resolution V separates main effects and two factor interactions from one another, with remaining confounding involving higher order effects.
The sparsity principle assumes relatively few important effects; hierarchy and heredity provide additional modeling guidance. These are scientific working assumptions, not guarantees. A strong screening interpretation states them and identifies which follow-up would challenge them. Cancellation between aliases can hide a real mechanism, and reinforcement can exaggerate a single apparent effect. A small screening estimate does not prove that every member of its alias chain is negligible.
06 · Protect inference and execution
Protect inference and execution
Randomize feasible run order and block nuisance conditions with a known confounding structure. A block generator can sacrifice an effect that the project cares about, so inspect block aliases before running. Independent replication estimates error; within run repeated readings estimate measurement precision. Center points can check aggregate curvature but do not remove factorial aliases or identify separate pure quadratic terms.
With unreplicated screening, special effect estimation methods depend on assumptions about inactive effects. Report those assumptions and avoid precise conventional intervals obtained by an unexplained pooling of selected effects. Record actual factor settings, missed runs, and departures from the planned matrix. Missing observations can damage orthogonality and create additional instability. Analyze the actual matrix, not the ideal template shown on a planning slide.
07 · Judge screening quality and use in DMAIC
Judge screening quality and use in DMAIC
A strong application prioritizes a few candidate drivers, reports alias combinations explicitly, uses mechanisms to plan augmentation, and confirms interventions independently. A weak application labels all apparent main effects or interactions uniquely causal, or assumes a higher resolution number means every question is answered. Another weak application drops uninteresting aliases from the report and loses the very limitation that justified further experimentation.
Analyze uses the fraction to reduce a wide candidate list responsibly. Improve resolves consequential ambiguities and verifies feasible settings. Control receives only the effects and operating rules supported by confirmation, along with remaining uncertainty. Preserve the generator, defining relation, signed alias chains, design matrix, model assumptions, and augmentation plan. Those records explain exactly what the experiment learned and what it could not identify.
08 · Separate resolving aliases from estimating noise
Separate resolving aliases from estimating noise
Adding the complementary fraction supplies the contrasts needed to separate AB and CD. It does not automatically produce independent replication at the sixteen full factorial combinations. If a saturated full model is fitted to one observation per combination, residual degrees of freedom remain zero. Additional replicated runs or a defensible simpler model are still needed for conventional error estimation.
Plan augmentation with both objectives in mind: which ambiguous effects must be separated, and how uncertainty will be estimated. Repeating the original fraction helps estimate error but leaves its exact aliases intact. Conversely, adding unique combinations can resolve aliases without estimating pure error. These are different information gains. Preserve stage indicators when augmentation occurs on later days, and retain shared check conditions where needed to distinguish treatment information from a shift between stages.
State the sign of each generator. The fractions D = ABC and D = −ABC have different signed alias relations even though their resolution labels match. A factor effect estimated under one sign convention can therefore look different under another. The complete signed design matrix is the authoritative record for interpreting coefficients and planning follow-up runs.
Connections
Related methods and phase use
Placed under Analyze because of the outcome it supports. Use the method in other phases when its purpose and assumptions fit the work; the phase list is a guide to selection, not a compulsory sequence.
Sources
Professional foundations and further reading
The explanation and fictional teaching examples are written for LeanTrainingLab. Linked primary sources provide professional foundations; example values and local conventions are stated in the guide. The accepted toolset draws on Six Sigma+Lean Toolset, second edition (Lunau, editor; Springer, 2013). No book text or examples are reproduced here.