01 · Use combinations to investigate interacting causes
Use combinations to investigate interacting causes
A full factorial design includes every combination of selected factor levels. For k two level factors, the basic design has 2k combinations. Three factors need eight, four need sixteen, and six need sixty four. Independent replication, blocks, and center points add runs. The design is full relative to the chosen levels; it does not map every possible continuous setting.
A warehouse may examine batch size, route policy, and scanner prompt. The important advantage is observing their combinations. A route policy may help only at large batch sizes, an interaction that changing one input at a time can miss. Select factors from process mechanisms and choose realistic low and high levels. Keep safety constraints and product quality responses visible alongside cycle time.
02 · Code factors and distinguish effects from coefficients
Code factors and distinguish effects from coefficients
Code low as −1 and high as +1. A three factor model can be written ŷ = b0 + bAA + bBB + bCC + bABAB + bACAC + bBCBC + bABCABC. Products represent interaction columns. The intercept is the grand mean in a balanced design. A main effect is the high mean minus the low mean, equal to twice its coded coefficient.
For an interaction column, its factorial effect is also twice its coefficient. In a two factor slice, the difference between the A effect at B high and at B low is four times bAB. Do not confuse this difference of differences with the factorial interaction effect. State conventions and factor coding when reporting effects so analysts can reconcile graphs and equations.
03 · Worked replicated warehouse design
Worked replicated warehouse design
Use fictional factors A for batch size, B for route policy, and C for scanner prompt. Suppose the cell mean model is ȳ = 20−2A−B−0.5C−1.5AB minutes. For standard combination order (−,−,−), (−,−,+), (−,+,−), (−,+,+), (+,−,−), (+,−,+), (+,+,−), (+,+,+), the means are 22, 21, 23, 22, 21, 20, 16, and 15.
At every combination, observe two independently assigned runs, one a minute below and one a minute above its displayed mean. There are sixteen observations. Each cell contributes two squared residual units, so SSE = 16. Fitting all eight coefficients leaves eight residual degrees of freedom; MSE = 2. Orthogonal coded columns each have squared length sixteen, giving coefficient standard error √(2/16) = 0.3536. An effect standard error is twice that, 0.7071.
Estimated effects are A = −4, B = −2, C = −1, and AB = −3 minutes; AC, BC, and ABC are zero. Using t0.975,8 = 2.3060 gives effect intervals of estimate ± 1.6306 minutes. A, B, and AB exclude zero; C does not. The A effect at B low is 2(−2 + 1.5) = −1 minute; at B high it is 2(−2−1.5) = −7. The route and batch settings therefore need a joint operational interpretation.
04 · Interpret combinations and uncertainty
Interpret combinations and uncertainty
The smallest fitted mean is fifteen at all three high settings. That is a candidate for confirmation, not an automatic production recommendation. The one minute C effect is uncertain, so the team may prefer a cheaper prompt setting if quality and practical constraints allow. Compare predicted combinations with intervals and business costs rather than declaring every significant factor worth changing.
Look at interaction plots and cell distributions. A main effect averages over other factor levels represented in the experiment. If the production mix differs, that average may not match actual deployment. Preserve hierarchy when retaining interactions, and state the levels for condition specific contrasts. If order classes modify the effects, block or expand the design appropriately rather than assuming the eight means cover all warehouse conditions.
05 · Randomize and replicate correctly
Randomize and replicate correctly
Randomize feasible run order so congestion, learning, or equipment drift do not align with factor columns. Block on nuisance conditions such as day when useful, while checking which effects become confounded with blocks. Independent replicates should recreate the treatment on separate experimental units. Multiple observations from one treated batch improve within batch measurement but do not provide the same evidence as independently assigned batches.
An unreplicated three factor full model also has eight parameters and eight observations, leaving no independent error estimate. Methods for screening unreplicated factorials rely on additional assumptions, such as most effects being negligible. Those assumptions must be stated. Replication protects against silently treating higher order interactions as error and permits clearer lack of fit or pure error assessments when suitable additional points are included.
06 · Check curvature and model adequacy
Check curvature and model adequacy
Two levels estimate linear contrasts across the chosen range but cannot identify pure quadratic effects. Center point replicates can detect a combined departure from the corner average. They do not separate individual quadratic coefficients because each squared coded factor has the same pattern at corners and centers. For optimization with curvature, add an appropriate response surface design.
Inspect residuals against fitted values and run order, identify unusual runs, and examine response variance by combination. A mean improving setting may worsen spread or error risk. Binary outcomes need a binomial model, and counts need a suitable count model. Avoid forcing every response into the same normal error analysis. If missing runs destroy balance, refit using the actual design matrix and report estimability limitations.
07 · Judge quality and use in DMAIC
Judge quality and use in DMAIC
A strong full factorial study selects meaningful factors, crosses them completely, respects randomization, and obtains enough independent error information for the desired conclusions. It reports effect units, conditional comparisons, uncertainty, and confirmation results. A weak study runs the matrix in standard order during a drifting shift, calls subsamples replicates, or interprets high level as inherently better without checking the real settings.
Analyze uses the design to verify interacting explanations for poor performance. Improve confirms a feasible combination under representative conditions and checks safeguards. Control specifies the settings and operating window, then monitors both mean and variation. Save the coded and actual levels, run order, blocks, raw responses, model terms, and confirmation protocol. Those details are what make the factorial evidence reproducible and transferable.
08 · Read the interaction graph against its averaging rule
Read the interaction graph against its averaging rule
The interaction plot averages across scanner levels. At B low, the low and high A means are 21.5 and 20.5 minutes. At B high, they are 22.5 and 15.5. These are actual marginal means from the displayed eight combinations, not additional observations. Their slopes show why a main A effect of −4 does not capture the operating decision by itself.
The AB interaction is useful if the underlying mechanism is credible and the joint change is feasible. A larger batch with the new route might avoid duplicate travel, whereas a larger batch with the old route might create congestion. That explanation should be checked through observation and confirmation. Keep the C level in any final prediction: averaging over C provides a graph of the interaction, but production must choose a specific prompt and assess its time, quality, and cost.
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.