01 · Let the operational structure choose the design
Let the operational structure choose the design
Advanced design choices arise when a completely randomized factorial does not match how work can be performed. A warehouse may need to account for day effects, restrict expensive conveyor changes, or optimize proportions of packing materials that must sum to one. These are different problems and require different design and analysis structures.
Define the question, experimental unit, randomization restrictions, feasible settings, and response model before selecting a named design. A sophisticated label does not repair a mismatch between assignment and analysis. Identify what can change independently and what remains shared across runs. This determines the correct error terms and how many genuinely independent pieces of information the study supplies.
02 · Use blocking for a nuisance condition
Use blocking for a nuisance condition
A randomized complete block design runs every treatment within each block and randomizes its order there. Days can be blocks when workload or staff varies between days. For fictional old and new release policies, day pairs are 20 versus 16, 22 versus 19, 18 versus 13, and 21 versus 17 minutes. Daily savings are 4, 3, 5, and 4, with mean four.
The standard error of the mean difference is √(2/3)/√4 = 0.4082. A 95% paired interval uses t with three degrees of freedom and gives 2.7008 to 5.2992 minutes. Blocking compares policies under each day's conditions rather than letting a bad day inflate treatment error. It cannot separate a treatment by day interaction from arbitrary noise with only one observation per treatment per day. If that interaction matters, obtain appropriate replication.
Incomplete blocks can be useful when a day cannot accommodate all treatments. Ensure the design connects treatments through shared blocks so relevant contrasts remain estimable. A Latin square can block two nuisance dimensions under restrictive balance and interaction assumptions. Choose these structures for real nuisance mechanisms, not simply to make an analysis table look orderly.
03 · Use split plots for hard to change inputs
Use split plots for hard to change inputs
Suppose conveyor speed requires a lengthy setup, while scanner prompts can change between orders. Assign speed to whole plots, such as independently reset conveyor sessions, and randomize prompts within each session. Randomize whole plot assignment as well as subplot order where feasible. A session is the independent unit for speed; individual orders are units for prompts.
A model can include yijk = μ + Ai + Bj + (AB)ij + uik + εijk, where A is speed, B is prompt, u is a shared whole plot random effect, and ε is subplot error. The precise indexing depends on blocking and replication. There is no single universal error term for every effect. More orders within a session improve subplot information but do not substitute for more independently assigned speed sessions.
For an illustrative calculation, assume whole plot variance is nine minute squared and subplot variance is one, with four orders per session. Session mean variance is 9 + 1/4 = 9.25. With four independent sessions per speed, variance of their difference is 9.25(1/4+1/4) = 4.625 and standard error is 2.1506 minutes. Pretending the sixteen orders per speed were independent gives √[10(1/16+1/16)] = 1.1180, severely understating uncertainty. These are assumed variance components demonstrating the design effect, not estimates from a fitted study.
04 · Use mixture designs for proportions
Use mixture designs for proportions
If packing materials A and B have proportions q and 1−q, increasing one necessarily decreases the other. They cannot be varied independently as ordinary unconstrained factors. A two component quadratic mixture model is ŷ = bAq + bB(1−q) + bABq(1−q), where y might be a damage score.
For fictional responses six at pure A, ten at pure B, and five at a half and half blend, bA = 6, bB = 10, and 5 = 3+5+0.25bAB, so bAB = −12. Substitution gives ŷ = 10−16q+12q². The derivative −16+24q is zero at q = 2/3, with fitted score 14/3 = 4.6667. The positive second derivative suggests a model minimum.
Three unreplicated blend points identify three coefficients but provide no independent error estimate or lack of fit test. Replicate, add intermediate blends, and confirm the candidate. If q is constrained to at most 0.6, the unconstrained optimum is infeasible; the constrained minimum on that interval is at 0.6, with fitted score 4.72. More components require simplex or constrained mixture geometry, not a rectangular factorial grid.
05 · Choose custom and sequential designs thoughtfully
Choose custom and sequential designs thoughtfully
A custom optimal design selects feasible candidate points to estimate a specified model efficiently. D optimality focuses on overall coefficient information; other criteria focus on prediction. Its usefulness depends on the assumed terms, candidate region, and randomization structure. If the actual process needs an interaction omitted during design selection, the chosen points may not estimate it well.
Sequential augmentation can add points to resolve aliases, estimate curvature, or repair lost runs. Preserve the original assignment and account for blocks between stages. A second stage conducted in a different season can be confounded with the augmentation unless overlap or controls connect the stages. Repeatedly adding runs until a preferred p value appears requires valid sequential inference or transparent exploratory treatment.
06 · Judge advanced designs in DMAIC
Judge advanced designs in DMAIC
A strong application selects a design because of the real operational constraint and analyzes its assignment structure. A weak application calls every clustered study blocked, treats hard to change factor observations as independent orders, or fits unconstrained ordinary coefficients to proportions summing to one. Another weak application trusts a computer generated design without checking estimable effects and feasible execution.
Analyze uses these designs to verify mechanisms under realistic constraints. Improve confirms feasible settings with enough independent units at each level. Control records both settings and the conditions that made them useful. Preserve the assignment hierarchy, block definitions, mixture limits, variance structure, intended model, and confirmation data. These details explain the precision and limits of the resulting evidence.
07 · Report replication at every assignment level
Report replication at every assignment level
For the split plot example, list the number of independent conveyor sessions as well as the number of orders. If only one session occurs at each speed, speed is inseparable from those two session histories no matter how many orders are measured. More orders improve a session mean but cannot establish what would happen across independently reset sessions. Plan enough whole plots for the speed comparison before allocating large numbers of subplots.
For mixture studies, report total material quantity along with proportions when both could affect the response. A half and half mix of one hundred grams need not behave like a half and half mix of one kilogram. Total amount can be a separate process variable, crossed with the feasible mixture region in a suitable design. State what was held fixed so a blend recommendation does not silently become a claim about every packaging quantity or shipment type.
Special designs require practical execution discipline. Record actual whole plot resets, session order, blend quantities, and deviations from planned settings. If operators quietly restrict a supposedly randomized factor, the executed study can have a different error structure from the planned study. Analyze assignment as performed and disclose material departures rather than keeping the original model merely because it was specified first.
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
- NIST: Randomized block designs
- NIST: Restricted randomization and nested variation
- Minitab: Split plot designs
- NIST: What is a mixture design?
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.