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8.6 Two-Level Fractional Factorial Experiments and Confounding

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1 Chapter 1: Introduction & CSSBB Exam Strategy

1.1 CSSBB Exam Format, Scoring, and Testing Windows 1.2 Eligibility, Project Affidavits, Application, and Recertification 1.3 Open-Book Strategy, Reference Materials, and a 16-Week Study Plan

2 Chapter 2: Organization-Wide Planning and Deployment

2.1 Fundamentals of Six Sigma and Lean Methodologies 2.2 Selecting the Right Methodology: DMAIC, DFSS, and Lean 2.3 Business Systems, Processes, and Stakeholder Relationships 2.4 Strategic Planning and Deployment of Six Sigma Initiatives 2.5 Six Sigma Roles and Responsibilities 2.6 Organizational Barriers and Common Causes of Six Sigma Failure 2.7 Change Management: Stakeholder Analysis, Readiness, and Communication

3 Chapter 3: Organizational Process Management and Measures

3.1 Impact of Six Sigma Projects on Customers, Suppliers, and Stakeholders 3.2 Benchmarking Types and Performance Goal Setting 3.3 Performance Measures: Balanced Scorecard, KPIs, KBIs, OKRs, and Line of Sight 3.4 Leading and Lagging Indicators and Performance Dashboards 3.5 Financial Measures: Revenue Growth, Margin, NPV, ROI, and Cost-Benefit Analysis 3.6 Cost of Poor Quality and Hard versus Soft Savings

4 Chapter 4: Team Management

4.1 Team Types and Constraints 4.2 Team Roles, Responsibilities, and Member Selection 4.3 Team Success Factors and Ground Rules 4.4 Motivational Techniques and Overcoming Demotivators 4.5 Stages of Team Development and Conflict Resolution 4.6 Team Communication Plans 4.7 Team Leadership Models: Direct, Coach, Support, and Delegate 4.8 Group Behaviors, Conflict Resolution, and Meeting Management 4.9 Facilitation Techniques and Team Decision-Making Methods 4.10 Team Training: Needs Assessment, Delivery, and Evaluation

5 Chapter 5: Define Phase

5.1 Customer Identification and Segmentation 5.2 Customer Data Collection: Methods, Validity, and Reliability 5.3 Customer Requirements: CTX, CTQ Trees, and Operational Definitions 5.4 The Kano Model and Quality Function Deployment 5.5 SIPOC and Process Mapping 5.6 Business Case Justification 5.7 The Project Charter and the Problem Statement 5.8 Project Scope and Boundaries 5.9 SMART Goals and Project Performance Measurements 5.10 Project Charter Reviews and Toll-Gate Governance 5.11 Project Management Tools: Gantt Charts, Toll-Gate Reviews, WBS, and RACI 5.12 Analytical Tools Used Throughout the DMAIC Cycle

6 Chapter 6: Measure Phase

6.1 Process Flow Metrics and the Hidden Factory 6.2 Process Analysis Tools and Value Stream Mapping 6.3 Types of Data and Measurement Scales 6.4 Sampling Concepts, Methods, and Sample Size for Estimation 6.5 Data Collection Plans and Common Pitfalls 6.6 Gage R&R for Variable Measurement Systems 6.7 Attribute Agreement Analysis and Audit MSA 6.8 Measurement Systems Across the Organization, and Metrology 6.9 Population Parameters, Sample Statistics, and Valid Statistical Conclusions 6.10 Descriptive Statistics: Central Tendency, Dispersion, and Shape 6.11 Graphical Methods for Exploratory Data Analysis 6.12 Probability Concepts, Common Distributions, and the Central Limit Theorem 6.13 Additional Distributions: Hypergeometric, Exponential, Lognormal, Weibull, and Bivariate 6.14 Process Capability and Performance Indices 6.15 Capability Studies, Attributes Capability, Yield Metrics, and Long-Term Capability

7 Chapter 7: Analyze Phase

7.1 The Correlation Coefficient, Its Confidence Interval, and Causation 7.2 Linear Regression and Residual Analysis 7.3 Multivariate Tools: Factor Analysis, Discriminant Analysis, and MANOVA 7.4 Hypothesis Testing Terminology, Errors, and Single-Sample Tests 7.5 Statistical Versus Practical Significance 7.6 Sample Size for Hypothesis Tests 7.7 Point Estimates, Confidence, Prediction, and Tolerance Intervals 7.8 Tests for Means, Variances, and Proportions 7.9 Analysis of Variance 7.10 Chi-Square Goodness-of-Fit and Tests of Independence 7.11 Types of Risk and Failure Mode and Effects Analysis 7.12 Gap Analysis and Root Cause Analysis 7.13 Waste Analysis: The Seven Classic Wastes and Resource Under-Utilization

8 Chapter 8: Improve Phase

8.1 Design of Experiments: Objectives, Terminology, and Effects 8.2 DOE Design Principles: Randomization, Blocking, Replication, and Resolution 8.3 Planning and Evaluating Designed Experiments 8.4 One-Factor Experiments: Completely Randomized, Randomized Block, and Latin Square Designs 8.5 Full Factorial Experiments and Design Selection 8.6 Two-Level Fractional Factorial Experiments and Confounding 8.7 Response Surface Methodology and Optimization 8.8 Waste Elimination: Pull Systems, Kanban, 5S, and Poka-Yoke 8.9 Cycle-Time Reduction: Continuous Flow, SMED, and Heijunka 8.10 Kaizen, Kaizen Blitz, Theory of Constraints, and OEE 8.11 Implementation Planning, Piloting, and Solution Selection

9 Chapter 9: Control Phase

9.1 SPC Objectives, Variable Selection, and Rational Subgrouping 9.2 Control Charts for Variables Data 9.3 Control Charts for Attributes Data 9.4 Short-Run SPC and Moving Average Charts 9.5 Control Chart Analysis: Common Cause, Special Cause, and Run Rules 9.6 Total Productive Maintenance and Visual Controls 9.7 Measurement System Reanalysis 9.8 Control Plan Development and Process Owner Handoff 9.9 Lessons Learned, Documentation, and Training for Sustainment 9.10 Ongoing Evaluation of the Improved Process

10 Chapter 10: Design for Six Sigma (DFSS)

10.1 DFSS Frameworks: DMADV, DMADOV, and Design Transfer 10.2 Design for X (DFX) 10.3 Robust Design, Parameter Design, and Verification 10.4 Tolerance Design and Statistical Tolerancing

8.6 Two-Level Fractional Factorial Experiments and Confounding

Key Takeaways Explain This Topic AI

A two-level fractional factorial runs 2 to the power (k minus p) of the 2 to the k full factorial combinations.

The generator defines which effects are deliberately aliased, and the defining relation lists every alias chain.

Design resolution is the length of the shortest word in the defining relation and states which effects are confounded.

The sparsity of effects, hierarchy, and heredity principles justify sacrificing high-order interactions.

A confounded pair can be separated by folding over the design, which doubles the runs and clears the main effects.

Last updated: August 2026

Why fractionate

A full two-level factorial in k k k factors needs 2 k 2^k 2 k runs, which grows impossibly fast.

  • Factors k k k Full factorial runs Effects estimated Of which main effects
  • 3 8 7 3
  • 5 32 31 5
  • 7 128 127 7
  • 10 1,024 1,023 10
  • Ten factors would consume 1,024 runs to estimate 1,023 effects, of which only 10 are main effects and
  • 45 are two-factor interactions. The remaining 968 are three-factor and higher interactions that are

almost always negligible. A fractional design spends the run budget on the effects that matter.

Three principles justify this:

Sparsity of effects. Most processes are driven by a few main effects and a small number of

two-factor interactions.

Hierarchy. Lower-order effects are more likely to be important than higher-order ones, and are

usually larger.

Heredity. An interaction is unlikely to be active unless at least one of its parent main effects

is active.

Notation

A fractional factorial is written 2 R k − p 2^{k-p}_{R} 2 R k − p ​ :

  • k k k = number of factors
  • p p p = degree of fractionation ( 1 / 2 p 1/2^p 1/ 2 p of the full design)
  • R R R = resolution, in Roman numerals

2 V 5 − 1 2^{5-1}_{V} 2 V 5 − 1 ​ means five factors in 2 4 = 16 2^4 = 16 2 4 = 16 runs, a half fraction, at Resolution V.

  • Design Runs Fraction Typical resolution
  • 2 5 − 1 2^{5-1} 2 5 − 1 16 1/2 V
  • 2 6 − 2 2^{6-2} 2 6 − 2 16 1/4 IV
  • 2 7 − 3 2^{7-3} 2 7 − 3 16 1/8 IV
  • 2 7 − 4 2^{7-4} 2 7 − 4 8 1/16 III
  • 2 15 − 11 2^{15-11} 2 15 − 11 16 1/2048 III
  • Generators, defining relations, and aliases

Build a 2 5 − 1 2^{5-1} 2 5 − 1 by writing the full 2 4 2^4 2 4 design in factors A, B, C, D, then assigning the fifth factor

to the highest-order interaction column: E = A B C D (the generator) E = ABCD \quad \text{(the generator)} E = A BC D (the generator)

Multiply both sides by E E E and use the rule that any column times itself gives the identity column I I I : I = A B C D E (the defining relation) I = ABCDE \quad \text{(the defining relation)} I = A BC D E (the defining relation)

Every effect's alias is found by multiplying it by the defining relation: A × A B C D E = B C D E , A B × A B C D E = C D E A \times ABCDE = BCDE, \qquad AB \times ABCDE = CDE A × A BC D E = BC D E , A B × A BC D E = C D E

So A A A is aliased with the four-factor interaction B C D E BCDE BC D E , and A B AB A B is aliased with the three-factor

interaction C D E CDE C D E . Since four-factor and three-factor interactions are almost always negligible, both

main effects and two-factor interactions can be interpreted cleanly. That is Resolution V.

Contrast with a 2 7 − 4 2^{7-4} 2 7 − 4 in 8 runs, whose defining relation contains several words of length 3, so main

effects are aliased with two-factor interactions: Resolution III.

Resolution from the defining relation

Resolution equals the length of the shortest word in the defining relation. An effect of order p p p is aliased with effects of order R − p R - p R − p .

  • Resolution Shortest word Main effects aliased with Two-factor interactions aliased with
  • III 3 letters Two-factor interactions Main effects
  • IV 4 letters Three-factor interactions Other two-factor interactions
  • V 5 letters Four-factor interactions Three-factor interactions
  • Interpreting a confounded result

Suppose a 2 I V 6 − 2 2^{6-2}_{IV} 2 I V 6 − 2 ​ screening design shows a large effect on the alias chain A B + C E AB + CE A B + CE . The

experiment cannot tell you which of the two it is. Four ways forward:

Process knowledge. If C and E are unrelated physically while A and B are known to interact, AB is

the plausible attribution. This is a judgment, and it should be recorded as one.

Heredity. If A and B are both active main effects while C and E are inactive, AB is far more

likely.

Fold over. Run a second fraction with the signs of one or all factors reversed. Combining the two

fractions breaks the alias chain.

Run the confirming pair. A small follow-up design over just those factors resolves it directly.

Fold-over designs

Fold-over type Effect

Full fold-over (reverse all factor signs) Raises a Resolution III design to Resolution IV; all main effects become clear of two-factor interactions

Single-factor fold-over (reverse one factor's signs) Frees that one factor and all its two-factor interactions from aliasing

A fold-over doubles the run count, so the sequential strategy of a small Resolution III screen followed

by a fold-over only where needed is often cheaper than starting with a larger design.

Worked example

A team screens seven factors in a 2 I I I 7 − 4 2^{7-4}_{III} 2 III 7 − 4 ​ design, 8 runs. Effects on the alias chains show three

large values: A, D, and the chain B D + C E + A F BD + CE + AF B D + CE + A F .

Main effects A and D are aliased with two-factor interactions, so even those attributions are not

certain at Resolution III.

The three-way chain is uninterpretable as it stands.

  • The team runs a full fold-over , 8 more runs, giving a 16-run Resolution IV design. Now all seven main
  • effects are clear of two-factor interactions, and A and D are confirmed as genuinely active. Two-factor
  • interactions remain aliased in pairs, but with A and D active and the other five factors inactive, the

heredity principle points at A D AD A D , which a small follow-up factorial in A and D confirms.

Total cost: 16 screening runs plus a small confirmation, against 128 for the full factorial.

Plackett-Burman designs

Plackett-Burman designs screen up to N − 1 N-1 N − 1 factors in N N N runs, where N N N is a multiple of 4 (12, 20, 24,

28). They are Resolution III but with partial (complex) aliasing : each main effect is aliased with a

fraction of many two-factor interactions rather than fully with one.

Advantage : extremely efficient. Eleven factors in 12 runs.

Disadvantage : interactions are smeared across main effects, and the alias structure is not clean,

so interpretation is riskier and they cannot be folded over as simply.

They are appropriate as a first pass when the factor list is long and interactions are believed weak.

Choosing the design

  • Situation Choice
  • More than 6 factors, interactions unknown, want few runs Resolution III or Plackett-Burman, plan to fold over
  • 5-7 factors, main effects must be trustworthy Resolution IV
  • 4-5 factors, need two-factor interactions Resolution V or full factorial
  • 2-3 factors Full factorial; there is no reason to fractionate
  • The standard sequential strategy is to spend no more than about a quarter of the experimental budget on

the first screen, then commit the rest based on what it finds.