Design of Experiments (DOE) Cheat Sheet

Updated 2026-05-28

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Design of Experiments (DOE) is a systematic statistical methodology for planning, conducting, and analyzing controlled experiments to understand and optimize complex processes and systems. Originally developed by Ronald A. Fisher in the 1920s for agricultural research, DOE has evolved into an indispensable tool across manufacturing, pharmaceuticals, materials science, biotechnology, and virtually every field requiring empirical optimization. DOE enables experimenters to efficiently extract maximum information from minimum experimental runs by strategically manipulating multiple factors simultaneously, rather than the wasteful one-factor-at-a-time approach. The fundamental insight underlying DOE is the principle of effect sparsity—relatively few factors and interactions dominate system behavior, allowing fractional designs to capture essential relationships while dramatically reducing experimental cost and time.

What This Cheat Sheet Covers

This topic spans 24 focused tables and 166 indexed concepts, 149 flashcards. Below is a complete table-by-table outline of this topic, spanning foundational concepts through advanced details.

Table 1: Fundamental Concepts & TerminologyTable 2: Basic Experimental DesignsTable 3: Two-Level Factorial DesignsTable 4: Fractional Factorial DesignsTable 5: Screening DesignsTable 6: Response Surface Methodology (RSM)Table 7: Central Composite Designs (CCD)Table 8: Box-Behnken and Other RSM DesignsTable 9: Mixture DesignsTable 10: Optimal DesignsTable 11: Taguchi Methods & Robust DesignTable 12: Blocking & Randomization StructuresTable 13: Analysis TechniquesTable 14: Model Diagnostics & ValidationTable 15: Optimization & Multiple Response MethodsTable 16: Space-Filling & Computer Experiment DesignsTable 17: Sequential & Adaptive StrategiesTable 18: Analysis of Variance ModelsTable 19: Multiple Comparison ProceduresTable 20: Effect Estimation & InterpretationTable 21: Model Assumptions & RequirementsTable 22: Experimental Planning & ExecutionTable 23: Advanced Designs & Special CasesTable 24: Software Tools & Implementation

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Table 1: Fundamental Concepts & Terminology

Before running any experiment, fluency with DOE vocabulary is essential — misusing terms like replication versus repetition or confusing a factor with a covariate leads to incorrect designs and flawed analyses. These definitions form the shared language across all DOE methods.

Concept	Example	Description
Factor	Temperature, Pressure, Catalyst	• Controllable input variable studied for its effect on the response • can be continuous, discrete, or categorical.
Level	Low (−1), High (+1)	Specific setting or value at which a factor is held during an experimental run.
Response	Yield (%), Strength (MPa)	Measured outcome variable (dependent variable) that depends on factor settings.
Treatment	T=200°C, P=5 bar, Catalyst A	Specific combination of factor levels applied to an experimental unit.
Experimental Unit	Batch, Specimen, Patient	• Smallest entity to which a treatment is independently applied • defines the replication level.
Run	Single test execution	One execution of a specific treatment combination in the experiment.
Replication	3 batches per condition	• Independent repetition of treatment combinations • provides estimate of pure experimental error.
Randomization	Random run order assignment	• Random assignment of treatments to units and run order • controls for unknown nuisance variables.
Blocking	Batch, Day, Operator	• Grouping experimental units by known nuisance factors to reduce variability • "block what you can, randomize what you cannot."
Main Effect	Average effect of Temperature	Change in mean response caused by changing a factor, averaged over all levels of other factors.
Interaction Effect	Temperature × Pressure	Non-additive effect where the effect of one factor depends on the level of another factor.
Confounding (Aliasing)	A = BC in fractional factorial	• Linear dependence of effects • two or more effects estimated by the same contrast, making them indistinguishable.
Effect Sparsity	Few of 20 factors active	• Principle that relatively few factors and interactions dominate system behavior • justifies fractional designs.
Effect Heredity	Include A,B if A×B significant	Principle that higher-order interactions are active only if parent main effects or lower-order interactions are active.
Experimental Error	σ² = 2.3 units²	• Variability in response due to uncontrolled factors and measurement imprecision • estimated via replication or pooling.

Table 2: Basic Experimental Designs

These classical designs form the foundation of experimental planning: they determine how treatments are assigned to units, how nuisance variation is controlled, and at what level randomization occurs. Choosing the right base design prevents confounding between treatments and uncontrolled variation before any factor-level decisions are made.

Design	Example	Description
Completely Randomized Design (CRD)	Assign 20 units randomly to 4 treatments	• Simplest design; all units homogeneous, treatments assigned purely at random. • Analyzed with one-way ANOVA. • Assumes no systematic variability among units.

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Table 3: Two-Level Factorial Designs

Two-level factorials are the workhorse of process improvement: they are efficient, fully interpretable, and scale naturally into fractional factorials. Understanding coded variables, contrasts, and the Yates algorithm gives you the mental model for everything that follows in fractional and response surface work.

Design	Example	Description
2^k Full Factorial	$2^5 = 32$ runs for 5 factors	• All combinations of k factors at two levels (low/high) • estimates all main effects and interactions. • Runs grow exponentially with factors.

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Table 4: Fractional Factorial Designs

Fractional factorials are the most commonly used DOE family in industrial practice: they cut experimental effort by powers of two while preserving the ability to estimate the effects that matter most. Mastering resolution, generators, and the defining relation is essential to selecting and interpreting any fractional design.

Design	Example	Description
2^(k-p) Fractional Factorial	$2^{7-3} = 16$ runs, 7 factors	• Fraction (1/2^p) of full factorial • studies k factors in fewer runs by deliberately confounding higher-order interactions.

The rest of this section is part of the full cheat sheet

Table 1: Fundamental Concepts & Terminology

Concept	Example	Description
Factor	Temperature, Pressure, Catalyst	• Controllable input variable studied for its effect on the response • can be continuous, discrete, or categorical.
Level	Low (−1), High (+1)	Specific setting or value at which a factor is held during an experimental run.
Response	Yield (%), Strength (MPa)	Measured outcome variable (dependent variable) that depends on factor settings.
Treatment	T=200°C, P=5 bar, Catalyst A	Specific combination of factor levels applied to an experimental unit.
Experimental Unit	Batch, Specimen, Patient	• Smallest entity to which a treatment is independently applied • defines the replication level.
Run	Single test execution	One execution of a specific treatment combination in the experiment.
Replication	3 batches per condition	• Independent repetition of treatment combinations • provides estimate of pure experimental error.
Randomization	Random run order assignment	• Random assignment of treatments to units and run order • controls for unknown nuisance variables.
Blocking	Batch, Day, Operator	• Grouping experimental units by known nuisance factors to reduce variability • "block what you can, randomize what you cannot."
Main Effect	Average effect of Temperature	Change in mean response caused by changing a factor, averaged over all levels of other factors.
Interaction Effect	Temperature × Pressure	Non-additive effect where the effect of one factor depends on the level of another factor.
Confounding (Aliasing)	A = BC in fractional factorial	• Linear dependence of effects • two or more effects estimated by the same contrast, making them indistinguishable.
Effect Sparsity	Few of 20 factors active	• Principle that relatively few factors and interactions dominate system behavior • justifies fractional designs.
Effect Heredity	Include A,B if A×B significant	Principle that higher-order interactions are active only if parent main effects or lower-order interactions are active.
Experimental Error	σ² = 2.3 units²	• Variability in response due to uncontrolled factors and measurement imprecision • estimated via replication or pooling.

Table 2: Basic Experimental Designs

Design	Example	Description
Completely Randomized Design (CRD)	Assign 20 units randomly to 4 treatments	• Simplest design; all units homogeneous, treatments assigned purely at random. • Analyzed with one-way ANOVA. • Assumes no systematic variability among units.

Table 3: Two-Level Factorial Designs

Design	Example	Description
2^k Full Factorial	$2^5 = 32$ runs for 5 factors	• All combinations of k factors at two levels (low/high) • estimates all main effects and interactions. • Runs grow exponentially with factors.

Table 4: Fractional Factorial Designs

Design	Example	Description
2^(k-p) Fractional Factorial	$2^{7-3} = 16$ runs, 7 factors	• Fraction (1/2^p) of full factorial • studies k factors in fewer runs by deliberately confounding higher-order interactions.