Design of Experiments (DOE) Tutorial - Elneuro | Factorial Design, RSM Academic Guide

1. Introduction to Design of Experiments

Design of Experiments (DOE) is a systematic approach to planning experiments to efficiently explore the relationship between input variables (factors) and output variables (responses). Pioneered by Sir Ronald A. Fisher in the 1920s and later developed by George Box, DOE provides a rigorous framework for scientific investigation and process optimization.

DOE offers several advantages over one-factor-at-a-time (OFAT) experimentation:

Efficiency: Fewer experiments needed to obtain equivalent information
Interactions: Can detect and quantify factor interactions
Optimization: Enables systematic process optimization
Robustness: Results valid over the entire experimental region

2. Fundamental Principles

Randomization

Random assignment of experimental units to treatments ensures that the effects of unknown or uncontrollable factors are distributed randomly across treatment groups, preventing systematic bias and validating statistical inference.

Replication

Repeating experimental runs provides an estimate of experimental error (pure error), enables assessment of statistical significance, and improves precision of effect estimates.

Blocking

Grouping experimental units into homogeneous blocks reduces variability by accounting for known sources of variation (e.g., day-to-day variation, batch effects), increasing the precision of factor effect estimates.

Fisher's Key Insight

"Block what you can, randomize what you cannot." This principle guides the design of efficient experiments that maximize information while controlling for extraneous sources of variation.

3. DOE Terminology

Factor: An input variable that may affect the response (e.g., temperature, pressure)
Level: A specific value or setting of a factor
Treatment: A specific combination of factor levels
Response: The output variable being measured
Run: A single experimental observation
Effect: Change in response due to change in factor level
Main Effect: The average effect of a factor across all levels of other factors
Interaction: When the effect of one factor depends on the level of another

4. Full Factorial Designs

A full factorial design includes all possible combinations of factor levels. For k factors each at n levels, the design requires n^k runs.

Full Factorial Size

N = n₁ × n₂ × ... × n_k

For balanced designs with each factor at the same number of levels: N = n^k

Example: 3² Full Factorial

Two factors (Temperature, Pressure) each at 3 levels:

Temperature: Low (100°C), Medium (150°C), High (200°C)
Pressure: Low (1 atm), Medium (2 atm), High (3 atm)

Total runs: 3² = 9 unique treatment combinations

5. Two-Level Factorial Designs (2^k)

Two-level factorials are particularly useful for screening experiments and investigating first-order effects. Factors are coded to -1 (low) and +1 (high).

Coded Factor Values

x = (X - X̄) / (X_high - X_low)/2

Where X is the natural value, X̄ is the center point, converting factors to [-1, +1] scale.

2² Factorial Design Matrix

Run	A	B	AB
1	-1	-1	+1
2	+1	-1	-1
3	-1	+1	-1
4	+1	+1	+1

Effect Calculations for 2^k Designs

Main Effect of Factor A

Effect_A = Ȳ_A+ - Ȳ_A-

The average response at high level minus average response at low level.

Interaction Effect AB

Effect_AB = (1/2)[(Ȳ_A+B+ - Ȳ_A-B+) - (Ȳ_A+B- - Ȳ_A-B-)]

Measures whether the effect of A depends on the level of B.

6. Fractional Factorial Designs

When full factorials become too large, fractional factorial designs provide a subset that estimates main effects and lower-order interactions with fewer runs.

Fractional Factorial Notation

2^k-p design

k factors in 2^k-p runs. The fraction is 1/2^p of the full factorial.

Resolution

Design resolution indicates the degree of aliasing:

Resolution III: Main effects aliased with two-factor interactions
Resolution IV: Main effects clear of two-factor interactions; 2FIs aliased with each other
Resolution V: Main effects and 2FIs clear of each other; 2FIs aliased with 3FIs

Example: 2^3-1 Design (Half-Fraction)

Three factors in 4 runs instead of 8. Generator: C = AB

Defining relation: I = ABC

Alias structure: A = BC, B = AC, C = AB

This is a Resolution III design.

7. ANOVA for Factorial Designs

Analysis of Variance (ANOVA) partitions total variation into components attributable to factors, interactions, and error.

Total Sum of Squares Decomposition (Two-Factor)

SS_Total = SS_A + SS_B + SS_AB + SS_Error

Sum of Squares for Factor A

SS_A = bn ∑_i (Ȳ_i.. - Ȳ_...)²

Where b = number of levels of B, n = replicates, Ȳ_i.. = mean at level i of A.

F-Tests for Significance

F = MS_Effect / MS_Error

Compare to F_{α, df_Effect, df_Error}. Reject H₀ if F > F_critical.

Source	SS	df	MS	F
Factor A	SS_A	a-1	SS_A/(a-1)	MS_A/MS_E
Factor B	SS_B	b-1	SS_B/(b-1)	MS_B/MS_E
A × B	SS_AB	(a-1)(b-1)	SS_AB/df	MS_AB/MS_E
Error	SS_E	ab(n-1)	SS_E/df	-
Total	SS_T	abn-1	-	-

8. Effect Estimation and Significance

Contrast Method

For 2^k designs, effects are estimated using contrasts:

Effect Estimate

Effect = (2/N) × Contrast = (2/N) ∑ (sign) × y_i

Standard Error of Effects

With replication:

SE_Effect = √(4MS_E / nN)

Where n = replicates per treatment, N = total unique treatments.

Normal Probability Plot

For unreplicated designs, plot ordered effects against normal quantiles. Significant effects will deviate from the line formed by negligible (noise) effects.

9. Response Surface Methodology (RSM)

RSM is used to optimize processes by fitting polynomial models to experimental data and finding factor settings that maximize or minimize the response.

First-Order Model

y = β₀ + ∑ β_ix_i + ε

Second-Order Model

Full Quadratic Model

y = β₀ + ∑ β_ix_i + ∑ β_iix_i² + ∑∑ β_ijx_ix_j + ε

Includes linear, quadratic, and interaction terms to model curvature.

Central Composite Design (CCD)

CCD is the most popular RSM design, consisting of:

Factorial points: 2^k or 2^k-p corners
Center points: n₀ runs at the center (coded 0,0,...,0)
Axial points: 2k points at distance α from center along each axis

Rotatability Condition

α = (2^k)^1/4

For rotatable designs, prediction variance is constant at all points equidistant from the center.

Box-Behnken Design

An alternative RSM design that does not include corner points, useful when extreme factor combinations should be avoided. Requires fewer runs than CCD but does not estimate all model terms as efficiently.

10. Taguchi Methods

Developed by Genichi Taguchi, these methods focus on robust design - making processes insensitive to noise factors.

Signal-to-Noise Ratios

Smaller-is-Better

S/N = -10 log₁₀[(1/n) ∑ y_i²]

Larger-is-Better

S/N = -10 log₁₀[(1/n) ∑ (1/y_i²)]

Nominal-is-Best

S/N = 10 log₁₀(Ȳ² / s²)

Maximizes the ratio of mean to variance.

Orthogonal Arrays

Taguchi uses standardized orthogonal arrays (L₈, L₉, L₁₆, etc.) as fractional factorial designs for efficient experimentation.

11. Optimal Design

Optimal designs use mathematical criteria to select the best experimental points.

D-Optimality

Maximizes |X^TX|, minimizing the volume of the confidence ellipsoid for parameter estimates.

I-Optimality

Minimizes average prediction variance over the design region.

A-Optimality

Minimizes the trace of (X^TX)^-1, the average variance of parameter estimates.

When to Use Optimal Designs

Irregular experimental regions
Constraints on factor combinations
Mixture experiments
Limited resources for experimentation

References and Further Reading

Montgomery, D.C. (2017). Design and Analysis of Experiments, 9th Edition. Wiley.
Box, G.E.P., Hunter, J.S., & Hunter, W.G. (2005). Statistics for Experimenters, 2nd Edition. Wiley.
Myers, R.H., Montgomery, D.C., & Anderson-Cook, C.M. (2016). Response Surface Methodology, 4th Edition. Wiley.
Taguchi, G. (1986). Introduction to Quality Engineering. Asian Productivity Organization.
Wu, C.F.J. & Hamada, M.S. (2021). Experiments: Planning, Analysis, and Optimization, 3rd Edition. Wiley.

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Design of Experiments (DOE)