Optimal Designs

Computer-generated experimental designs optimized for specific statistical criteria, offering maximum efficiency for custom experimental constraints and objectives.

Optimal Design Overview & Philosophy

What they accomplish:

Optimal designs use computer algorithms to select experimental points that maximize statistical efficiency according to specific mathematical criteria. They provide customized solutions when standard designs are inadequate due to constraints, irregular regions, or unusual objectives.

Advantages:

• • Maximizes statistical efficiency
• • Handles complex constraints naturally
• • Customizable optimization criteria
• • Works with irregular design regions
• • Minimal run count for given precision

Requirements:

• • Computer algorithm generation
• • Model specification in advance
• • Understanding of optimality criteria
• • Constraint specification

Optimality Criteria & Applications

Common Optimality Criteria:

D-Optimal:

• • Minimizes volume of confidence ellipsoid
• • Best for parameter estimation precision
• • Most commonly used criterion
• • Good general-purpose choice

I-Optimal (G-Optimal when uniform):

• • Minimizes average prediction variance
• • Best for prediction over design region
• • Useful for response surface modeling
• • Requires specification of prediction region

A-Optimal:

• • Minimizes average variance of parameter estimates
• • Focus on overall parameter precision
• • Good when all parameters equally important
• • Less commonly used than D-optimal

Custom Criteria:

• • V-optimal: Minimizes average prediction variance
• • E-optimal: Maximizes minimum eigenvalue
• • Compound criteria: Multiple objectives
• • User-defined objective functions

Reading the Optimal (D / I) Design Design Layout

Design Evaluation & Validation

Efficiency Metrics:

D-Efficiency:

• • Relative to best possible D-optimal design
• • Typically aim for > 80% efficiency
• • Higher values indicate better precision
• • Most common efficiency metric

Prediction Properties:

• • Fraction of Design Space plots
• • Average and maximum prediction variance
• • Prediction variance maps
• • G-efficiency assessment

Design Diagnostics:

• Alias Matrix: Check confounding patterns

• Leverage Values: Identify influential points

• Condition Number: Assess numerical stability

• Power Analysis: Detection capabilities for effects

Comparison Guidelines:

• vs Standard Designs: Efficiency gains and trade-offs

• Robustness: Performance under model changes

• Practical Constraints: Implementation feasibility

• Sequential Flexibility: Ability to add runs later

Implementation Process & Algorithms

Design Generation Steps:

1. Model Specification:

• Define expected model terms (linear, quadratic, interactions)

• Specify factor ranges and types

• Identify any known constraints

2. Criterion Selection:

• Choose optimality criterion (D, I, A, custom)

• Define region of interest if needed

• Set relative importance of objectives

3. Algorithm Execution:

• Point exchange algorithms

• Coordinate exchange methods

• Genetic algorithm optimization

• Multiple starting points for global optimum

Platform Implementation:

The sixsigmacenter.com platform provides state-of-the-art optimal design generation using advanced algorithms, handles complex constraints automatically, evaluates design efficiency, and compares alternative designs with detailed diagnostics.

When to Use Optimal Designs

Ideal Applications:

Constrained Experiments:

• • Irregular feasible regions
• • Linear/nonlinear constraints
• • Mixture design constraints
• • Equipment/safety limitations

Efficiency Requirements:

• • Extremely limited run budget
• • Expensive experiments
• • Time-critical studies
• • Maximum precision needed

Design Scenarios:

• Augmentation: Adding runs to existing designs optimally

• Model Discrimination: Distinguishing between competing models

• Robust Design: Accounting for model uncertainty

• Sequential Design: Adaptive experimental strategies

Example Data Layout for D-Optimal Design (constrained region)

D-optimal design for quadratic model with constraints X1 + X2 ≤ 1.5 and X3² + X1 ≤ 1. Sample Size Recommendation: Minimum (p+2) runs for p parameters, typically add 20-50% for power and robustness.

This is how your spreadsheet or CSV file should be structured before you upload it. Match the column names, keep one row per observation, and leave the analysis to the platform.