15.2 Alphabetic optimality criteria (A, D, E, G-optimality)

Optimal design theory helps researchers create efficient experiments. Alphabetic optimality criteria, like A, D, E, and G, provide different ways to measure a design's quality. Each criterion focuses on specific aspects of precision in parameter estimation or prediction.

These criteria help balance trade-offs in experimental design. A-optimality minimizes average variance, D-optimality maximizes overall precision, E-optimality focuses on worst-case scenarios, and G-optimality improves prediction accuracy across the design space.

Information-based Optimality Criteria

Trace Criterion (A-optimality)

• A-optimality minimizes the average variance of the parameter estimates
• Aims to minimize the trace of the inverse of the information matrix $tr(X^TX)^{-1}$
• Equivalent to minimizing the average variance of the parameter estimates
• Focuses on the precision of the parameter estimates
• Useful when all parameters are of equal importance (treatment effects in a clinical trial)

Determinant Criterion (D-optimality)

• D-optimality maximizes the determinant of the information matrix $det(X^TX)$
• Equivalent to minimizing the generalized variance of the parameter estimates
• Aims to minimize the volume of the joint confidence ellipsoid of the parameters
• Focuses on the overall precision of the parameter estimates
• Useful when the overall precision of the estimates is important (response surface modeling)

Eigenvalue Criterion (E-optimality)

• E-optimality maximizes the minimum eigenvalue of the information matrix $\lambda_{min}(X^TX)$
• Aims to minimize the maximum variance of the parameter estimates
• Focuses on the worst-case precision of the parameter estimates
• Useful when the worst-case precision is important (ensuring all parameters are estimated with a minimum precision)

Prediction-based Optimality Criteria

Maximum Prediction Variance (G-optimality)

• G-optimality minimizes the maximum prediction variance over the design space
• Aims to minimize the worst-case prediction variance $\max_{x \in \mathcal{X}} Var(\hat{y}(x))$
• Focuses on the precision of the predicted response over the entire design space
• Useful when the goal is to make precise predictions throughout the design space (response surface modeling, process optimization)
• Equivalent to D-optimality for linear models (General Equivalence Theorem)

Minimax Prediction Variance

• Minimax prediction variance minimizes the maximum prediction variance over a specific set of points
• Aims to minimize the worst-case prediction variance over a subset of the design space $\max_{x \in \mathcal{X}_0} Var(\hat{y}(x))$
• Focuses on the precision of the predicted response over a subset of the design space
• Useful when precise predictions are required at specific locations (critical points in a process)

Advanced Optimality Concepts

General Equivalence Theorem

• Establishes the equivalence between D-optimality and G-optimality for linear models
• States that a design is D-optimal if and only if it is G-optimal
• Provides a unified framework for information-based and prediction-based optimality criteria
• Allows for the verification of optimality using the equivalence theorem
• Enables the construction of optimal designs using iterative algorithms (Fedorov-Wynn algorithm)

Compound Criteria

• Compound criteria combine multiple optimality criteria into a single objective function
• Allow for the balancing of different optimality goals (precision of estimates and predictions)
• Examples include the weighted sum of A-optimality and D-optimality $w_A tr(X^TX)^{-1} + w_D det(X^TX)^{-1/p}$
• Enables the construction of designs that satisfy multiple optimality criteria simultaneously
• Useful when multiple objectives need to be considered (balancing parameter estimation and prediction precision)