15.2 Alphabetic optimality criteria (A, D, E, G-optimality)
3 min read•august 7, 2024
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 . minimizes average variance, maximizes overall precision, focuses on , and 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 tr(XTX)−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(XTX)
Equivalent to minimizing the of the parameter estimates
Aims to minimize the volume of the of the parameters
Focuses on the overall precision of the parameter estimates
Useful when the overall precision of the estimates is important ()
Eigenvalue Criterion (E-optimality)
E-optimality maximizes the of the information matrix λmin(XTX)
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 over the design space
Aims to minimize the worst-case prediction variance maxx∈XVar(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, )
Equivalent to D-optimality for linear models ()
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 maxx∈X0Var(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
Allows for the verification of optimality using the equivalence theorem
Enables the construction of optimal designs using (Fedorov-Wynn algorithm)
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 wAtr(XTX)−1+wDdet(XTX)−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)