15.4 Robust optimal designs

Robust optimal designs tackle uncertainty in experimental planning. They aim to create designs that perform well across different scenarios, whether it's model uncertainty or parameter uncertainty. This approach ensures experiments are effective even when we're not sure about the underlying model or parameter values.

Minimax, maximin efficiency, and compound optimal designs address model uncertainty. Bayesian optimal designs and sensitivity analysis tackle parameter uncertainty. These methods help researchers create experiments that are resilient to various unknowns, improving the reliability of results.

Robust Designs for Model Uncertainty

Minimax and Maximin Efficiency Designs

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• Model uncertainty occurs when there is doubt about the true underlying model structure or form
• Minimax designs aim to minimize the maximum loss or risk across all possible models under consideration
  • Useful when the goal is to protect against the worst-case scenario
  • Ensures the design performs reasonably well even under the least favorable model
• Maximin efficiency designs maximize the minimum efficiency across all candidate models
  • Efficiency measures how well a design performs relative to the optimal design for each model
  • Seeks to find a design that has good performance for all models, rather than being optimal for one specific model

Compound Optimal Designs

• Compound optimal designs are a compromise between different optimality criteria or models
• Constructed by combining multiple optimality criteria or models into a single objective function
  • Example: weighted sum of efficiencies for different models
• Allows for balancing the performance across different scenarios or objectives
• Provides a way to incorporate multiple sources of uncertainty or multiple design goals simultaneously

Robust Designs for Parameter Uncertainty

Bayesian Optimal Designs

• Parameter uncertainty refers to the lack of precise knowledge about the true values of model parameters
• Bayesian optimal designs incorporate prior information about the parameters into the design process
  • Prior information is represented by a probability distribution over the parameter space
• Bayesian designs aim to maximize the expected utility or minimize the expected loss, averaged over the prior distribution
  • Takes into account the uncertainty in the parameter values
  • Provides designs that are robust to parameter misspecification

Sensitivity Analysis and Design Robustness

• Sensitivity analysis assesses how sensitive the optimal design is to changes in the parameter values or assumptions
  • Involves perturbing the parameters or assumptions and evaluating the impact on the design performance
  • Helps identify the critical parameters or assumptions that have a significant influence on the design
• Design robustness refers to the ability of a design to maintain good performance despite variations in the parameters or assumptions
  • A robust design is relatively insensitive to parameter uncertainty or model misspecification
  • Can be assessed through sensitivity analysis or by evaluating the design performance under different scenarios
• Techniques for improving design robustness include:
  • Using robust optimality criteria that account for parameter uncertainty (Bayesian optimality)
  • Incorporating parameter uncertainty directly into the design optimization process
  • Constructing designs that are efficient across a range of plausible parameter values