Optimal design theory helps researchers choose the best experimental setup. It's about picking the right factor combinations to get the most information from your study. This chapter dives into the nuts and bolts of making smart design choices.

We'll look at different types of designs, how to measure their effectiveness, and what makes a design "optimal." Understanding these concepts will help you create experiments that are more efficient and give you better results.

Optimal Design Fundamentals

Key Concepts in Optimal Design

• Optimal design involves selecting design points from a design space to maximize the information obtained from an experiment
• Design space represents the set of all possible combinations of factor levels that can be used in an experiment
• Design points are specific combinations of factor levels chosen from the design space to be included in the experiment
• Continuous designs allow factor levels to take on any value within a specified range (temperature between 20°C and 30°C)
• Exact designs specify the precise factor level combinations and the number of replicates for each design point

Types of Designs

• Optimal designs can be classified into different categories based on their properties and characteristics
  • Continuous designs are suitable when factors can be varied continuously over a range of values
  • Exact designs are appropriate when factors have discrete levels and the number of replicates for each design point is fixed
• The choice between continuous and exact designs depends on the nature of the factors and the experimental constraints
  • Continuous designs offer more flexibility in factor level selection but may be more challenging to implement in practice
  • Exact designs provide a specific blueprint for the experiment but may be less efficient if the optimal factor levels are not included

Information and Efficiency

Quantifying Information in Optimal Designs

• The information matrix quantifies the amount of information provided by a design about the parameters of interest
• Fisher information measures the expected amount of information that an observable random variable carries about an unknown parameter
• The information matrix is often used to evaluate and compare different designs in terms of their information content
  • Designs with larger information matrices are generally preferred as they provide more precise parameter estimates
  • The determinant or trace of the information matrix can be used as summary measures of the overall information provided by a design

Efficiency of Optimal Designs

• Efficiency is a measure of how close a design is to the optimal design in terms of the information it provides
• Efficiency is typically expressed as a percentage, with 100% indicating an optimal design
• Designs with higher efficiency are preferred as they require fewer experimental runs to achieve the same level of precision
  • For example, a design with 90% efficiency would require 10% more runs than the optimal design to obtain the same information
• Efficiency can be used to compare and rank different designs based on their performance relative to the optimal design

Design Criteria and Optimality

Defining Design Criteria

• A design criterion is a mathematical function that quantifies the desirability of a design based on its properties
• Common design criteria include D-optimality, A-optimality, and E-optimality, which focus on different aspects of the information matrix
  • D-optimality maximizes the determinant of the information matrix, which corresponds to minimizing the generalized variance of the parameter estimates
  • A-optimality minimizes the trace of the inverse of the information matrix, which represents the average variance of the parameter estimates
  • E-optimality maximizes the minimum eigenvalue of the information matrix, ensuring that all parameters are estimated with a certain level of precision
• The choice of design criterion depends on the specific goals and priorities of the experiment

Assessing Optimality of Designs

• Optimality refers to the property of a design being the best possible design according to a given criterion
• A design is considered optimal if it maximizes or minimizes the chosen design criterion
• Checking for optimality involves comparing the performance of a design to theoretical bounds or other candidate designs
  • For D-optimality, the determinant of the information matrix can be compared to the maximum possible value achievable within the design space
  • For A-optimality and E-optimality, similar comparisons can be made based on the respective criteria
• Verifying optimality ensures that the selected design is indeed the best choice for the given experimental objectives and constraints