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Atkinson: Optimum Experimental Design
SAS programs
These SAS programs accompany Atkinson, Donev, and Tobias (2007), reproducing the calculations underlying most of the tables and figures in the book. The programs use the following SAS products for the purposes listed:
- Base SAS Data processing, manipulation, and display
- SAS/STAT Fitting and analyzing statistical models
- SAS/QC Constructing experimental designs, including optimal exact designs
- SAS/IML Matrix calculations and optimization, including finding optimal continuous designs
- SAS/ETS Fitting differential equation models
You will need all these products in order to run all of the SAS programs. Note that most universities and business organizations license a bundle of SAS products that includes these tools.
The programs were developed using SAS 9.1, released in 2004, but most of the SAS facilities used are available in any recent version of the software. The most significant exception is the ODS Graphics facility, demonstrated in Chapter 8.
Chapter 3 - Experimental Strategies
Chapter 4 - The Choice of a Model
Chapter 5 - Models and Least Squares
Chapter 6 - Criteria for a Good Experiment
- Computations for Figures 6.1 & 6.2 - Confidence ellipses
- Computations for Figures 6.3 & 6.4 - Standardized variances for simple regression
- Computations for Figures 6.5 & 6.6 - Standardized variances for quadratic regression
- Computations for Figures 6.7 & 6.8 - Standardized variances for two-factor designs
- Computations for Figure 6.9 - Variance dispersion graph
Chapter 8 - The Analysis of Experiments
Chapter 9 - Optimum Design Theory
- Example 9.1 - Quadratic regression through the origin
- Example 9.2 - Quadratic regression: Variance function
- Optimum design measure for quadratic design on [-1,1]
- Example 9.3 - Quadratic regression: Variance function
This is a complicated program because of its generality: the same code can be used with minimal changes for finding any optimal design for a single factor, and the approach is also useful for designs with more factors. IML functions are defined to compute the criterion for both w and x, for w with x fixed, and for x with w fixed, and these functions are iteratively optimized to find both the optimum w and the optimum x.
This program performs the calculation two different ways and compares the results: (1) directly with matrix code and (2) indirectly with SAS design and analysis PROCs.
In order to modify this program to study designs for higher-order polynomials, add corresponding columns to the Zg matrix over the predictive grid, and perhaps also decrease the increment in the definition of the Grid data set.
Chapter 10 - Criteria of Optimality
- Figure 10.1 - Quadratic regression through the origin
This program is largely the same as Program 9.1, which see for comments on the method. The computation at the end of the program of the variance function for the Ds criterion is the only difference.
Chapter 11 - D-optimum Designs
- Example 11.1 - Cubic regression through the origin
- Example 11.1 - Cubic regression through the origin
- Table 11.2 - Efficiencies for Example 1.1
- Table 11.3 - Polynomial regression in one variable
- Table 11.7 - Optimal quadratic designs
The program uses OPTEX and compares the results to those given in Atkinson and Donev (1992).
Chapter 12 - Algorithms for the Construction of Exact D-optimum Designs
- Example 12.1 - Exact optimal quadratic designs on square
- Equation 12.2 - Exact design weights over [-1,0,1]2
- Example 12.1 - Exact optimal quadratic designs
- Example 12.2 - Performance of an Internal Combustion Engine
Same as Program 13.2 - SAS Task 13.2
Same as Program 13.1 - SAS Task 13.1
Chapter 13 - Optimum Experimental Design with SAS
- SAS Task 13.1 - Exact optimal quadratic designs
- SAS Task 13.2 - Exact optimal quadratic designs on square
- SAS Task 13.3 - Exact optimal quadratic designs on square
Same as Program 12.3 - Example 12.1
Same as Program 12.1 - Example 12.1
Compare to Program 12.1, Program 13.2
Chapter 14 - Experiments with Both Qualitative and Quantitative Factors
- Example 14.1 - Second-order response surface with one qualitative factor
- Table 14.1 - D-optimum exact N-trial designs for a second-order model in m quantitative factors with one qualitative factor at B levels with one qualitative factor
- Example 14.3 - Second-order response surface with two qualitative factors
Took ~3 minutes to run on server-class PC in May, 2007.
See also SAS Task 14.1
Chapter 15 - Blocking Response Surface Designs
- Example 15.1 - Second-order block designs with variable block sizes
- Example 15.2 - Second-order designs with random effects Took ~8.5 minutes to run on server-class PC in May, 2007.
- SAS Task 15.1 - Blocked second-order designs Uses a macro that is similar to, but not the same as, the one in Program 15.1.
- SAS Task 15.2 - Trend-free second-order design Uses a macro that is similar to, but not the same as, the one in Program 15.1.
- SAS Task 5.3 - Second-order designs with random effects
- Section 15.5.5 - Db-optimum design for clinical trial
Took ~2.5 minutes to run on server-class PC in May, 2007.
Chapter 16 - Mixture Experiments
- Figures 16.1 & 16.2 - Optimal mixture designs
- Example 16.1 Modification of an Acrylonitrile Powder
- Table 16.2. Example 16.2: mixture experiment with linear constraints
- Example 16.3: quadratic constraints for a mixture experiment
- Fig. 16.5. Example 16.4: second-order model for a three-component mixture with a qualitative factor at three levels
- Fig. 16.6. Three-component mixture in two blocks
- Fig. 16.7 & 16.8 - 16-trial D-optimum designs in 6 blocks for a 4-component mixture
- Fig. 16.9-16.11 - Optimally blocked and orthogonally blocked mixture designs
Uses the ADX macros to create simplex lattice designs. Also addresses SAS Task 16.3.
Uses the ADX macros to create XVERT points.
Uses the ADX macros to create XVERT points. Also addresses SAS Tasks 16.4, 16.5, and 16.7.
Uses the ADX macros to create simplex lattice designs. Also addresses SAS Task 16.6.
Uses the ADX macros to create simplex lattice designs. Also addresses SAS Tasks 16.8 and 16.9.
Uses the ADX macros to create simplex lattice designs.
Uses the ADX macros to create simplex lattice designs.
Uses the ADX macros to create simplex lattice designs.
Chapter 17 - Non-linear Models
- Fig. 17.2 & 17.4 - Examples 17.1 and 17.3: Computing Jacobians for directly specified nonlinear models
- Fig. 17.6 - Optimum design for two consecutive first-order models
- Fig. 17.7 & 17.8 - Sampling window for two consecutive first-order reactions
- Tables 17.1 and 17.2 - D- and c-optimum designs for a compartmental model
- Example 17.5 - Sequential nonlinear optimum design
- Example 17.6 and SAS Task 17.5 - Generalized consecutive reaction
- Example 17.7 - Multivariate nonlinear design
- SAS Tasks 17.1 & 17.2 - Nonlinear fitting
Uses NLIN to create the derivatives as callable code, in order to find optimum design support and weights using only the input models.
Chapter 18 - Bayesian Optimum Designs
- Example 18.1: Truncated quadratic model
- Example 18.2: Exponential decay
- Tables 18.5 - 18.7: Exponential decay, optimum designs for various priors
- Tables 18.8-18.11 - Bayesian D- and c-optimum designs for compartmental model
- Tables 18.12 - Local and Bayesian D-optimum designs for reversible reaction
- how the computations are performed,
- how the optimization is initialized, and
- what values are used for consolidation
Took ~40 seconds to run on server-class PC in May, 2007.
Uses the QUAD subroutine in IML to perform double integration over the prior distribution for the parameters. Took ~15 minutes to run on server-class PC in May, 2007.
The flatness of this optimization problem makes it very sensitive numerically. Different machines will get different answers, and even on the same machine, slight changes in
Chapter 19 - Design Augmentation
- Example 19.1: Augmentation of a second-order design to third order
- Table 19.1 - Augmenting the design of Fig. 19.3
- Tables 19.2 & 19.3 - Second order response surface, constrained design region
- SAS Task 19.1 - Augmenting a second-order design to third order
- SAS Task 19.2 - Augmenting the design of Fig. 19.3
- SAS Task 19.1 - Augmenting a second-order design to third order
Chapter 20 - Model Checking and Designs for Discriminating Between Models
- Example 20.1 & SAS Task 20.1 - Checking for second-order departures from a first-order model
- Example 2.2 & SAS Task 20.2 - Second-order response surface: constrained design region
- Example 20.3 & SAS Task 20.3 - A Non-linear Model for crop yield and plant density
- SAS Task 20.4 - Resolution IV designs as Bayesian D-optimum
- Figure 20.4 - Derivative function for T-optimum design
- Figure 20.6 - Sequential T-optimum design
- SAS Task 20.4 - Resolution IV designs as Bayesian D-optimum
This program generates optimum continuous designs by optimizing only weights for given support, comparing results to optimum replications for large exact designs over the same support.
This program generates optimum continuous designs by optimizing only weights for given support, comparing results to optimum replications for large exact designs over the same support.
Using both large exact designs with OPTEX and nonlinear optimization with IML. Incorporates a two-factor version of the now-familiar template for finding an optimum continuous design by nonlinear optimization.
Chapter 21 - Compound Design Criteria
- Section 21.4 and SAS Task 21.1 - Compound optimum designs for polynomials in one factor
- Table 21.2 - Linear or quadratic regression
- Example 21.2 and SAS Task 21.2-21.3: Response surface in four factors
- Example 21.3: First-order growth model
- Example 21.4: Two linear models
- Example 21.6: Constant or quadratic model?
- Example 21.7: Compartmental model
Uses an IML-implemented exchange search. Took ~9 minutes to run on server-class PC in May, 2007.
Chapter 22 - Generalized Linear Models
- Sections 22.4.2 and 22.4.3 and SAS Tasks 22.1-22.4 - One-variable regression
- Two-variable logistic regression
- Second-order response surface
- Section 22.5 - Optimum design for gamma models
Took ~1 minute to run on server-class PC in May, 2007.
Uses a two-factor version of the familiar template for finding a continuous optimal design by nonlinear optimization.
Chapter 23 - Response Transformation and Structured Variances
- Example 23.1 - Box and Cox wool data
- Example 23.2 - Exponential decay
Uses a three-factor version of the familiar template for finding a continuous optimal design by nonlinear optimization, with initialization by sequential addition by nonlinear optimization. Took ~3 minutes to run on server-class PC in May, 2007.
Chapter 24 - Time-dependent Models with Correlated Observations
- Example 24.1 - Two consecutive first-order reactions
- Example 24.2 - Compartmental model
Took ~2.5 minutes to run on server-class PC in May, 2007.
Not quite the designs given in the text, but equivalent with respect to the optimality criterion, as demonstrated by the printed efficiency comparisons.
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