Analyzing Supersaturated Designs Using Biased Estimation

1. FAMU-FSU College of Engineering, Department of Industrial Engineering Analyzing Supersaturated Designs Using Biased Estimation QPRC 2003ByAdnan Bashir andJames Simpson May 23,2003

2. FAMU-FSU College of Engineering, Department of Industrial Engineering Outline • Introduction • Motivation example • Research objectives • Proposed analysis method • Multicollinearity & ridge • Best subset model • Simulated case studies • Example • Results • Conclusion & recommendations • Future research

3. FAMU-FSU College of Engineering, Department of Industrial Engineering Introduction • Many studies and experiments contain a large number of variables • Fewer variables are significant • Which are those few factors? How do we find those factors? • Screening experiments (Design & Analysis) are used to find those important factors • Several methods & techniques (Design & Analysis) are available to screen

4. Motivation exampleComposites Production FAMU-FSU College of Engineering, Department of Industrial Engineering Raw Materials INPUTS (Factors) Resin Flow Rate (x1) Type of Resin (x2) Gate Location (x3) Fiber Weave (x4) Mold Complexity (x5) Fiber Weight (x6) Curing Type (x7) Pressure (x8) OUTPUTS (Responses) Fiber Permeability Product Quality Tensile Strength Process Noise

5. Motivation example (continued) FAMU-FSU College of Engineering, Department of Industrial Engineering Response y = Tensile strength Each experiment costs $500, requires 8 hours, budget $3,000 (6 experiments) 1: High level -1: Low level • Supersaturated Designs: number of factors m ≥ number of runs n • Columns are not Orthogonal

6. FAMU-FSU College of Engineering, Department of Industrial Engineering Research Objectives • Propose an efficient technique to screen the important factors in an experiment with fewer number of runs • Construct improved supersaturated designs • Develop an accurate, reliable and efficient technique to analyze supersaturated designs

7. FAMU-FSU College of Engineering, Department of Industrial Engineering Analysis of SSDs – Current Methods • Stepwise regression, most commonly used • Lin (1993, 1995), Wang (1995), Nguyen (1996) • All possible regressions • Abraham, Chipman, and Vijayan (1999) • Bayesian method • Box and Meyer (1993) Investigated techniques • Principle components, partial least squares and flexible regression methods (MARS & CART)

8. FAMU-FSU College of Engineering, Department of Industrial Engineering Analysis of SSDs – Proposed Method • Modified best subset via ridge regression (MBS-RR) • Ridge regression for multicollinearity • Best subset for variable selection in each model • Criterion based selection to identify best model

9. FAMU-FSU College of Engineering, Department of Industrial Engineering Ridge Regression Motivation Ordinary Least Squares Ridge Regression Consider adding k > 0 to each diagonal of X*'X* , say k = 0.1 Consider a centered, scaled matrix, X*

10. FAMU-FSU College of Engineering, Department of Industrial Engineering Ridge Regression • Ridge regression estimates where k is referred to as a shrinkage parameter • Thus, Geometric interpretation of ridge regression

11. FAMU-FSU College of Engineering, Department of Industrial Engineering Ridge Regression, (continued)Shrinkage parameter • Hoerl and Kennard (1975) suggest • where p is number of parameter • are found from the least squares solution

12. FAMU-FSU College of Engineering, Department of Industrial Engineering Shrinkage ParameterRidge Trace Ridge trace for nine regressors (Adapted from Montgomery, Peck, & Vining; 2001)

13. FAMU-FSU College of Engineering, Department of Industrial Engineering Proposed Analysis Method Read X, Y Cont’d. Select the best 1-factor model By OLS (k=0) Calculate k, and find the best 2-factor model by all possible subsets Adding 1 factor at a time to the best 2-factor model, from the remaining variables to get the best 3-factor model

14. FAMU-FSU College of Engineering, Department of Industrial Engineering Proposed Analysis Method Is the stopping rule satisfied? Yes No Adding 1 factor at a time to the best 3-factor model, from the remaining variables to get the best 4-factor model Yes Is the stopping rule satisfied? No Adding 1 factor at a time to the best 7-factor model, from the remaining variables to get the best 8-factor model Final Model with Min. Cp

15. FAMU-FSU College of Engineering, Department of Industrial Engineering Selecting the Best Model Where diff: user defined tolerance Cp

16. FAMU-FSU College of Engineering, Department of Industrial Engineering Method Comparison-Monte CarloSimulation & Design of Experiments Factors considered in the simulation study III Fractional Factorial Design Matrix

17. FAMU-FSU College of Engineering, Department of Industrial Engineering Analysis Method Comparison • The performance measures, Type I and Type II errors

18. FAMU-FSU College of Engineering, Department of Industrial Engineering Case Studies with Corresponding Models

19. FAMU-FSU College of Engineering, Department of Industrial Engineering Method Comparison Results, Type I errors

20. FAMU-FSU College of Engineering, Department of Industrial Engineering Method Comparison Results, Type II errors

21. FAMU-FSU College of Engineering, Department of Industrial Engineering Factors Contributing to Method PerformanceType II Errors Stepwise Method var

22. FAMU-FSU College of Engineering, Department of Industrial Engineering Factors Contributing to Method PerformanceType II Errors Proposed Method var

23. FAMU-FSU College of Engineering, Department of Industrial Engineering Summary Results A: No. of runs B: No. of factors C: Multicollinearity D: Error variance E: No. of Sig. factors

24. FAMU-FSU College of Engineering, Department of Industrial Engineering Conclusions & Recommendations SSDs Analysis: Best Subset Ridge Regression • Use ridge regression estimation • Best subset variable selection method outperforms stepwise regression

25. FAMU-FSU College of Engineering, Department of Industrial Engineering Future Research Analyzing SSDs • Multiple criteria in selecting the best model • All possible subset, 3 factor model • Streamline program code • Real-life case studies • Genetic algorithm for variable selection

26. FAMU-FSU College of Engineering, Department of Industrial Engineering Acknowledgement • Dr. Carroll Croarkin, chair of selection committee for Mary G. Natrella • Selection Committee for Mary G. Natrella Scholarship • Dr. Simpson, Supervisor