1. Experimental Design Tutorial Presented By Michael W. Totaro Wireless Research Group Center for Advanced Computer Studies University of Louisiana at Lafayette

2. Topics  Introduction  2 k Factorial Designs  Factors/Responses  Effects  Factor Interaction  Quantifying the Effects  Proper Perspective

3. Topics  Introduction  2 k Factorial Designs  Factors/Responses  Effects  Factor Interaction  Quantifying the Effects  Proper Perspective

4. Introduction  Broad goal of simulation projects is to learn how the inputs affect the outputs  Kinds of factors (input parameters) Quantitative vs. Qualitative Controllable vs. Uncontrollable  In modeling, everything is controllable  Simulation output performance measures are the responses

5. Analogy to Traditional Physical Experiments Laboratory Agricultural Industrial

6. Goal  In simulation, experimental design provides a way of deciding before the runs are made which particular configurations to simulate so that the desired information can be obtained with the least amount of simulating.

7. Possible Factors/Responses Usually, there are many possible factors and responses

8. Setting Factor Levels  There is no real prescription for setting factor levels (i.e., values they can take on) Qualitative—may be clear from context Quantitative—may set at “reasonable” levels; however, that might push the boundaries

9. Opportunities  Special opportunities in simulation- based experiment Everything is controllable Control source of randomness, and exploit for variance reduction No need to randomize assignment of treatments to experimental results

10. Topics  Introduction  2 k Factorial Designs  Factors/Responses  Effects  Factor Interaction  Quantifying the Effects  Proper Perspective

11. Feasible Design  Example of a design that is feasible in many simulations: 2 k factorial design  Have k factors (inputs), each at just two levels  Number of possible combinations of factors—usually called design points— is 2 k

12. Topics  Introduction  2 k Factorial Designs  Factors/Responses  Effects  Factor Interaction  Quantifying the Effects  Proper Perspective

13. Single Factor vs. Multiple Factors  Case of single factor (k = 1) Vary the factor (maybe at more than two levels), make plots, and so on  In general, assume k ≥ 2 factors— want to know about: Effect on response(s) of each factor Possible interactions between factors— effect of one factor depends on the level of some of the other factors

14. 2 k Factorial Design—Process  Code each factor to a “+” and a “-” level  Design matrix: All possible combinations of factor levels  Example for k = 3 factors: Make the 8 simulation runs, and measure the effects of the factors!

15. Topics  Introduction  2 k Factorial Designs  Factors/Responses  Effects  Factor Interaction  Quantifying the Effects  Proper Perspective

16. Main Effect of a Factor Main effect of a factor is the average difference in the response when this factor is at its “+” level as opposed to its “-” level:

17. Main Effect of a Factor – cont’d The main effects measure the average change in the response due to a change in an individual factor, with this average being taken over all possible combinations of the other k-1 factors (numbering 2 k-1 ).

18. Main Effect of a Factor – cont’d We can rewrite the above as “Factor 1” column ● “Response” column / 2 k-1 -R 1 + R 2 – R 3 + R 4 – R 5 + R 6 – R 7 + R 8 e 1 = 4

19. Topics  Introduction  2 k Factorial Designs  Factors/Responses  Effects  Factor Interaction  Quantifying the Effects  Proper Perspective

20. Factor Interaction  Two factors A and B are said to interact if the effect of one depends upon the level of the other  Conversely, these two factors, A and B, are said to be noninteracting if the performance of one is not affected by the level of the other  We shall look at examples of interacting factors and noninteracting factors

21. Examples of Noninteracting and Interacting Factors A1A1 A2A2 B1B1 35 B2B2 68 Noninteracting Factors Interacting Factors A1A1 A2A2 B1B1 35 B2B2 69 As the factor A is changed from level A 1 to level A 2, the performance increases by 2 regardless of the level of factor B As the factor A is changed from level A 1 to level A 2, the performance increases either by 2 or 3 depending upon whether B is at level B 1 or level B 2, respectively

22. Examples of Noninteracting and Interacting Factors—cont’d Performance Graphical representation of interacting and noninteracting factors. 6 2 8 A1A1 A2A2 B2B2 B1B1 Performance 6 2 8 B1B1 A2A2 A1A1 B2B2 (a) No Interaction Performance 6 2 8 A1A1 A2A2 B2B2 B1B1 6 2 8 B1B1 A2A2 A1A1 B2B2 (b) Interaction

23. Interaction Effects 1 x 3 interaction effect: “Factor 1” ● “Factor 3” ● “Response” / 2 k-1 R 1 - R 2 + R 3 - R 4 – R 5 + R 6 – R 7 + R 8 e 13 = 4  Addresses the question: “Does the effect of a factor depend on level of others?”  Sign of effect indicates direction of effect on response of moving that factor from its “-” to its “+” level

24. Topics  Introduction  2 k Factorial Designs  Factors/Responses  Effects  Factor Interaction  Quantifying the Effects  Proper Perspective

25. Quantifying the Effects  Statistical significance of effects estimates (i.e., are they real?)  A luxury in simulation-based experiments: Replicate the whole design n times Get n observations on each effect Compute sample mean, sample variance, confidence interval, etc., on expected effects—effect is “real” if confidence interval misses 0

26. Quantifying the Effects--Example  Example of 2 6 Factorial Design  In addition to above, machine suffers breakdowns, and thus must undergo repair  Response: Average time in system of a part (called the makespan)

27. Quantifying the Effects—Example (cont’d)  Factors and coding:  Full 2 6 factorial design involves 64 factor combinations  Entire design is replicated n = 5 times; thus, this is a 2 6 5 factorial experimental design

28. Quantifying the Effects—Example (cont’d) The figures below plot 90% confidence intervals of the expected main effects and two-way way interactions for both responses, obtained by the five replications of the entire design We see that factor 2 (inspection time) has a large negative effect on makespan—thus, “improving” it to “+” level would be the single most worthwhile step to take to reduce makespan. (Put another way, faster inspections would provide the greatest improvement.) Improving factor 5 (probability of failing inspection) would have the next-most-important effect on makespan

29. Topics  Introduction  2 k Factorial Designs  Factors/Responses  Effects  Factor Interaction  Quantifying the Effects  Proper Perspective

30. Keep a Proper Perspective  Results are relative to the particular values chosen for the factors, and cannot necessarily be extrapolated to other regions in the factor space  It is probably not good to choose the “-” and “+” levels of a factor to be extremely far apart from each other Could result in experiments for factor levels that are unrealistic in the problem context Might not get information on “interior” of design space between the factor levels; thus, we may not see interactions that might be present there

31. Sources  Simulation Modeling and Analysis, Third Ed., by Averill M. Law and W. David Kelton, The McGraw-Hill Companies, Inc., 2000.  The Art of Computer Systems Performance Analysis: Techniques for Experimental Design, Measurement, Simulation, and Modeling, by Raj Jain, John Wiley & Sons, Inc., New York, 1991.