Saturated Designs

1. Saturated Designs … are designs in which the effects of a specified number of factors are investigated using the minimum number of runs • focus is on single-factor effects • ignore interaction effects (will be confounded with main effects) • Resolution III designs Why? • example 27 -4III design provides info on 7 main effects in 8 runs • these are screening designs to provide info on main effects • Screen out effects that aren’t important and then plan some additional experiments to learn about interactions for important factors K. McAuley - Fall 2012

2. Plackett-Burman Designs - use when the number of runs you can afford isn’t a power of 2 - look them up in a textbook or the 1946 paper - designs with 4, 8, 12, 16, 20, 24 ….92 plus some for higher numbers of runs - XTX is diagonal but patterns are not obvious Examples - • 12 run design - use for 7-11 factors • 16 run design - use for 12 - 15 factors • Same as 215 -11 design for 15 factors K. McAuley - Fall 2012

3. 12 Run Plackett-Burman Design K. McAuley - Fall 2012

4. Calculating Effects for Fractional Factorial Designs Same procedure as for the full 2-level Fractional Factorial designs • Calculate effects directly • Use regression The key difference is in the interpretation - • calculated effect is the effect of the factor PLUS the effects of all factors aliased with that factor • e.g., in half fraction of 23 design, x1 is aliased with x2x3 so the main effect calculated for x1 is the effect of x1 + the effect of x2 x3 K McAuley - Fall 2012

5. Precision of Calculated Effects Precision is handled in the same way as for full factorial designs - obtain estimate of noise variance from • replicate runs (typically, centre-point runs) • external estimate from previous runs • mean squared error (insignificant effects) Key distinction is in the number of runs being used K. McAuley 2012

6. Precision of Calculated Effects Fractional factorial designs are still balanced • equal number of high and low levels of factors Variance of calculated effect can be developed by • using formal defn -> avg. of the responses at high level (half of the runs) - avg. of the responses at low level • there are 2k-p/2 runs at each level, and variances are additive • variance of calculated effects in a 2k-p fractional factorial design is: • using regression approach • effect is 2*estimated parameter • variance is 4*variance of estimated parameter K. McAuley - Fall 2012

7. Experimental Design - The Home Stretch… • Building on Fractional Factorial Designs • Sequential experiments • Examining Nonlinear Factor Effects -- Higher-Order Designs K. McAuley- Fall 2012

8. Building on Fractional Factorial Designs After an initial fractional factorial design has been implemented, we can obtain better resolution by conducting additional designs. Two approaches - • repeat design with levels of ONE factor reversed - keep same pattern for other factors • reverse patterns of levels for ALL factors - “Foldover Design” K. McAuley - Fall 2012

9. Reversing the Levels of a Single Factor Example - 26 -3 Resolution III design • x1x2 x4 = x1x3 x5 = x2x3 x6 (= x2x3x4x5 = x1x3x4x6 = x1x2x5x6 = x4x5x6 ) =+1 • phase 1 - conduct this experiment - now for Phase 2? K. McAuley - Fall 2012

10. Reversing the Levels of a Single Factor Imagine that we want to know more about the effect of x1 (no confounding with 2nd order interactions) Phase 2 - reverse the levels of factor #1 and do more experiments • other factor levels are fixed, x1 is reversed • For these new experiments: x1x2x4 = -1 , x1x3x5 = -1 K. McAuley - Fall 2012

11. Reversing Levels of a Single Factor Calculated main effect for x1 : • is effect of (x1 + x2x4 + x3x5 ) for Phase 1design • is effect of (x1 - x2x4 - x3x5 ) for Phase 2 design • assuming higher-order interaction effects are negligible Averaging the calculated main effects for x1 over both experimental phases yields 1/2{ (x1 + x2x4 + x3x5 ) + (x1 - x2x4 - x3x5 ) } = x1 Combining the runs from both experimental phases yields eliminates terms aliased with x1 Aliasing of other main effects with two-factor interactions involving x1 is also eliminated. K. McAuley - Fall 2012

12. Reversing the Levels of a Single Factor General Result - a sequence of two 2k -pIII designs in which the second design differs from the first only in the reversal of the pattern of levels for one factor yields unaliased estimates of the main effect of that factor, and all 2-factor interaction effects involving that factor Why is this true? K. McAuley - Fall 2012

13. Reversing Levels of a Single Factor Let’s calculate some interaction effects after fold-over using x1 for our fold-over design: 1/2{ (x1 + x2x4 + x3x5 ) - (x1 - x2x4 - x3x5 ) } = (x2x4 + x3x5 ) 1/2{ (x2 + x1x4 + x3x6 ) - (x2 - x1x4 + x3x6 ) } = x1x4 If we do this for all six effects, we obtain unaliased estimates of two-factor interactions involving x1 we eliminate aliasing of other interaction terms with x1xj terms K. McAuley - Fall 2012

14. Another Clever Foldover Design Example – a 27 -4 Resolution III design • x1x2x5 = x2x3x7 = x1x3x6 = x1x2x3x4 (= x1x3x5x7 = x2x3x5x6 = x3x4x5 = x1x2x6x7 = x1x4x7 = x2x4x6) = +1 • Phase 1 experimental program K. McAuley - Fall 2012

15. Another Clever Foldover Design Phase 2 – let’s reverse the levels of ALL seven factors • Impact - the defining relations for 3 factors become • x1x2x5 = x2x3x7 = x1x3x6 = (x3x4x5 = x1x4x7 = x2x4x6 ) = -1 • the defining relations for 4 factors remain = +1 K. McAuley - Fall 2012

16. Another Clever Foldover Design Check out what happens if we average the calculated effects for x1 from the two phases: (ignoring 3-factor interactions or greater) 1/2{ (x1 + x2x5 + x3x6 + x4x7 ) + (x1 - x2x5 - x3x6 - x4x7 ) } = x1 The same result is obtained for EACH main factor Result - a Resolution III design is transformed into a Resolution IV design. Phase 1 Phase 2 K. McAuley - Fall 2012

17. Foldover Designs Impact on Calculated 2-Factor Interaction Effects: • examine difference between calculated main effect for x1 for Phases 1 and 2 1/2{ (x1 + x2x5 + x3x6 + x4x7 ) - (x1 - x2x5 - x3x6 - x4x7 ) } = x2x5 + x3x6 + x4x7 • aliasing of the combination of 2-factor interaction terms with main effects is eliminated • aliasing between 2-factor interaction terms is NOT eliminated K. McAuley - Fall 2012

18. Foldover Designs General Statement - Foldover Designs can be used to improve initial Resolution III designs to Resolution IV designs • Foldover Designs CANNOT be used to improve Resolution IV designs to Resolution V designs • Foldover Designs CANNOT be used to improve Resolution III designs to Resolution V designs K. McAuley - Fall 2012

19. More on Blocking (with multiple blocking factors)Blocking of a 24 Design We can only do 8 runs per day - need 2 blocks • think of 24 design as 25 -1 half fraction (5th factor is “day”) • a good blocking variable - > x1x2 x3 x4x5 = +1 • Day 1 runs -> x5 = +1 • Day 2 runs -> x5 = -1 • check alias structure - what is aliased with “day”? How can this approach be generalized to deal with blocking into more than two blocks or blocking of fractional factorial designs? K. McAuley - Fall 2012

20. Higher-Order Designs How should we design experiment when we want to estimate a full second-order model (with x2 terms) in k factors? • we require more than two levels in each factor (Why?) Options: • 3-level factorial design -- 3k • large numbers of runs • all combinations of high, middle and low factor levels for all factors • Central Composite Design • Spherical, Rotatable, Face-Centred • Box-Behnken Designs K. McAuley - Fall 2012

21. Central Composite Designs Start with a 2k design - • add centre-point runs • add “star points” • usual star point distances are k1/2 to get a “spherical” design or (2k) 1/4 to get a “rotatable” design • Uncertainty of model predictions from a rotatable design depends only on the distance from the centre point, not the direction Total number of points = 2k + 2k + m for k factors with m runs at centre point What is this design good for? K. McAuley - Fall 2012

22. Face-Centered Central Composite Some choices of star points can take us outside our desired experimental region. Solution - clamp star points at +/-1 K. McAuley - Fall 2012

23. Box-Behnken Designs … offer another alternative for keeping the runs within the high/low boundaries • design is formed by combining 2-level factorial designs with extra runs with settings of -1, 0, 1 for various factors • look them up for different numbers of factors • available for 3 factors or more • resulting designs are rotatable, or nearly rotatable • like central composite designs, they are used for fitting quadratic models K. McAuley - Fall 2012

24. Estimating Second-Order Models Use the standard multiple linear regression approach • models are still linear in the parameters Full second-order model includes • intercept • main effects terms • two-factor interaction terms • quadratics terms each factor Look at the correlation matrix for the parameters to understand which parameters are correlated with each other • Covariance matrix is: • How do we get the correlation matrix? K. McAuley - Fall 2012

25. Other Experimental Designs you may see • Taguchi designs • Fewer runs than Box-Behnken or Central Composite designs but: • Precision of parameters and predicted responses is not as good • Quality of prediction depends on distance from centre point and direction From Wikipedia: Professional statisticians have welcomed the goals and improvements brought about by Taguchi methods, particularly by Taguchi's development of designs for studying variation, but have criticized the inefficiency of some of Taguchi's proposals K. McAuley - Fall 2012