EXPERIMENTAL DESIGNS Prof.Dr.Cevdet Demir cevdet @ uludag.tr

1. EXPERIMENTAL DESIGNS Prof.Dr.Cevdet Demir cevdet@uludag.edu.tr

2. MOTIVATIONS FOR DESIGN • Screening • Saving time • Quantitative modelling • Optimisation

3. WHY DESIGN EXPERIMENTS? Example : Optimisation of a reaction with pH and temperature.

4. IMAGINARY RESPONSE SURFACE • We want to find optimum • Response surface unknown • Mathematical model may not be of interest in its own right • Not necessarily interested in underlying molecular mechanism • Reproducibility and flat optimum

6. One factor at a time strategy : may miss optima

7. DIFFICULTY Interactions – the response for each factor is not independent. The optimum temperature at pH 5 differs from that at pH 6.

8. How to be on the safe side? • Grid search. 10 pHs, 10 temperatures, 100 experiments. • Big grid. Then smaller grid.

9. PROBLEMS • Time consuming and expensive. • Many experiments we are almost certain are not near at optimum so are obviously a waste of time • Reproducibility and experimental error

10. WHAT DO WE DO? We need rules! Formal experimental design

11. Screening • Factorial designs • Partial factorials and Plackett-Burman designs • Modelling and optimisation • Response surface designs

12. FACTORIAL DESIGNS

13. SIMPLEST IDEA • List factors • High and low levels • 2k experiments

14. Can be used to look at how significant each factor is • Can be used to form a quantitative model between response (e.g. taste, texture, colour or even market acceptability) and value of the factors • Can use categorical factors e.g. supplier or whether one process used or not • Takes all interactions into account

15. PROBLEM • For many factors a lot of experiments. • 10 factors = 1024 experiments • Screening many factors not really good • How can we reduce the number of experiments?

16. FRACTIONAL FACTORIAL DESIGNS Remove some of the experiments

17. Which to remove? Fractional factorial. Takes out certain experiments.

18. Fractional factorials from 2k to 2k-f E.g. from 8 to 4 (half) from 32 to 16 (half) from 32 to 8 (quarter)

19. Lose information about higher order interactions. Call this confounding. Must have at least k + 1 experiments But a little inefficient. What happens if 19 factors? Do we need to do 32 experiments?

20. Factors PLACKETT BURMAN DESIGNS Number of experiments equals 4N so number of factors equals 4N-1

21. Factors 1 2 3 4 5 6 7 8 9 10 11 1 - - - - - - - - - - - 2 + - + - - - + + + - + 3 + + - + - - - + + + - Experiments 4 - + + - + - - - + + + 5 + - + + - + - - - + + 6 + + - + + - + - - - + 7 + + + - + + - + - - - 8 - + + + - + + - + - - 9 - - + + + - + + - + - 10 - - - + + + - + + - + 11 + - - - + + + - + + - + 12 - + - - - + + + - + • Typical design • First row 0s. • First column experiments 2 to 13 : generators

22. Plackett-Burman designs for 4N-1 factors • Dummy factors • Equivalence PB and fractional factorials

23. NUMBER OF EXPERIMENTS Factorials : 2k Fractional factorials : 2k-f Plackett-Burman : 4N NUMBER OF EXPERIMENTS Example : 10 factors Factorials : 1024 Fractional factorials : 16 (or 32 etc.) Plackett-Burman : 12

24. AS THE NUMBER OF EXPERIMENTS IS REDUCED SO THE INFORMATION E.G. ABOUT INTERACTIONS IS REDUCED Use screening experiments to reduce the number of factors and then settle on key factors for detailed models.

25. MODELLING • Optimisation : what factors result in the best taste? • Sometimes several criteria e.g. taste, texture, cost, so several optima. • Quantitative modelling : can we relate the manufacturing process to the %protein in wheat? • Many factors such as the origins of the materials as well as the production, not looking for an optimum.

26. CENTRAL COMPOSITE DESIGNS

27. Problems with factorial designs • Two levels and so no squared terms. • No replicates.

30. Relatively detailed models between response e.g. taste and factors e.g. ingredients. Replicate errors can be calculated. Optima can be calculated. Interactions can be calculated. Number of experiments : 2k+ 2k + 6 3 factors : 8 + 6 + 6 = 12 4 factors : 16 + 8 + 6 = 30

31. Characteristics • Number of replicates in the centre • Position of star or axial points (1 is not the only choice)

32. Reduce further by fractional factorials, for example, 5 factors • Quarter factorial : 8 experiments • Axial points : 10 experiments • Central points : 6 experiments • (Not good for interactions, half factorial better, but might screen for interactions).