1. Dr. Gary Blau, Sean HanMonday, Aug 13, 2007 Statistical Design of Experiments SECTION IV FACTORIAL EXPERIMENTATION

2. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han FULL FACTORIAL EXPERIMENT Definition A set of experimental runs such that all levels of a given factor are combined with all levels of every other factor.

3. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han TWO FACTORS A two level full factorial experiment in 2 factors consists of four runs: (low, low) (low, high) (high, high) (high, low)

4. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han EXAMPLE OF A TWO FACTOR FACTORIAL EXPERIMENT Design a full factorial experiment in a tablet press to study the effect of Pressure (P) and Punch Distance (D) on the percent dissolution of tablets after 45 minutes. FactorsLevel Pressure (P).5 Ton - 1 Ton Punch Distance (D) 1 mm - 2 mm Experimental runs P(ton) D(mm) 1.51 2 11 3.52 4 12

5. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han GEOMETRIC REPRESENTATION (high) Punch Distance (low) (.5,2) (1,2) (.5,1) (1,1) (low) (high) Pressure

6. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han CODING In order to normalize the data and eliminate unit confusion, it is common practice to code the levels Coding Values: Coded value = (original value – mean) /(range/2) Example: To code the original values of Pressure: low coded value=(.5-.75)/(.5/2) = -1 or – high coded value=(1-.75)/(.5/2) =1 or +

7. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han TREATMENT COMBINATION A convenient approach to identify levels of a factor is to use an algebraic letter when the factor is at its high level. If all the levels in a treatment are low, we denote it by (1). A B (-1, -1) = (1) (+1, -1) = a (-1, +1) = b (+1, +1) = ab

8. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han GENERAL ALGEBRAIC / GEOMETRIC REPRESENTATION For Two Factor A and B Experiment:

9. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han FACTORIAL EXPERIMENTS WITH THREE FACTORS A two level full factorial experiment in three factors A, B and C consists of eight experiments: Level Coding Treatment Comb. (low, low, low) (-1, -1, -1) (1) (high, low, low) (+1, -1, -1) a (low, high, low) (-1, +1, -1) b (high, high, low) (+1, +1, -1) ab (low, low, high) (-1, -1, +1) c (high, low, high) (+1, -1, +1) ac (low, high, high) (-1, +1, +1) bc (high, high, high) (+1, +1, +1) abc

10. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han GEOMETRIC REPRESENTATION Three Factor Experiment

11. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han THREE FACTOR EXAMPLE Design a full factorial experiment in a Tablet Press to study the effect of Pressure(P), Punch Distance(D) and API/Binder Ratio(R) on the percent dissolution of tablets after 45 minutes

12. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han THREE FACTOR EXAMPLE FactorsLevel Pressure (P).5 ton - 1 ton Punch Distance (D) 1 mm - 2 mm API/Binder Ratio (R).05 -.15 Experimental runsPressure(P) Distance(D) Ratio(R) 1.51.05 2 11.05 3.52.05 4 12.05 5.51.15 6 11.15 7.52.15 8 12.15

13. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han THREE FACTOR EXAMPLE Geometric Representation Associated mathematical model Y = D + P + R + error

14. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han NOTATION Coded ( +, -) Original P D R Treatment Exp Press Dist Ratio (A) (B) (C) Combination 1.5 1.05 - -- (1) 2 1 1.05 + -- a 3.5 2.05 - + - b 4 1 2.05 + + - ab 5.5 1.15 - - + c 6 1 1.15 + - + ac 7.5 2.15 - ++ bc 8 1 2.15 + + + abc

15. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han CALCULATING MAIN EFFECTS Main Effect of a factor – is the change in response produced by the change in the level of a factor. When a factor is examined at two levels only, the main effect of the factor is the average of the differences of the responses between the high level and the low level of the factor.

16. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han EXAMPLE FOR CALCULATING MAIN EFFECTS Example 1:In two factors: Treatment Combination(P) (D) %D issolution (1)- - 61 a+ - 86 b- + 76 ab+ + 86

17. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han EXAMPLE FOR CALCULATING MAIN EFFECTS There are two measurements of the main effect of A: (response at high value of A) – (response at low value of A) Dissolution(a)Dissolution(1) 86-61= 25 Dissolution(ab)Dissolution(b) 86 76 = 10 The overall main effect is an average of these differences: (25+10)/2=17.5 Similarly, the main effect of B: ((76-61)+(86-86))/2=7.5

18. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han EXAMPLE FOR CALCULATING MAIN EFFECTS Example 2 - In three factors Treatment Combination(P)(D)(R) %Dissolution (1) --- 61 a +-- 86 b -+-76 ab ++-86 c --+56 ac +-+83 bc -++74 abc +++95

19. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han EXAMPLE FOR CALCULATING MAIN EFFECTS How many measurements exist for the main effect of A? 4 What are they? a- (1) ab-b ac-c abc-bc

20. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han EXAMPLE FOR CALCULATING MAIN EFFECTS Main effects of A

21. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han INTERACTION AB Interaction is the difference between the main effect of A at the high level of B and the main effect of A at the low level of B.

22. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han TYPE OF INTERACTION No Interaction Negative Interaction Strong Negative Interaction Positive Interaction

23. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han CALCULATING SIGNS To calculate the sign of the interaction, simply multiply the signs of the factors in the interaction. e.g. AB for A(+), B(-) is: (+) * (-) = (-)

24. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han SIGNS COMPUTING TABLE

25. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han MODELS

26. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han NUMBER OF INTERACTIONS For Factors at Two Levels:

27. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han EXAMPLE FOR INTERACTION IN THREE FACTOR EXPERIMENT

28. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han EXAMPLE FOR INTERACTION IN THREE FACTOR EXPERIMENT Calculate the AB interaction in three factors example:

29. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han CHOICE OF LEVELS The choice of levels has considerable impact on the chances of detecting important effects If the range of levels is not wide enough, the important effects will not be detected!! Determine the smallest change in the levels of the factors that produces a noticeable change in the response Space the levels as far apart as reasonable to improve detection and estimation of effects

30. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han REPLICATION Without a message of reproducibility, the significance of the effects cannot be evaluated. This message can come from replicated points, previous measures, or “hidden” measures in the data The single most efficient replication point is usually the “center point” of the experimental region Complete replication of the design improves the chances of detecting an effect

31. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han RANDOMIZATION AND BLOCKING In order to reduce the effect of the experimental order, the selection of the experimental runs should be randomized. The center points should be randomized and evenly distributed throughout Where appropriate, blocking may be used to improve sensitivity, i.e., blocking removes a source of variation from the total error.

32. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han EXAMPLE FOR RANDOMIZATION

33. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han SUMMARY Enable the main effects of every factor to be estimated independently of one another Enable the dependence of the effect of every factor upon the levels of the others (interaction) to be determined Supply an estimate of experimental error for the purpose of assessing the significance of effects and enable confidence limits to be determined