Two-factor experiment Tâm Minh Lê, Ph.D Faculty of Food Technology, HUFI Content A systematic approach to analyze two-factors experiment Predictions from the model Two-factor experiments with interactions Three-factor experiment Tâm Minh Lê Design of experiment 2.01 A systematic approach to analyze two-factor experiment Tâm Minh Lê Design of experiment Remind  Factors: things influencing the outcome that we want to change  Also called variables  Quantitative = Numeric  Qualitative = Categorical Tâm Minh Lê Design of experiment Context: popcorn Tâm Minh Lê Design of experiment Issue: unpopped popcorn Tâm Minh Lê Design of experiment A systematic approach What is my outcome? What is my objective? Which factors? At what levels? Plan the experiment Implement the experiment Analyze the results Tâm Minh Lê Design of experiment A systematic approach What is my outcome ? What is my objective ? Outcome and Objective Tâm Minh Lê Design of experiment A systematic approach What is my outcome ? What is my objective ? Outcome and Objective Outcome:  number of popped corn Objective:  maximizing number of popped corn Tâm Minh Lê Design of experiment A systematic approach What is my outcome ? What is my objective ? Which factors ? At what levels ? Factors and Levels  A: time on the stove (160 sec or 200 sec.)  B: type of kernel used (white corn or yellow corn) ‒ + + ‒ 𝑘 Tâm Minh Lê number of factors number of levels for the factors Design of experiment 10 Results (3/5): Predictive model + ‒ 4 ‒ C 𝐂 𝐞𝐟𝐟𝐞𝐜𝐭 ∶ 33 T 30 + + S A + + ‒ ‒ B + ‒ + ‒ − ≡ 200 + ≡ 400 𝑪+ − 𝑪− − 33 = −29 − 30 = −27 − = −1 − = −1 ‒ [C] = −14.5 ⇒ 𝑏𝐶 = −7.25 Tâm Minh Lê Design of experiment 76 Results (3/5): Predictive model without interactions 𝑦 = 11.25 + 6.25𝑥𝐶 + 0.75𝑥𝑇 − 7.25𝑥𝑆 + 33 T ‒ Exercise: calculate 𝑦2 + S ‒ C 30 + ‒ 𝑦2 = 11.25 + 6.25 +1 + 0.75 −1 − −7.25 −1 = 24 Tâm Minh Lê Design of experiment 77 2.04 Calculate interactions for the threefactor experiments Tâm Minh Lê Design of experiment 78 Issue How can we sure about the existence of interactions in an experiment? Tâm Minh Lê Design of experiment 79 Issue Predicted values are not good + 33 B ‒ 𝑦 = 11.25 + 6.25𝑥𝐶 + 0.75𝑥𝑇 − 7.25𝑥𝑆 + C ‒ 30 A + Tâm Minh Lê 𝑦2 = 24 ‒ Design of experiment 80 Issue Standard order Actual order C=A T=B S=C Pollutant amount (lbs) 8 ‒ + ‒ + ‒ + ‒ + ‒ ‒ + + ‒ ‒ + + ‒ ‒ ‒ ‒ + + + + 30 33 Tâm Minh Lê Design of experiment 81 Issue + 33 4 [AC] interaction is important B ‒ The two levels of one factor react differently to a change in level of another factor + C ‒ 30 A + Tâm Minh Lê ‒ Design of experiment 82 Calculate the average of [AC] interaction  + 33 B ‒ Average of [AC] interaction =  At [𝐵]+ : 𝐵+ + 𝐵− + C ‒ 30 A + Tâm Minh Lê ‒ Design of experiment 83 Calculate the average of [AC] interaction  400 +  C ‒ 100 ‒P Tâm Minh Lê  Average of [AC] interaction = At [𝐵]+ :  Calculate [AC] interaction:  33 A +Q 𝐴 𝐴 [𝐴𝐶] = Design of experiment 𝐵+ + 𝐵− = 33 − = 27 𝐶 + = − = −1 𝐶− 𝐴 𝐶+ − 𝐴 𝐶− −1 − 27 = = −14 85 Calculate the average of [AC] interaction  400 ‒ 100 ‒P 33 +   𝐵+ + 𝐵− Average of [AC] interaction = At [𝐵]+ , [AC] interaction = -14 At [𝐵]− , [AC] interaction = ? C Tâm Minh Lê A +Q Design of experiment 86 Calculate the average of [AC] interaction  400 + Tâm Minh Lê  Average of [AC] interaction = At [𝐵]+ , [AC] interaction = -14 At [𝐵]− :  C ‒ 100 ‒P  𝐵+ + 𝐵−  30 A +Q 𝐴 𝐴 [𝐴𝐶] = Design of experiment = 30 − = 25 𝐶 + = − = −1 𝐶− 𝐴 𝐶+ − 𝐴 𝐶− −1 − 25 = = −13 87 Calculate the average of [AC] interaction  400 + C ‒ 100 ‒P Tâm Minh Lê   ⇒ 30 A +Q 𝐵+ +𝐵− Average of [AC] interaction = At [𝐵]+ , [AC] interaction = -14 At [𝐵]− , [AC] interaction = -13 −14 + (−13) Average of AC interaction = = −13.5 ⇒ 𝑏𝐴𝐶 −13.5 = = −6.75 Design of experiment 88 Calculate the average of [AC] interaction Predicted the value:  +  33 B ‒  chemical Q (A+) 72 degrees F (B-) 200 RPM stirring speed (C−) + C ‒ 30 A + Tâm Minh Lê ‒ 𝑦 = 11.25 + 6.25𝑥𝐴 + 0.75𝑥𝐵 − 7.25𝑥𝐶 − 6.75𝑥𝐴 𝑥𝐶 𝑦2 = 30.75 Design of experiment 89 Predictive model with all interactions 𝑦 = 11.25 + 6.25𝑥𝐶 +0.75𝑥𝑇 −7.25𝑥𝑆 −6.75𝑥𝐶 𝑥𝑆 + 0.25𝑥𝐶 𝑥𝑇 − 0.25𝑥𝑇 𝑥𝑆 −0.25𝑥𝐶 𝑥𝑇 𝑥𝑆 Occam's razor ≡ The simpler is the better 𝑦 = 11.25 + 6.25𝑥𝐶 +0.75𝑥𝑇 −7.25𝑥𝑆 −6.75𝑥𝐶 𝑥𝑆 Tâm Minh Lê Design of experiment 91 Understanding the interaction + 33 4 T ‒ 𝑦 = 11.25 + 6.25𝑥𝐶 +0.75𝑥𝑇 −7.25𝑥𝑆 −𝟔 𝟕𝟓𝑥𝐶 𝑥𝑆 + S ‒ C Tâm Minh Lê 30 + ‒  Dilution time for Q chemical (C+)?  What if the requirement is: the outcome < 10 units? And  What if the cost of chemical P is double expensive than the cost of chemical Q? Design of experiment 92 ... Tâm Minh Lê Design of experiment 22 Results (5/5) –Interaction plot 80 yellow + 80 Corn 62 62 52 ‒ white ‒ white 74 52 ‒ 160 s Tâm Minh Lê Corn + yellow 74 Time + 20 0 s Design of experiment 23 ... Analyze the results A effect: 52 ‒ 160 s 74 A = Time +  A B− = 74 − 52 = 22  A B+ = 80 − 62 = 18 20 0 s Average of A effect = Tâm Minh Lê Design of experiment 22 + 18 = 20 19 Results (3/5) – The... white ‒ 160 s Tâm Minh Lê Time + 20 0 s Design of experiment 21 Results (5/5) –Interaction plot 80 yellow + 62 80 74 Corn 62 52 ‒ 160 s ‒ white 52 ‒ 160 s Time + 24 0 s 74 Time + 20 0 s Exercise: