实验设计与方差分析 Chap9_handout_

Chapter 9 Two level Factorial Designs Jingyuan Liu SOE and WISE Xiamen University DOE and ANOVA Chapter9Chapter 9 Two level Factorial Designs Introduction of Two level Factorial Designs Factorial experiments are widely used in DOE involving k factorial factors where it is necessary to study their joint eff ects main and interaction on the response We will talk about a special factorial experiment k factors each at only two levels A complete replicate of such a design requires 2kcells and is called 2kfactorial design This design is of great practical value esp in early stage factor screening experiments and enjoys a special analysis method thus we extract it from general factorial experiments Throughout this chapter we assume The factors are fi xed eff ect The designs are completely randomized so the DOE is trivial The usual normality assumptions are satisfi ed Ref Ch6 in the book DOE and ANOVA Chapter9Chapter 9 Two level Factorial Designs The 22Design The easiest 2kfactorial design is when k 2 The levels of the factors may be arbitrarily called low and high Example Study the eff ect of the concentration of reactant factor A 15 vs 25 and the amount of catalyst factor B 1 pound vs 2 pounds on the yield in a chemical process The experiment is replicated 3 times so N 12 runs in total The design is DOE and ANOVA Chapter9Chapter 9 Two level Factorial Designs The 22Design Data Structure After the experiment the data are collected as follows DOE and ANOVA Chapter9Chapter 9 Two level Factorial Designs The 22 Design Factor Eff ect Estimation With n replications in each cell here n 3 and N 4n 12 we can estimate the main eff ect of factor A the main eff ect of factor B and the interaction eff ect between factor A and factor B where the interaction eff ect AB is defi ned as the average diff erence between the eff ect of A at the high level of B and the eff ect of A at the low level of B which is equivalent to the average of the right to left diagonal treatment combinations minus the average of the left to right diagonal treatment combinations DOE and ANOVA Chapter9Chapter 9 Two level Factorial Designs The 22 Design Factor Eff ect Estimation Back to the example The eff ect of A reactant concentration is positive this suggests that increasing A from the low level 15 to the high level 25 will increase the yield The eff ect of B catalyst is negative this suggests that increasing the amount of catalyst added to the process will decrease the yield The interaction eff ect is small relative to the main eff ects DOE and ANOVA Chapter9Chapter 9 Two level Factorial Designs The 22 Design Factor Eff ect Estimation Remarks Here we use capital letters A B AB etc to denote the estimated factor eff ects when there is no confusion When there is lA lBand lABare more frequently used Distinguish these main treat other eff ects as error and redo the analysis Refer to the example later DOE and ANOVA Chapter9Chapter 9 Two level Factorial Designs The General 2kDesign Example Example A chemical product is produced in a pressure vessel A 24factorial design is carried out in the pilot plant to study four factors for the fi ltration rate of this product temperature A pressure B concentration of formaldehyde C and stirring rate D The 16 24 runs are made in random order with no replication n 1 N 16 DOE and ANOVA Chapter9Chapter 9 Two level Factorial Designs The General 2kDesign Example DOE and ANOVA Chapter9Chapter 9 Two level Factorial Designs The General 2kDesign Example The geometric presentation of the design is DOE and ANOVA Chapter9Chapter 9 Two level Factorial Designs The General 2kDesign Example To estimate the eff ects fi rst construct the interaction columns DOE and ANOVA Chapter9Chapter 9 Two level Factorial Designs The General 2kDesign Example Then the factor eff ect estimates and sums of squares are DOE and ANOVA Chapter9Chapter 9 Two level Factorial Designs The General 2kDesign Example Important main eff ect and interaction plots are illustrated below DOE and ANOVA Chapter9Chapter 9 Two level Factorial Designs The General 2kDesign Example Some conclusions All the three important main eff ect A C and D are positive so if we consider only the main eff ects we would run all three factors at the high level to maximize the fi ltration rate However the main eff ects do not have much meaning when they are involved in signifi cant interactions like AC and AD The AC interaction indicates that the temperature A eff ect is very small when the concentration C is high and very large when the concentration is low with the best results obtained with low concentration and high temperature The AD interaction indicates that stirring rate D has little eff ect at low temperature but a large positive eff ect at high temperature Also temperature A has little eff ect for low D but large positive eff ect for high D Thus the best fi ltration rates would appear for high A low C and high D DOE and ANOVA Chapter9Chapter 9 Two level Factorial Designs The General 2kDesign Example Since factor B is not important we may compute the following ANOVA table with A C D and their interactions but treat all other SS s as error DOE and ANOVA Chapter9Chapter 9 Two level Factorial Designs The General 2kDesign Example Based on the above analysis we can obtain a regression model to predict fi ltration rate y 0 1xA 3xC 4xD 13xAxC 14xAxD where xA xCand xDare coded variables 1 and 1 for A C and D Since the signifi cant factor eff ects are A 21 625 C 9 875 D 14 625 AC 18 125 and AD 16 625 the estimated fi ltration rate is Why y 70