Factorial Experiments

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1 Factorial Experiments

2 96 97 ack to two-factor experiment 98 99

3 DOE terminology main effects: Interaction effects 00 0 Three factors z z2 z3 z2 z3 z23 z

4 Two-factor interaction Three-factor interaction Or: using + and in columns: 06 07

ube plot 08 09 Four factors example Full

5 ube plot Four factors example Full Factorial Design Factors: 4 ase Design: 4; 6 Runs: 6 Replicates: locks: enter pts (total): 0 ll terms are free from aliasing. 0

6 Factorial Fit: Y versus ; ; ; D Estimated Effects and oefficients for Y (coded units) Term Effect oef onstant 72,250-8,000-4,000 24,000 2,000-2,250 -,25 D -5,500-2,750 *,000 0,500 * 0,750 0,375 *D -0,000-0,000 * -,250-0,625 *D 4,500 2,250 *D -0,250-0,25 ** -0,750-0,375 **D 0,500 0,250 **D -0,250-0,25 **D -0,750-0,375 ***D -0,250-0,25 nalysis of Variance for Y (coded units) Source DF Seq SS dj SS dj MS F P Main Effects 4 270,25 270,25 675,33 * * 2-Way Interactions 6 93,75 93,75 5,625 * * 3-Way Interactions 4 5,75 5,75,438 * * 4-Way Interactions 0,25 0,25 0,250 * * Residual Error 0 * * * Total 5 280,00 S = * 2 3 Normal Probability Plot of the Effects (response is Y, lpha =,05) Pareto hart of the Effects (response is Y, lpha =,05) Percent D D Effect Type Not Significant Significant Factor Name D D Term D D D D D D D D 2,89 Factor D Name D -0-5 Lenth's PSE =, Effect Lenth's PSE =, Effect

Main Effects Plot (data means) for Y Interaction Plot (data means) for Y 84 78 - - - 80 70-72 60 Mean of Y 66 60 84 78 - - D 80 70 60 80 70 - - 72 60 66 60 D - - 6 7 Example: Three factors and

7 Main Effects Plot (data means) for Y Interaction Plot (data means) for Y Mean of Y D D Example: Three factors and replicate Factorial Fit: Y versus ; ; Estimated Effects and oefficients for Y (coded units) Term Effect oef SE oef T P onstant 64,250 0,707 90,86 0,000 23,000,500 0,707 6,26 0,000-5,000-2,500 0,707-3,54 0,008,500 0,750 0,707,06 0,320 *,500 0,750 0,707,06 0,320 * 0,000 5,000 0,707 7,07 0,000 * 0,000 0,000 0,707 0,00,000 ** 0,500 0,250 0,707 0,35 0,733 S = 2,82843 R-Sq = 97,63% R-Sq(adj) = 95,55% nalysis of Variance for Y (coded units) Source DF Seq SS dj SS dj MS F P Main Effects , ,00 74,667 92,7 0,000 2-Way Interactions 3 409,00 409,00 36,333 7,04 0,00 3-Way Interactions,00,00,000 0,3 0,733 Residual Error 8 64,00 64,00 8,000 Pure Error 8 64,00 64,00 8,000 Total ,00 8 9

8 Term 2,3 Pareto hart of the Standardized Effects (response is Y, lpha =,05) Factor Name Percent Normal Probability Plot of the Standardized Effects (response is Y, lpha =,05) Effect Type Not Significant Significant Factor Name Standardized Effect Standardized Effect locking in 2^k experiments Full experiment: Two blocks: Use column as generator, i.e. lock consists of experiments with = - lock 2 consists of experiments with = The interaction is confounded ( mixed ) with the block effect. This means that the value of the estimated coefficient of can be due to both interaction effect and block-effect. Suppose all Y in block 2 are increased by a value h. Then the estimated effect of will increase by h. ut one cannot know from observations whether this is due to the interaction or the block effect. On the other hand, the estimated main effects,, and the two-factor interactions,, are not changed by the h. These are of most importance to estimate, so the choice of blocking seems reasonable

9 Four blocks in 2^3 experiment lock structure is as follows: Need two columns of +/- to define 4 blocks. Turns out that the best option is to use two two-factor interactions, e.g. and (which is default in MINIT lock : Experiments where = = - lock 2: Experiments where = -, = lock 3: Experiments where =, = - lock 4: Experiments where = = 24 Interaction effects and are confounded with the block effect, since they are generators for the blocks. In addition, their product * = = is confounded with the block effect (Note: the column is constant within each block. dding h2 to block 2, h3 to block 3, h4 to block 4 does not change estimated effects of,,, and also does not change the third order interaction. However, e.g. will change by 2h3+2h4-2h2 and we do not know whether this is due to an interaction effect or blocking effect: This is ONFOUNDING. 25 How to determine which columns to use for blocking? Idea: Try to leave estimates for main effects and low order interactions unchanged by blocking. Note: I = = = where I is a column of s Find the blocks for a 2^3 experiment using columns and. The interaction between and is * = ** = which is a main effect, which hence is confounded with the block effect (in addition to and ) 26 27

10 Example obligatory project

11 MINIT plots ssuming third and fourth order interactions are Interaction plots 34 35

13 From Exam in TM4260 Industrial Statistics, december 2003, Exercise 2 company decides to investigate the hardening process of a ballbearing production. The following four factors are chosen: : content of added carbon : Hardening temperature : Hardening time D: ooling temperature. b) What is the variance of the main effect and the interaction? ssume that the st deviation sigma has been estimated from other experiments, by s = 0.32 with 9 degrees of freedom (in the exam, this had been done in Ex.) Use this estimate to find out whether the interaction is significantly different from 0 (i.e. active ) Use 5% significance level. What is the conclusion of the experiment so far? Design and results are given below: a) What is the generator and the defining relation of the design, and what is the design s resolution? Write down the alias structure. Find the estimates of the main effect of and the interaction effect The company is well satisfied with the results so far and they decide to carry out also the other half fraction. The result of the other half fraction is given below. Use this to find unconfounded estimates for the main effects and the two-factor interactions. ssume that one would like to estimate the variance of the effects from the higher order interactions. Explain how this can be done, and find the estimate. Is it wise to include the four-factor interaction in this calculation? Why (not)? Later, one of the operators that participated in the experiments asked whether one could have carried out the first half fraction in (a) in two blocks. This would, he said, have simplified considerably the performance of the experiments. What answer would you give to the operator? 42 43

14 From Exam in SIF 5066 Experimental design and, May 2003, Exercise company making ballbearings experienced problems with the lifetimes of the products. In an experiments that they carried out they considered the factors : type of ball standard (-) or modified (+) : type of cage - standard (-) or modified (+) : type of lubricate - standard (-) or modified (+) D: quantity of lubricate normal (-) or large (+) The repsonse was the lifetime of the ballbearing in an accelerated life testing experiment. The results are given on the next page. 44 : type of ball : type of cage : type of lubricate D: quantity of lubricate a) What type of experiment is this? What is the defining relation? What is the resolution? alculate estimates of the main effect of and the two-factor interaction. b) Estimated contrasts for,,d,,d are, respectively, 0.60, 0.3, 0.22, -0., What can you say about the estimated effects for D, D,. D, D, D,? ssume that factors and D do not influence the response. Explain why this is then a 2^2 experiment with replicate. alculate an estimate for the variance of the effects, and find out whether, and are now significant. c) Give an interpretation of the results. The experiment was in fact carried out in two blocks, where experiments -4 was one block and 5-8 the other. How is this blocking constructed? How will we need to modify the analysis of significance in (b)? (ssume again that,d do not influence the response) 45

CSCI 688 Homework 6. Megan Rose Bryant Department of Mathematics William and Mary

CSCI 688 Homework 6. Megan Rose Bryant Department of Mathematics William and Mary CSCI 688 Homework 6 Megan Rose Bryant Department of Mathematics William and Mary November 12, 2014 7.1 Consider the experiment described in Problem 6.1. Analyze this experiment assuming that each replicate