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Chapter 13 Analysis of Multifactor Experiment

Chapter 13 Analysis of Multifactor Experiment. Dec 7th, 2006.  Our Group . Group Members I. Shengnan Cai : Why we work on this topic? What can it be used? Tingting He : Who developed related technology? Weixin Guo : Theory about Two-factor experiments

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Chapter 13 Analysis of Multifactor Experiment

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  1. Chapter 13 Analysis of Multifactor Experiment Dec 7th, 2006

  2. Our Group

  3. Group Members I • Shengnan Cai: Why we work on this topic? What can it be used? • Tingting He: Who developed related technology? • Weixin Guo: Theory about Two-factor experiments • Yinghua Li: Theory about 2^k experiment

  4. Group Members IIData Examples • Yi Su: Parameter Estimates • Siuying: 22 factors experiment • Sandy: 23Experiment • Yi Zhang: 2k factors examples

  5. Group Members III • Tianyi Zhang: Model Diagnostics and SAS Programming • Ling Leng: Regression Approach; Conclusion

  6. The Goal of Analysis of Multifactor Experiment Shengnan Cai • Why we work on this topic? • What can it be used?

  7. About factors • A Factor is a linked set of experimental conditions we may wish to compare. e.g. Levels of temperature different methods to teach group from different academic background • Factors are also sometimes called “independent variables.”

  8. Two-factor experiments We have seen how to use one-way ANOVA related to samples designs to compare responses for a factor in the previous chapter. We need not to restrict ourselves to just one factor. Several different factors can be studied in a single experiment. We combine their levels to provide treatment combinations which can be compared in either related or unrelated samples.

  9. The methods developed for two factors can be generalized to three or more factors Two-factor experiments 2^k factors experiments

  10. Why the multifactor analysis is important? • Multifactor world • Multifactor problems Example: Meteorologists wants to know what influences the amount of the snowfall in Long Island. The influences could be temperature, moisture capacity, wind speed etc. These factors could form a multifactor system.

  11. Three sections • Two-factor Experiments with Fixed Crossed Factors • 2^3 Factorial Experiments • 2^k Factorial Experiments

  12. History of this Techonology Tingting He • Introduction to ANOVA

  13. ANOVA---Analysis of variance • A collection of statistical models and their associated procedures which compare means by splitting the overall observed variance into different parts. • The initial techniques of the analysis of variance were pioneered by the statistician and geneticist R.A.Fisher in the 1920s and 1930s.

  14. Two-Way ANOVAfor Balanced Design Assumptions • The populations from which the samples were obtained must be normally or approximately normally distributed. • The samples must be independent. • The variances of the populations must be equal • The groups must have the same sample size

  15. Hypotheses The null hypotheses for each of the sets are given below. • The population means of the first factor are equal. • The population means of the second factor are equal. • There is no interaction between the two factors.

  16. Breakdown of variability TOTAL SS Between Treatments SS Residual SS Between Subjects SS Factor 1 Factor 2 Interaction Extent to which factors influence each other – important information Main effects

  17. F-test There is an F-test for each of the hypotheses, and the F-test is the mean square for each main effect and the interaction effect divided by the within variance. The numerator degrees of freedom come from each effect, and the denominator degrees of freedom is the degrees of freedom for the within variance in each case.

  18. Two-Way ANOVA table ( for an a*b factorial experiment)

  19. Test Supplementing ANOVA Necessary condition for pairwise comparisons When the interactions are nonsignificant ( H0AB is not rejected) pairwise comparisons between the row main effects and/or between the column main effects are generally of interest. Method apply: Tukey method, recommended by Tukey, is highly valued in statistics. We find better and accurate confidence intervals by this method.

  20. Things I haven’t told you • What happens if you have unequal sample sizes. Answer is that the method of calculation is modified 2 What happens if sample is not normal? Don’t worry too much. ANOVA is robust and can endure violations of assumptions. However, you might consider transforming

  21. What happens if samples do not have same variance? Again, ANOVA is robust and can deal with this (to some extent). If homogeneity of variance is seriously violated, then Howell advises using ‘Welch’s Procedure’.

  22. Theory Derivation I Weixin Guo Theory derivation about Two-factor experiments

  23. The parameter of interest

  24. The Sets of testing hypothesis

  25. The first Chi Square Variable

  26. The second Kai Square Variable

  27. The pivotal quantity

  28. F-test

  29. Theory Derivation II Yinghua Li Theory derivation about 2^k experiment

  30. Data Example I Yi Su An Example on Parameter Estimates

  31. Example 13.1 Parameter Estimates & ANOVA Bonding Strength of Capacitors Capacitors are bonded to a circuit board used in high voltage electronic equipment. Engineers designed and carried out an experiment to study how the mechanical bonding strength of capacitors depends on the type of substrate (factor A) and the bonding material (factor B). There were 3 types of substrates: aluminum oxide (Al2O3) with bracket, Al2O3 without bracket, and beryllium oxide (BeO) without bracket. Four types of bonding materials were used: Epoxy I, Epoxy II, Solder I and Solder II. Four capacitors were tested at each factor level combination. Calculate the estimates of the parameters for these data.

  32. Bonding Strength of Capacitors

  33. Parameters & Estimates yijk: the kthobservation on the (i, j)th treatment combination the mean of cell (i, j) i.i.d random error, normal distribution : i th row main effect : j th column main effect : (i, j)th row-column interaction

  34. Parameters & Estimates : sample mean of the (i, j)th cell; least square estimate of

  35. Parameters & Estimates Sample variance for the (i, j)th cell is The pooled estimate of is

  36. Parameters estimates: Bonding Strength of Capacitors Sample Means

  37. Parameters estimates:Bonding Strength of Capacitors The cell sample SD’s are: s11=0.217 s12=0.138 s13=0.139 s14=0.321 s21=0.124 s22=0.131 s23=0.044 s24=0.122 s31=0.208 s32=0.209 s33=0.240 s34=0.192 The pooled sample SD=0.187 with 36 d.f.

  38. Parameters estimates: Bonding Strength of Capacitors The Estimates of Model Parameters for Capacitor Bonding Strength Data

  39. Analysis of Variance ANOVA Table for Crossed Two-Way Layout

  40. Bonding Strength of Capacitors: ANOVA Analysis of Variance for Bonding Strength Data Conclusion: The main effect of bonding material and the interaction between the bonding material and substrate are both highly significant, but the main effect of substrate is NOT significant at the .05 level.

  41. Data Example II Siuying • An example on 22 factorial experiment

  42. Calculate the estimated main effects A and B, and the interaction AB. Factor B Low High Low y11=10 y12=15 ỹ1.=12.5 Factor A High y21=20 y22=35ỹ2.=27.5 ỹ.1=15ỹ.2=25ỹ..=40

  43. The estimated main effects are: A ={(y22-y12)+(y21-y11)}/2 ={(35-15)+(20-10)}/2 = 15 B ={(y22-y21)+(y12-y11)}/2 ={(35-20)+(15-10)}/2 = 10 The estimated interaction effect is: AB ={(y22-y12)-(y21-y11)}/2 ={(35-15)-(20-10)}/2 = 5

  44. ANOVA Table (Two-Way Layout with Fixed Factors) Source d.f. SS MS F A B AB ___________________________________________ Total

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