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Stat 470-11

Stat 470-11. Today: More Chapter 3. Analysis of Location and Dispersion Effects. The epitaxial growth layer experiment is a 2 4 factorial design Have looked at ways to analyze response of a factorial experiment Plotting effects on a normal probability plot Regression

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Stat 470-11

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  1. Stat 470-11 • Today: More Chapter 3

  2. Analysis of Location and Dispersion Effects • The epitaxial growth layer experiment is a 24 factorial design • Have looked at ways to analyze response of a factorial experiment • Plotting effects on a normal probability plot • Regression • May wish to model mean and also the variance

  3. Analysis of Location and Dispersion Effects • Recall, from Section 3.2, the quadratic loss function • The expected loss E(y,t)=cVar(y)+c(E(y)-t)2 suggested • Selecting levels of some factors to minimize V(y) • Selecting levels of other factors to adjust the mean as close as possible to the target, t. • Need a model for the variance (dispersion)

  4. Analysis of Location and Dispersion Effects • Let be the sample mean of observations taken at the ith treatment of the experiment • Let si2 be the sample variance of observations taken at the ith treatment of the experiment • That is, • Can model both the mean and variance using regression

  5. Analysis of Location and Dispersion Effects • Would like to model the variance as a function of the factors • Regression assumes that quantities measured at each treatment be normally distributed • Is it likely that is normally distributed?

  6. Example: Original Growth Layer Experiment

  7. Example: Original Growth Layer Experiment

  8. Example: Original Growth Layer Experiment • Model Matrix for a single replicate:

  9. Example: Original Growth Layer Experiment • Effect Estimates and QQ-Plot:

  10. Example: Original Growth Layer Experiment • Regression equation for the mean response:

  11. Example: Original Growth Layer Experiment • Dispersion analysis:

  12. Example: Original Growth Layer Experiment • Regression equation for the ln(s2) response:

  13. Example: Original Growth Layer Experiment • Suggested settings for the process:

  14. Example: Original Growth Layer Experiment • Suggested settings for the process in the original units of the factors:

  15. Location-Dispersion Modeling • Steps:

  16. Example • An experiment was conducted to improve a heat treatment process on truck leaf springs • The heat treatment process, which forms the curvature of the leaf spring, consists of • Heating in a furnace • Processing by machine forming • Quenching in an oil bath • The height of an unloaded spring, known as the free height, is the quality characteristic of interest and has a target of 8 inches

  17. Example • The experiment goals are to • Minimize the variability about the target • Keep the process mean as close to the target of 8 inches as possible • A 24 factorial experiment was conducted with factors: • A. Furnace Temperature • B. Heating Time • C. Transfer Time • Q. Quench Oil Temperature • There were 3 replicates of the experiment

  18. Example • Data

  19. Example • Data

  20. Example: Location Model

  21. Example: Location Model

  22. Example • Regression equation for the mean response:

  23. Example: Dispersion Model

  24. Example: Dispersion Model

  25. Example • Regression equation for the dispersion responses:

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