Stat 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 May wish to model mean and also the variance

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 1.Selecting levels of some factors to minimize V(y) 2.Selecting levels of other factors to adjust the mean as close as possible to the target, t. Need a model for the variance (dispersion)

Analysis of Location and Dispersion Effects Let be the sample mean of observations taken at the i th treatment of the experiment Let s i 2 be the sample variance of observations taken at the i th treatment of the experiment That is, Can model both the mean and variance using regression

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?

Example: Original Growth Layer Experiment

Model Matrix for a single replicate:

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

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

Example: Original Growth Layer Experiment Dispersion analysis:

Example: Original Growth Layer Experiment Regression equation for the ln(s 2 ) response:

Example: Original Growth Layer Experiment Suggested settings for the process:

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

Location-Dispersion Modeling Steps:

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 1.Heating in a furnace 2.Processing by machine forming 3.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

Example The experiment goals are to 1.Minimize the variability about the target 2.Keep the process mean as close to the target of 8 inches as possible A 2 4 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

Example Data

Example Data

Example: Location Model

Example Regression equation for the mean response:

Example: Dispersion Model

Example Regression equation for the dispersion responses: