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Unformatted text preview: Recitation for week #12 (starting Sunday, Apr 17, 2011) and Assignment #12 Design of Experiments OR&IE 3120, Spring 2011 This assignment contains a computer lab and homework problems. All problems should be handed in as Homework #12 which is due at noon on Wed, Apr 27, 2011. 1 A Nested Design with Random Effects The engineers at QualTech, a company making computer chips, noticed too much variation in the thickness of lines etched in their chips by one of their etchers. To study the sources of the variation, they performed an experiment. The experiment used 12 runs of this etcher. In each run, there were 6 wafers and on each wafer there were 3 chips, so there is a total of 216 observations of thickness, one per chip. The data are in the file QualTech.csv on Blackboard. There are four columns: run, wafer, chip, and thickness. Because of the way that the experiment was designed, the factor wafer is nested in run and chip is nested in wafer . Read the data into R and look at several different plots of thickness with these com- mands: wdata = read.csv(file="QualTech.csv",header=TRUE) par(mfrow=c(1,3)) boxplot(wdata$thickness) qqnorm(wdata$thickness) plot(wdata$thickness) The code par(mfrow=c(1,3)) opens up a 1 × 3 graphics window, par changes graphics parameters, mfrow means multiple figures that are filled row by row (though there is only one row here), and = c(1,3) specifies the dimensions of the window. You will notice that the plots look very strange. Something is wrong. Most values of thickness are similar to each other, but one is quite different. It is in the 49th row. This shows why you should ALWAYS LOOK AT YOUR DATA . The value “20” to the left of the decimal must be an error. Probably “20” should have been “10.” Clearly, you need to do something. If you could consult with the group that collected the data to discover the problem, that would be ideal. However, that group is not available. Change the “20” to a “10” with the following command: wdata$thickness = wdata$thickness - 10 Now rerun the plotting commands given above. 1 Problem 1 Discuss whether there are any problems with the variable thickness after observations #49 has been changed. Include both sets of plots (before and after changing “20” to “10”) in your homework. Now fit a random effects model with wafer nested in run with the following code. (Notice that the second command takes two lines. R understands this because the first line is not a complete command.) library(nlme) chip.lme = lme(fixed = thickness ~ 1,data=wdata,random= ~ 1| run/wafer) VarCorr(chip.lme) fixef(chip.lme) chip.lme$residual[1:5,] The model being fit is thickness i,j,k = μ + u i + v i,j + i,j,k where u i is the effect of the i th run, v i,j is the effect of the j th wafer in the i th run, and i,j,k the effect of the k th chip in the j th wafer in the i th run. Further, it is assumed that the u i are independent N (0 , σ 2 u ), the v ij, are independent N (0 , σ 2 v ), and the...
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- Spring '09
- Normal Distribution, Random effects model, Wafer, random effects