Recitation #12 and Homework #10: DOX I
OR&IE 3120, Spring 2009
Lab for the week of April 13–17, 2009
This assignment contains a computer lab and homework problems. All problems should
be handed in as Homework #10 which is due Thursday, April 23, 2008, at 4pm.
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
. 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 the com
mands.
wdata = read.csv(file=’QualTech.csv’,header=TRUE)
par(mfrow=c(2,2))
boxplot(wdata$thickness)
qqnorm(wdata$thickness)
plot(wdata$thickness)
The code
par(mfrow=c(2,2))
opens up a 2
×
2 graphics window,
par
changes graphics
parameters,
mfrow
means multiple figures that are filled row by row, and
= c(2,2)
specifies
the dimensions of the window.
You will notice that the plots look very strange. Something is wrong. Most values of
thickness
are similar to each other, but one is quite different. It is in the 49th row. This
shows why you should
ALWAYS LOOK AT YOUR DATA
. The value “20” to the left
of the decimal must be an error. Probably “20” should have been “10.” Clearly, you need
to do something. If you could consult with the group that collected the data, that would be
ideal. However, that group is not available. Change the “20” to a “10” with the following
command:
wdata$thickness[49] = wdata$thickness[49]  10
Now rerun the plotting commands given above.
1
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Problem 1
Discuss whether there are any problems with the variable
thickness
after
observations #49 has been changed. Include both sets of plots (before and after changing
“20” to “10”) in your homework.
Now fit a random effects model with
wafer
nested in
run
.
library(nlme)
chip.lme = lme(fixed = thickness ~ 1,data=wdata,random= ~ 1
run/wafer)
VarCorr(chip.lme)
fixef(chip.lme)
chip.lme$residual[1:5,]
The model being fit is
thickness
i,j,k
=
μ
+
u
i
+
v
i,j
+
²
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
²
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
²
i,j,k
are independent
N
(0
, σ
2
²
). The parameter
μ
is called a fixed effect since it does not vary randomly and
u
i
,
v
i,j
, and
²
i,j,k
are called random effects since they vary randomly across runs, wafers with
runs, and chips within wafers, respectively.
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 Spring '09
 JACKSON
 Regression Analysis, Randomness, Wafer, Shelf life

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