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are as many parameters as cases.
It s important to understand the coding here, so look at the X-matrix.
> model.matrix(g)
(Intercept) h d l b j f n a i e m c k g o
1 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
...etc...
16.6. FACTORIAL EXPERIMENTS 193
We see that + is coded as 0 and - is coded as 1. This unnatural ordering is because of their order in
the ASCII alphabet.
We don t have any degrees of freedom so we can t make the usual F-tests. We need a different method.
Suppose there were no significant effects and the errors are normally distributed. The estimated effects
would then just be linear combinations of the errors and hence normal. We now make a normal quantile plot
of the main effects with the idea that outliers represent significant effects. Theqqnorm()function is not
suitable because we want to label the points.
> coef
> i
> plot(qnorm(1:15/16),coef[i],type="n",xlab="Normal Quantiles",
ylab="Effects")
> text(qnorm(1:15/16),coef[i],names(coef)[i])
g e
b
j
no
g
iml
kahd
f
c
c
f
k
bdha
i
m
l
e jon
-1.5 -0.5 0.5 1.5 0.0 0.5 1.0 1.5
Normal Quantiles Half-Normal Quantiles
Figure 16.10: Fractional Factorial analysis
See Figure 16.6. Notice that e and possibly g are extreme. Since the e effect is negative, the +
level of e increases the response. Since shrinkage is a bad thing, increasing the response is not good so
we d prefer what ever wire braid type corresponds to the - level of e. The same reasoning for g leads us to
expect that a larger (assuming that is +) would decrease shrinkage.
A half-normal plot is better for detecting extreme points. This plots the sorted absolute values against
1
¦ n i 2n 1 . Thus it compares the absolute values of the data against the upper half of a normal
distribution. We don t particularly care if the coefficients are not normally distributed, it s just the extreme
cases we want to detect. Because the half-normal folds over the ends of a QQ plot it doubles our resolution
for the detection of outliers.
> coef
> i
> plot(qnorm(16:30/31),coef[i],type="n",xlab="Half-Normal Quantiles",
Effects
Effects
-0.2
-0.1
0.0
0.1
0.00
0.10
0.20
16.6. FACTORIAL EXPERIMENTS 194
ylab="Effects")
> text(qnorm(16:30/31),coef[i],names(coef)[i])
We might now conduct another experiment focusing on the effect of e and g .
Appendix A
Recommended Books
A.1 Books on R
There are currently no books written specifically for R , although several guides can be downloaded from
the R web site.
R is very similar to S-PLUS so most material on S-PLUS applies immediately to R . I highly recom-
mend Venables and Ripley (1999). Alternative introductory books are Spector (1994) and Krause and Olson
(2000). You may also find Becker, Chambers, and Wilks (1998) and Chambers and Hastie (1991), useful
references to the S language. Ripley and Venables (2000) is a more advanced test on programming in S or
R .
A.2 Books on Regression and Anova
There are many books on regression analysis. Weisberg (1985) is a very readable book while Sen and
Srivastava (1990) contains more theoretical content. Draper and Smith (1998) is another well-known book.
One popular textbook is Kutner, Nachtschiem, Wasserman, and Neter (1996). This book has everything
spelled out in great detail and will certainly strengthen your biceps (1400 pages) if not your knowledge of
regression.
195
Appendix B
R functions and data
This book uses some functions and data that are not part of base R . You may wish to download these
functions from the R web site. The additional packages used are
MASS
leaps
xgobi
ellipse
nlme
The MASS functions are part of the VR package that comes with the book Venables and Ripley (1999). The
xgobi data visualization application will also need to be installed. This is an X-windows application but it
is possible with some work to make it work under Windows. See the documentation that comes with the
package for details. This is not essential so don t sweat it if you can t install it. In addition, you will need
the splines, mva and lqs packages but these come with basic R installation so no extra work is necessary.
I have packaged the data and functions that I have used in this book as an R package that you may obtain
from my web site www.stat.lsa.umich.edu/Üfaraway. The functions available are
halfnorm Half normal plot
Cpplot Cp plot
qqnorml Case-labeled Q-Q plot
maxadjr Models with maximum adjusted RÆ2
vif Variance Inflation factors
prplot Partial residual plot
These are very simple functions and no documentation has been written for them. In addition the following
datasets are used:
breaking Breaking strengths of material by day, supplier, operator
cathedral Cathedral nave heights and lengths in England
chicago Chicago insurance redlining
chiczip Chicago zip codes north/south
chmiss Chicago data with some missing values
coagulation Blood coagulation times by diet
196
APPENDIX B. R FUNCTIONS AND DATA 197
corrosion Corrosion loss in Cu-Ni alloys
eco Ecological regression example
gala Species diversity on the Galapagos Islands
odor Odor of chemical by production settings
penicillin Penicillin yields by block and treatment
rabbit Rabbit weight gain by diet and litter
rats Rat survival times by treatment and poison
savings Savings rates in 50 countries
speedo Speedometer cable shrinkage
star Star light intensities and temperatures
strongx Strong interaction experiment data
twins Twin IQs from Burt
Again no documentation has been written other than that found in the text.
Where add-on libraries are needed in the text, you will find the appropriate library() command.
However, I have assumed that thefarawaylibrary is always loaded.
Appendix C
Quick introduction to R
C.1 Reading the data in
The first step is to read the data in. You can use theread.table()orscan()function to read data in
from outside R . You can also use thedata()function to access data already available within R .
> data(stackloss)
> stackloss
Air.Flow Water.Temp Acid.Conc. stack.loss
1 80 27 89 42
2 80 27 88 37
... stuff deleted ...
21 70 20 91 15
Type
> help(stackloss)
We can check the dimension of the data:
> dim(stackloss)
[1] 21 4
C.2 Numerical Summaries
One easy way to get the basic numerical summaries is:
> summary(stackloss)
Air.Flow Water.Temp Acid.Conc. stack.loss
Min. :50.0 Min. :17.0 Min. :72.0 Min. : 7.0
1st Qu.:56.0 1st Qu.:18.0 1st Qu.:82.0 1st Qu.:11.0
Median :58.0 Median :20.0 Median :87.0 Median :15.0
Mean :60.4 Mean :21.1 Mean :86.3 Mean :17.5
3rd Qu.:62.0 3rd Qu.:24.0 3rd Qu.:89.0 3rd Qu.:19.0
Max. :80.0 Max. :27.0 Max. :93.0 Max. :42.0
198
C.2. NUMERICAL SUMMARIES 199
We can compute these numbers seperately also:
> stackloss$Air.Flow
[1] 80 80 75 62 62 62 62 62 58 58 58 58 58 58 50 50 50 50 50 56 70
> mean(stackloss$Ai)
[1] 60.429
> median(stackloss$Ai)
[1] 58
> range(stackloss$Ai)
[1] 50 80
> quantile(stackloss$Ai)
0% 25% 50% 75% 100%
50 56 58 62 80
We can get the variance and sd:
> var(stackloss$Ai)
[1] 84.057
> sqrt(var(stackloss$Ai))
[1] 9.1683
We can write a function to compute sd s:
> sd
> sd(stackloss$Ai)
[1] 9.1683
We might also want the correlations:
> cor(stackloss)
Air.Flow Water.Temp Acid.Conc. stack.loss
Air.Flow 1.00000 0.78185 0.50014 0.91966
Water.Temp 0.78185 1.00000 0.39094 0.87550
Acid.Conc. 0.50014 0.39094 1.00000 0.39983
stack.loss 0.91966 0.87550 0.39983 1.00000
Another numerical summary with a graphical element is the stem plot:
> stem(stackloss$Ai)
The decimal point is 1 digit(s) to the right of the |
5 | 000006888888
6 | 22222
7 | 05
8 | 00
C.3. GRAPHICAL SUMMARIES 200
C.3 Graphical Summaries
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zanotowane.pldoc.pisz.plpdf.pisz.plkwiatpolny.htw.pl
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Dobre pomysły nie mają przeszłości, mają tylko przyszłość. Robert Mallet
De minimis - o najmniejszych rzeczach.
Dobroć jest ważniejsza niż mądrość, a uznanie tej prawdy to pierwszy krok do mądrości. Theodore Isaac Rubin
Dobro to tylko to, co szlachetne, zło to tylko to, co haniebne.
Dla człowieka nie tylko świat otaczający jest zagadką; jest on nią sam dla siebie. I z obu tajemnic bardziej dręczącą wydaje się ta druga. Antoni Kępiński (1918-1972)
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