![93. Statistics in the 21st Century Adrian E. Raftery, Martin A. Tanner, and Martin T. Wells (2001)
94. Accelerated Life Models: Modeling and Statistical Analysis
Vilijandas Bagdonavicius and Mikhail Nikulin (2001)
95. Subset Selection in Regression, Second Edition Alan Miller (2002)
96. Topics in Modelling of Clustered Data Marc Aerts, Helena Geys, Geert Molenberghs, and Louise M. Ryan (2002)
97. Components of Variance D.R. Cox and P.J. Solomon (2002)
98. Design and Analysis of Cross-Over Trials, 2nd Edition Byron Jones and Michael G. Kenward (2003)
99. Extreme Values in Finance, Telecommunications, and the Environment
Bärbel Finkenstädt and Holger Rootzén (2003)
100. Statistical Inference and Simulation for Spatial Point Processes
Jesper Møller and Rasmus Plenge Waagepetersen (2004)
101. Hierarchical Modeling and Analysis for Spatial Data
Sudipto Banerjee, Bradley P. Carlin, and Alan E. Gelfand (2004)
102. Diagnostic Checks in Time Series Wai Keung Li (2004)
103. Stereology for Statisticians Adrian Baddeley and Eva B. Vedel Jensen (2004)
104. Gaussian Markov Random Fields: Theory and Applications +Ý
DYDUG5XHDQG/HRQKDUG+HOG(2005)
105. Measurement Error in Nonlinear Models: A Modern Perspective, Second Edition
Raymond J. Carroll, David Ruppert, Leonard A. Stefanski, and Ciprian M. Crainiceanu (2006)
106. *HQHUDOL]HG/LQHDU0RGHOVZLWK5DQGRP(IIHFWV8QLÀHG$QDOVLVYLD+OLNHOLKRRG
Youngjo Lee, John A. Nelder, and Yudi Pawitan (2006)
107. Statistical Methods for Spatio-Temporal Systems
Bärbel Finkenstädt, Leonhard Held, and Valerie Isham (2007)
108. Nonlinear Time Series: Semiparametric and Nonparametric Methods Jiti Gao (2007)
109. Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis
Michael J. Daniels and Joseph W. Hogan (2008)
110. Hidden Markov Models for Time Series: An Introduction Using R
Walter Zucchini and Iain L. MacDonald (2009)
111. ROC Curves for Continuous Data Wojtek J. Krzanowski and David J. Hand (2009)
112. Antedependence Models for Longitudinal Data Dale L. Zimmerman and Vicente A. Núñez-Antón (2009)
113. Mixed Effects Models for Complex Data Lang Wu (2010)
114. Intoduction to Time Series Modeling Genshiro Kitagawa (2010)
115. Expansions and Asymptotics for Statistics Christopher G. Small (2010)
116. Statistical Inference: An Integrated Bayesian/Likelihood Approach Murray Aitkin (2010)
117. Circular and Linear Regression: Fitting Circles and Lines by Least Squares Nikolai Chernov (2010)
118. Simultaneous Inference in Regression Wei Liu (2010)
119. Robust Nonparametric Statistical Methods, Second Edition
Thomas P. Hettmansperger and Joseph W. McKean (2011)
120. Statistical Inference: The Minimum Distance Approach
Ayanendranath Basu, Hiroyuki Shioya, and Chanseok Park (2011)
121. Smoothing Splines: Methods and Applications Yuedong Wang (2011)
122. Extreme Value Methods with Applications to Finance Serguei Y. Novak (2012)
123. Dynamic Prediction in Clinical Survival Analysis Hans C. van Houwelingen and Hein Putter (2012)
124. Statistical Methods for Stochastic Differential Equations
Mathieu Kessler, Alexander Lindner, and Michael Sørensen (2012)
125. Maximum Likelihood Estimation for Sample Surveys
R. L. Chambers, D. G. Steel, Suojin Wang, and A. H. Welsh (2012)
126. Mean Field Simulation for Monte Carlo Integration Pierre Del Moral (2013)
127. Analysis of Variance for Functional Data Jin-Ting Zhang (2013)
128. Statistical Analysis of Spatial and Spatio-Temporal Point Patterns, Third Edition Peter J. Diggle (2013)
129. Constrained Principal Component Analysis and Related Techniques Yoshio Takane (2014)
130. Randomised Response-Adaptive Designs in Clinical Trials Anthony C. Atkinson and Atanu Biswas (2014)
131. Theory of Factorial Design: Single- and Multi-Stratum Experiments Ching-Shui Cheng (2014)](https://image.slidesharecdn.com/2366154-250601201558-3d550fa4/85/Theory-Of-Factorial-Design-Single-And-Multistratum-Experiments-Cheng-9-320.jpg)
![8 INTRODUCTION
We end up with 504 combinations of sub-subplots and strips:
Each of the 126 treatment combinations appears once in each of the four blocks. To
summarize, we have four replicates of a complete 3×2×3×7 factorial experiment
with the block structure 4/[(3/2/3)×7]. Both subplots in the same plot are assigned
the same variety/herbicide combination, all the sub-subplots in the same subplot are
assigned the same herbicide application time, and all the sub-subplot and strip inter-
sections in the same strip are assigned the same weed species. Various aspects of the
analysis of this design will be discussed in Sections 12.1, 12.9, 12.10, and 13.10.
In Example 1.1, there are 18 sub-subplots in each block. If different soybean
varieties were to be assigned to neighboring sub-subplots, then 17 buffer zones
would be needed in each block. With only two buffer zones per block under the
proposed design, comparisons of soybean varieties are based on between-plot com-
parisons, which are expected to be more variable than those between subplots and
sub-subplots. The precision of the estimates of such effects is sacrificed in order to
satisfy the practical constraints.
Example 1.2. McLeod and Brewster (2004) discussed an experiment for identifying
key factors that would affect the quality of a chrome-plating process. Suppose six
two-level treatment factors are to be considered in the experiment: A, chrome con-
centration; B, chrome to sulfate ratio; C, bath temperature; S, etching current density;
T, plating current density; and U, part geometry. The response variables include, e.g.,
the numbers of pits and cracks. The chrome plating is done in a bath (tank), which
contains several rectifiers, but only two will be used. On any given day the levels of
A, B, and C cannot be changed since they represent characteristics of the bath. On the
other hand, the levels of factors S, T, and U can be changed at the rectifier level. The
experiment is to be run on 16 days, with four days in each of four weeks. Therefore
there are a total of 32 runs with the block structure (4 weeks)/(4 days)/(2 runs), and](https://image.slidesharecdn.com/2366154-250601201558-3d550fa4/85/Theory-Of-Factorial-Design-Single-And-Multistratum-Experiments-Cheng-27-320.jpg)
![ESTIMATION OF σ2 17
It is common to refer to cT
θ
θ
θ as the best linear unbiased estimator, or BLUE, of
cTθ
θ
θ.
For any estimable function cTθ
θ
θ, we have
var
cT
θ
θ
θ
= σ2
cT
(XT
X)−
c,
where (XT X)−
is any generalized inverse of XT X. We call XT X the information
matrix for θ
θ
θ.
Suppose the model matrix X is design dependent. We say that a design is D-, A-,
or E-optimal if it minimizes, respectively, the determinant, trace, or the largest eigen-
value of the covariance matrix of
θ
θ
θ among all the competing designs. By (2.5), these
optimality criteria, which are referred to as the D-, A-, and E-criterion, respectively,
are equivalent to maximizing det(XT X), minimizing tr(XT X)−1, and maximizing the
smallest eigenvalue of XT X, respectively.
2.2 Estimation of σ2
We have
y = X
θ
θ
θ +(y−X
θ
θ
θ), (2.7)
where X
θ
θ
θ and y−X
θ
θ
θ are orthogonal to each other. By the Pythagorean Theorem,
y
2
=
X
θ
θ
θ
2
+
y−X
θ
θ
θ
2
.
The last term in this identity,
y−X
θ
θ
θ
2
, is called the residual sum of squares. Since
y−X
θ
θ
θ = PR(X)⊥ y,
E
y−X
θ
θ
θ
2
= E
PR(X)⊥ y
T
PR(X)⊥ y
= E yT
PR(X)⊥ y
= [E(y)]T
PR(X)⊥ [E(y)]+σ2
tr PR(X)⊥
= σ2
tr PR(X)⊥
= σ2
dim R(X)⊥
= σ2
(N −r),
where the second equality follows from (A.2) and (A.3) in the Appendix, the third
equality follows from (A.5), the fourth equality holds since E(y) ∈ R(X) and R(X) ⊥
R(X)⊥
, and the fifth equality follows from (A.4).
Let s2 =
y−X
θ
θ
θ
2
/(N − r). Then s2, called the residual mean square, is an
unbiased estimator of σ2. The dimension of R(X)⊥
, N − r, is called the degrees of
freedom associated with the residual sum of squares.](https://image.slidesharecdn.com/2366154-250601201558-3d550fa4/85/Theory-Of-Factorial-Design-Single-And-Multistratum-Experiments-Cheng-36-320.jpg)
![20 LINEAR MODEL BASICS
is B/(t−1)
W/(N−t) . In the application to completely randomized experiments, the between-
group sum of squares is also called the treatment sum of squares.
These results can be summarized in Table 2.1, called an ANOVA (Analysis of
Variance) table.
Table 2.1 ANOVA table for one-way layout
source sum of squares d.f. mean square
Between groups ∑
t
l=1 rl(yl· −y··)2 t −1 1
t−1 ∑
t
l=1 rl(yl· −y··)2
Within groups ∑
t
l=1 ∑
rl
h=1(ylh −yl·)2 N −t 1
N−t ∑
t
l=1 ∑
rl
h=1(ylh −yl·)2
Total ∑
t
l=1 ∑
rl
h=1(ylh −y··)2 N −1
Remark 2.1. If we write the model as E(ylh) = μ +αl, then the model matrix X has
an extra column of 1’s and p = t + 1. Since rank(X) = t p, the parameters them-
selves are not identifiable. By using (2.6), one can verify that ∑
t
l=1 clαl is estimable if
and only if ∑
t
l=1 cl = 0. Such functions are called contrasts. The pairwise differences
αl − αl , 1 ≤ l = l
≤ t, are examples of contrasts and are referred to as elemen-
tary contrasts. Because of the constraint ∑
t
l=1 cl = 0, the treatment contrasts form a
(t −1)-dimensional vector space that is generated by the elementary contrasts. Func-
tions such as α1 − 1
2 (α2 +α3) and 1
2 (α1 +α2)− 1
3 (α3 +α4 +α5) are also contrasts. If
∑
t
l=1 clαl is a contrast, then ∑
t
l=1 clαl = ∑
t
l=1 cl(μ + αl), and it can be shown that, as
before, ∑
t
l=1 cl
αl = ∑
t
l=1 clyl·. Therefore, if the interest is in estimating the contrasts,
then it does not matter whether E(ylh) is written as αl or μ +αl. A contrast ∑
t
l=1 clαl
is said to be normalized if ∑
t
l=1 c2
l = 1.
2.5 Estimation of a subset of parameters
Suppose θ
θ
θ is partitioned as θ
θ
θ = (θ
θ
θT
1 θ
θ
θT
2 )T , where θ
θ
θ1 is q × 1 and θ
θ
θ2 is (p − q) × 1;
e.g., the parameters in θ
θ
θ2 are nuisance parameters, or the components of θ
θ
θ1 and θ
θ
θ2
are effects of the levels of two different factors. Partition X as [X1 X2] accordingly.
Then (2.2) can be written as
y = X1θ
θ
θ1 +X2θ
θ
θ2 +ε
ε
ε, (2.12)
and (2.4) is the same as
XT
1 X1
θ
θ
θ1 +XT
1 X2
θ
θ
θ2 = XT
1 y, (2.13)
and
XT
2 X1
θ
θ
θ1 +XT
2 X2
θ
θ
θ2 = XT
2 y. (2.14)
If R(X1) and R(X2) are orthogonal, (XT
1 X2 = 0), then
θ
θ
θ1 and
θ
θ
θ2 can be computed
by solving XT
1 X1
θ
θ
θ1 = XT
1 y and XT
2 X2
θ
θ
θ2 = XT
2 y separately. In this case, least squares
estimators of estimable functions of θ
θ
θ1 are the same regardless of whether θ
θ
θ2 is](https://image.slidesharecdn.com/2366154-250601201558-3d550fa4/85/Theory-Of-Factorial-Design-Single-And-Multistratum-Experiments-Cheng-39-320.jpg)
![22 LINEAR MODEL BASICS
Theorem 2.3. Under (2.12) and (2.3), a linear function cT
1 θ
θ
θ1 of θ
θ
θ1 is estimable
if and only if c1 is a linear combination of the column vectors of C1. If cT
1 θ
θ
θ1 is
estimable, then cT
1
θ
θ
θ1 is its best linear unbiased estimator, and
var
cT
1
θ
θ
θ1
= σ2
cT
1 C−
1 c1,
where
θ
θ
θ1 is any solution to (2.17), and C−
1 is any generalized inverse of C1.
In particular, if C1 is invertible, then all the parameters in θ
θ
θ1 are estimable, with
cov
θ
θ
θ1
= σ2
C−1
1 . (2.20)
The matrix C1 is called the information matrix for θ
θ
θ1.
Suppose the parameters in θ
θ
θ2 are nuisance parameters, and we are only interested
in estimating θ
θ
θ1. Then a design is said to be Ds-, As-, or Es-optimal if it minimizes,
respectively, the determinant, trace, or the largest eigenvalue of the covariance ma-
trix of
θ
θ
θ1 among all the competing designs. By (2.20), these optimality criteria are
equivalent to maximizing det(C1), minimizing tr(C−1
1 ), and maximizing the smallest
eigenvalue of C1, respectively. Here s refers to a subset of parameters.
2.6 Hypothesis testing for a subset of parameters
Under (2.12) and (2.3), suppose we further assume that ε
ε
ε has a normal distribution.
Consider testing the null hypothesis
H0: E(y) = X2θ
θ
θ2.
Under (2.12), E(y) ∈ R(X1)+R(X2), and under H0, E(y) ∈ R(X2). By (2.11), the
F-test statistic in this case is
F =
P[R(X1)+R(X2)]R(X2)y
2
/dim([R(X1)+R(X2)]R(X2))
P[R(X1)+R(X2)]⊥ y
2
/(N −dim[R(X1)+R(X2)])
.
This is based on the decomposition
RN
= R(X2)⊕([R(X1)+R(X2)]R(X2))⊕[R(X1)+R(X2)]⊥
. (2.21)
It can be shown that
P[R(X1)+R(X2)]R(X2)y
2
=
θ
θ
θ
T
1 Q1, (2.22)
and
dim([R(X1)+R(X2)]R(X2)) = rank(C1). (2.23)
We leave the proofs of these as an exercise.
A test of the null hypothesis H0: E(y) = X1θ
θ
θ1 is based on the decomposition
RN
= R(X1)⊕([R(X1)+R(X2)]R(X1))⊕[R(X1)+R(X2)]⊥
. (2.24)](https://image.slidesharecdn.com/2366154-250601201558-3d550fa4/85/Theory-Of-Factorial-Design-Single-And-Multistratum-Experiments-Cheng-41-320.jpg)
![ADJUSTED ORTHOGONALITY 23
In this case,
P[R(X1)+R(X2)]R(X1)y
2
=
θ
θ
θ
T
2 Q2,
where
Q2 = XT
2
I−X1 XT
1 X1
−
XT
1 y.
When XT
1 X2 = 0, both (2.21) and (2.24) reduce to
RN
= R(X1)⊕R(X2)⊕[R(X1)⊕R(X2)]⊥
.
2.7 Adjusted orthogonality
Consider the model
y = X1θ
θ
θ1 +X2θ
θ
θ2 +X3θ
θ
θ3 +ε
ε
ε, (2.25)
with
E(ε
ε
ε) = 0, and cov(ε
ε
ε) = σ2
IN.
Suppose the parameters in θ
θ
θ2 are nuisance parameters. To estimate θ
θ
θ1 and θ
θ
θ3, we
eliminate θ
θ
θ2 by projecting y onto R(X2)⊥
. This results in a reduced normal equation
for
θ
θ
θ1 and
θ
θ
θ3:
X̃T
1 X̃1 X̃T
1 X̃3
X̃T
3 X̃1 X̃T
3 X̃3
θ
θ
θ1
θ
θ
θ3
=
X̃T
1 ỹ
X̃T
3 ỹ
,
where
X̃i = PR(X2)⊥ Xi,i = 1,3, and ỹ = PR(X2)⊥ y.
If X̃T
1 X̃3 = 0, then
θ
θ
θ1 and
θ
θ
θ3 can be computed by solving the equations X̃T
1 X̃1
θ
θ
θ1 =
X̃T
1 ỹ and X̃T
3 X̃3
θ
θ
θ3 = X̃T
3 ỹ separately. In this case, least squares estimators of estimable
functions of θ
θ
θ1 under (2.25) are the same regardless of whether θ
θ
θ3 is in the model.
Likewise, dropping θ
θ
θ1 from (2.25) does not change the least squares estimators of
estimable functions of θ
θ
θ3. Loosely we say that θ
θ
θ1 and θ
θ
θ3 are orthogonal adjusted for
θ
θ
θ2.
Note that X̃T
1 X̃3 = 0 is equivalent to
PR(X2)⊥ [R(X1)] is orthogonal to PR(X2)⊥ [R(X3)]. (2.26)
Under normality, the F-test statistic for the null hypothesis that E(y) = X2θ
θ
θ2 +X3θ
θ
θ3
is
P[R(X1)+R(X2)+R(X3)][R(X2)+R(X3)]y
2
/dim([R(X1)+R(X2)+R(X3)][R(X2)+R(X3)])
P[R(X1)+R(X2)+R(X3)]⊥ y
2
/(N −dim[R(X1)+R(X2)+R(X3)])
.
If (2.26) holds, then we have
R(X1)+R(X2)+R(X3) = R(X2)⊕PR(X2)⊥ [R(X1)]⊕PR(X2)⊥ [R(X3)].](https://image.slidesharecdn.com/2366154-250601201558-3d550fa4/85/Theory-Of-Factorial-Design-Single-And-Multistratum-Experiments-Cheng-42-320.jpg)
![24 LINEAR MODEL BASICS
In this case,
[R(X1)+R(X2)+R(X3)][R(X2)+R(X3)] = PR(X2)⊥ [R(X1)]
= [R(X1)+R(X2)]R(X2).
So the sum of squares in the numerator of the F-test statistic is equal to the quantity
θ
θ
θ
T
1 Q1 given in (2.22), with its degrees of freedom equal to rank(C1), where C1 and
Q1 are as in (2.18) and (2.19), respectively. A similar conclusion can be drawn for
testing the hypothesis that E(y) = X1θ
θ
θ1 +X2θ
θ
θ2.
2.8 Additive two-way layout
In a two-way layout, the observations are classified according to the levels of two
factors. Suppose the two factors have t and b levels, respectively. At each level com-
bination (i, j), 1 ≤ i ≤ t, 1 ≤ j ≤ b, there are nij observations yijh, 0 ≤ h ≤ nij, such
that
yijh = αi +βj +εijh, (2.27)
where the εijh’s are uncorrelated random variables with zero mean and constant vari-
ance σ2. We require ∑
b
j=1 nij 0 for all i, and ∑
t
i=1 nij 0 for all j, so that there
is at least one observation on each level of the two factors, but some nij’s may be
zero. This is called an additive two-way layout model, which is commonly used for
analyzing block designs, where each yijh is an observation on the ith treatment in the
jth block. With this in mind, we call the two factors treatment and block factors, and
denote them by T and B, respectively; then α1,...,αt are the treatment effects and
β1,...,βb are the block effects. Let N = ∑
t
i=1 ∑
b
j=1 nij, and think of the observations
as taken on N units that are grouped into b blocks. Define an N ×t matrix XT with
0 and 1 entries such that the (v,i)th entry of XT , 1 ≤ v ≤ N, 1 ≤ i ≤ t, is 1 if and
only if the ith treatment is assigned to the vth unit. Similarly, let XB be the N × b
matrix with 0 and 1 entries such that the (v, j)th entry of XB, 1 ≤ v ≤ N, 1 ≤ j ≤ b,
is 1 if and only if the vth unit is in the jth block. The two matrices XT and XB are
called unit-treatment and unit-block incidence matrices, respectively. Then we can
write (2.27) as
y = XT α
α
α +XBβ
β
β +ε
ε
ε,
where α
α
α = (α1,...,αt)T and β
β
β = (β1,...,βb)T .
We use the results in Sections 2.5 and 2.6 to derive the analysis for such models.
Then we apply the results in Section 2.7 to show in Section 2.9 that the analysis can
be much simplified when certain conditions are satisfied.
Let N be the t × b matrix whose (i, j)th entry is nij, ni+ = ∑
b
j=1 nij, and n+j =
∑
t
i=1 nij. For block designs, ni+ is the number of observations on the ith treatment, and
n+j is the size of the jth block. Also, let yi++ = ∑
b
j=1 ∑h yijh and y+j+ = ∑
t
i=1 ∑h yijh,
the ith treatment total and jth block total, respectively. Then XT
T XT is the diagonal
matrix with diagonal entries n1+,...,nt+, XT
BXB is the diagonal matrix with diagonal
entries n+1,...,n+b, XT
T y = (y1++,...,yt++)T , XT
By = (y+1+,...,y+b+)T , and
XT
T XB = N. (2.28)](https://image.slidesharecdn.com/2366154-250601201558-3d550fa4/85/Theory-Of-Factorial-Design-Single-And-Multistratum-Experiments-Cheng-43-320.jpg)
![THE CASE OF PROPORTIONAL FREQUENCIES 27
Table 2.2 ANOVA table for an additive two-way layout
source sum of squares d.f. mean square
Blocks ∑j n+ j(y· j· −y...)2
b−1 1
b−1 ∑j n+ j(y· j· −y...)2
(ignoring treatments)
Treatments
αT
QT t −1 1
t−1
αT
QT
(adjusted for blocks)
Residual By subtraction N −b−t +1 1
N−b−t+1 (Residual SS)
Total ∑i ∑j ∑h(yijh −y...)2
N −1
2.9 The case of proportional frequencies
As explained in Remark 2.1, model (2.27) can be written as
yijh = μ +αi +βj +εijh (2.30)
without changing the least squares estimators of estimable functions of (α1,...,αt)T ,
least squares estimators of estimable functions of (β1,...,βb)T , and the reduced
normal equations. Write (2.30) in the form of (2.25) with θ
θ
θ1 = (α1,...,αt)T ,θ
θ
θ3 =
(β1,...,βb)T , and θ
θ
θ2 = μ. Then X1 = XT , X3 = XB, and X2 = 1N. In this case, the
adjusted orthogonality condition (2.26) is that
PR(1N)⊥ [R(XT )] is orthogonal to PR(1N)⊥ [R(XB)]. (2.31)
Since only one treatment can be assigned to each unit, each row of XT has ex-
actly one entry equal to 1, and all the other entries are zero. It follows that the
sum of all the columns of XT is equal to 1N. Therefore R(1N) ⊆ R(XT ), and hence
PR(1N)⊥ [R(XT )] = R(XT )R(1N). Likewise, PR(1N)⊥ [R(XB)] = R(XB)R(1N).
We say that T and B satisfy the condition of proportional frequencies if
nij =
ni+n+j
N
for all i, j; (2.32)
here nij/ni+ does not depend on i, and nij/n+j does not depend on j. In particular, if
nij is a constant for all i and j, then (2.32) holds; in the context of block designs, this
means that all the treatments appear the same number of times in each block.
Theorem 2.5. Factors T and B satisfy the condition of proportional frequencies if
and only if R(XT )R(1N) is orthogonal to R(XB)R(1N).
Proof. We first note that (2.31) is equivalent to
PR(1N)⊥ XT
T
PR(1N)⊥ XB = 0. (2.33)](https://image.slidesharecdn.com/2366154-250601201558-3d550fa4/85/Theory-Of-Factorial-Design-Single-And-Multistratum-Experiments-Cheng-46-320.jpg)
![28 LINEAR MODEL BASICS
Since PR(1N) = 1N(1T
N1N)−11T
N = 1
N JN, where JN is the N ×N matrix of 1’s, we have
PR(1N)⊥ = I− 1
N JN. Then (2.33) is the same as
XT
T
I−
1
N
JN
XB = 0
or
XT
T XB =
1
N
XT
T JNXB. (2.34)
By (2.28), the left-hand side of (2.34) is N. The (i,j)th entry of the right-hand side is
ni+n+j/N. Therefore (2.33) holds if and only if the two factors satisfy the condition
of proportional frequencies.
By the results in Section 2.7 and Theorem 2.5, if the treatment and block factors
satisfy the condition of proportional frequencies, then the least squares estimator of
any contrast ∑
t
i=1 ciαi is the same as that under the one-way layout
yijh = μ +αi +εijh.
So ∑
t
i=1 ci
αi = ∑
t
i=1 ciyi··, with variance σ2(∑
t
i=1 c2
i /ni+). Likewise, the least squares
estimator of any contrast ∑
b
j=1 djβj is ∑
b
j=1 djy· j·, with variance σ2(∑
b
j=1 d2
j /n+j).
In this case, we have the decomposition
RN
= R(1N)⊕[R(XT )R(1N)]⊕[R(XB)R(1N)]⊕[R(XT )+R(XB)]⊥
.
Thus
y−PR(1N)y = PR(XT )R(1N)y+PR(XB)R(1N)y+P[R(XT )+R(XB)]⊥ y.
Componentwise, we have
yijh −y... = (yi·· −y...)+(y· j· −y...)+(yijh −yi·· −y· j· +y...).
The identity
y−PR(1N)y
2
=
PR(XT )R(1N)y
2
+
PR(XB)R(1N)y
2
+
P[R(XT )+R(XB)]⊥ y
2
gives
∑
i
∑
j
∑
h
(yijh −y...)2
= ∑
i
ni+(yi·· −y...)2
+∑
j
n+j(y· j· −y...)2
+∑
i
∑
j
∑
h
(yijh −yi·· −y· j· +y...)2
.
This leads to a single ANOVA table.](https://image.slidesharecdn.com/2366154-250601201558-3d550fa4/85/Theory-Of-Factorial-Design-Single-And-Multistratum-Experiments-Cheng-47-320.jpg)
![30 LINEAR MODEL BASICS
(a) Show that if nij = 0 or 1 for all i, j, then the ith diagonal entry of CT is
equal to (k −1)qi/k, and the (i,i
)th off-diagonal entry is equal to −λii /k,
where qi is the number of times the ith treatment appears, and λii is the
number of blocks in which both the ith and i
th treatments appear.
(b) In addition to nij = 0 or 1 for all i, j, suppose qi = q for all i, and λii = λ
for all 1 ≤ i = i
≤ t. Such designs are called balanced incomplete block
designs; see Section A.8. Use Exercise 2.3 to show that (t −1)λ = (k−1)q.
(c) Show that under a balanced incomplete block design, k
tλ It is a generalized
inverse of CT . Use this to provide an expression for the least squares es-
timator of a pairwise comparison αi − αi of treatment effects, and show
that var(
αi −
αi ) = 2k
tλ σ2 for all 1 ≤ i = i
≤ t.
(d) Show that the properties in (c) and (d) of Exercise 2.4 also hold for
balanced incomplete block designs, and hence for given b, t, and k,
if there exists a balanced incomplete block design, then it minimizes
∑1≤ii≤tvar(
αi −
αi ) among all the block designs with n+j = k for all
1 ≤ j ≤ b. [Kiefer (1958, 1975)]
2.6 (Löwner ordering of matrices and comparison of experiments) Given two sym-
metric matrices A1 and A2, we say that A1 ≥ A2 if A1 − A2 is nonnegative
definite. Consider two linear models
model 1: E(y) = X1θ
θ
θ, cov(y) = σ2IN,
model 2: E(y) = X2θ
θ
θ, cov(y) = σ2IN.
Show that XT
1 X1≥ XT
2 X2 if and only if
(a) all linear functions cTθ
θ
θ of θ
θ
θ that are estimable under model 2 are also
estimable under model 1, and
(b) for any cTθ
θ
θ that is estimable under model 2, the variance of its least
squares estimator under model 2 is at least as large as that under model
1. [Ehrenfeld (1956)]
2.7 Consider the linear models
model 1: E(y) = X1θ
θ
θ1 +X2θ
θ
θ2, cov(y) = σ2IN,
model 2: E(y) = X1θ
θ
θ1 +X2θ
θ
θ2 +X3θ
θ
θ3, cov(y) = σ2IN.
Let C1 and C∗
1 be the information matrices for θ
θ
θ1 under models 1 and 2, re-
spectively. Show that C1 ≥ C∗
1, and C1 = C∗
1 if (2.26) holds.
2.8 Consider the linear model
E(y) = Xθ, cov(y) = σ2
IN,
where X is N × p and θ
θ
θ is p×1. Suppose x ≤ a for all columns x of X, where
x
2
= xT x. Show that for any unknown parameter θi, var(
θi) ≥ 1
a2 σ2, and that
the equality is attained if XT X = a2Ip. In particular, if |xij| ≤ 1 for all the entries
xij of X, then var(
θi) ≥ 1
N σ2, and the equality is attained if XT X = NIp. Such
a matrix must have all its entries equal to 1 or −1. (An N ×N matrix X with 1
and −1 entries such that XT X = NIN is called a Hadamard matrix.)](https://image.slidesharecdn.com/2366154-250601201558-3d550fa4/85/Theory-Of-Factorial-Design-Single-And-Multistratum-Experiments-Cheng-49-320.jpg)
Theory Of Factorial Design Single And Multistratum Experiments Cheng
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- 5. Theory of Factorial Design Single- and Multi-Stratum Experiments Ching-Shui Cheng Monographs on Statistics and Applied Probability 131
- 6. Theory of Factorial Design Single- and Multi-Stratum Experiments
- 7. MONOGRAPHS ON STATISTICS AND APPLIED PROBABILITY General Editors F. Bunea, V. Isham, N. Keiding, T. Louis, R. L. Smith, and H. Tong 1. Stochastic Population Models in Ecology and Epidemiology M.S. Barlett (1960) 2. Queues D.R. Cox and W.L. Smith (1961) 3. Monte Carlo Methods J.M. Hammersley and D.C. Handscomb (1964) 4. The Statistical Analysis of Series of Events D.R. Cox and P.A.W. Lewis (1966) 5. Population Genetics W.J. Ewens (1969) 6. Probability, Statistics and Time M.S. Barlett (1975) 7. Statistical Inference S.D. Silvey (1975) 8. The Analysis of Contingency Tables B.S. Everitt (1977) 9. Multivariate Analysis in Behavioural Research A.E. Maxwell (1977) 10. Stochastic Abundance Models S. Engen (1978) 11. Some Basic Theory for Statistical Inference E.J.G. Pitman (1979) 12. Point Processes D.R. Cox and V. Isham (1980) 13. ,GHQWLÀFDWLRQRI2XWOLHUVD.M. Hawkins (1980) 14. Optimal Design S.D. Silvey (1980) 15. Finite Mixture Distributions B.S. Everitt and D.J. Hand (1981) 16. ODVVLÀFDWLRQA.D. Gordon (1981) 17. Distribution-Free Statistical Methods, 2nd edition J.S. Maritz (1995) 18. 5HVLGXDOVDQG,QÁXHQFHLQ5HJUHVVLRQR.D. Cook and S. Weisberg (1982) 19. Applications of Queueing Theory, 2nd edition G.F. Newell (1982) 20. Risk Theory, 3rd edition R.E. Beard, T. Pentikäinen and E. Pesonen (1984) 21. Analysis of Survival Data D.R. Cox and D. Oakes (1984) 22. An Introduction to Latent Variable Models B.S. Everitt (1984) 23. Bandit Problems D.A. Berry and B. Fristedt (1985) 24. Stochastic Modelling and Control M.H.A. Davis and R. Vinter (1985) 25. The Statistical Analysis of Composition Data J. Aitchison (1986) 26. Density Estimation for Statistics and Data Analysis B.W. Silverman (1986) 27. Regression Analysis with Applications G.B. Wetherill (1986) 28. Sequential Methods in Statistics, 3rd edition G.B. Wetherill and K.D. Glazebrook (1986) 29. Tensor Methods in Statistics P. McCullagh (1987) 30. Transformation and Weighting in Regression R.J. Carroll and D. Ruppert (1988) 31. Asymptotic Techniques for Use in Statistics O.E. Bandorff-Nielsen and D.R. Cox (1989) 32. Analysis of Binary Data, 2nd edition D.R. Cox and E.J. Snell (1989) 33. Analysis of Infectious Disease Data N.G. Becker (1989) 34. Design and Analysis of Cross-Over Trials B. Jones and M.G. Kenward (1989) 35. Empirical Bayes Methods, 2nd edition J.S. Maritz and T. Lwin (1989) 36. Symmetric Multivariate and Related Distributions K.T. Fang, S. Kotz and K.W. Ng (1990) 37. Generalized Linear Models, 2nd edition P. McCullagh and J.A. Nelder (1989) 38. Cyclic and Computer Generated Designs, 2nd edition J.A. John and E.R. Williams (1995) 39. Analog Estimation Methods in Econometrics C.F. Manski (1988) 40. Subset Selection in Regression A.J. Miller (1990) 41. Analysis of Repeated Measures M.J. Crowder and D.J. Hand (1990) 42. Statistical Reasoning with Imprecise Probabilities P. Walley (1991) 43. Generalized Additive Models T.J. Hastie and R.J. Tibshirani (1990) 44. Inspection Errors for Attributes in Quality Control N.L. Johnson, S. Kotz and X. Wu (1991) 45. The Analysis of Contingency Tables, 2nd edition B.S. Everitt (1992)
- 8. 46. The Analysis of Quantal Response Data B.J.T. Morgan (1992) 47. Longitudinal Data with Serial Correlation—A State-Space Approach R.H. Jones (1993) 48. Differential Geometry and Statistics M.K. Murray and J.W. Rice (1993) 49. Markov Models and Optimization M.H.A. Davis (1993) 50. Networks and Chaos—Statistical and Probabilistic Aspects O.E. Barndorff-Nielsen, J.L. Jensen and W.S. Kendall (1993) 51. Number-Theoretic Methods in Statistics K.-T. Fang and Y. Wang (1994) 52. Inference and Asymptotics O.E. Barndorff-Nielsen and D.R. Cox (1994) 53. Practical Risk Theory for Actuaries C.D. Daykin, T. Pentikäinen and M. Pesonen (1994) 54. Biplots J.C. Gower and D.J. Hand (1996) 55. Predictive Inference—An Introduction S. Geisser (1993) 56. Model-Free Curve Estimation M.E. Tarter and M.D. Lock (1993) 57. An Introduction to the Bootstrap B. Efron and R.J. Tibshirani (1993) 58. Nonparametric Regression and Generalized Linear Models P.J. Green and B.W. Silverman (1994) 59. Multidimensional Scaling T.F. Cox and M.A.A. Cox (1994) 60. Kernel Smoothing M.P. Wand and M.C. Jones (1995) 61. Statistics for Long Memory Processes J. Beran (1995) 62. Nonlinear Models for Repeated Measurement Data M. Davidian and D.M. Giltinan (1995) 63. Measurement Error in Nonlinear Models R.J. Carroll, D. Rupert and L.A. Stefanski (1995) 64. Analyzing and Modeling Rank Data J.J. Marden (1995) 65. Time Series Models—In Econometrics, Finance and Other Fields D.R. Cox, D.V. Hinkley and O.E. Barndorff-Nielsen (1996) 66. Local Polynomial Modeling and its Applications J. Fan and I. Gijbels (1996) 67. Multivariate Dependencies—Models, Analysis and Interpretation D.R. Cox and N. Wermuth (1996) 68. Statistical Inference—Based on the Likelihood A. Azzalini (1996) 69. Bayes and Empirical Bayes Methods for Data Analysis B.P. Carlin and T.A Louis (1996) 70. Hidden Markov and Other Models for Discrete-Valued Time Series I.L. MacDonald and W. Zucchini (1997) 71. Statistical Evidence—A Likelihood Paradigm R. Royall (1997) 72. Analysis of Incomplete Multivariate Data J.L. Schafer (1997) 73. Multivariate Models and Dependence Concepts H. Joe (1997) 74. Theory of Sample Surveys M.E. Thompson (1997) 75. Retrial Queues G. Falin and J.G.C. Templeton (1997) 76. Theory of Dispersion Models B. Jørgensen (1997) 77. Mixed Poisson Processes J. Grandell (1997) 78. Variance Components Estimation—Mixed Models, Methodologies and Applications P.S.R.S. Rao (1997) 79. Bayesian Methods for Finite Population Sampling G. Meeden and M. Ghosh (1997) 80. Stochastic Geometry—Likelihood and computation O.E. Barndorff-Nielsen, W.S. Kendall and M.N.M. van Lieshout (1998) 81. Computer-Assisted Analysis of Mixtures and Applications—Meta-Analysis, Disease Mapping and Others D. Böhning (1999) 82. ODVVLÀFDWLRQQGHGLWLRQA.D. Gordon (1999) 83. Semimartingales and their Statistical Inference B.L.S. Prakasa Rao (1999) 84. Statistical Aspects of BSE and vCJD—Models for Epidemics C.A. Donnelly and N.M. Ferguson (1999) 85. Set-Indexed Martingales G. Ivanoff and E. Merzbach (2000) 86. The Theory of the Design of Experiments D.R. Cox and N. Reid (2000) 87. Complex Stochastic Systems O.E. Barndorff-Nielsen, D.R. Cox and C. Klüppelberg (2001) 88. Multidimensional Scaling, 2nd edition T.F. Cox and M.A.A. Cox (2001) 89. Algebraic Statistics—Computational Commutative Algebra in Statistics G. Pistone, E. Riccomagno and H.P. Wynn (2001) 90. Analysis of Time Series Structure—SSA and Related Techniques N. Golyandina, V. Nekrutkin and A.A. Zhigljavsky (2001) 91. Subjective Probability Models for Lifetimes Fabio Spizzichino (2001) 92. Empirical Likelihood Art B. Owen (2001)
- 9. 93. Statistics in the 21st Century Adrian E. Raftery, Martin A. Tanner, and Martin T. Wells (2001) 94. Accelerated Life Models: Modeling and Statistical Analysis Vilijandas Bagdonavicius and Mikhail Nikulin (2001) 95. Subset Selection in Regression, Second Edition Alan Miller (2002) 96. Topics in Modelling of Clustered Data Marc Aerts, Helena Geys, Geert Molenberghs, and Louise M. Ryan (2002) 97. Components of Variance D.R. Cox and P.J. Solomon (2002) 98. Design and Analysis of Cross-Over Trials, 2nd Edition Byron Jones and Michael G. Kenward (2003) 99. Extreme Values in Finance, Telecommunications, and the Environment Bärbel Finkenstädt and Holger Rootzén (2003) 100. Statistical Inference and Simulation for Spatial Point Processes Jesper Møller and Rasmus Plenge Waagepetersen (2004) 101. Hierarchical Modeling and Analysis for Spatial Data Sudipto Banerjee, Bradley P. Carlin, and Alan E. Gelfand (2004) 102. Diagnostic Checks in Time Series Wai Keung Li (2004) 103. Stereology for Statisticians Adrian Baddeley and Eva B. Vedel Jensen (2004) 104. Gaussian Markov Random Fields: Theory and Applications +Ý DYDUG5XHDQG/HRQKDUG+HOG(2005) 105. Measurement Error in Nonlinear Models: A Modern Perspective, Second Edition Raymond J. Carroll, David Ruppert, Leonard A. Stefanski, and Ciprian M. Crainiceanu (2006) 106. *HQHUDOL]HG/LQHDU0RGHOVZLWK5DQGRP(IIHFWV8QLÀHG$QDOVLVYLD+OLNHOLKRRG Youngjo Lee, John A. Nelder, and Yudi Pawitan (2006) 107. Statistical Methods for Spatio-Temporal Systems Bärbel Finkenstädt, Leonhard Held, and Valerie Isham (2007) 108. Nonlinear Time Series: Semiparametric and Nonparametric Methods Jiti Gao (2007) 109. Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis Michael J. Daniels and Joseph W. Hogan (2008) 110. Hidden Markov Models for Time Series: An Introduction Using R Walter Zucchini and Iain L. MacDonald (2009) 111. ROC Curves for Continuous Data Wojtek J. Krzanowski and David J. Hand (2009) 112. Antedependence Models for Longitudinal Data Dale L. Zimmerman and Vicente A. Núñez-Antón (2009) 113. Mixed Effects Models for Complex Data Lang Wu (2010) 114. Intoduction to Time Series Modeling Genshiro Kitagawa (2010) 115. Expansions and Asymptotics for Statistics Christopher G. Small (2010) 116. Statistical Inference: An Integrated Bayesian/Likelihood Approach Murray Aitkin (2010) 117. Circular and Linear Regression: Fitting Circles and Lines by Least Squares Nikolai Chernov (2010) 118. Simultaneous Inference in Regression Wei Liu (2010) 119. Robust Nonparametric Statistical Methods, Second Edition Thomas P. Hettmansperger and Joseph W. McKean (2011) 120. Statistical Inference: The Minimum Distance Approach Ayanendranath Basu, Hiroyuki Shioya, and Chanseok Park (2011) 121. Smoothing Splines: Methods and Applications Yuedong Wang (2011) 122. Extreme Value Methods with Applications to Finance Serguei Y. Novak (2012) 123. Dynamic Prediction in Clinical Survival Analysis Hans C. van Houwelingen and Hein Putter (2012) 124. Statistical Methods for Stochastic Differential Equations Mathieu Kessler, Alexander Lindner, and Michael Sørensen (2012) 125. Maximum Likelihood Estimation for Sample Surveys R. L. Chambers, D. G. Steel, Suojin Wang, and A. H. Welsh (2012) 126. Mean Field Simulation for Monte Carlo Integration Pierre Del Moral (2013) 127. Analysis of Variance for Functional Data Jin-Ting Zhang (2013) 128. Statistical Analysis of Spatial and Spatio-Temporal Point Patterns, Third Edition Peter J. Diggle (2013) 129. Constrained Principal Component Analysis and Related Techniques Yoshio Takane (2014) 130. Randomised Response-Adaptive Designs in Clinical Trials Anthony C. Atkinson and Atanu Biswas (2014) 131. Theory of Factorial Design: Single- and Multi-Stratum Experiments Ching-Shui Cheng (2014)
- 10. Monographs on Statistics and Applied Probability 131 Theory of Factorial Design Single- and Multi-Stratum Experiments Ching-Shui Cheng University of California, Berkeley, USA and Academia Sinica, Taiwan
- 11. CRC Press Taylor Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2014 by Taylor Francis Group, LLC CRC Press is an imprint of Taylor Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20131017 International Standard Book Number-13: 978-1-4665-0558-2 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information stor- age or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copy- right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro- vides licenses and registration for a variety of users. For organizations that have been granted a pho- tocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
- 12. Dedicated to the memory of Leonard Chung-Wei Cheng
- 13. Contents Preface xv 1 Introduction 1 2 Linear Model Basics 15 2.1 Least squares 15 2.2 Estimation of σ2 17 2.3 F-test 18 2.4 One-way layout 19 2.5 Estimation of a subset of parameters 20 2.6 Hypothesis testing for a subset of parameters 22 2.7 Adjusted orthogonality 23 2.8 Additive two-way layout 24 2.9 The case of proportional frequencies 27 3 Randomization and Blocking 31 3.1 Randomization 31 3.2 Assumption of additivity and models for completely randomized designs 32 3.3 Randomized block designs 33 3.4 Randomized row-column designs 34 3.5 Nested row-column designs and blocked split-plot designs 35 3.6 Randomization model∗ 36 4 Factors 39 4.1 Factors as partitions 39 4.2 Block structures and Hasse diagrams 40 4.3 Some matrices and spaces associated with factors 42 4.4 Orthogonal projections, averages, and sums of squares 44 4.5 Condition of proportional frequencies 45 4.6 Supremums and infimums of factors 46 4.7 Orthogonality of factors 47 5 Analysis of Some Simple Orthogonal Designs 51 5.1 A general result 51 5.2 Completely randomized designs 55 ix
- 14. x 5.3 Null ANOVA for block designs 57 5.4 Randomized complete block designs 59 5.5 Randomized Latin square designs 60 5.6 Decomposition of the treatment sum of squares 62 5.7 Orthogonal polynomials 63 5.8 Orthogonal and nonorthogonal designs 65 5.9 Models with fixed block effects 67 6 Factorial Treatment Structure and Complete Factorial Designs 71 6.1 Factorial effects for two and three two-level factors 71 6.2 Factorial effects for more than three two-level factors 75 6.3 The general case 77 6.4 Analysis of complete factorial designs 81 6.5 Analysis of unreplicated experiments 83 6.6 Defining factorial effects via finite geometries 84 6.7 Defining factorial effects via Abelian groups 87 6.8 More on factorial treatment structure∗ 90 7 Blocked, Split-Plot, and Strip-Plot Complete Factorial Designs 93 7.1 An example 93 7.2 Construction of blocked complete factorial designs 95 7.3 Analysis 98 7.4 Pseudo factors 99 7.5 Partial confounding 99 7.6 Design keys 100 7.7 A template for design keys 104 7.8 Construction of blocking schemes via Abelian groups 106 7.9 Complete factorial experiments in row-column designs 108 7.10 Split-plot designs 110 7.11 Strip-plot designs 115 8 Fractional Factorial Designs and Orthogonal Arrays 117 8.1 Treatment models for fractional factorial designs 117 8.2 Orthogonal arrays 118 8.3 Examples of orthogonal arrays 122 8.4 Regular fractional factorial designs 124 8.5 Designs derived from Hadamard matrices 125 8.6 Mutually orthogonal Latin squares and orthogonal arrays 128 8.7 Foldover designs 128 8.8 Difference matrices 130 8.9 Enumeration of orthogonal arrays 133 8.10 Some variants of orthogonal arrays∗ 134
- 15. xi 9 Regular Fractional Factorial Designs 139 9.1 Construction and defining relation 139 9.2 Aliasing and estimability 142 9.3 Analysis 145 9.4 Resolution 147 9.5 Regular fractional factorial designs are orthogonal arrays 147 9.6 Foldovers of regular fractional factorial designs 151 9.7 Construction of designs for estimating required effects 155 9.8 Grouping and replacement 157 9.9 Connection with linear codes 161 9.10 Factor representation and labeling 162 9.11 Connection with finite projective geometry∗ 164 9.12 Foldover and even designs revisited∗ 166 10 Minimum Aberration and Related Criteria 169 10.1 Minimum aberration 169 10.2 Clear two-factor interactions 170 10.3 Interpreting minimum aberration 171 10.4 Estimation capacity 173 10.5 Other justifications of minimum aberration 178 10.6 Construction and complementary design theory 179 10.7 Maximum estimation capacity: a projective geometric approach∗ 183 10.8 Clear two-factor interactions revisited 185 10.9 Minimum aberration blocking of complete factorial designs 187 10.10 Minimum moment aberration 188 10.11 A Bayesian approach 190 11 Structures and Construction of Two-Level Resolution IV Designs 195 11.1 Maximal designs 195 11.2 Second-order saturated designs 196 11.3 Doubling 199 11.4 Maximal designs with N/4+1 ≤ n ≤ N/2 202 11.5 Maximal designs with n = N/4+1 204 11.6 Partial foldover 207 11.7 More on clear two-factor interactions 209 11.8 Applications to minimum aberration designs 211 11.9 Minimum aberration even designs 213 11.10 Complementary design theory for doubling 216 11.11 Proofs of Theorems 11.28 and 11.29∗ 220 11.12 Coding and projective geometric connections∗ 221 12 Orthogonal Block Structures and Strata 223 12.1 Nesting and crossing operators 223 12.2 Simple block structures 228 12.3 Statistical models 230
- 16. xii 12.4 Poset block structures 232 12.5 Orthogonal block structures 233 12.6 Models with random effects 234 12.7 Strata 236 12.8 Null ANOVA 238 12.9 Nelder’s rules 239 12.10 Determining strata from Hasse diagrams 242 12.11 Proofs of Theorems 12.6 and 12.7 244 12.12 Models with random effects revisited 245 12.13 Experiments with multiple processing stages 247 12.14 Randomization justification of the models for simple block structures* 251 12.15 Justification of Nelder’s rules* 253 13 Complete Factorial Designs with Orthogonal Block Structures 257 13.1 Orthogonal designs 257 13.2 Blocked complete factorial split-plot designs 259 13.3 Blocked complete factorial strip-plot designs 263 13.4 Contrasts in the strata of simple block structures 265 13.5 Construction of designs with simple block structures 269 13.6 Design keys 271 13.7 Design key templates for blocked split-plot and strip-plot designs 273 13.8 Proof of Theorem 13.2 278 13.9 Treatment structures 279 13.10 Checking design orthogonality 280 13.11 Experiments with multiple processing stages: the nonoverlapping case 282 13.12 Experiments with multiple processing stages: the overlapping case 288 14 Multi-Stratum Fractional Factorial Designs 291 14.1 A general procedure 291 14.2 Construction of blocked regular fractional factorial designs 292 14.3 Fractional factorial split-plot designs 295 14.4 Blocked fractional factorial split-plot designs 300 14.5 Fractional factorial strip-plot designs 302 14.6 Design key construction of blocked strip-plot designs 305 14.7 Post-fractionated strip-plot designs 306 14.8 Criteria for selecting blocked fractional factorial designs based on modified wordlength patterns 308 14.9 Fixed block effects: surrogate for maximum estimation capacity 310 14.10 Information capacity and its surrogate 312 14.11 Selection of fractional factorial split-plot designs 317 14.12 A general result on multi-stratum fractional factorial designs 319 14.13 Selection of blocked fractional factorial split-plot designs 321
- 17. xiii 14.14 Selection of blocked fractional factorial strip-plot designs 322 14.15 Geometric formulation∗ 323 15 Nonregular Designs 329 15.1 Indicator functions and J-characteristics 329 15.2 Partial aliasing 331 15.3 Projectivity 332 15.4 Hidden projection properties of orthogonal arrays 334 15.5 Generalized minimum aberration for two-level designs 338 15.6 Generalized minimum aberration for multiple and mixed levels 340 15.7 Connection with coding theory 341 15.8 Complementary designs 343 15.9 Minimum moment aberration 345 15.10 Proof of Theorem 15.18∗ 347 15.11 Even designs and foldover designs 348 15.12 Parallel flats designs 349 15.13 Saturated designs for hierarchical models: an application of algebraic geometry 353 15.14 Search designs 355 15.15 Supersaturated designs 356 Appendix 365 A.1 Groups 365 A.2 Finite fields 365 A.3 Vector spaces 367 A.4 Finite Euclidean geometry 368 A.5 Finite projective geometry 368 A.6 Orthogonal projections and orthogonal complements 369 A.7 Expectation of a quadratic form 369 A.8 Balanced incomplete block designs 370 References 371 Index 389
- 18. Preface Factorial designs are widely used in many scientific and industrial investigations. The objective of this book is to provide a rigorous, systematic, and up-to-date treatment of the theoretical aspects of this subject. Despite its long history, research in factorial design has grown considerably in the past two decades. Several books covering these advances are available; nevertheless new discoveries continued to emerge. There are also old useful results that seem to have been overlooked in recent literature and, in my view, deserve to be better known. Factorial experiments with multiple error terms (strata) that result from compli- cated structures of experimental units are common in agriculture. In recent years, the design of such experiments also received much attention in industrial applications. A theory of orthogonal block structures that goes back to John Nelder provides a unify- ing framework for the design and analysis of multi-stratum experiments. One feature of the present book is to present this elegant and general theory which, once under- stood, is simple to use, and can be applied to various structures of experimental units in a unified and systematic way. The mathematics required to understand this theory is perhaps what, in Rosemary Bailey’s words, “obscured the essential simplicity” of the theory. In this book, I tried to minimize the mathematics needed, and did not present the theory in the most general form as developed by Bailey and her coauthors. To prepare readers for the general theory, a unified treatment of some simple designs such as completely randomized designs, block designs, and row-column designs is presented first. Therefore the book also covers these elementary non-factorial-design topics. It is suitable as a reference book for researchers and as a textbook for grad- uate students who have taken a first course in the design of experiments. Since the book is self-contained and includes many examples, it should also be accessible to readers with minimal previous exposure to experimental design as long as they have good mathematical and statistical backgrounds. Readers are required to be familiar with linear algebra. A review of linear model theory is given in Chapter 2, and a brief survey of some basic algebraic results on finite groups and fields can be found in the Appendix. Sections that can be skipped, at least on the first reading, without affecting the understanding of the material in later parts of the book are marked with stars. In addition to a general theory of multi-stratum factorial design, the book covers many other topics and results that have not been reported in books. These include, among others, the useful method of design key for constructing multi-stratum facto- rial designs, the methods of partial foldover and doubling for constructing two-level resolution IV designs, some results on the structures of two-level resolution IV de- signs taken from the literature of projective geometry, the extension of minimum xv
- 19. xvi PREFACE aberration to nonregular designs, and the minimum moment aberration criterion, which is equivalent to minimum aberration. The book does not devote much space to the analysis of factorial designs due to its theoretical nature, and also because excellent treatment of strategies for data analysis can be found in several more applied books. Another subject that does not receive a full treatment is the so-called nonregular designs. It is touched upon in Chapter 8 when orthogonal arrays are introduced, and some selected topics are sur- veyed in Chapter 15. The research on nonregular designs is still very active and expands rapidly. It deserves another volume. The writing of this book originated from a ten-lecture workshop on “Recent De- velopments in Factorial Design” I gave in June 2002 at the Institute of Statistical Science, Academia Sinica, in Taiwan. I thank Chen-Hsin Chen, Director of the insti- tute at the time, for his invitation. The book was written over a long period of time while I taught at the University of California, Berkeley, and also during visits to the National Center for Theoretical Sciences in Hsinchu, Taiwan, and the Issac Newton Institute for Mathematical Sciences in Cambridge, United Kingdom. The support of these institutions and the US National Science Foundation is acknowledged. The book could not have been completed without the help of many people. It contains re- sults from joint works with Rosemary Bailey, Dursun Bulutoglu, Hegang Chen, Lih- Yuan Deng, Mike Jacroux, Bobby Mee, Rahull Mukerjee, Nam-Ky Nguyen, David Steinberg, Don Sun, Boxin Tang, Pi-Wen Tsai, Hongquan Xu, and Oksoun Yee. I had the privilege of working with them. I also had the fortune to know Rosemary Bailey early in my career. Her work has had a great impact on me, and this book uses the framework she had developed. Boxin Tang read the entire book, and both Rosemary Bailey and Don Ylvisaker read more than half of it. They provided numerous de- tailed and very helpful comments as well as pointing out many errors. Hegang Chen, Chen-Tuo Liao, and Hongquan Xu helped check the accuracy of some parts of the book. As a LaTex novice, I am very grateful to Pi-Wen Tsai for her help whenever I ran into problems with LaTex. She also read and commented on earlier versions of several chapters. Yu-Ting Chen and Chiun-How Kao helped fix some figures. I would also like to acknowledge our daughter Adelaide for her endearing love and support as well as her upbeat reminder to always see the bright side. Last but not least, I am most grateful to my wife Suzanne Pan for her thankless support and care over the years and for patiently reading this “Tian Shu” from cover to cover. Additional material for the book will be maintained at http://www.crcpress.com/ product/isbn/9781466505575/ and http://www.stat.sinica.edu.tw/factorial-design/.
- 20. Chapter 1 Introduction Many of the fundamental ideas and principles of experimental design were devel- oped by Sir R. A. Fisher at the Rothamsted Experimental Station (Fisher, 1926). This agricultural background is reflected in some terminology of experimental design that is still being used today. Agricultural experiments are conducted, e.g., to compare different varieties of a certain crop or different fertilizers. In general, those that are under comparison in an experiment are called treatments. Manufacturing processes in industrial experiments and drugs in pharmaceutical studies are examples of treat- ments. In an agricultural experiment, the varieties or fertilizers are assigned to plots, and the yields are compared after harvesting. Each plot is called an experimental unit (or unit). In general, an experimental unit can be defined as the smallest division of the experimental material such that different units may receive different treatments (Cox, 1958, p. 2). At the design stage, a treatment is chosen for each experimental unit. One fundamental difficulty in such comparative experiments is inherent variabil- ity of the experimental units. No two plots have exactly the same soil quality, and there are other variations beyond the experimenter’s control such as weather condi- tions. Consequently, effects of the treatments may be biased by uncontrolled vari- ations. A solution is to assign the treatments randomly to the units. In addition to guarding against potential systematic biases, randomization also provides a basis for appropriate statistical analysis. The simplest kind of randomized experiment is one in which treatments are as- signed to units completely at random. In a completely randomized experiment, the precision of a treatment comparison depends on the overall variability of the experi- mental units. When the experimental units are highly variable, the treatment compar- isons do not have good precision. In this case, the method of blocking is an effective way to reduce experimental error. The idea is to divide the experimental units into more homogeneous groups called blocks. When the treatments are compared on the units within each block, the precision is improved since it depends on the smaller within-block variability. Suppose the experimental units are grouped into b blocks of size k. Even though efforts are made for the units in the same block to be as alike as possible, they are still not the same. Given an initial assignment of the treatments to the bk unit labels based on statistical, practical and/or other considerations, randomization is carried 1
- 21. 2 INTRODUCTION out by randomly permuting the unit labels within each block (done independently from block to block), and also randomly permuting the block labels. The additional step of randomly permuting block labels is to assure that an observation intended on a given treatment is equally likely to occur at any of the experimental units. Under a completely randomized experiment, the experimental units are consid- ered to be unstructured. The structure of the experimental units under a block design is an example of nesting. Suppose there are b blocks each consisting of k units; then each experimental unit can be labeled by a pair (i, j), i = 1,...,b, j = 1,...,k. This involves two factors with b and k levels, respectively. Here if i = i , unit (i, j) bears no relation to unit (i , j); indeed, within-block randomization renders positions of the units in each block immaterial. We say that the k-level factor is nested in the b-level factor, and denote this structure by b/k or block/unit if the two factors involved are named “block” and “unit,” respectively. Another commonly encountered structure of the experimental units involves two blocking factors. For example, in agricultural experiments the plots may be laid out in rows and columns, and we try to eliminate from the treatment comparisons the spa- tial variations due to row-to-row and column-to-column differences. In experiments that are carried out on several different days and in several different time slots on each day, the observed responses might be affected by day-to-day and time-to-time variations. In this case each experimental run can be represented by a cell of a rect- angular grid with those corresponding to experimental runs taking place on the same day (respectively, in the same time slot) falling in the same row (respectively, the same column). In general, suppose rc experimental units can be arranged in r rows and c columns such that any two units in the same row have a definite relation, and so do those in the same column. Then we have an example of crossing. This structure of experimental units is denoted by r×c or row × column if the two factors involved are named “row” and “column,” respectively. In such a row-column experiment, given an initial assignment of the treatments to the rc unit labels, randomization is carried out by randomly permuting the row labels and, independently, randomly permuting the column labels. This assures that the structure of the experimental units is preserved: two treatments originally assigned to the same row (respectively, column) remain in the same row (respectively, column) after randomization. For example, suppose there are four different manufacturing processes compared in four time slots on each of four days. With the days represented by rows and times represented by columns, a possible design is 1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 where the four numbers 1, 2, 3, and 4 are labels of the treatments assigned to the units represented by the 16 row-column combinations. We see that each of the four numbers appears once in each row and once in each column. Under such a design, called a Latin square, all the treatments can be compared on each of the four days
- 22. INTRODUCTION 3 as well as in each of the four time slots. If the random permutation is such that the first, second, third, and fourth rows of the Latin square displayed above are mapped to the first, fourth, second, and third rows, respectively, and the first, second, third, and fourth columns are mapped to the fourth, third, first, and second columns, re- spectively, then it results in the following Latin square to be used in actual experi- mentation. 3 4 2 1 1 2 4 3 2 1 3 4 4 3 1 2 The structures of experimental units are called block structures. Block and row- column designs are based on the two simplest block structures involving nesting and crossing, respectively. Nelder (1965a) defined simple block structures to be those that can be obtained by iterations of nesting (/) and crossing (×) operators. For ex- ample, n1/(n2 ×n3) represents the block structure under a nested row-column design, where n1n2n3 experimental units are grouped into n1 blocks of size n2n3, and within each block the n2n3 units are arranged in n2 rows and n3 columns. Randomization of such an experiment can be done by randomly permuting the block labels and car- rying out the appropriate randomization for the block structure n2 × n3 within each block, that is, randomly permuting the row labels and column labels separately. In an experiment with the block structure n1/(n2/n3), n1n2n3 experimental units are grouped into n1 blocks, and within each block the n2n3 units are further grouped into n2 smaller blocks (often called whole-plots) of n3 units (often called subplots). To randomize such a blocked split-plot experiment, we randomly permute the block la- bels and carry out the appropriate randomization for the block structure n2/n3 within each block, that is, randomly permute the whole-plot labels within each block, and randomly permute the subplot labels within each whole-plot. Note that n1/(n2/n3) is the same as (n1/n2)/n3. In general, randomization of an experiment with a simple block structure is car- ried out according to the appropriate randomization for nesting or crossing at each stage of the block structure formula. Like experimental units, the treatments may also have a structure. One can com- pare treatments by examining the pairwise differences of treatment effects. When the treatments do not have a structure (for example, when they are different varieties of a crop), one may be equally interested in all the pairwise comparisons of treatment effects. However, if they do have a certain structure, then some comparisons may be more important than others. For example, suppose one of the treatments is a control. Then one may be more interested in the comparisons between the control and new treatments. In this book, treatments are to have a factorial structure: each treatment is a com- bination of multiple factors (variables) called treatment factors. Suppose there are n treatment factors and the ith factor has si values or settings to be studied. Each of these values or settings is called a level. The treatments, also called treatment com-
- 23. 4 INTRODUCTION binations in this context, consist of all s1 ··· sn possible combinations of the factor levels. The experiment is called an s1 × ··· × sn factorial experiment, and is called an sn experiment when s1 = ··· = sn = s. For example, a fertilizer may be a combi- nation of the levels of three factors N (nitrogen), P (phosphate), and K (potash), and a chemical process might involve temperature, pressure, concentration of a catalyst, etc. Fisher (1926) introduced factorial design to agricultural experiments, and Yates (1935, 1937) made significant contributions to its early development. When the treatments have a factorial structure, typically we are interested in the effects of individual factors, as well as how the factors interact with one another. Special functions of the treatment effects, called main effects and interactions, can be defined to represent such effects of interest. We say that the treatment factors do not interact if, when the levels of a factor are changed while those of the other factors are kept constant, the changes in the treatment effects only depend on the levels of the varying factor. In this case, we can separate the effects of individual factors, and the effect of each treatment combination can be obtained by summing up these individual effects. Under such additivity of the treatment factors, for example, to determine the combination of N, P, and K with the highest average yield, one can simply find the best level of each of the three factors separately. Otherwise, the factors need to be considered simultaneously. Roughly speaking, the main effect of a treatment factor is its effects averaged over the levels of the other factors, and the interaction effects measure departures from additivity. Precise definitions of these effects, collectively called factorial effects, will be given in Chapter 6. A factorial experiment with each treatment combination observed once is called a complete factorial experiment. We also refer to it as a single-replicate complete factorial experiment. The analysis of completely randomized experiments in which each treatment combination is observed the same number of times, to be presented in Chapter 6, is straightforward. It becomes more involved if the experimental units have a more complicated block structure and/or if not all the treatment combinations can be observed. When a factorial experiment is blocked, with each block consisting of one repli- cate of all the treatment combinations, the analysis is still very simple. As will be discussed in Chapter 6, in this case all the treatment main effects and interactions can be estimated in the same way as if there were no blocking, except that the vari- ances of these estimators depend on the within-block variability instead of the overall variability of the experimental units. Since the total number of treatment combina- tions increases rapidly as the number of factors becomes large, a design that accom- modates all the treatment combinations in each block requires large blocks whose homogeneity is difficult to control. In order to achieve smaller within-block vari- ability, we cannot accommodate all the treatment combinations in the same block and must use incomplete blocks. It may also be impractical to carry out experiments in large blocks. Then, since not all the treatment combinations appear in the same block, the estimates of some treatment factorial effects cannot be based on within- block comparisons alone. This may result in less precision for such estimates. For example, suppose an experiment on two two-level factors A1 and A2 is to be run on two different days with the two combinations (0,0) and (1,1) of the levels of A1 and
- 24. INTRODUCTION 5 A2 observed on one day, and the other two combinations (0,1) and (1,0) observed on the other day, where 0 and 1 are the two factor levels. Then estimates of the main effect (comparison of the two levels) of factor A1 and the main effect of A2 are based on within-block comparisons, but as will be seen in Chapter 7, the interaction of the two factors would have to be estimated by comparing the observations on the first day with thoseon the second day, resulting in less precision. We say that this two-factor interaction is confounded with blocks. When a factorial experiment must be run in incomplete blocks, we choose a de- sign in such a way that only those factorial effects that are less important or are known to be negligible are confounded with blocks. Typically the main effects are deemed more important, and one would avoid confounding them with blocks. How- ever, due to practical constraints, sometimes one must confound certain main effects with blocks. For instance, it may be difficult to change the levels of some factors. In the aforementioned example, if a factor must be kept at the same level on each day, then the main effect of that factor can only be estimated by a more variable between-day comparison. Often the number of treatment combinations is so large that it is practically pos- sible to observe only a small subset of the treatment combinations. This is called a fractional factorial design. Then, since not all the treatment combinations are ob- served, some factorial effects are mixed up and cannot be distinguished. We say that they are aliased. For example, when only 16 treatment combinations are to be ob- served in an experiment involving six two-level factors, there are 63 factorial effects (6 main effects, 15 two-factor interactions, 20 three-factor interactions, 15 four-factor interactions, 6 five-factor interactions, and 1 six-factor interaction), but only 15 de- grees of freedom are available for estimating them. This is possible if many of the factorial effects are negligible. One design issue is which 16 of the 64 treatment combinations are to be selected. An important property of a fractional factorial design, called resolution, pertains to the extent to which the lower-order effects are mixed up with higher-order effects. For example, under a design of resolution III, no main effect is aliased with other main effects, but some main effects are aliased with two-factor interactions; under a design of resolution IV, no main effect is aliased with other main effects or two- factor interactions, but some two-factor interactions are aliased with other two-factor interactions; under a design of resolution V, no main effects and two-factor inter- actions are aliased with one another. When the experimenter has little knowledge about the relative importance of the factorial effects, it is common to assume that the lower-order effects are more important than higher-order effects (the main effects are more important than interactions, and two-factor interactions are more important than three-factor interactions, etc.), and effects of the same order are equally impor- tant. Under such a hierarchical assumption, it is desirable to have a design with high resolution. A popular criterion of selecting fractional factorial designs and a refine- ment of maximum resolution, called minimum aberration, is based on the idea of minimizing the aliasing among the more important lower-order effects. When the experimental units have a certain block structure, in addition to pick- ing a fraction of the treatment combinations, we also have to decide how to assign
- 25. 6 INTRODUCTION the selected treatment combinations to the units. In highly fractionated factorial ex- periments with complicated block structures, we have complex aliasing of treatment factorial effects as well as multiple levels of precision for their estimates. The bulk of this book is about the study of such designs, including their analysis, selection, and construction. The term “multi-stratum” in the subtitle of the book refers to multiple sources of errors that arise from complicated block structures, while “single-stratum” is synonymous with “complete randomization” where there is one single error term. Treatment and block structures are two important components of a randomized experiment. Nelder (1965a,b) emphasized their distinction and developed a theory for the analysis of randomized experiments with simple block structures. Simple block structures cover most, albeit not all the block structures that are commonly encountered in practice. Speed and Bailey (1982) and Tjur (1984) further developed the theory to cover the more general orthogonal block structures. This theory, an account of which can be found in Bailey (2008), provides the basis for the approach adopted in this book. We turn to five examples of factorial experiments to motivate some of the topics to be discussed in the book. The first three examples involve simple block structures. The block structures in Examples 1.4 and 1.5 are not simple block structures, but the theory developed by Speed and Bailey (1982) and Tjur (1984) is applicable. We will return to these examples from time to time in later chapters to illustrate applications of the theory as it is developed. Our first example is a replicated complete factorial experiment with a relatively complicated block structure. Example 1.1. Loughin (2005) studied the design of an experiment on weed control. Herbicides can kill the weeds that reduce soybean yields, but they can also kill soy- beans. On the other hand, soybean varieties can be bred or engineered to be resistant to certain herbicides. An experiment is to be carried out to study what factors in- fluence weed control and yield of genetically altered soybean varieties. Four factors studied in the experiment are soybean variety/herbicide combinations in which the herbicide is safe for the soybean variety, dates and rates of herbicide application, and weed species. There are three variety/herbicide combinations, two dates (early and late), three rates (1/4, 1/2, and 1), and seven weed species, giving a total of 126 treatments with a 3 × 2 × 3 × 7 factorial structure. Soybeans and weeds are planted together and a herbicide safe for the soybean variety is sprayed at the designated time and rate. Then weed properties (numbers, density, mass) and soybean yields are measured. However, there are some practical constraints on how the experiment can be run. Due to herbicide drift, different varieties cannot be planted too close together and buffer zones between varieties are needed, but the field size is not large enough to allow for 126 plots of adequate size with large buffers between each pair of adjacent plots. Therefore, for efficient use of space, one needs to plant all of a given soybean variety contiguously so that fewer buffers are needed. Additional drift concerns lead to a design described as follows. First the field is divided into four blocks to accom- modate four replications:
- 26. INTRODUCTION 7 Each block is split into three plots with two buffer zones, and the variety/herbicide combinations are randomly assigned to the plots within blocks: Var 1 Var 3 Var 2 Each plot is then split into two subplots, with application times randomly assigned to subplots within plots: Late Early Early Late Late Early Furthermore each subplot is split into three sub-subplots, with application rates ran- domly assigned to sub-subplots within subplots: 1 2 1 4 1 1 1 4 1 2 1 2 1 1 4 1 1 2 1 4 1 1 2 1 4 1 4 1 1 2 Each block is divided into seven horizontal strips, with the weed species randomly assigned to the strips within blocks:
- 27. 8 INTRODUCTION We end up with 504 combinations of sub-subplots and strips: Each of the 126 treatment combinations appears once in each of the four blocks. To summarize, we have four replicates of a complete 3×2×3×7 factorial experiment with the block structure 4/[(3/2/3)×7]. Both subplots in the same plot are assigned the same variety/herbicide combination, all the sub-subplots in the same subplot are assigned the same herbicide application time, and all the sub-subplot and strip inter- sections in the same strip are assigned the same weed species. Various aspects of the analysis of this design will be discussed in Sections 12.1, 12.9, 12.10, and 13.10. In Example 1.1, there are 18 sub-subplots in each block. If different soybean varieties were to be assigned to neighboring sub-subplots, then 17 buffer zones would be needed in each block. With only two buffer zones per block under the proposed design, comparisons of soybean varieties are based on between-plot com- parisons, which are expected to be more variable than those between subplots and sub-subplots. The precision of the estimates of such effects is sacrificed in order to satisfy the practical constraints. Example 1.2. McLeod and Brewster (2004) discussed an experiment for identifying key factors that would affect the quality of a chrome-plating process. Suppose six two-level treatment factors are to be considered in the experiment: A, chrome con- centration; B, chrome to sulfate ratio; C, bath temperature; S, etching current density; T, plating current density; and U, part geometry. The response variables include, e.g., the numbers of pits and cracks. The chrome plating is done in a bath (tank), which contains several rectifiers, but only two will be used. On any given day the levels of A, B, and C cannot be changed since they represent characteristics of the bath. On the other hand, the levels of factors S, T, and U can be changed at the rectifier level. The experiment is to be run on 16 days, with four days in each of four weeks. Therefore there are a total of 32 runs with the block structure (4 weeks)/(4 days)/(2 runs), and
- 28. INTRODUCTION 9 one has to choose 32 out of the 26 = 64 treatment combinations. Weeks, days, and runs can be considered as blocks, whole-plots, and subplots, respectively. The three factors A, B, and C must have constant levels on the two experimental runs on the same day, and are called whole-plot treatment factors. The other three factors S, T, and U are not subject to this constraint and are called subplot treatment factors. We will return to this example in Sections 12.9, 13.2, 13.4, 13.5, 13.7, 14.4, and 14.13. Example 1.3. Miller (1997) described a laundry experiment for investigating meth- ods of reducing the wrinkling of clothes. Suppose the experiment is to be run in two blocks, with four washers and four dryers to be used. After four cloth samples have been washed in each washer, the 16 samples are divided into four groups with each group containing one sample from each washer. Each of these groups is then assigned to one dryer. The extent of wrinkling on each sample is evaluated at the end of the experiment. This results in 32 experimental runs that can be thought to have the 2/(4×4) block structure shown in Figure 1.1, where each cell represents a cloth sample, rows represent sets of samples that are washed together, and columns repre- sent sets of samples that are dried together. There are ten two-level treatment factors, Figure 1.1 A 2/(4×4) block structure six of which (A, B, C, D, E, F) are configurations of washers and four (S, T, U, V) are configurations of dryers. One has to choose 32 out of the 210 = 1024 treatment combinations. Furthermore, since the experimental runs on the cloth samples in the same row are conducted in the same washing cycle, each of A, B, C, D, E, F must have a constant level in each row. Likewise, each of S, T, U, V must have a constant level in each column. Thus in each block, four combinations of the levels of A, B, C, D, E, F are chosen, one for each row, and four combinations of the levels of S, T, U, V are chosen, one for each column. The four combinations of washer settings are then coupled with the four combinations of dryer settings to form 16 treatment com- binations of the ten treatment factors in the same block. An experiment run in this way requires only four washer loads and four dryer loads in each block. If one were to do complete randomization in each block, then four washer loads and four dryer loads could produce only four observations. The trade-off is that the main effect of each treatment factor is confounded with either rows or columns. Construction and analysis of designs for such blocked strip-plot experiments will be discussed in Sec- tions 12.2, 12.9, 12.10, 13.3, 13.4, 13.5, 13.6, 13.7, 14.5, 14.6, and 14.14. Federer and King (2006) gave a comprehensive treatment of split-plot and strip-
- 29. 10 INTRODUCTION plot designs and their many variations. In this book, we present a unifying theory that can be systematically applied to a very general class of multi-stratum experiments. Example 1.3 is an experiment with two processing stages: washing and drying. Many industrial experiments involve a sequence of processing stages, with the levels of various treatment factors assigned and processed at different stages. At each stage the experimental units are partitioned into disjoint classes. Those in the same class, which will be processed together, are assigned the same level of each of the treatment factors that are to be processed at that stage. We call the treatment factors processed at the ith stage the ith-stage treatment factors. In Example 1.3, levels of the six washer factors are set at the first stage and those of the four dryer factors are set at the second stage. So the washer configurations are first-stage treatment factors and the dryer configurations are second-stage treatment factors. Such an experiment with two processing stages can be thought to have experimental units with a row-column structure. In Examples 1.1–1.3, the experimental units can be represented by all the level combinations of some unit factors. In the next two examples, we present experiments in which the experimental units are a fraction of unit-factor level combinations. Example 1.4. Mee and Bates (1998) discussed designs of experiments with multiple processing stages in the fabrication of integrated circuits. Suppose that at the first stage 16 batches of material are divided into four groups of equal size, with the same level of each first-stage treatment factor assigned to all the batches in the same group. At the second stage they are rearranged into another four groups of equal size, again with the same level of each second-stage treatment factor assigned to all the batches in the same group. As in Example 1.3, the groupings at the two stages can be represented by rows and columns. Then each of the first-stage groups and each of the second-stage groups have exactly one batch in common. This is a desirable property whose advantage will be explained in Section 12.13. Now suppose there is a third stage. Then we need a third grouping of the batches. One possibility is to group according to the numbers in the Latin square shown earlier: 1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 One can assign the same level of each third-stage treatment factor to all the units (batches) corresponding to the same number in the Latin square. One advantage is that each of the third-stage groups has exactly one unit in common with any group at the first two stages. If a fourth stage is needed, then one may group according to the following Latin square:
- 30. INTRODUCTION 11 1 2 3 4 4 3 2 1 2 1 4 3 3 4 1 2 This and the previous Latin square have the property that when one is superimposed on the other, each of the 16 pairs of numbers (i, j), 1 ≤ i, j ≤ 4, appears in exactly one cell. We say that these two Latin squares are orthogonal to each other. If the fourth-stage grouping is done according to the numbers in the second Latin square, then each of the fourth-stage groups also has exactly one unit in common with each group at any of the first three stages. This kind of block structure cannot be obtained by iterations of nesting and crossing operators. To be a simple block structure, with four groups at each of three or four stages, one would need 43 = 64 or 44 = 256 units, respectively. Thus the 16 units can be regarded as a quarter or one-sixteenth fraction of the combinations of three or four 4-level factors, respectively. The following is a complete 24 design which can be used for experiments in which the levels of the four treatment factors are set at four stages, one factor per stage: the first factor has a constant level in each row, the second factor has a constant level in each column, the third factor has a constant level in each cell occupied by the same number in the first Latin square, and the fourth factor has a constant level in each cell occupied by the same number in the second Latin square. 0000 0011 0101 0110 0010 0001 0111 0100 1001 1010 1100 1111 1011 1000 1110 1101 We will return to this example in Sections 12.5, 12.10, 12.13, and 13.11. Example 1.5. Bingham, Sitter, Kelly, Moore, and Olivas (2008) discussed experi- ments with multiple processing stages where more groups are needed at each stage, which makes it impossible for all the groups at different stages to share common units. For example, in an experiment with two processing stages, suppose 32 exper- imental units are to be partitioned into 8 groups of size 4 at each of the two stages. One possibility is to partition the 32 units as in Figure 1.1. The eight rows of size 4, four of which from each of the two blocks, together constitute the eight first-stage groups, and the eight columns in the two blocks together constitute the eight second- stage groups. As shown in the following figure, the 32 starred experimental units are a fraction of the 64 units in a completely crossed 8×8 square.
- 31. 12 INTRODUCTION ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ As in Example 1.4, the 32 units do not have a simple block structure. An important difference, however, is that in the current setting not all the first-stage groups can meet with every second-stage group, causing some complications in the design and analysis (to be discussed in Section 13.12). The figure shows that the 32 units are divided into two groups of size 16, which we call pseudo blocks since they are not part of the originally intended block structure. We will revisit this example in Sec- tions 12.13 and 13.12, and show that it can be treated as an experiment with the block structure 2/(4×4). A similar problem was studied in Vivacqua and Bisgaard (2009), to be discussed in Section 14.7. An overview Some introductory material is presented in Chapters 2–5. Chapter 2 is a review of some results on linear models, with emphasis on one-way and two-way layout models and a geometric characterization of the condition of proportional frequencies between two factors. Under the assumption of treatment-unit additivity, randomiza- tion models are developed in Chapter 3 for some designs with simple block struc- tures, including block designs and row-column designs. In Chapter 4, the condition of proportional frequencies is extended to a notion of orthogonal factors that plays an important role in the block structures studied in this book. Some mathematical results on factors that are needed throughout the book are also gathered there. A condition that entails simple analysis of a randomized design (Theorem 5.1) is established in Chapter 5. This result is used to present a unified treatment of the analyses of three classes of orthogonal designs (completely randomized designs, complete block de- signs, and Latin square designs) under the randomization models derived in Chapter 3. It is also a key result for developing, in later chapters, a general theory of orthog- onal designs for experiments with more complicated block structures. The treatment factorial structure is introduced in Chapter 6. It is shown how cer- tain special functions of the treatment effects can be defined to represent main effects and interactions of the treatment factors. Unless all the treatment factors have two levels, the choices of such functions are not unique. Several methods of construct- ing them based on orthogonal polynomials, finite Euclidean geometry, and Abelian groups are presented. The discussion of complete factorial designs is continued in Chapter 7, for experiments that are conducted in incomplete blocks or row-column
- 32. INTRODUCTION 13 layouts, including split-plot and strip-plot designs. In this case, there is more than one error term and some factorial effects are confounded with blocks, rows, or columns. A construction method based on design keys is presented in addition to a commonly used method, and is shown to enjoy several advantages. Fractional factorial designs under complete randomization are treated in Chap- ters 8–11. In Chapter 8, the important combinatorial structure of orthogonal arrays is introduced. Some basic properties of orthogonal arrays as fractional factorial de- signs, including upper bounds on the number of factors that can be accommodated by orthogonal arrays of a given run size, are derived. We also present several methods of constructing orthogonal arrays, in particular the foldover method and the con- struction via difference matrices. The chapter is concluded with a brief discussion of applications of orthogonal arrays to computer experiments and three variants of orthogonal arrays recently introduced for this purpose. The emphasis of this book is mainly on the so-called regular fractional factorial designs, which are easy to con- struct and analyze, and have nice structures and a rich theory. In Chapter 9 we provide a treatment of their basics, including design construction, aliasing and estimability of factorial effects, resolution, a search algorithm for finding designs under which some required effects can be estimated, and the connection with the linear codes of coding theory. The criterion of minimum aberration and some related criteria for selecting regular fractional factorial designs are discussed in Chapter 10. The statistical mean- ing of minimum aberration is clarified via its implications on the aliasing pattern of factorial effects. It is shown that this criterion produces designs with good properties under model uncertainty and good lower-dimensional projections. The connection to coding theory provides two powerful tools for constructing minimum aberration designs: the MacWilliams identities can be used to establish a complementary de- sign theory that is useful for determining minimum aberration designs when there are many factors; the Pless power moment identities lead to the criterion of mini- mum moment aberration, which is equivalent to minimum aberration. Besides the theoretical interest, this equivalence is useful for analytical characterization and al- gorithmic construction of minimum aberration designs. A Bayesian approach to the design and analysis of factorial experiments, also applicable to nonregular designs, is presented at the end of Chapter 10. Regular designs are also closely related to finite projective geometries. The connection is made in two optional sections in Chapter 9, and is used to characterize and construct minimum aberration designs in Chapter 10. The geometric connection culminates in an elegant theory of the construction and structures of resolution IV designs in Chapter 11. While foldover is a well-known method of constructing resolution IV designs, many resolution IV designs cannot be constructed by this method. We translate the geometric results into design language, and among other topics, present the methods of doubling and partial foldover for constructing them. In Chapters 12–14, we turn to factorial designs with more complicated block structures called multi-stratum designs. Some basic results on Nelder’s simple block structures and the more general orthogonal block structures are derived in Chapter 12. A general theory for the design and analysis of orthogonal multi-stratum com- plete factorial designs is developed in Chapter 13. This theory is applied to several
- 33. 14 INTRODUCTION settings, including blocked split-plot designs, blocked strip-plot designs, and design of experiments with multiple processing stages. Chapter 14 is devoted to the con- struction of multi-stratum fractional factorial designs and criteria for their selection under model uncertainty in the spirit of minimum aberration. The five motivating examples presented above are revisited. We survey a few nonregular design topics in Chapter 15. Under nonregular de- signs, the factorial effects are aliased in a complicated way, but their run sizes are more flexible than regular designs. At the initial stage of experimentation, often only a small number of the potential factors are important. Due to their run-size economy, nonregular designs are suitable for conducting factor screening experiments under the factor sparsity principle. In this context, it is useful to study the property of the design when it is projected onto small subsets of factors. We also discuss the rele- vant topics of search designs and supersaturated designs. The objective of a search design is to identify and discriminate nonnegligible effects under the assumption that the number of nonnegligible effects is small. Supersaturated designs have more un- known parameters than the degrees of freedom available for estimating them and are useful for screening active factors. In addition to these and other miscellaneous top- ics, we show how some of the results presented in earlier chapters can be extended to nonregular designs. For example, coding theory again proves useful for providing a way to extend minimum aberration to nonregular designs. Throughout this book, the starred sections can be skipped, at least on the first reading. Relevant exercises are also marked with stars.
- 34. Chapter 2 Linear Model Basics In this chapter we review some basic results from linear model theory, including least squares estimation, the Gauss–Markov Theorem, and tests of linear hypothe- ses, with applications to the analysis of fixed-effect one-way and additive two-way layout models. We show how the analysis of an additive two-way layout model is simplified when certain vector spaces associated with the two factors are orthogonal. This geometric condition is shown to be equivalent to that the two factors satisfy the condition of proportional frequencies. 2.1 Least squares Consider the linear model yi = p ∑ j=1 xijθj +εi, i = 1,...,N, (2.1) where xij,1 ≤ i ≤ N,1 ≤ j ≤ p, are known constants, θ1,...,θp are unknown param- eters, and ε1,...,εN are uncorrelated random variables with zero mean and common variance σ2. Let y = (y1,...,yN)T , θ θ θ = (θ1,...,θp)T , ε ε ε = (ε1,...,εN)T , and X be the N × p matrix with the (i, j)th entry equal to xij, where T stands for “transpose.” Then (2.1) can be written as y = Xθ θ θ +ε ε ε, (2.2) with E(ε ε ε) = 0, and cov(ε ε ε) = σ2 IN, (2.3) where E(ε ε ε) = (E(ε1),..., E(εN))T , cov(ε ε ε) is the covariance matrix of ε ε ε, 0 is the vector of zeros, and IN is the identity matrix of order N. We call X the model matrix. Least squares estimators θ1,..., θp of θ1,...,θp are obtained by minimizing ∑ N i=1(yi −∑ p j=1 xijθj)2 = y−Xθ θ θ 2 . Let E(y) = (E(y1),..., E(yN))T . Then under (2.2) and (2.3), E(y) = Xθ θ θ. This im- plies that E(y) is a linear combination of the column vectors of X. Let R(X), called the column space of X, be the linear space generated by the column vectors of X. Then E(y) ∈ R(X). The least squares method is to find a vector y = X θ θ θ in R(X) such that y− y is minimized. This is achieved if y is the orthogonal projection of y onto 15
- 35. 16 LINEAR MODEL BASICS R(X). Denoting the orthogonal projection matrix onto a space V by PV , we have y = X θ θ θ = PR(X)y. Here y−X θ θ θ, called the residual, is the orthogonal projection of y onto R(X)⊥ , where R(X)⊥ = {x ∈ RN : x is orthogonal to all the vectors in R(X)} is the orthogonal complement of R(X) in RN. Therefore y−X θ θ θ is orthogonal to all the column vectors of X, and it follows that XT y−X θ θ θ = 0 or XT X θ θ θ = XT y. (2.4) Equation (2.4) is called the normal equation. If XT X is invertible, then θ θ θ = (XT X)−1XT y, with E( θ θ θ) = θ θ θ and cov( θ θ θ) = σ2 (XT X)−1 . (2.5) Let the rank of X be r. Then XT X is invertible if and only if r = p. Unless r = p, not all the parameters in θ θ θ are identifiable, and solutions to (2.4) are not unique. In this case, (2.4) can be solved by using generalized inverses. A generalized inverse of a matrix A is defined as any matrix A− such that AA− A = A. For any generalized inverse (XT X)− of XT X, θ θ θ = (XT X)− XT y is a solution to the normal equation. Even though θ θ θ may not be unique, since y has a unique orthogonal projection onto R(X), X θ θ θ = X(XT X)− XT y is unique and does not depend on the choice of (XT X)− . A byproduct of this is an explicit expression for the orthogonal projection matrix onto the column space of any matrix. Theorem 2.1. The orthogonal projection matrix onto R(X) is X(XT X)− XT , where (XT X)− is any generalized inverse of XT X. A linear function cTθ θ θ = ∑ p i=1 ciθi of the unknown parameters is said to be es- timable if there exists an N × 1 vector a such that E(aT y) = cTθ θ θ for all θ θ θ. Such an estimator of cTθ θ θ is called a a linear unbiased estimator. Since E(aT y) = aT Xθ θ θ, it is equal to cTθ θ θ for all θ θ θ if and only if c = XT a. This shows that cTθ θ θ is estimable if and only if c ∈ R(XT ). From matrix algebra, it is known that R(XT ) = R(XT X). Therefore, cT θ θ θ is estimable if and only if c ∈ R(XT X). (2.6) If cTθ θ θ is estimable, then cT θ θ θ does not depend on the solution to the normal equation. Theorem 2.2. (Gauss–Markov Theorem) Under (2.2)–(2.3), if cTθ θ θ is estimable, then for any solution θ θ θ to (2.4), cT θ θ θ has the smallest variance among all the linear unbi- ased estimators of cTθ θ θ.
- 36. ESTIMATION OF σ2 17 It is common to refer to cT θ θ θ as the best linear unbiased estimator, or BLUE, of cTθ θ θ. For any estimable function cTθ θ θ, we have var cT θ θ θ = σ2 cT (XT X)− c, where (XT X)− is any generalized inverse of XT X. We call XT X the information matrix for θ θ θ. Suppose the model matrix X is design dependent. We say that a design is D-, A-, or E-optimal if it minimizes, respectively, the determinant, trace, or the largest eigen- value of the covariance matrix of θ θ θ among all the competing designs. By (2.5), these optimality criteria, which are referred to as the D-, A-, and E-criterion, respectively, are equivalent to maximizing det(XT X), minimizing tr(XT X)−1, and maximizing the smallest eigenvalue of XT X, respectively. 2.2 Estimation of σ2 We have y = X θ θ θ +(y−X θ θ θ), (2.7) where X θ θ θ and y−X θ θ θ are orthogonal to each other. By the Pythagorean Theorem, y 2 = X θ θ θ 2 + y−X θ θ θ 2 . The last term in this identity, y−X θ θ θ 2 , is called the residual sum of squares. Since y−X θ θ θ = PR(X)⊥ y, E y−X θ θ θ 2 = E PR(X)⊥ y T PR(X)⊥ y = E yT PR(X)⊥ y = [E(y)]T PR(X)⊥ [E(y)]+σ2 tr PR(X)⊥ = σ2 tr PR(X)⊥ = σ2 dim R(X)⊥ = σ2 (N −r), where the second equality follows from (A.2) and (A.3) in the Appendix, the third equality follows from (A.5), the fourth equality holds since E(y) ∈ R(X) and R(X) ⊥ R(X)⊥ , and the fifth equality follows from (A.4). Let s2 = y−X θ θ θ 2 /(N − r). Then s2, called the residual mean square, is an unbiased estimator of σ2. The dimension of R(X)⊥ , N − r, is called the degrees of freedom associated with the residual sum of squares.
- 37. 18 LINEAR MODEL BASICS Under the assumption that y has a normal distribution, 1 σ2 y−X θ θ θ 2 has a χ2- distribution with N −r degrees of freedom. Thus, for any estimable function cTθ θ θ, cT θ θ θ s cT (XT X)−c has a t-distribution with N −r degrees of freedom. Therefore a 100(1 − α)% confidence interval for cTθ θ θ is cT θ θ θ ± tN−r;1−α/2s · cT (XT X)−c, where tN−r;1−α/2 is the (1−α/2)th quantile of the t-distribution with N −r degrees of freedom. In the rest of the book, we often abbreviate degrees of freedom, sum of squares, and mean square as d.f., SS, and MS, respectively. 2.3 F-test We have seen that under linear model (2.2)–(2.3), E(y) ∈ R(X). Suppose V is a sub- space of R(X) with dim(V) = q. Let R(X) V = {y ∈ R(X) : y is orthogonal to V} be the orthogonal complement of V relative to R(X). Then R(X)V has dimension r −q, and X θ θ θ = PR(X)y can be decomposed as PR(X)y = PV y+PR(X)V y. (2.8) By combining (2.7) and (2.8), we have y−PV y = PR(X)V y+PR(X)⊥ y, (2.9) where the two components on the right side are orthogonal. Thus y−PV y 2 = PR(X)V y 2 + PR(X)⊥ y 2 . (2.10) Under the assumption that y has a normal distribution, a test of the null hypothesis H0: E(y) ∈ V against the alternative hypothesis that E(y) / ∈ V is based on the ratio F = PR(X)V y 2 /(r −q) PR(X)⊥ y 2 /(N −r) . (2.11) It can be shown that the likelihood ratio test is to reject H0 for large values of F. Under H0, F has an F-distribution with r−q and N −r degrees of freedom. Therefore the null hypothesis is rejected at level α if F Fr−q,N−r;1−α , where Fr−q,N−r;1−α is the (1−α)th quantile of the F-distribution with r −q and N −r degrees of freedom. The left side of (2.10) is the residual sum of squares when H0 is true, and the second term on the right side is the residual sum of squares under the full model (2.2). Thus the sum of squares PR(X)V y 2 that appears in the numerator of the test statistic in (2.11) is the difference of the residual sums of squares under the full model and the reduced model specified by H0.
- 38. ONE-WAY LAYOUT 19 2.4 One-way layout Let y = (y11,...,y1r1 ,...,yt1,...,ytrt )T , where y11,...,y1r1 ,...,yt1,...,ytrt are uncor- related random variables with constant variance σ2, and E(ylh) = αl, for all 1≤ h ≤ rl, 1 ≤ l ≤ t. Then we can express y as in (2.2)–(2.3), with θ θ θ = (α1,...,αt)T , and X = ⎡ ⎢ ⎢ ⎢ ⎣ 1r1 0 ··· 0 0 1r2 ··· 0 . . . . . . ... . . . 0 0 0 1rt ⎤ ⎥ ⎥ ⎥ ⎦ , where 1rl is the rl × 1 vector of ones. This model arises, for example, when yl1,...,ylrl are a random sample from a population with mean αl and variance σ2. It is also commonly used as a model for analyzing a completely randomized exper- iment, where the lth treatment, 1 ≤ l ≤ t, is replicated rl times, yl1,...,ylrl are the observations on the lth treatment, and α1,...,αt are effects of the treatments. Clearly XT X = diag(r1,...,rt), the t × t diagonal matrix with r1,...,rt as the diagonal entries; therefore (2.4) has a unique solution: αl = yl·, where yl· = 1 rl ∑ rl h=1 ylh. For any function ∑ t l=1 clαl, we have ∑ t l=1 cl αl = ∑ t l=1 clyl·, with var t ∑ l=1 cl αl = σ2 t ∑ l=1 c2 l rl . The projection PR(X)y = X θ θ θ has its first r1 components equal to y1·, the next r2 com- ponents equal to y2·,..., etc. Therefore the residual sum of squares can be expressed as y−X θ θ θ 2 = t ∑ l=1 rl ∑ h=1 (ylh −yl·)2 . Call this the within-group sum of squares and denote it by W. We have N = ∑ t l=1 rl, and rank(X) = t. Therefore the residual sum of squares has N −t degrees of freedom, and so if s2 = W/(N −t), then E(s2) = σ2. Now we impose the normality assumption and consider the test of H0: α1 = ··· = αt. Under H0, E(y) ∈ V, where V is the one-dimensional space consisting of all the vectors with constant entries. Then PV y is the vector with all the entries equal to the overall average y·· = 1 N ∑ t l=1 ∑ rl h=1 ylh. Componentwise, (2.9) can be expressed as ylh −y·· = (yl· −y··)+(ylh −yl·) and, in the present context, (2.10) reduces to t ∑ l=1 rl ∑ h=1 (ylh −y··)2 = t ∑ l=1 rl(yl· −y··)2 + t ∑ l=1 rl ∑ h=1 (ylh −yl·)2 . Thus PR(X)V y 2 = ∑ t l=1 rl(yl· − y··)2, which has t − 1 degrees of freedom and is called the between-group sum of squares. Denote it by B. Then the F-test statistic
- 39. 20 LINEAR MODEL BASICS is B/(t−1) W/(N−t) . In the application to completely randomized experiments, the between- group sum of squares is also called the treatment sum of squares. These results can be summarized in Table 2.1, called an ANOVA (Analysis of Variance) table. Table 2.1 ANOVA table for one-way layout source sum of squares d.f. mean square Between groups ∑ t l=1 rl(yl· −y··)2 t −1 1 t−1 ∑ t l=1 rl(yl· −y··)2 Within groups ∑ t l=1 ∑ rl h=1(ylh −yl·)2 N −t 1 N−t ∑ t l=1 ∑ rl h=1(ylh −yl·)2 Total ∑ t l=1 ∑ rl h=1(ylh −y··)2 N −1 Remark 2.1. If we write the model as E(ylh) = μ +αl, then the model matrix X has an extra column of 1’s and p = t + 1. Since rank(X) = t p, the parameters them- selves are not identifiable. By using (2.6), one can verify that ∑ t l=1 clαl is estimable if and only if ∑ t l=1 cl = 0. Such functions are called contrasts. The pairwise differences αl − αl , 1 ≤ l = l ≤ t, are examples of contrasts and are referred to as elemen- tary contrasts. Because of the constraint ∑ t l=1 cl = 0, the treatment contrasts form a (t −1)-dimensional vector space that is generated by the elementary contrasts. Func- tions such as α1 − 1 2 (α2 +α3) and 1 2 (α1 +α2)− 1 3 (α3 +α4 +α5) are also contrasts. If ∑ t l=1 clαl is a contrast, then ∑ t l=1 clαl = ∑ t l=1 cl(μ + αl), and it can be shown that, as before, ∑ t l=1 cl αl = ∑ t l=1 clyl·. Therefore, if the interest is in estimating the contrasts, then it does not matter whether E(ylh) is written as αl or μ +αl. A contrast ∑ t l=1 clαl is said to be normalized if ∑ t l=1 c2 l = 1. 2.5 Estimation of a subset of parameters Suppose θ θ θ is partitioned as θ θ θ = (θ θ θT 1 θ θ θT 2 )T , where θ θ θ1 is q × 1 and θ θ θ2 is (p − q) × 1; e.g., the parameters in θ θ θ2 are nuisance parameters, or the components of θ θ θ1 and θ θ θ2 are effects of the levels of two different factors. Partition X as [X1 X2] accordingly. Then (2.2) can be written as y = X1θ θ θ1 +X2θ θ θ2 +ε ε ε, (2.12) and (2.4) is the same as XT 1 X1 θ θ θ1 +XT 1 X2 θ θ θ2 = XT 1 y, (2.13) and XT 2 X1 θ θ θ1 +XT 2 X2 θ θ θ2 = XT 2 y. (2.14) If R(X1) and R(X2) are orthogonal, (XT 1 X2 = 0), then θ θ θ1 and θ θ θ2 can be computed by solving XT 1 X1 θ θ θ1 = XT 1 y and XT 2 X2 θ θ θ2 = XT 2 y separately. In this case, least squares estimators of estimable functions of θ θ θ1 are the same regardless of whether θ θ θ2 is
- 40. ESTIMATION OF A SUBSET OF PARAMETERS 21 in the model. Likewise, dropping θ θ θ1 from (2.12) does not change the least squares estimators of estimable functions of θ θ θ2. If R(X1) and R(X2) are not orthogonal, then for estimating θ θ θ1 one needs to adjust for θ θ θ2, and vice versa. By (2.14), θ θ θ2 can be written as θ θ θ2 = XT 2 X2 − XT 2 y−XT 2 X1 θ θ θ1 . (2.15) We eliminate θ θ θ2 by substituting the right side of (2.15) for the θ θ θ2 in (2.13). Then θ θ θ1 can be obtained by solving XT 1 X1 −XT 1 X2 XT 2 X2 − XT 2 X1 θ θ θ1 = XT 1 y−XT 1 X2 XT 2 X2 − XT 2 y or XT 1 I−X2 XT 2 X2 − XT 2 X1 θ θ θ1 = XT 1 I−X2 XT 2 X2 − XT 2 y. (2.16) This is called the reduced normal equation for θ θ θ1. We write the reduced normal equation as C1 θ θ θ1 = Q1, (2.17) where C1 = XT 1 I−X2 XT 2 X2 − XT 2 X1, (2.18) and Q1 = XT 1 I−X2 XT 2 X2 − XT 2 y. (2.19) Since X2(XT 2 X2)− XT 2 is the orthogonal projection matrix onto R(X2), I− X2(XT 2 X2)− XT 2 is the orthogonal projection matrix onto R(X2)⊥ , and C1 = XT 1 PR(X2)⊥ X1. If we put X̃1 = PR(X2)⊥ X1, then we can express C1 as C1 = X̃T 1 X̃1, and the reduced normal equation (2.16) can be written as X̃T 1 X̃1 θ θ θ1 = X̃T 1 ỹ, where ỹ = PR(X2)⊥ y. Therefore the least squares estimators of estimable functions of θ θ θ1 are functions of ỹ = PR(X2)⊥ y: to eliminate θ θ θ2, (2.12) is projected onto R(X2)⊥ to become ỹ = X̃1θ θ θ1 +ε̃, where ε̃ ε ε = PR(X2)⊥ε ε ε.
- 41. 22 LINEAR MODEL BASICS Theorem 2.3. Under (2.12) and (2.3), a linear function cT 1 θ θ θ1 of θ θ θ1 is estimable if and only if c1 is a linear combination of the column vectors of C1. If cT 1 θ θ θ1 is estimable, then cT 1 θ θ θ1 is its best linear unbiased estimator, and var cT 1 θ θ θ1 = σ2 cT 1 C− 1 c1, where θ θ θ1 is any solution to (2.17), and C− 1 is any generalized inverse of C1. In particular, if C1 is invertible, then all the parameters in θ θ θ1 are estimable, with cov θ θ θ1 = σ2 C−1 1 . (2.20) The matrix C1 is called the information matrix for θ θ θ1. Suppose the parameters in θ θ θ2 are nuisance parameters, and we are only interested in estimating θ θ θ1. Then a design is said to be Ds-, As-, or Es-optimal if it minimizes, respectively, the determinant, trace, or the largest eigenvalue of the covariance ma- trix of θ θ θ1 among all the competing designs. By (2.20), these optimality criteria are equivalent to maximizing det(C1), minimizing tr(C−1 1 ), and maximizing the smallest eigenvalue of C1, respectively. Here s refers to a subset of parameters. 2.6 Hypothesis testing for a subset of parameters Under (2.12) and (2.3), suppose we further assume that ε ε ε has a normal distribution. Consider testing the null hypothesis H0: E(y) = X2θ θ θ2. Under (2.12), E(y) ∈ R(X1)+R(X2), and under H0, E(y) ∈ R(X2). By (2.11), the F-test statistic in this case is F = P[R(X1)+R(X2)]R(X2)y 2 /dim([R(X1)+R(X2)]R(X2)) P[R(X1)+R(X2)]⊥ y 2 /(N −dim[R(X1)+R(X2)]) . This is based on the decomposition RN = R(X2)⊕([R(X1)+R(X2)]R(X2))⊕[R(X1)+R(X2)]⊥ . (2.21) It can be shown that P[R(X1)+R(X2)]R(X2)y 2 = θ θ θ T 1 Q1, (2.22) and dim([R(X1)+R(X2)]R(X2)) = rank(C1). (2.23) We leave the proofs of these as an exercise. A test of the null hypothesis H0: E(y) = X1θ θ θ1 is based on the decomposition RN = R(X1)⊕([R(X1)+R(X2)]R(X1))⊕[R(X1)+R(X2)]⊥ . (2.24)
- 42. ADJUSTED ORTHOGONALITY 23 In this case, P[R(X1)+R(X2)]R(X1)y 2 = θ θ θ T 2 Q2, where Q2 = XT 2 I−X1 XT 1 X1 − XT 1 y. When XT 1 X2 = 0, both (2.21) and (2.24) reduce to RN = R(X1)⊕R(X2)⊕[R(X1)⊕R(X2)]⊥ . 2.7 Adjusted orthogonality Consider the model y = X1θ θ θ1 +X2θ θ θ2 +X3θ θ θ3 +ε ε ε, (2.25) with E(ε ε ε) = 0, and cov(ε ε ε) = σ2 IN. Suppose the parameters in θ θ θ2 are nuisance parameters. To estimate θ θ θ1 and θ θ θ3, we eliminate θ θ θ2 by projecting y onto R(X2)⊥ . This results in a reduced normal equation for θ θ θ1 and θ θ θ3: X̃T 1 X̃1 X̃T 1 X̃3 X̃T 3 X̃1 X̃T 3 X̃3 θ θ θ1 θ θ θ3 = X̃T 1 ỹ X̃T 3 ỹ , where X̃i = PR(X2)⊥ Xi,i = 1,3, and ỹ = PR(X2)⊥ y. If X̃T 1 X̃3 = 0, then θ θ θ1 and θ θ θ3 can be computed by solving the equations X̃T 1 X̃1 θ θ θ1 = X̃T 1 ỹ and X̃T 3 X̃3 θ θ θ3 = X̃T 3 ỹ separately. In this case, least squares estimators of estimable functions of θ θ θ1 under (2.25) are the same regardless of whether θ θ θ3 is in the model. Likewise, dropping θ θ θ1 from (2.25) does not change the least squares estimators of estimable functions of θ θ θ3. Loosely we say that θ θ θ1 and θ θ θ3 are orthogonal adjusted for θ θ θ2. Note that X̃T 1 X̃3 = 0 is equivalent to PR(X2)⊥ [R(X1)] is orthogonal to PR(X2)⊥ [R(X3)]. (2.26) Under normality, the F-test statistic for the null hypothesis that E(y) = X2θ θ θ2 +X3θ θ θ3 is P[R(X1)+R(X2)+R(X3)][R(X2)+R(X3)]y 2 /dim([R(X1)+R(X2)+R(X3)][R(X2)+R(X3)]) P[R(X1)+R(X2)+R(X3)]⊥ y 2 /(N −dim[R(X1)+R(X2)+R(X3)]) . If (2.26) holds, then we have R(X1)+R(X2)+R(X3) = R(X2)⊕PR(X2)⊥ [R(X1)]⊕PR(X2)⊥ [R(X3)].
- 43. 24 LINEAR MODEL BASICS In this case, [R(X1)+R(X2)+R(X3)][R(X2)+R(X3)] = PR(X2)⊥ [R(X1)] = [R(X1)+R(X2)]R(X2). So the sum of squares in the numerator of the F-test statistic is equal to the quantity θ θ θ T 1 Q1 given in (2.22), with its degrees of freedom equal to rank(C1), where C1 and Q1 are as in (2.18) and (2.19), respectively. A similar conclusion can be drawn for testing the hypothesis that E(y) = X1θ θ θ1 +X2θ θ θ2. 2.8 Additive two-way layout In a two-way layout, the observations are classified according to the levels of two factors. Suppose the two factors have t and b levels, respectively. At each level com- bination (i, j), 1 ≤ i ≤ t, 1 ≤ j ≤ b, there are nij observations yijh, 0 ≤ h ≤ nij, such that yijh = αi +βj +εijh, (2.27) where the εijh’s are uncorrelated random variables with zero mean and constant vari- ance σ2. We require ∑ b j=1 nij 0 for all i, and ∑ t i=1 nij 0 for all j, so that there is at least one observation on each level of the two factors, but some nij’s may be zero. This is called an additive two-way layout model, which is commonly used for analyzing block designs, where each yijh is an observation on the ith treatment in the jth block. With this in mind, we call the two factors treatment and block factors, and denote them by T and B, respectively; then α1,...,αt are the treatment effects and β1,...,βb are the block effects. Let N = ∑ t i=1 ∑ b j=1 nij, and think of the observations as taken on N units that are grouped into b blocks. Define an N ×t matrix XT with 0 and 1 entries such that the (v,i)th entry of XT , 1 ≤ v ≤ N, 1 ≤ i ≤ t, is 1 if and only if the ith treatment is assigned to the vth unit. Similarly, let XB be the N × b matrix with 0 and 1 entries such that the (v, j)th entry of XB, 1 ≤ v ≤ N, 1 ≤ j ≤ b, is 1 if and only if the vth unit is in the jth block. The two matrices XT and XB are called unit-treatment and unit-block incidence matrices, respectively. Then we can write (2.27) as y = XT α α α +XBβ β β +ε ε ε, where α α α = (α1,...,αt)T and β β β = (β1,...,βb)T . We use the results in Sections 2.5 and 2.6 to derive the analysis for such models. Then we apply the results in Section 2.7 to show in Section 2.9 that the analysis can be much simplified when certain conditions are satisfied. Let N be the t × b matrix whose (i, j)th entry is nij, ni+ = ∑ b j=1 nij, and n+j = ∑ t i=1 nij. For block designs, ni+ is the number of observations on the ith treatment, and n+j is the size of the jth block. Also, let yi++ = ∑ b j=1 ∑h yijh and y+j+ = ∑ t i=1 ∑h yijh, the ith treatment total and jth block total, respectively. Then XT T XT is the diagonal matrix with diagonal entries n1+,...,nt+, XT BXB is the diagonal matrix with diagonal entries n+1,...,n+b, XT T y = (y1++,...,yt++)T , XT By = (y+1+,...,y+b+)T , and XT T XB = N. (2.28)
- 44. ADDITIVE TWO-WAY LAYOUT 25 By (2.17), (2.18), and (2.19), the reduced normal equations for α α α and β β β are, respec- tively, CT α α α = QT (2.29) and CB β β β = QB, where CT = ⎡ ⎢ ⎣ n1+ ··· 0 . . . ... . . . 0 ··· nt+ ⎤ ⎥ ⎦−N ⎡ ⎢ ⎢ ⎣ 1 n+1 ··· 0 . . . ... . . . 0 ··· 1 n+b ⎤ ⎥ ⎥ ⎦NT , QT = ⎡ ⎢ ⎣ y1++ . . . yt++ ⎤ ⎥ ⎦−N ⎡ ⎢ ⎢ ⎣ 1 n+1 ··· 0 . . . ... . . . 0 ··· 1 n+b ⎤ ⎥ ⎥ ⎦ ⎡ ⎢ ⎣ y+1+ . . . y+b+ ⎤ ⎥ ⎦, CB = ⎡ ⎢ ⎣ n+1 ··· 0 . . . ... . . . 0 ··· n+b ⎤ ⎥ ⎦−NT ⎡ ⎢ ⎣ 1 n1+ ··· 0 . . . ... . . . 0 ··· 1 nt+ ⎤ ⎥ ⎦N, and QB = ⎡ ⎢ ⎣ y+1+ . . . y+b+ ⎤ ⎥ ⎦−NT ⎡ ⎢ ⎣ 1 n1+ ··· 0 . . . ... . . . 0 ··· 1 nt+ ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ y1++ . . . yt++ ⎤ ⎥ ⎦. It can be verified that both CT and CB have zero column sums. Therefore rank(CT ) ≤ t −1, rank(CB) ≤ b−1, and, by Theorem 2.3, if ∑ t i=1 ciαi is estimable, then ∑ t i=1 ci = 0. All such contrasts are estimable if and only if rank(CT ) = t −1. Sim- ilarly, if ∑ b j=1 djβj is estimable, then ∑ b j=1 dj = 0, and all such contrasts are estimable if and only if rank(CB) = b−1. Theorem 2.4. All the contrasts of α1,...,αt are estimable if and only if for any 1 ≤ i = i ≤ t, there is a sequence i1, j1, i2, j2, ..., ik, jk, ik+1 such that i1 = i, ik+1 = i , and for all 1 ≤ s ≤ k, nis js 0 and nis+1, js 0. Proof. Suppose the condition in the theorem holds. We need to show that all the contrasts of α1,...,αt are estimable. Since the space of all the contrasts is generated by the pairwise differences, it suffices to show that all the pairwise differences αi − αi are estimable.
- 45. 26 LINEAR MODEL BASICS Let yij be any of the observations at the level combination (i, j). Then E(yi1 j1 −yi2 j1 +yi2 j2 −···+yik jk −yik+1 jk ) = (αi1 +βj1 )−(αi2 +βj1 )+(αi2 +βj2 )−···+(αik +βjk )−(αik+1 +βjk ) = αi1 −αik+1 = αi −αi . This shows that αi −αi is estimable. Conversely, if the condition in the theorem does not hold, then the treatments can be partitioned into two disjoint sets such that any treatment from one set never appears in the same block with any treatment from the other set. Then it can be seen that CT is of the form C1 0 0 C2 . Since CT has zero column sums, both C1 and C2 also have zero column sums. It follows that rank(CT ) = rank(C1) + rank(C2) ≤ t −2; therefore not all the contrasts of α1,...,αt are estimable. If any two treatments can be connected by a chain of alternating treatments and blocks as in Theorem 2.4, then any two blocks can also be connected by such a se- quence. Therefore the condition in Theorem 2.4 is also a necessary and sufficient condition for all the contrasts of β1,...,βb to be estimable. In particular, all the con- trasts of α1,...,αt are estimable if and only if all the contrasts of β1,...,βb are estimable. Throughout the rest of this section, we assume that the condition in Theorem 2.4 holds; therefore rank(CT ) = t −1 and rank(CB) = b−1. This is the case, for example, when there is at least one observation at each level combination of the two factors. In view of Theorem 2.4, designs with rank(CT ) = t −1 are called connected designs. Suppose we would like to test the hypothesis that α1 = ··· = αt. Under the null hypothesis, let the common value of the αi’s be α; then E(yijh) = α +βj. This reduces (2.27) to a one-way layout model. Absorb α into βj (see Remark 2.1); then it is the same as to test that E(y) = XBβ β β. Therefore the results in Section 2.6 can be applied. In particular, the sum of squares that appears in the numerator of the F-test statistic is equal to α α α T QT = α α α T CT α α α, with t − 1 degrees of freedom. The residual sum of squares can be computed by subtracting α α α T QT from the residual (within-group) sum of squares under the one-way layout model containing block effects only. Let yi·· = 1 ni+ ∑ b j=1 ∑h yijh, y· j· = 1 n+j ∑ t i=1 ∑h yijh, and y··· = 1 N ∑ t i=1 ∑ b j=1 ∑h yijh be the ith treatment mean, jth block mean, and overall mean, respectively. Then we have the ANOVA in Table 2.2. A test of the hypothesis that E(y) = XT α α α can be based on an ANOVA similar to that in Table 2.2 with the roles of treatments and blocks reversed.
- 46. THE CASE OF PROPORTIONAL FREQUENCIES 27 Table 2.2 ANOVA table for an additive two-way layout source sum of squares d.f. mean square Blocks ∑j n+ j(y· j· −y...)2 b−1 1 b−1 ∑j n+ j(y· j· −y...)2 (ignoring treatments) Treatments αT QT t −1 1 t−1 αT QT (adjusted for blocks) Residual By subtraction N −b−t +1 1 N−b−t+1 (Residual SS) Total ∑i ∑j ∑h(yijh −y...)2 N −1 2.9 The case of proportional frequencies As explained in Remark 2.1, model (2.27) can be written as yijh = μ +αi +βj +εijh (2.30) without changing the least squares estimators of estimable functions of (α1,...,αt)T , least squares estimators of estimable functions of (β1,...,βb)T , and the reduced normal equations. Write (2.30) in the form of (2.25) with θ θ θ1 = (α1,...,αt)T ,θ θ θ3 = (β1,...,βb)T , and θ θ θ2 = μ. Then X1 = XT , X3 = XB, and X2 = 1N. In this case, the adjusted orthogonality condition (2.26) is that PR(1N)⊥ [R(XT )] is orthogonal to PR(1N)⊥ [R(XB)]. (2.31) Since only one treatment can be assigned to each unit, each row of XT has ex- actly one entry equal to 1, and all the other entries are zero. It follows that the sum of all the columns of XT is equal to 1N. Therefore R(1N) ⊆ R(XT ), and hence PR(1N)⊥ [R(XT )] = R(XT )R(1N). Likewise, PR(1N)⊥ [R(XB)] = R(XB)R(1N). We say that T and B satisfy the condition of proportional frequencies if nij = ni+n+j N for all i, j; (2.32) here nij/ni+ does not depend on i, and nij/n+j does not depend on j. In particular, if nij is a constant for all i and j, then (2.32) holds; in the context of block designs, this means that all the treatments appear the same number of times in each block. Theorem 2.5. Factors T and B satisfy the condition of proportional frequencies if and only if R(XT )R(1N) is orthogonal to R(XB)R(1N). Proof. We first note that (2.31) is equivalent to PR(1N)⊥ XT T PR(1N)⊥ XB = 0. (2.33)
- 47. 28 LINEAR MODEL BASICS Since PR(1N) = 1N(1T N1N)−11T N = 1 N JN, where JN is the N ×N matrix of 1’s, we have PR(1N)⊥ = I− 1 N JN. Then (2.33) is the same as XT T I− 1 N JN XB = 0 or XT T XB = 1 N XT T JNXB. (2.34) By (2.28), the left-hand side of (2.34) is N. The (i,j)th entry of the right-hand side is ni+n+j/N. Therefore (2.33) holds if and only if the two factors satisfy the condition of proportional frequencies. By the results in Section 2.7 and Theorem 2.5, if the treatment and block factors satisfy the condition of proportional frequencies, then the least squares estimator of any contrast ∑ t i=1 ciαi is the same as that under the one-way layout yijh = μ +αi +εijh. So ∑ t i=1 ci αi = ∑ t i=1 ciyi··, with variance σ2(∑ t i=1 c2 i /ni+). Likewise, the least squares estimator of any contrast ∑ b j=1 djβj is ∑ b j=1 djy· j·, with variance σ2(∑ b j=1 d2 j /n+j). In this case, we have the decomposition RN = R(1N)⊕[R(XT )R(1N)]⊕[R(XB)R(1N)]⊕[R(XT )+R(XB)]⊥ . Thus y−PR(1N)y = PR(XT )R(1N)y+PR(XB)R(1N)y+P[R(XT )+R(XB)]⊥ y. Componentwise, we have yijh −y... = (yi·· −y...)+(y· j· −y...)+(yijh −yi·· −y· j· +y...). The identity y−PR(1N)y 2 = PR(XT )R(1N)y 2 + PR(XB)R(1N)y 2 + P[R(XT )+R(XB)]⊥ y 2 gives ∑ i ∑ j ∑ h (yijh −y...)2 = ∑ i ni+(yi·· −y...)2 +∑ j n+j(y· j· −y...)2 +∑ i ∑ j ∑ h (yijh −yi·· −y· j· +y...)2 . This leads to a single ANOVA table.
- 48. THE CASE OF PROPORTIONAL FREQUENCIES 29 source sum of squares d.f. mean square Blocks ∑j n+ j(y· j· −y...)2 b−1 1 b−1 ∑j n+ j(y· j· −y...)2 Treatments ∑i ni+(yi·· −y...)2 t −1 1 t−1 ∑i ni+(yi·· −y...)2 Residual By subtraction N −b−t +1 1 N−b−t+1 (Residual SS) Total ∑i ∑j ∑h(yijh −y...)2 N −1 Exercises 2.1 Prove (2.22) and (2.23). 2.2 Suppose θ θ θ1 and θ θ θ2 are the least squares estimators of θ θ θ1 and θ θ θ2 under the model y = X1θ θ θ1 + X2θ θ θ2 +ε ε ε, and θ θ θ∗ 1 is the least squares estimator of θ θ θ1 under y = X1θ θ θ1 +ε ε ε. Show that X1θ θ θ∗ 1, the prediction of y based on the model y = X1θ θ θ1 +ε ε ε, is equal to X1 θ θ θ1 +X1 XT 1 X1 − XT 1 X2 θ θ θ2. Comment on this result. 2.3 Show that the two matrices CT and CB defined in Section 2.8 have zero col- umn sums. Therefore, if ∑ t i=1 ciαi is estimable, then ∑ t i=1 ci = 0, and if ∑ b j=1 djβj is estimable, then ∑ b j=1 dj = 0. 2.4 (Optimality of complete block designs) In the setting of Section 2.8, suppose n+j = k for all j = 1,...,b. Such a two-way layout can be considered as a block design with b blocks of constant size k. Suppose rank(CT ) = t −1. (a) Let CT = ∑ t−1 i=1 μiξ ξ ξiξ ξ ξT i , where μ1,...,μt−1 are the nonzero eigenvalues of CT , and ξ ξ ξ1,...,ξ ξ ξt−1 are the associated orthonormal eigenvectors. Show that ∑ t−1 i=1 μ−1 i ξ ξ ξiξ ξ ξT i is a generalized inverse of CT . (b) Use the generalized inverse given in (a) to show that ∑ 1≤ii≤t var αi − αi = σ2 t t−1 ∑ i=1 μ−1 i . (c) Consider the case k = t. In this case, a design with nij = 1 for all i, j is called a complete block design. Show that for a complete block design, μ1 = ··· = μt−1. (d) Show that a complete block design maximizes tr(CT ) = ∑ t−1 i=1 μi among all the block designs with n+j = k (= t) for all 1 ≤ j ≤ b. (e) Use (b), (c), (d), and the fact that f(x) = x−1 is a convex function to show that a complete block design minimizes ∑1≤ii≤tvar( αi − αi ) among all the block designs with n+j = k (= t) for all 1 ≤ j ≤ b. 2.5 (Balanced incomplete block designs and their optimality) Continuing Exercise 2.4, suppose k t.
- 49. 30 LINEAR MODEL BASICS (a) Show that if nij = 0 or 1 for all i, j, then the ith diagonal entry of CT is equal to (k −1)qi/k, and the (i,i )th off-diagonal entry is equal to −λii /k, where qi is the number of times the ith treatment appears, and λii is the number of blocks in which both the ith and i th treatments appear. (b) In addition to nij = 0 or 1 for all i, j, suppose qi = q for all i, and λii = λ for all 1 ≤ i = i ≤ t. Such designs are called balanced incomplete block designs; see Section A.8. Use Exercise 2.3 to show that (t −1)λ = (k−1)q. (c) Show that under a balanced incomplete block design, k tλ It is a generalized inverse of CT . Use this to provide an expression for the least squares es- timator of a pairwise comparison αi − αi of treatment effects, and show that var( αi − αi ) = 2k tλ σ2 for all 1 ≤ i = i ≤ t. (d) Show that the properties in (c) and (d) of Exercise 2.4 also hold for balanced incomplete block designs, and hence for given b, t, and k, if there exists a balanced incomplete block design, then it minimizes ∑1≤ii≤tvar( αi − αi ) among all the block designs with n+j = k for all 1 ≤ j ≤ b. [Kiefer (1958, 1975)] 2.6 (Löwner ordering of matrices and comparison of experiments) Given two sym- metric matrices A1 and A2, we say that A1 ≥ A2 if A1 − A2 is nonnegative definite. Consider two linear models model 1: E(y) = X1θ θ θ, cov(y) = σ2IN, model 2: E(y) = X2θ θ θ, cov(y) = σ2IN. Show that XT 1 X1≥ XT 2 X2 if and only if (a) all linear functions cTθ θ θ of θ θ θ that are estimable under model 2 are also estimable under model 1, and (b) for any cTθ θ θ that is estimable under model 2, the variance of its least squares estimator under model 2 is at least as large as that under model 1. [Ehrenfeld (1956)] 2.7 Consider the linear models model 1: E(y) = X1θ θ θ1 +X2θ θ θ2, cov(y) = σ2IN, model 2: E(y) = X1θ θ θ1 +X2θ θ θ2 +X3θ θ θ3, cov(y) = σ2IN. Let C1 and C∗ 1 be the information matrices for θ θ θ1 under models 1 and 2, re- spectively. Show that C1 ≥ C∗ 1, and C1 = C∗ 1 if (2.26) holds. 2.8 Consider the linear model E(y) = Xθ, cov(y) = σ2 IN, where X is N × p and θ θ θ is p×1. Suppose x ≤ a for all columns x of X, where x 2 = xT x. Show that for any unknown parameter θi, var( θi) ≥ 1 a2 σ2, and that the equality is attained if XT X = a2Ip. In particular, if |xij| ≤ 1 for all the entries xij of X, then var( θi) ≥ 1 N σ2, and the equality is attained if XT X = NIp. Such a matrix must have all its entries equal to 1 or −1. (An N ×N matrix X with 1 and −1 entries such that XT X = NIN is called a Hadamard matrix.)
- 50. Chapter 3 Randomization and Blocking We begin with a discussion of randomization and blocking, and present statistical models for completely randomized designs, randomized block designs, row-column designs, nested row-column designs, and blocked split-plot designs. Based on an as- sumption of additivity between treatments and experimental units, these models can be justified by appropriate randomizations that preserve the block structures (Grundy and Healy, 1950; Nelder, 1965a; Bailey, 1981, 1991). The same approach can be ap- plied to experiments with more general block structures, including all the simple block structures, to be discussed in Chapter 12. 3.1 Randomization Throughout this book, we denote the set of experimental units by Ω and the set of treatments by T. We also denote the number of treatments and the number of experimental units by t and N, respectively. The units can be labeled by integers 1,...,N, and the treatments are labeled by 1,...,t. For any finite set A, we denote the number of elements in A by |A|. Under complete randomization, the allocation of treatments to the units is com- pletely random. Suppose the lth treatment is to be assigned to rl units, where l = 1,...,t, with ∑ t l=1 rl = N. One can choose r1 of the N units randomly and assign to them the first treatment, then choose r2 of the remaining N − r1 units randomly and assign to them the second treatment, etc. A mathematically equivalent method that can be generalized conveniently to more complicated block structures is to start with an arbitrary initial design in which the lth treatment is assigned to rl unit labels, l = 1,...,t. One permutation of the N unit labels is then drawn randomly from the N! possible permutations. The chosen random permutation is used to determine which rl units will actually receive the lth treatment. In general, given an initial assignment of the treatment labels to unit labels, a random permutation of the unit labels is applied to obtain a randomized experimen- tal plan for use in actual experimentation. The experimental units are considered to be unstructured under complete randomization. When they have a certain structure, we restrict to permutations of unit labels that preserve the structure. Such permuta- tions are called allowable permutations. Randomization is carried out by drawing a permutation randomly from the set of allowable permutations only. Each initial design can be described by an N ×t unit-treatment incidence matrix 31
- 51. 32 RANDOMIZATION AND BLOCKING XT whose rows correspond to unit labels and columns correspond to treatment la- bels, with the (w,l)th entry equal to 1 if the lth treatment label is assigned to the wth unit label, and 0 otherwise. For example, under the completely randomized design described above, XT is an arbitrary N ×t matrix with 0 and 1 entries such that there is one 1 in each row and rl 1’s in the lth column. 3.2 Assumption of additivity and models for completely randomized designs We make the assumption of additivity between treatments and units: an observation on unit w when treatment l is applied there, 1 ≤ w ≤ N, 1≤ l ≤ t, is assumed to be αl +δw, (3.1) where αl is an unknown constant representing the effect of the lth treatment, and δw is the effect of the wth unit. The treatment effects α1,...,αt are parameters of inter- est. We allow δw to be a random variable, incorporating measurement errors, uncon- trolled variations, etc. It is also assumed that δw has a finite variance and cov(δw,δw ) only depends on the units. Let φ(w),1 ≤ φ(w) ≤ t, be the treatment applied to the wth unit, and yw be the observation on that unit. Then yw can be expressed as yw = αφ(w) +δw,w = 1,...,N. (3.2) Some simplifying assumptions on the joint distribution of the δw’s are needed to carry out statistical inference for the treatment effects. A common assumption for completely randomized designs is that δ1,...,δN are uncorrelated and identically distributed. Under this assumption, with μ = E(δw),σ2 = var(δw), and εw = δw − μ, we have yw = μ +αφ(w) +εw,w = 1,...,N, (3.3) where E(εw) = 0 and var(εw) = σ2 , cov(εw,εw ) = 0 for all w = w . (3.4) This is the usual one-way layout model discussed in Section 2.4. A more general model replaces the assumption that the εw’s are uncorrelated with cov(εw,εw ) = ρσ2 for all w = w . In matrix notation, we have y = μ1N +XT α α α +ε ε ε, (3.5) and E(ε ε ε) = 0,V = cov(ε ε ε) = σ2 (1−ρ)IN +ρσ2 JN. (3.6) In the future, we will omit the subscripts in IN and JN when the dimension is obvious from the context. Note that model (3.3)–(3.4) is a special case with ρ = 0. Model (3.5)–(3.6) can be justified by complete randomization; see Section 3.6.
- 52. RANDOMIZED BLOCK DESIGNS 33 3.3 Randomized block designs Suppose the experimental units are grouped into b blocks of size k. Given an ini- tial assignment of the treatments to the bk unit labels, randomization is carried out by randomly permuting the unit labels within each block (done independently from block to block), and also randomly permuting the block labels. This is equivalent to drawing a permutation randomly, not from all the N! permutations of the N = bk unit labels as in the case of complete randomization, but from the (b!)(k!)b allowable permutations that preserve the structure of the units: two unit labels are in the same block if and only if they remain in the same block after permutation. A commonly used linear model for block designs assumes that the δw’s in (3.2) are uncorrelated random variables with a constant variance and that E(δw) only de- pends on the block to which the unit belongs. Let yij, i = 1, ..., b, j = 1, ..., k, be the observation on the jth unit in the ith block. Then such a model can be expressed as yij = μ +αφ(i,j) +βi +εij, (3.7) where φ(i, j) is the treatment applied to the jth unit in the ith block, β1,··· ,βb are unknown constants, and {εij} are mutually uncorrelated random variables with E(εij) = 0 and var(εij) = σ2 . (3.8) This is the usual fixed-effect additive two-way layout model discussed in Section 2.8. Another model assumes that E(δw) = 0 and var(δw) = σ2, but cov(δw,δw ) = ρ1σ2 for any two units w and w in the same block, and cov(δw,δw ) = ρ2σ2 if units w and w are in different blocks. Such a model can be written as yij = μ +αφ(i,j) +δij, (3.9) where E(δij) = 0, cov(δij,δi j ) = ⎧ ⎪ ⎨ ⎪ ⎩ σ2, if i = i , j = j ; ρ1σ2, if i = i , j = j ; ρ2σ2, if i = i . (3.10) Typically the units in the same block are more alike than those in different blocks; then one would have ρ1 ρ2. (3.11) The patterns-of-covariance form of (3.10) arises, e.g., when {βi} and {εij} in (3.7) are mutually uncorrelated random variables with E(βi) = E(εij) = 0, and var(βi) = σ2 B, var(εij) = σ2 E . (3.12) Let δij = βi +εij. Then (3.9) and (3.10) hold with σ2 = σ2 B +σ2 E , ρ1 = σ2 B σ2 B +σ2 E , and ρ2 = 0. (3.13) In this case, we do have (3.11). We show in Section 3.6 that model (3.9)–(3.10) can be justified by randomization.
- 53. 34 RANDOMIZATION AND BLOCKING Remark 3.1. Both the model induced by randomization and model (3.7) with ran- dom effects satisfying (3.12) can be expressed in the same mathematical form as in (3.9) and (3.10). However, they are philosophically different. For example, under the latter, we have ρ1 ρ2. Such a constraint, however, may not hold for a general randomization model. Remark 3.2. The between-block randomization is possible only if all the blocks are of the same size. For unequal block sizes, (3.10) cannot be justified by randomization and has to be assumed. 3.4 Randomized row-column designs Suppose the experimental units are arranged in r rows and c columns. Given an initial assignment of the treatment labels to the rc unit labels, randomization can be carried out by randomly permuting the row labels and, independently, randomly permuting the column labels. This is equivalent to drawing a permutation of the unit labels randomly from the r!c! allowable permutations: two unit labels are in the same row (column) if and only if they remain in the same row (column, respectively) after permutation. As in the case of block designs, we present two commonly used models for row- column designs, one with fixed and the other with random row and column effects. Let yij, i = 1, ..., r, j = 1, ..., c, be the observation on the unit at the intersection of the ith row and jth column. Then a model with fixed row and column effects assumes that yij = μ +αφ(i,j) +βi +γj +εij, (3.14) where φ(i, j) is the treatment assigned to the unit at the intersection of the ith row and the jth column, βi and γj are unknown constants, and {εij} are mutually uncorrelated random variables with E(εij) = 0 and var(εij) = σ2 . (3.15) Another model assumes that yij = μ +αφ(i,j) +δij, (3.16) where E(δij) = 0, cov(δij,δi j ) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ σ2, if i = i , j = j ; ρ1σ2, if i = i , j = j ; ρ2σ2, if i = i , j = j ; ρ3σ2, if i = i , j = j . (3.17) That is, all the observations have the same variance, and there are three correlations ρ1, ρ2, and ρ3 depending on whether the two observations involved are in the same row and different columns, same column and different rows, or different rows and different columns. Typically, we expect to have ρ1 ρ3 and ρ2 ρ3. (3.18)
- 54. NESTED ROW-COLUMN AND BLOCKED SPLIT-PLOT DESIGNS 35 For example, if {βi}, {γj}, and {εij} in (3.14) are mutually uncorrelated random variables with E(βi) = E(γj) = E(εij) = 0, var(βi) = σ2 R, var(γj) = σ2 C, var(εij) = σ2 E , and δij = βi +γj +εij, then (3.16) and (3.17) hold with σ2 = σ2 R +σ2 C +σ2 E , ρ1 = σ2 R σ2 R +σ2 C +σ2 E , ρ2 = σ2 C σ2 R +σ2 C +σ2 E , and ρ3 = 0. (3.19) In this case, we do have (3.18). We show in Section 3.6 that (3.16)–(3.17) can be justified by randomization. 3.5 Nested row-column designs and blocked split-plot designs Consider a randomized experiment with the block structure n1/(n2 ×n3). Let yijk be the observation on the unit at the intersection of the jth row and kth column in the ith block. We assume the following model for such a randomized experiment. yijk = μ +αφ(i,j,k) +εijk,1 ≤ i ≤ n1,1 ≤ j ≤ n2,1 ≤ k ≤ n3, (3.20) where φ(i, j,k) is the treatment applied to unit (i, j,k), and E(εijk) = 0, cov(εijk,εi jk ) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ σ2, if i = i , j = j , k = k ; ρ1σ2, if i = i , j = j , k = k ; ρ2σ2, if i = i , j = j , k = k ; ρ3σ2, if i = i , j = j , k = k ; ρ4σ2, if i = i . (3.21) Thus there are four correlations depending on whether the two observations involved are in the same row and different columns of the same block, the same column and different rows of the same block, different rows and different columns of the same block, or different blocks. These reflect four different relations between any pair of experimental units. On the other hand, in an experiment with the block structure (n1/n2)/n3, let yijk be the observation on the kth subplot of the jth whole-plot in the ith block. Then a commonly used model assumes that yijk = μ +αφ(i,j,k) +εijk,1 ≤ i ≤ n1,1 ≤ j ≤ n2,1 ≤ k ≤ n3, (3.22) where φ(i, j,k) is the treatment applied to unit (i, j,k), and E(εijk) = 0, cov(εijk, εi jk ) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ σ2, if i = i , j = j , k = k ; ρ1σ2, if i = i , j = j , k = k ; ρ2σ2, if i = i , j = j ; ρ3σ2, if i = i . (3.23)
- 55. Another Random Document on Scribd Without Any Related Topics
- 56. — Mihin minä johdan? — kysyi Lajevski. — Jos teille sanoo esimerkiksi: kuinka viinirypäleterttu on kaunis! niin te tokaisette: mutta kuinka ruma se on, kun se pureksitaan ja sulatetaan vatsassa. Mitä toimittaa sellainen puhuminen? Ei se ole mitään uutta … yleensä vain kummallinen tapa. Lajevski tiesi, ettei hän ollut von Corenin suosiossa, ja sentähden pelkäsi häntä ja tunsi hänen ollessaan läsnä ikäänkuin kaikkien olisi ollut ahdasta ja ikäänkuin selän takana olisi seisonut joku. Hän ei vastannut mitään, vaan astui syrjään ja katui, että oli lähtenyt mukaan. — Herrat, mars hakemaan risuja valkeaan! — komensi Samoilenko. Kaikki hajaantuivat, kuka minnekin, ja paikalle jäivät vain Kirilin, Atshmianov ja Nikodim Aleksandritsh. Kerbalai toi tuoleja, levitti nurmelle maton ja laski sille muutamia viinipulloja. Kirilin, pitkäkasvuinen, muhkea mies, joka ilmoista huolimatta käytti palttinamekkonsa päällä upseeriviittaa, muistutti ylpeällä ryhdillään, arvokkaalla astunnallaan sekä matalalla, hieman käheällä äänellään nuorenpuoleisia pikkukaupungin poliisimestareita. Hänen kasvonsa näyttivät murheellisilta ja unisilta, ikäänkuin hänet vastikään olisi vasten tahtoaan herätetty. — Mitä sinä toit sieltä, aasi? — kysyi hän Kerbalailta, lausuen verkkaan joka sanan. — Käskin sinun tuoda kvareli-viiniä, mutta mitä sinä toit, tataarilaiskuono? Häh? — Meillä on paljon omaa viiniä, Jegor Alekseitsh, — huomautti arasti ja kohteliaasti Nikodim Aleksandritsh.
- 57. — Kuinka? Mutta minä tahoon, että minunkin viiniäni pitää olla. Otan osaa kekkereihin ja otaksun siis, että minullakin on oikeus maksaa osaltani. Niin otaksun! Tuo kymmenen pulloa kvarelia! — Miksi niin paljon? — ihmetteli Nikodim Aleksandritsh tietäen, ettei Kirinillä ollut rahaa. — Kaksikymmentä pulloa! Kolmekymmentä! — huusi Kirilin. — Antaa olla, tuokoon, — kuiskasi Atshmianov Nikorim Aleksandritshille, — minä maksan. Nadeshda Feodorovna oli iloisella, vallattomalla päällä. Häntä halutti hyppiä, nauraa, huutaa, ärsyttää, keikailla. Yllään huokeahintainen, sinisilmäisestä pumpulikankaasta tehty puku, jalassa punaiset tohvelit ja ennenmainittu olkihattu päässä hän oli mielestänsä pieni, yksinkertainen, kevyt ja leijaileva kuin perhonen. Hän juoksi notkuvalle sillalle ja katseli kotvasen veteen, jotta olisi alkanut päätä huimata, sitten hän huudahti ja juoksi nauraen toiselle rannalle kuivuuvajan luo, ja hänestä näytti, että kaikki mieshenkilöt, yksinpä Kerbalaikin, ihaillen katselivat häntä. Kun äkkiä yllättäneessä hämärässä puut ja vuoret, hevoset ja ajoneuvot sulivat yhteen ja tataarilaismökin ikkunoista pilkahti esille tulta, kiipesi hän kivien ja okaisten pensasten välitse polveilevaa polkua pitkin vuorelle ja istahti kivelle. Alhaalla paloi jo nuotio. Sen ääressä liikkui, hihat ylös käärittyinä, diakoni, ja hänen pitkä, musta varjonsa kierteli säteenä valkean ympärillä, kun hän heitteli nuotioon risuja ja hämmenteli pataa pitkään keppiin sidotulla lusikalla. Samoilenko, kasvot kuparinpunaisina, hääräili siellä myös kuin omassa keittiössään ja huusi tuikeasti:
- 58. — Missä suola on, hyvät herrat? Unohtui tietenkin? Ja mitä te kaikki siinä istua nökötätte kuin mitkäkin paroonit, ja minä saan tässä yksin puuhata? Tuulenkaatopuulla istuivat rinnakkain Lajevski ja Nikodim Aleksandritsh ja katselivat miettiväisinä. Maria Konstantinovna, Katja ja Kostja ottivat esille korista teevehkeet ja lautaset. Von Coren käsivarret ristissä ja toinen jalka kivellä seisoi rannalla lähellä vedenrajaa ja mietti jotakin. Nuotion luomat punaiset läikät yhdessä varjojen kanssa kiertelivät maanpinnalla tummien ihmishahmojen vaiheilla, värähtelivät vuorilla, puissa, sillalla; toisella puolella jyrkkä, veden uurtama rantaäyräs oli kokonaan valaistu, näytti iskevän silmää ja kuvastui veteen, ja vuolaasti virtaava, myllertävä vesi repi kappaleiksi heijastuksen. Diakoni lähti noutamaan kaloja, joita Kerbalai rannalla puhdisti ja pesi, mutta puolitiessä hän pysähtyi ja katseli ympärilleen. Voi sentään, kuinka ihanaa! ajatteli hän. Ihmiset, kivet, tuli, hämärä, maassa lojuva puu — ei mitään sen enempää, mutta kuinka ihanaa silti! Toisella rannalla ilmestyi kuivuuvajan luo joitakin tuntemattomia ihmisiä. Kun valo häilähteli ja savu painautui sille puolelle, ei näitä ihmisiä voinut heti täysin eroittaa, vaan siellä näkyi milloin karvalakki ja harmaa parta, milloin sininen paita, milloin ryysyt olkapäästä polviin ja tikari poikkipuolin vatsalla, milloin taas nuoret, tummaveriset kasvot mustine kulmakarvoineen, jotka olivat niin tuuheat ja silmäänpistävät kuin hiilellä piirustetut. Viisi heistä istahti piiriin nurmelle, toiset viisi menivät kuivuuvajaan. Yksi pysähtyi ovessa selin tuleen ja pannen kädet selän taakse alkoi kertoa jotakin hyvin mielenkiintoista, sillä Samoilenkon heitettyä risuja tuleen, joka
- 59. siitä leimahtaen räiskäytti säkeniä ja kirkkaasti valaisi kuivuuvajan, näkyi ovesta tuijottamassa kaksi rauhallista naamaa, ilmaisten innostunutta tarkkaavaisuutta, ja piiriin istuutuneet kääntyivät taaksepäin kuuntelemaan kertomusta. Vähän myöhemmin virittivät piirissä istujat hiljaisin äänin soinnukkaan pitkäveteisen laulun, joka muistutti suuren paaston aikana kirkossa veisattavaa virttä… Kuunnellessaan heitä diakoni ajatteli, mikä hän on miehiään kymmenen vuoden kuluttua, kun hän palaa tutkimusretkeltä: nuori lähetyspappi, kirjailija, jolla on kuuluisa nimi ja kunniakas menneisyys; hänestä tehdään arkkimandriitti, sitten piispa; hän toimittaa tuomiokirkossa jumalanpalvelusta: päässä kultahiippa hän astuu pyhimyskuvaa kantaen alttarin eteen lavalle, korottaa kolmihaaraisen ja kaksihaaraisen kynttilänjalan, valaisee niillä suurta kansanpaljoutta ja julistaa: Holho taivaasta, Herra, tätä viinipensasta, katso lempeästi sen puoleen ja lähesty sitä, ja oikea kätesi istuttakoon sen! Ja enkelimäisin äänin lapset veisaavat vastaan: Pyhä jumala… — Diakoni, missä kalat ovat? — kuului Samoilenkon ääni. Palattuaan nuotion luo diakoni kuvitteli, että kuumana heinäkuun päivänä astuu pölyistä tietä kirkollinen juhlakulkue: edellä talonpojat kantavat kirkkolippuja, eukot ja tytöt pyhäinkuvia, heti seuraavat kuoripojat ja lukkari poski sidottuna ja oljenkorsia hiuksissa, sitten järjestyksessä hän, diakoni, hänen jäljestään pappi kalotti päässä ristiä kantaen, ja takana pöllyttää joukko ukkoja, akkoja, poikia; joukossa ovat papin ja diakonin vaimot huivit päässä. Kuoripojat laulavat, lapset kirkuvat, peltopyyt piipottavat, leivot visertävät… Kulkue pysähtyy, ja siunattua vettä vihmotaan karjan päälle… Kuljetaan edelleen ja polvistumalla rukoillaan sadetta. Sitten syödään, jutellaan…
- 60. Niin, hyvää on sekin… ajatteli diakoni.
- 61. VII. Kirilin ja Atshmianov kiipesivät vuorelle polkua myöten, Atshmianov jättäytyi jälkeen ja pysähtyi, mutta Kirilin astui Nadeshda Feodorovnan luo. — Hyvää iltaa! — virkkoi hän tervehtien sotilaan tapaan. — Hyvää iltaa. — Niinpä niin! — lausui Kirilin, katsoen taivasta kohden ja miettien. — Mitä niinpä niin? — kysyi Nadeshda Feodorovna, oltuaan hetken vaiti ja huomattuaan, että Atshmianov piti heitä kumpaakin silmällä. — Niinpä siis, — puheli upseeri verkkaan, — rakkautenne kuihtuu ennenkuin on ehtinyt kukkaan puhjeta, niin sanoakseni. Miten on tämä ymmärrettävä? Oliko se teidän puoleltanne keimailua tavallaan, vai pidättekö minua tyhjäntoimittajana, jota kohtaan voi menetellä mielensä mukaan? — Se oli hairahdus! Jättäkää minut! — lausui Nadeshda Feodorovna jyrkästi, katsoen häneen pelokkaasti tänä ihanana,
- 62. hurmaavana iltana ja kysyen itseltään hämillään: onko todella ollut hetki, jolloin tuo mies minua miellytti ja pääsi niin läheiseksi? — Niin vainen! — virkkoi Kirilin; hän seisoi kotvan mätään puhumatta, mietti ja jatkoi: — Mitähän tuosta? Odottakaamme, kunnes tulette paremmalle päälle, mutta toistaiseksi rohkenen vakuuttaa teille, että olen kunnon ihminen enkä salli kenenkään sitä epäillä. Minun kanssani ei leikitellä! Hyvästi! Hän tervehti sotilaallisesti ja astui syrjään, raivaten tietä pensasten läpi. Hetken kuluttua lähestyi arastellen Atshmianov. — Tänään on kaunis ilta! — virkkoi hän hieman murtaen armenialaiseen tapaan. Hän oli koko pulska ulkomuodoltaan, kävi muodinmukaisesti puettuna, käyttäytyi luontevasti, kuten hyvinkasvatettu nuorukainen, mutta Nadeshda Feodorovna ei pitänyt hänestä sentähden, että hän oli velkaa tämän isälle kolmesataa ruplaa; myöskään ei hänestä ollut mieleen, että kekkereihin oli osalliseksi päästetty puotilainen, ja hänestä oli kiusallista, että Atshmianov lähestyi häntä juuri tänä iltana, jolloin hänen sielussaan oli niin puhdasta. — Kekkerit ovat onnistuneet ylimalkaan hyvin, — sanoi Atshmianov hetkisen vaitiolon jälkeen. — Niin, — myönsi Nadeshda Feodorovna, ja ikäänkuin juuri tällä hetkellä olisi muistunut hänelle mieleen, virkkoi hän yliolkaisesti: — Niin tuota, sanokaapa siellä kaupassanne, että Ivan Andreitsh käy näinä päivinä suorittamassa ne kolmesataa ruplaa … tai minkäverran sitä lienee.
- 63. — Olen valmis antamaan vielä kolmesataa, kunhan ette joka päivä muistuta tuosta velasta. Jättäkäämme tuo proosa! Nadeshda Feodorovna naurahti; hänelle tuli mieleen naurettava ajatus, että jos hän olisi ollut riittämättömän siveellinen ja tahtonut, olisi hän hetkessä voinut vapautua velastaan. Jospa esimerkiksi tältä kauniilta, nuorelta houkalta panisi pään pyörälle! Kuinka se itse asiassa olikin naurettavaa, tyhmää ja hurjaa! Ja hän tunsi äkkiä halua hurmata, riistää puhtaaksi, hyljätä ja sitten katsoa, mitä tuosta tulisi. — Sallikaa minun antaa teille yksi neuvo, — virkkoi Atshmianov arasti. — Varokaa Kiriliniä. Hän kertoo kaikkialla teistä mitä kamalimpia juttuja. — Minua ei huvita tietää, mitä mikin aasinpää minusta kertoo, — vastasi Nadeshda Feodorovna kylmästi, ja hänet valtasi äkkiä levottomuus, ja naurettava ajatus pitää tuota nuorta, kaunista Atshmianovia lelunaan oli kadottanut viehätyksensä. — Täytyy jo mennä alas, — sanoi hän. — Minua kutsutaan. Alhaalla oli kalakeitto jo valmiina. Sitä kaadettiin lautasille ja syötiin niin hartaasti kuin vain ulkoilmakekkereissä on tavallista; ja kaikista maistui kalakeitto erittäin hyvältä, ja kaikki olivat sitä mieltä, etteivät he olleet kotona milloinkaan syöneet mitään niin herkullista. Kuten tällaisissa kekkereissä on tavallista, sai ruokaliinojen, myttyjen, tarpeettomien, tuulessa liehuvien rasvaisten paperien paljous sellaista sekasotkua aikaan, etteivät osanottajat tienneet missä kenenkin lasi ja leipäpala olivat; he kaatoivat viiniä matolle tai polvilleen, pudottivat maahan suolaa, ja ympärillä oli pimeä eikä nuotiokaan enää loimunnut niin kirkkaasti eikä kukaan viitsinyt
- 64. nousta heittämään risuja tuleen. Kaikki joivat viiniä ja Kostjalle ja Katjallekin annettiin puoli juomalasillista. Nadeshda Feodorovna joi koko lasillisen, sitten toisen, päihtyi ja unohti koko Kirilinin. — Erinomaiset kekkerit, hurmaava ilta, — sanoi Lajevski riemastuen viinin vaikutuksesta, — mutta sittenkin pitäisin hyvää talvea kaikkea tätä parempana. Majavainen kaulus hopeaisna pakkasessa loistaa. — Kullakin oma makunsa, — huomautti von Coren. Lajevskin tuli paha olla: selkää paahtoi nuotiosta hohtava kuumuus, rintaa ja kasvoja von Corenin viha; ja tämä viha, lähtien viisaasta ja säällisestä miehestä, jolla luultavasti oli perusteellinen syykin vihata, alensi ja lannisti häntä, ja kykenemättä sitä vastustamaan hän lausui mielistelevällä äänellä: — Minä rakastan intohimoisesti luontoa ja säälin, etten ole luonnontutkija. Minä kadehdin teitä. — Minäpä en sääli enkä kadehdi, — virkkoi Nadeshda Feodorovna. — En ymmärrä, kuinka voi tosissaan työskennellä turilasten ja koppakuoriaisten tutkimisesssa sillä aikaa kun kansa kärsii. Lajevski kannatti hänen mielipidettään. Hän ei ollut yhtään perillä luonnontieteistä eikä sentähden koskaan ollut voinut kärsiä muurahaisten tuntosarvia ja torakoiden koipia tutkivien miesten mahtipontista äänensävyä ja viisasta, syvämielistä ulkomuotoa, ja häntä oli aina suututtanut se, että nämä ihmiset, tuntosarvien, koipien ja jonkin alkuliman (hän oli ties miksi kuvitellut tätä alkulimaa osterin muotoiseksi) nojalla ottavat ratkaistaksensa ihmisen alkuperää ja elämää koskevia kysymyksiä. Mutta Nadeshda
- 65. Feodorovnan sanoissa oli kuulostavinaan valhetta, ja vain inttääkseen vastaan hän lausui: — Tähdellistä eivät ole koppakuoriaiset, vaan johtopäätökset!
- 66. VIII. Myöhään, kellon jo käydessä kahtatoista, alettiin laittautua ajoneuvoihin kotimatkaa varten. Kaikki istuivat jo paikoillaan, eikä puuttunut muita kuin Nadeshda Feodorovna ja Atshmianov, jotka joen tuolla puolen juoksivat tippasilla ja nauroivat. — Arvoisa herrasväki, joutukaa! — huusi heille Samoilenko. — Ei pitäisi naisille antaa viiniä, — lausui von Coren hiljakseen. Kekkereiden, von Corenin vihan ja omien ajatusten väsyttämänä Lajevski lähti Nadeshda Feodorovnaa vastaan, ja kun tämä iloisena, tuntien itsensä keveäksi kuin höyhen, hengästyneenä ja nauraa hohottaen otti häntä molemmista käsistä ja laski päänsä hänen rintaansa vasten, peräytyi hän askelen ja virkkoi tylysti: — Sinä käyttäydyt ihan kuin … yleinen nainen. Sattui tulemaan kovin raa'asti sanotuksi, ja hänen tuli sääli Nadeshda Feodorovnaa. Hänen vihaisista, väsyneistä kasvoistaan Nadeshda Feodorovna voi lukea sekä vihaa että sääliä ja suuttumusta, ja äkkiä hänen mielensä masentui. Hän käsitti, että oli mennyt liiallisuuteen, käyttäytynyt liian vapaasti ja käyden surulliseksi, tuntien itsensä raskaaksi, paksuksi, töykeäksi ja pöhnäiseksi istahti ensinnä eteen sattuneihin
- 67. ajoneuvoihin yhdessä Atshmianovin kanssa. Lajevski istuutui samoihin ajoneuvoihin Kirilinin, eläintieteilijä Samoilenkon ja diakoni naisten seuraan, ja matkue lähti liikkeelle. — Sellaisia ne ovat koira-apinat … alkoi von Coren, kääriytyen levättiinsä ja peittäen silmänsä. — Kuulithan, hän ei tahtoisi tutkia turilaita ja koppakuoriaisia sentähden, että kansa kärsii. Samaan tapaan arvostelevat meikäläisiä kaikki koira-apinat. Viekas, kymmenessä polvessa ruoskalla ja nyrkillä pelotettu orjanheimo! Se vapisee, tuntee sääliä ja suitsuttaa vain väkivallan edessä, mutta päästä koira-apina vapaalle alueelle, missä sen ei tarvitse pelätä kenenkään niskaansa tarraavan, silloin se kyllä osaa rehennellä ja näyttää mikä hän on miehiään. Katso, kuinka rohkea se on taulunäyttelyissä, museoissa, teattereissa tahi arvostellessaan tiedettä: suurentelee, nousee takajaloilleen, sättii, arvostelee… Orjan piirre — täytyy välttämättömästi arvostella! Paneppa merkille: vapaiden ammattien harjoittajia sätitään useammin kuin petkuttajia — se johtuu siitä, että yleisö on kolmeksi neljännekseksi orjia, samanlaisia koira-apinoita. Sitä ei tapahdu, että orja ojentaisi sinulle kätensä ja lausuisi vilpittömän kiitoksensa, kun teet työtä. — En ymmärrä, mitä oikein tahdot! — sanoi Samoilenko haukotellen. — Naisparkaa halutti yksinkertaisuudessaan puhella kanssasi viisaista asioista, ja heti sinä teet johtopäätöksen. Olet suuttunut mieheen jostakin, ja summamutikassa saa naisparkakin kyytiä. Hän on oikein hyvä nainen. — Ole jo! Tavallinen leipäsusi, irstas ja paheellinen. Kuule, Aleksander Daviditsh, kun sinä kohtaat yksinkertaisen maalaisvaimon, joka ei asu miehensä luona, joka ei tee mitään ja joka vain hihittää ja hahattaa, niin sanot hänelle: mene työhön. Miksi
- 68. sitten tässä kohden arastelet ja pelkäät lausua totuutta? Senkötähden vain, että Nadeshda Feodorovna on virkamiehen eikä matruusien pitohelluna? — Mitä minun olisi hänelle tehtävä? — kivahti Samoilenko. — Selkäänkö annettava, vai? — Pahetta ei ole mairiteltava. Me tuomitsemme pahetta vain selän takana, mikä on samaa kuin heristää nyrkkiä taskussa. Minä olen eläintieteilijä tahi sosiologi, mikä on aivan samaa, sinä lääkäri; yhteiskunta luottaa meihin; velvollisuutemme on huomauttaa sille sitä hirmuista turmiota, joka uhkaa sitä ja tulevia sukupolvia Nadeshda Ivanovnan kaltaisten naikkosten olemassaolon takia. — Feodorovnan, — oikaisi Samoilenko. — Ja mitä yhteiskunnan pitäisi puolestaan tehdä? — Senkö? Se on sen oma asia. Minusta on suorin ja varmin keino — käyttää väkivaltaa. Hänet on järjestysvallan toimesta lähetettävä miehensä luo, ja jos tämä ei ota häntä vastaan, on hänet toimitettava pakkotyöhön tahi johonkin ojennuslaitokseen. — Oih! — huoahti Samoilenko; tovin vaiti oltuaan hän sitten kysyi hiljaisella äänellä: — Sanoit tässä eräänä päivänä, että sellaiset ihmiset, kuin Lajevski, ovat hävitettävät… Sanoppa, jos nyt esimerkiksi valtio tahi yhteiskunta antaisi sinulle toimeksi toimittaa hänet pois, tekisitkö sen? — Käsi ei suinkaan vavahtaisi.
- 69. IX. Kotiin tultua Lajevski ja Nadeshda Feodorovna astuivat pimeihin, tukahuttaviin huoneisiinsa. Molemmat olivat vaiti. Lajevski sytytti kynttilän, Nadeshda Feodorovna istuutui ja riisumatta viittaansa ja hattuansa katsoi häneen suruisin, syyllisin silmin. Lajevski ymmärsi hänen odottavan selitystä; mutta selityksen antaminen olisi ollut ikävää, hyödytöntä ja väsyttävää, ja sydäntä ahdisti tieto siitä, ettei hän ollut hillinnyt itseään, vaan oli sanonut hänelle raakuuden. Taskussa sattui hänelle käteen kirje, jonka hän oli joka päivä aikonut lukea Nadeshda Feodorovnalle, ja hän ajatteli, että jos hän nyt näyttää tämän kirjeen hänelle, johtaa se hänen huomionsa muuanne. On jo aika selvittää välimme, ajatteli hän. Annan hänelle; tulkoon mikä on tullakseen. Hän otti taskustaan kirjeen ja antoi sen Nadeshdalle. — Lue, se koskee sinua. Sen sanottuaan hän meni työhuoneeseensa ja kävi pimeässä sohvalle pitkäkseen ilman päänalusta. Nadeshda Feodorovna luki
- 70. kirjeen, ja hänestä näytti ikäänkuin katto olisi painunut ja seinät siirtyneet lähemmäksi. Kävi äkkiä ahtaaksi, pimeäksi, pelottavaksi. Hän risti nopeasti silmiään ja virkkoi: — Suo, Herra, rauha … suo, Herra, rauha… Ja ratkesi itkuun. — Vanja! — huusi hän. — Ivan Andreitsh! Vastausta ei tullut. Luullen Lajevskin tulleen huoneeseen ja seisovan hänen tuolinsa takana, hän nyyhkytti kuin lapsi ja puheli: — Mikset ennemmin sanonut minulle, että hän on kuollut? En olisi lähtenyt kekkereihin, en olisi nauranut niin paljon… Herrat lausuivat minulle typeryyksiä. Mikä synti! Pelasta minut, Vanja, pelasta minut… Olen menettänyt järkeni… Minä olen hukassa… Lajevski kuuli hänen nyyhkytyksensä. Hänen olonsa oli sietämättömän tukahuttava, ja sydän jyskytti hirmuisesti. Ahdistuksissaan hän nousi, seisoi kotvasen keskellä huonetta, hapuroi pimeässä pöydän luona nojatuolia ja istahti siihen. Tämä on vankila… ajatteli hän. Pitää lähteä pois… Minä en kestä… Oli jo myöhäistä mennä pelaamaan korttia, ravintoloita ei kaupungissa ollut. Hän kävi jälleen pitkälleen ja tukki korvansa, ettei olisi kuullut nyyhkytyksiä, mutta äkkiä hän muisti, että Samoilenkon luo sopi kyllä mennä. Jotta ei tarvitsisi kulkea Nadeshda Feodorovnan ohi, pujahti hän ikkunan kautta puutarhaan, kiipesi aidan yli ja lähti pitkin katua. Oli pimeä. Vastikään oli saapunut joku höyrylaiva, tulista päättäen iso matkustajalaiva ..
- 71. Ankkurikettinki ratisi. Rannasta lähti punainen tuli nopeasti lipumaan laivaa kohden: se oli tullivene. Siellä ne nukkuvat matkustajat hyteissään… ajatteli Lajevski, ja hänen kävi kateeksi toisten lepo. Ikkunat olivat Samoilenkon talossa auki. Lajevski katsoi yhdestä ikkunasta sisään, sitten toisesta: huoneissa oli pimeää ja hiljaista. — Aleksander Daviditsh, nukutko? — huhuili hän. — Aleksander Daviditsh! Kuului yskimistä ja huolestunut huudahdus: — Ken siellä? Mitä lempoa? — Minä se olen, Aleksander Daviditsh. Suo anteeksi. Hetkisen kuluttua avautui väliovi; pyhimyslampusta tuikahti pehmoinen valo, ja Samoilenkon valkopukuinen jättiläisvartalo, valkoinen yömyssy päässä, tuli näkyviin. — Mitä sinä haet? — kysyi hän unisena raskaasti hengittäen ja kyhnien itseään. — Maltas nyt, heti minä avaan. — Älä vaivaa itseäsi, minä kömmin ikkunasta… Lajevski kiipesi ikkunasta sisään ja lähestyttyään Samoilenkoa tarttui hänen käteensä. — Aleksander Daviditsh, — sanoi hän vapisevalla äänellä, — pelasta minut! Pyydän sinua, rukoilen, ymmärrä minua! Asemani on sietämätön. Jos se jatkuu vielä pari päivää, kuristan itseni kuin … kuin koiran!
- 72. — Maltahan… Mikäs sinua oikeastaan vaivaa? — Sytytä kynttilä. — Oh-oh… huokasi Samoileko, pannen kynttilään tulta. — Siunaa ja varjele… Kello käy jo kahta, veikkonen. — Suo anteeksi, mutta minä en voi istua kotona, — virkkoi Lajevski, tuntien valosta ja Samoilenkon läsnäolosta suurta huojennusta. — Sinä, Aleksander Daviditsh, olet ainoa, paras ystäväni… Kaikki toivoni on sinussa. Jos tahdot tahi et, pelasta Jumalan nimessä. Minun täytyy kaikin mokomin matkustaa täältä pois. Lainaa minulle rahaa! — Herra siunatkoon! — huokasi Samoilenko, kyhnien itseään. — Heräsin ja kuulen, laiva huutaa tuloaan, sitten sinä… Paljonkos tarvitset? — Vähintäänkin kolmesataa ruplaa. Hänelle on jätettävä sata ja itse tarvitsen matkalle kaksisataa… Olen sinulle velkaa jo lähes neljäsataa, mutta minä lähetän kaikki … kaikki. Samoilenko kourasi yhteen käteen molemmat poskipartansa, levitti jalat haralleen ja vaipui mietteisiin. — Ja-ah … urahti hän miettiväisenä. — Kolmesataa… Mutta minulla ei ole niin paljoa. Pitää ottaa joltakin lainaksi. — Lainaa Herran tähden! — virkkoi Lajevski, nähden Samoilenkon muodosta, että hän tahtoo antaa rahaa ja ehdottomasti antaakin. — Lainaa, minä varmasti maksan takaisin. Lähetän Pietarista, heti kun tulen sinne. Siitä saat olla huoleti. Kuulehan, Sasha, — virkkoi hän elpyneenä — juodaanpas viiniä!
- 73. — Miksei, juodaan vaan. Mentiin ruokasaliin. — Entä, kuinka käy Nadeshda Feodorovnan? — kysyi Samoilenko, tuoden pöytään kolme pulloa viiniä ja lautasellisen persikoita. — Jääkö hän tänne? — Kaikki järjestän, kaikki järjestän… — vastasi Lajevski, tuntien odottamattoman riemunpuuskan täyttävän rintansa. — Minä sitten lähetän hänelle rahaa, hän matkustaa luokseni… Siellä sitten selvitämme suhteemme. Terveydeksesi, jalo ystävä. — Älähän kiirehdi! — sanoi Samoilenko. — Ryyppäähän ensinnä tätä. Se on omasta viinitarhastani. Tämä pullo taas on Navaridzen viinitarhan tuotetta, tämä Ahatulovin… Maista kaikkia kolmea lajia ja sano peittelemättä… Minun on vähän niinkuin hapahkoa? Miltä sinusta tuntuu? Eikö ole? — Niin on. Hyvinpä minua lohdutit, Aleksander Daviditsh. Kiitos … oikein elvyin. — Onko hapahkoa? — Lempo hänet tietää, minä en. Mutta sinä olet mitä mainioin, erinomainen mies! Katsoen häntä kalpeihin, kiihtyneihin, hyvänsävyisiin kasvoihin Samoilenko muisti von Corenin mielipiteen, että tuollaiset on poistettava, ja Lajevski näytti hänestä heikolta, turvattomalta lapselta, jota jokainen voi loukata ja sortaa. — Kun sinä tulet sinne, niin sovi äitisi kanssa, — sanoi hän. — Näin ei ole hyvä.
- 74. — Kyllä, kyllä, välttämättä. Oltiin kotvasen vaiti. Kun ensimmäinen pullo oli juotu, virkkoi Samoilenko: — Voisit sopia von Coreninkin kanssa. Mitä mainioimpia, viisaimpia miehiä kumpainenkin, ja kyräilette toisiinne kuin sudet. — Kyllä hän on kaikkein viisain, mainioin mies, — myönsi Lajevski, tällä haavaa valmiina kaikkia kiittämään ja kaikille anteeksi antamaan. — Hän on merkillinen mies, mutta minun on mahdotonta hyväksyä hänen mielipiteitään. Ei, siksi erilaiset ovat luonteemme. Minä olen veltto, heikko, alistuvainen luonne; ehkä hyvänä hetkenä ojentaisinkin hänelle käteni, mutta hän kääntyisi kuitenkin minusta pois … halveksivasti. Lajevski hörppi viiniä, astahti nurkasta nurkkaan ja jatkoi, seisoen keskellä lattiaa: — Minä ymmärrän mainiosti von Corenia. Hän on kova, voimakas, despoottinen luonne. Olethan kuullut, hän hokee yhtenään tutkimusretkestä, eikä se ole tyhjää puhetta. Hänellä pitää olla erämaa, kuutamoyö: ympärillä teltoissa ja avotaivaan alla nukkuvat hänen nälkäiset ja sairaat, vaivaloisten päivämatkojen rasittamat kasakkansa, oppaansa, kantajansa, lääkärinsä, pappinsa, hän yksin vain ei nuku, vaan istuu kuten Stanley kokoonpantavalla tuolilla ja tuntee itsensä erämaan valtiaaksi ja näiden ihmisten herraksi. Hän kulkee kulkemistaan jonnekin, miehet vaikeroivat ja kuolevat toinen toisensa jälkeen, hän vain kulkee eteenpäin, viimein hän sortuu itsekin, mutta jää kuitenkin erämaan despootiksi ja valtiaaksi, koska hänen hautaristinsä on karavaanien nähtävänä kolmen-, neljänkymmenen peninkulman päässä ja vallitsee erämaata. Säälin,
- 75. ettei tämä mies ole sotapalveluksessa. Hänestä tulisi erinomainen, nerokas sotapäällikkö. Hänessä olisi miestä hukuttamaan ratsuväkensä virtaan ja laatimaan ruumiista sillan, ja sellainen rohkeus on sodassa enemmän tarpeen kuin linnoitustaidot ja taktiikat. Oi, minä ymmärrän hänet niin hyvin! Sano, minkätähden hän tuhlaa varojansa oleskelemalla täällä? Mitä hän täältä kaipaa? — Hän tutkii meren eläimistöä. — Ei, ei, veikko, ei! — huokasi Lajevski. — Laivalla kertoi minulle eräs matkustavainen tiedemies, että Musta meri on köyhä eläimistä ja että sen syvyydessä veden rikkivetyisyyden takia elimellinen elämä on mahdoton. Kaikki vakavat eläintieteilijät työskentelevät biologisilla asemilla Neapelissa tai Villefranchessa. Mutta von Coren on itsenäinen ja itsepäinen: hän työskentelee Mustalla merellä sentähden, että täällä ei kukaan työskentele; hän katkaisi välinsä yliopiston kanssa, ei tahdo tietää oppineista ja tovereista sentähden, että hän ennen kaikkea on hirmuvaltias ja vasta toisessa sijassa eläintieteilijä. Ja hän saa jotakin suurta aikaan, sen tulet näkemään. Jo nytkin hän haaveilee, että palattuaan tutkimusretkeltä hän karkoittaa yliopistoistamme vehkeilyn ja puolinaisuuden ja perinjuurin masentaa oppineet mahtimiehet. Hirmuvalta on tieteessäkin yhtä väkevä kuin sodassa. Tässä haisevassa kaupunkipahasessa hän asuu jo toista kesää sentähden, että on parempi olla ensimmäisenä kylässä kuin toisena kaupungissa. Täällä hän on kuningas ja kotka; hän pitää kaikkia kaupunkilaisia kurissa ja painaa heitä arvovallallaan. Hän on siepannut kouriinsa kaikki, sekaantuu vieraisiin asioihin, kaikki ovat hänelle tarpeen ja kaikki pelkäävät häntä. Minä luisun hänen kouransa ulottuvilta, hän tuntee sen eikä voi sietää minua. Eikö hän ole puhunut sinulle, että minut olisi hävitettävä tahi toimitettava yleiseen työhön?
- 76. — Kyllä, — naurahti Samoilenko. Lajevski naurahti samaten ja ryyppäsi viiniä. — Hänen ihanteensakin ovat despoottiset, — sanoi hän nauraen ja haukaten persikkaa. — Tavallisesti kuolevaiset, jos tekevät työtä yhteiseksi hyväksi, tarkoittavat sillä lähimmäistään: minua, sinua, sanalla sanoen ihmistä. Von Corenista ihmiset ovat koiranpentuja ja joutavuuksia, liian vähäpätöisiä kelvatakseen hänen elämänsä tarkoitusperäksi. Hän työskentelee, lähtee tutkimusretkelle ja taittaa siellä niskansa, ei lähimmäisenrakkauden nimessä, vaan sellaisten abstraktsionien takia kuin ihmiskunta, tulevat sukupolvet, ihanteellinen ihmisrotu. Hän huolehtii ihmisrodun parantamisesta ja siinä suhteessa me olemme hänen kannaltaan vain orjia, tykinruokaa, kuormajuhtia; toiset hän hävittäisi tahi pistäisi pakkotyöhän, toiset ruhjoisi rautaisella kurikalla, pakottaisi kuten Araktshejev nousemaan ja käymään levolle rummun pärrytyksen mukaan, panisi eunukit vartioimaan puhtauttamme ja siveellisyyttämme, käskisi ampumaan jokaisen, joka astuu ahtaan, vanhoillisen siveyskäsitepiirimme ulkopuolelle, ja kaikki tämä ihmisrodun parantamisen nimessä. Mutta mitä on ihmisrotu? Kuvitelma, kangastus… Hirmuvaltiaat ovat aina haaveilleet. Veikkonen, ymmärrän hänet niin hyvin. Pidän arvossa enkä suinkaan kiellä hänen merkitystään; sellaisten varassa, kuin hän on, tämä maailma pysyy pystyssä, ja jos maailma jätettäisiin yksistään meidän hoteisiimme, niin tekisimme siitä, kaikesta tahdostamme ja hyvistä aikeistamme huolimatta, saman kuin mitä kärpäset ovat tehneet tästä kuvataulusta. Juuri niin! Lajevski kävi Samoilenkon viereen istumaan ja lausui vilpittömän innostuksen vallassa:
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