•
J~"'I.
A TEST O'~SIGNIFlCANCE FOR COMPARING
TWO m:.FFEltENf'·SYSTEMS OF STRATIFYING
THE
SAME
POPULATION
OJ Yandle
'---..... ~_/>j
...
and R. J. Hader
Institute of Statistics
Mimeograph Series No. 456
Dec~
1965
iv
TABLE OF COl\"'TENTS
.Page
LIST OF TABLES • • • •
...
LIST OF ILLUSTRATIONS
1.
INTRODUCTION MID REVIEW OF LITERATURE
2.
GENERAL
2.1
2.2
2.3
2.4
2.5
3.
•
•
•
a
*
•
•
•
•
•
0
•
•
•
•
1
•
Q
•
10
•
Formulation of the Null Hypothesis • • • •
Linear Transformations
• • • •
Some Properties of the a.. • • • •
Transformation of Ho
lJ....
Derivation of the Test Statistic
...
22
28
The Case with Equal Cell Proportions
The Case with r=c=2
•••••••
...
....
GENERAL SOLUTION FOR r > 2 MID c = 2, ••• , _r
4.1
4.2
4.3
4.4
10
12
18
23
TWO SPEC IAL CASES OF THE GENERAL RESULT
3.1
3.2
4.
DEVELOP~NT
*
e
28
32
37
General Remarks •
• ••••
Explicit Algebraic Representation of the
Equation g: 'B-1UB-l£* = 0 • • • • • • • •
An Example Illustrating Use of the Methods
Numerical Solution for Problems of Size Greater
than r=c=4 • • • • • • •
37
38
43
...
47
. . . . . . .f. •
50
5.
EMPIRICAL SAMPLING
6.
SUMMARY A1'ID CONCLUSIONS •
66
7.
LIST OF REFERENCES
69
8.
APPEND ICES
8.1
8.2
8.3
8.4
Theorems
8.1.1
8.1.2
8.1.3
•
EXPER~NTS
•
...,
71
and Derivations
•
Theorem on U2 • • • • • • • •• • • • •
Theorem on the Determinant of IT •
Relationship Between the Sums of Principal
Minors of a Matrix and Those of Its
Inverse • • • • • • • ••
• • • •
8.1.4 Theorem on the Inverse of B
• • • •
Matrices and Sums of Principal Minors for Problem
Sizes to r=e=4 • • • • • • • • • •
• • • • ••
Computer Program for Transformations
• •
Computer Program to Obtain Polynomial Roots •
0
••
••••••
71
71
73
81
82
89
94
106
v
LIST OF T.AJ3LES
Page
4.1.
Summary of white oak sp. sawlog data
44
5.1.
Rowand column dimensions and sample sizes for
empirical sampling experiments • •
51
Transformation of rectangular distribution offuur-digit
random numbers to random normal deviates •
• • • ••
52
5.2.
8.1.
Example of computer program output
104
vi
LIST OF ILLUSTRATIONS
Page
5.1.
Empirical distributions for r=e=2 with equal cell sizes
55
Empirical distributions for r=3, c=2 with equal cell
sizes •
e
eo.
·..
56
5.3.
Empirical distributions for r=c=3 with equal cell sizes ••
57
5.4.
Empirical distributions for r=c=2 with unequal cell
sizes - part one
• • • • • • • • •
58
0
5.5.
5.6.
5.7.
5.8.
5.9.
5.10.
0
•
Q
0
GOG
0
Q
0
0
•
0
•
•
Empirical distributions for r=e=2 with unequal cell
sizes - part two
• • • • • • • • • •
59
Empirical and theoretical power curves for r=e=2 with
equal cell sizes
• • • • • • • •
• • ••
61
Empirical and theoretical power curves for r=3, c=2 with
equal cell sizes
62
········ ··..
Empirical and theoretical power curves for r=e=3 with
equal cell sizes
············..
Empirical and theoretical power curves for r=e =2 with
unequal cell sizes - part one
···········
Empirical and theoretical power curves for r=e=2 with
unequal cell sizes - part two
···········
63
64
65
1.
INTRODUCTION AND REYIEW OF LITERATURE
The problem considered in this dissertation was first brought
to the author's attention in the context of grades for sawlogs.
problem is posed by the question:
The
given two systems for grading saw-
logs, how should we obtain and use a sample from a population of
sawlogs to compare the two grading systems in order to make some
judgment as to which is the llbetter ll system?
In order to give perspective to the problem, we will discuss
briefly how logs are graded and the uses of log grades.
A set of specifications for grading logs is usually, of necessity, based on characteristics that are identifiable on the whole
log; whereas, the value of the log is the sum of the values of the
end products (lumber, veneer, etc.) less the cost of conversion.
The identifiable characteristics of a log are those of size, length,
and diameter; irregular geometry exhibited by sweep, crook, eccentricity, and excessive taper; and such defects as rot, stain,
seams, bumps, burls, knots, and others on the bark surfaces and
ends of the log.
In aa.dition, the relative positions of defects or
of clear areas between defects are visible characteristics.
The
set of specifications for a grading system sets forth the permissible characteristics and their range, either singly or in combination, for each of the grades in the system.
For illustration, we
reproduce one set of specifications for grading Southern pine logs
as given by Campbell (3, p.4):
2
Southern pine logs are graded in two steps. First they
are given a tentative g.cade based on diameter and K countY j
secondly, they are given a final g::::'ade based o~;other degrading factors. Step 1 consists of determining D~ and total K
count on all four faces. Establish a tentative grade according to the following tabulati,;)n:
Grade
I
II
III
rv
Minim-urn scaling
diameter CD)
Maximum Knot
count (K)
17
10
5
5
Dis
D/2
no limit
no limit
As step 2, determine in the sequence lj_sted:
Sweep.--Degrade any tentative I, II, or III grade log
one grade if sweep is at least 3 inches and equals or
exceeds D/3. (This is the final grade if the log has
no evidence of heart rot and no rotten or oversize knots.)
Heart rot.--Degrade any tentative I, II, or III grade log
one grade if conk, massed hyphae, or other evidence of
advanced heart rot is found. (This is the final grade if
the log has no unsound or oversize knots.)
Unsound or oversize knots.--Degrade any tentative grade
III log to grad.e rv if unsound or oversize knots are dispersed so that they cannot be contained in one quarter
face.
In virtually all the papers reviewed, the authors have indicated
the need for log grades in statements such as these by Petro (19,
p. 5):
"The steady rise in production costs and increased market
competition over the years, has provided an impetus to the need for
evaluating the quality of hard.wood sawlogs."
Y
K count == number of overgrown knots plus sum of diameters of
sound knots plus twice the sum of diameters of unsound knots.
~ D == average diameter at small end of log inside bark to nearest whole inch.
3
By Vaughan (23, pG 1):
l!Increasingly d.iversified hardwood utiliza-
tion further complicates the task of determining the highest use
for and appraising the value of hardwood ll)gs or standing timber.
Or more simply by Newport and O'Regan (18, p. 1):
tI
"Workable log
grading systems are needed as aids to marketing timber and other
phases of forest management."
port and OiRegan
appe~rs
However, a further statement by New-
to be unique in that it refers to log
grading by a term not used by other authors.
less of the manner of selecting
They state:
"Regard-
the grade specifications
they need testing to determine the effectiveness of stratification."
The key word, of course, is "stratification."
Many other papers
essentially discuss stratification of logs but without using the
term directly.
If we
conside~
log grades to be the definition of
strata in a given population of logs, then we can interpret the
need for log grades in a statistical sense as being the need to
increase precision by stratifying the population.
In this paper we will consider that a system of log grades,
having specifications that define a log as uniquely belonging to
one and only one grade, defines the stratification of a population
of logs.
Further) we will limit consideration to a single measurable
characteristic or response variable: log value.
Thus, we will treat
the problems of log grading as equivalent to those of stratified
sampling.
Newport e~ ale (17, p. 28) recognized three problem areas requiring further work in developing adequate log grading systems:
4
1. Techr.iques are needed for the testing a~d selection
of the controlling factors to be included in grade specifications. This includes the dete~mination of breaking
points between grades for certain factors such as knot
size.
2. Methods are needed for comparing the effectiveness
and reliability of grading systems and of grades within
a system.
3. Techniques are needed for the calculation of endproduct performance data.
First, let us dispense with problem area c.
This is the
problem of determining the appropriate estimators.
The best esti-
mator for a given situation will depend on many considerations in
addition to stratification,
~.§.,
we may at times be satisfied with
simple estimates for means or totals but at other times it may be
necessary to use ratio or regression estimators or other techniques
to satisfy our purpose.
Since in this paper we will be concerned
with a problem of stratification, we will not elaborate further on
estimation procedures.
Problem area a is essentially that of con-
structing optimum stratification which in the past fifteen years
has been the object of a considerable research effort by Dalenius
(5, 6, 7, 8), Dalenius and Gurney (9), Dalenius and Hodges (10, 11),
Ekman (12), Cochran (4), and others.
This is a very difficult
problem except under the simplest of conditions.
Dalenius first
showed how to determine optimlXill stratum boundaries for a fixed
number of strata when the frequency function is known.
Further
work has been primarily concerned with particular cases or with the
development of more rapid approximate methods for estimating the
stratum boundaries.
Cochran (4) compared four of these approximate
5
methods as applied to eight
degrees of skewness.
frequen~y
distributions having different
The r,:o;sults of this paper indicate (although
i t is not explicitly stated) that a gain in precision could be
realized by using the frequency distribution of the variate from a
recent survey.
A great amount of further work in this direction is
required before methods applicable to log grading or similar problems
will be available.
The use of principal components to construct strata in a situation having similarities to the log grading problem was reported by
Hagood and Bernert (16).
Analogous to the log characteristics pre-
viously discussed, they use twelve population and agricultural
•
variables which are termed "control" variables.
The main points of
the method are given as follows, (16, p. 335):
To utilize ir~ormation on all the selected control variables
in stratification, mutually uncorrelated component indexes
were employed. Each index is a linear function of the twelve
control variables , with the weights for the variables bei.nK
determined by component analysis of the matrix of intercorrelations of the variables.
tL~d further,
(16, p. 337):
The principles generally followed in the present stratification were: (1) to use an index for each component explaining more than 10 percent of the total variation of the control
variables; (2) to use more class intel~als for the index of
the component which explained the greatest proportion of the
variation of the control variables.
Log grading systems in use today can be broadly classified according
to the method of developing the system:
Judgment - Log grade is based on an estimate of performance.
The system specifications set forth the minimum amounts of key
tr
6
products to be produced from the leg.
The log grader must use his own
judgment in determ:::"ning whether or not a given leg meets the specification.
Thus the system is developed on the basis of statements of
desired performance and its success is dependent upon the experience
and skill of the grader.
It is obvious that such a system does not
uniquely define the stratification of a polulation of logs.
For-
tunately, systems of this kind are being replaced with others that
are better defined and less dependent on judgment.
Arbitrary definition - The set of specifications is based on
the visible log characteristics but defined arbitrarily, albeit by
men experienced in woods and mill operatj.ons.
•
AYJ.alytical development - 'I'he set of specifications is based on
the visible log characteristics after a study of data obtained expressly for the purpose of studying the relationship of log characteristics to product yields or values.
Either the second or third method
of developing log grades as just outlined above can lead to grading
specifications that uniquely define a system of stratifj.cation.
(Although the problem of misclassificatio~ will exist to some extent
even with the best defined system.)
It is systems such as these
that we consider in this paper.
Most of the log grading systems that have been developed
analytically have had that development take place through the use of
some form of regression or correlation analysis.
The system, pre-
viously outlined, by Campbell (3), was partially developed. by studying the multiple regression of log grade value on several quantitative log characteristics.
For a more thorough discussion of multiple
7
regression, including consideratian of appropriate models, in the
development of log grades see the paper by Newport and O'Regan (18).
The second major problem area outlined on page 4 is further
elaborated by Newport et al., (17, p. 30):
a great deal of effort will be spent on the comparison of existing grading systems in order to select
the most effective ones and on the testing of grading
systems for possible use on other species or in other
areas. Therefore techniques are needed for such testing
Rephrasing this statement in the more general context of sampling theory
we can say:
techniques are needed to compare the effectiveness of
two different stratifications of the same population.
In order to
learn the state of knowledge with respect to such techniques, we
have turned to the literature of sampling theory and methods with
the surprising result of finding that the problem seems to have been
completely ignored.
To be sure, this problem cannot be completely
divorced from other problems relating to stratification, particularly
the problem of finding optimum stratificationj however, the direction
is different and it is a lack of other work in this direction that is
striking.
It is the work toward a partial solution of this problem
with which we will be concerned in this paper.
In approaching this problem we must first have a suitable working definition of what j.s meant by the I!better l! of the two systems
of stratification to be compared.
First, we will consider some
standards for log grading systems as recommended by Newport et al.,
(17, po 18):
8
a. The grades in a grading system must group the logs
or trees so that the variability in value and/or product
yields is reduced to a reasonable limit.
b. The square root of the variance of value per unit
volume should be [no greater than] 7 percent of the mean
value per unit volume for each grade within a grading
system.
c. For a given log size one grade should differ from
another by not less than 10 percent of the mean value
of the higher of the two grades under consideration.
The difference in mean value between the several grades
should be approximately equal.
It is obvious that these standards are unsatisfactory:
a is in-
definite, b is an arbitrary standard which could be difficult to
obtain for some populations and even impossible for others, and
taken together band c could for some populations be contradictory.
We will adopt the following definition:
the "better" of two
systems of stratification is that one for which the weighted average
within-stratum variance is smaller.
'llhis is equivalent to defining
the "better" system to be that one which, when used together with
proportional allocation of the sample, would result in the smaller
variance for the estimated population mean or total.
Thus we will
ignore 'all costs and physical or administrative problems involved in
delineating the strata or obtaining samples.
The problem of comparing two systems of stratification will be
treated as one of devising a test of the hypothesis of equality of
the weighted average within-stratum variances for the two systems.
This test will be derived from the ratio of maximum likelihoods
AH = LO
Lftu ~
9
where L(w) is the likelihood f'unction in the space restricted by the
null hypothesis and L(O) is the likelihood function in the unrestricted space.
As will be apparent in the ensuing sections, the null
hypothesis is a
tests derived
non~linear
f~om
function of the parameters.
Examples of
likelihood ratios when the null hypothesis is non-
linear are given by (1) and (24).
Watson (24) in the investigation of
equatorial distributions on a sphere was able to formulate his problem
in such a way that the Lagrangian mUltiplier, appearing in the equations to be solved for the estimates of parameters in the restricted
space, can be shown to be the smallest root of a given matrix; hence,
the maximum likelihood solution is given by the characteristic vector
associated. with this least root.
A.nderson and Bancroft (1) give an
example in deriving a test of second-order interaction in a 2 X2 X2
contingency table.
In that problem, the restricted maximum likeli-
hood estimates are found as functions of the Lagrangian multiplier.
These functions must be substituted into the equation restriction,
1.~.,
null hypothesis, and a
Lagrangian multiplier.
nl~erical
solution obtained for the
This numerical solution is then used to ob-
tain the numerical estimates of the parameters.
Except for two
special cases, the prcblem being treated. here will require a numerical solution parallel to that described by
A.~derson
and Bancroft.
10
2.
2.1
GENERAL DKlELOPMEl\""I'
Formulation of the Null
Hypoth~s~s
The two systems of stratification that are to be compared will be
designateo_ as "R", composed of r
strata~
As a convention, we will set r
c.
~
and "C", composed of c strata.
The population to be sampled may
be considered as a two-way array with r rows andc columns being defined as the strata of systems Rand C respectively.
in this array defines a sub-population with mean
a proportion,
P1j,
of the entire population,
f.L1j
The (ij)th cell
and containing
i.e.~ ~ Pij
-
-
ij
= 1.
We wish to construct a statistic that will be a test of the
hypothesis that the average variance within strata of system R
(within rows) and the average variance within strata of system C
(within columns) are equal, l.~.,
(2.1.1)
If we consider the population in any stratum of either system to be a
composite of the populations of all cells contained in the stratum,
then the frequency function for any stratum can be expressed in terms
of the frequency functions of the cells, ~.~., for the i th stratum in
system R,
0) 10(YiO)
• 11 (Yl1) + ••• + (Pi
fi(Yi) :::: ( Pl1)
TI
-.'-'=' i
Pi··
(2.1.2)
11
The first two moments about the origin for f 1 (Yi) are:
=
-l
Pi·
[PH fl.1l
(2.1.3)
+ ••• + Pi c fl.1 c ]
and
(2.1.4)
'
Wl'th'In th e_"th stratum·as.·
Thus, we can wrl't e th e varlance
Similarly,
CT6 j = P~ j [P1
j
IJ.~ (13)
+
8
o.
+ Pr
j
IJ.~ (r
j )]
(2.1.6)
Upon substituting (2.1.5) and (2.106) and simplifying, equation (2.1.1)
becomes
I [p~. (~Pi
i
j
lJ.i j
)2J
12
We will see later (section 2.5) that a sample of' size n = n.. = ~ n i
j
1j
will be required to make the test and, f'urther, that the sample will
be proportional,
~.~.,
(2.1.8)
Thus, the equality in (2.1.7) can be written equivalently as:
or
(2.1.10)
2.2
Linear Transformations
In order to have a better framework within which to describe
certain relationships and to make many of the required derivations
more tractable, it is desirable to work with linear transforms of
both observations and the parameters.
Equation (2.1.10) can be rewritten as:
6.
R
-6.
C
=0
where
6.
R
=
S.S. Rows (unadj. for ColsJ, in terms of the
parameters,
6.
C
=
~ij
•
S.S. Cols. (unadj. for Rows)~ in terms of the
parameters,
~i j •
(2.2.1)
13
Since the r X.c array has unequal p:::'oportions in the cells, the
linear functions comprising the orthogonal set associated with 6 R are
wco
:\
this reason, we will find it convenient to make two parallel, nonorthogonal transformations, examine the relationships of one to the
other, and show how one set of the transformed parameters can be expressed in terms of the other.
Consider the vector of observations to be arranged as
=
¥,.r c
where
Y1.l1
[
Y1Jk
(n Xl)
j
14
The vector of
p~rameters
is then
~l
.
o
,~c
E(yJ
.
= ~ =
o
o
o
~1c
l:!'.rl
~rc
where
~i
j
= ~1
(2.2.5)
j..! •
Now, we will define the transformations
C1 (1 X n)
~
13 =
~
=
C:a(r - 1 X n)
2;3
Cs (c -1 X n)
~
C4 ((r - l)(C - 1) X n)
!6
C6(N - rcX n)
o
~ =
Cl!
15
and
Ql (1 X n)
':iJ.
y
=
~
==
~( c - 1 X n)
Y;3
~ ( r - 1 X n)
Y!J,
'4 «r
ys
Qs (N - rc X n)
H: = %
- l)(c - 1) X n)
where C and Q are each orthonormal matrices with
Cl - Ql,
and
C4 -
'4,
Cs -
~.
(2.2.8)
Similarly, for the observations, we define
and
1.
== Cl:
t
== ~
Further, C and Q are constructed such that the following statements
hold:
~
I
I
I
I
C1C1J! == J! Ql Ql~
f1. I
= U10l
I
==.Y1.YJ. == S.S. due to overall mean
~/C~C2~ == ~§:p == 6R
== S.S. Rows (unadj. for Cols.)
J! IC;C3~ == ~§.s
== S. S. Cols. (adj. for Rows)
~
I
I
I
Q2QeJ!:=::tPYF
== 6 C
== S.S. Cols.
~
I
QSI QaH:
== S.S. Rows
==
I
YsYs
(
)
unadj. for
Rows
(
adj. for)
Cols.
J!/C;C4~ == J!/'4'4!:!'. == §.;04= 1.;"14
== S.S. R X C (adj. Rows and Cols.)
!:!'./C~CS~ =~/Qs'Qs~ =~.§.p=~YJs
= S.S. Within Cells (= 0)
~/C/C ~ = ~/Q Q ~
= S.S.
Total
16
A parallel set of relationships holds using
course, S.S. Within Cells
Z~
g,
and t except, of
= Sw f o.
We will now examine a single term, V~1' from ~ , (a
(i
=
1, ... , c
~
=2
1)
or 3):
(2.2.11)
Since C is an orthonormal matrix, we have from (2.2.6)
(2.2.12)
which upon substitution into (2.2.11) gives
2
Vai
= 0 'C q
'C' 0
-"'O:'i 3a i
-
-
.9o-1.9a, i [C'1 c'2
00.
C6'J~
~
(2.2.13)
From equations (2.2.8) we know that all row vectors in C1
C4
,
,
and C6
are orthogonal to 3ai; therefore,
0
c.5!a1 .5!a' 1 C'
=
L..
0
0
0
0
0
c2.9ai .9a1·
' C'2
C
' C'
2.9ai.9ai 3
0
0
0
C3S!a1~iC~
C3~i.9ai Cs
0
0
, ,
0
0
0
0
0
0
0
0
0
0
(2.2.14)
17
If we let
0*
c*
=[:]
=[:]
,
(2.2.15)
,
(2.2.16)
and
Ai
= C*.9aiii c*'
then, Y~i is uniquely expressed in terms of 0- parameter set by
-0*' ~ -0*
Now, let
ai, r -1
=
, -1
~,r
%i '
(l"
= 1 ,.••• ,
r +·c - 2)
(2.2.19)
,
and
where as a convention we consider that the vectors, q
-"'OIi
,are arranged in
the sequence
.9.:31 , ·.9,a2' • • ., .9.2, C -1 ,
-Cb1
, -'h2' • • .,
(1~
~,
r-1
so that it will be clear that it is the vectors comprising
~
that are
under consideration when we restrict the range of the index to i = 1,
••• , c - 1 as will sometimes be necessary.
18
Using the definition given by (2.2.19) we can rewrite
[
I
I
.9a1.£21 ••• S!ai.£2, r_l
I
.9a1~1
I
•••
.9a1,Ss J 0-1
]
I
~l3ai
as
or
Ai
=
2
an
an a 12
a12 a f1
a12
2
ail ai, r+0-.2
a12 a i, r+o-2
(2.2.21)
a"r+0_2 a 12
ai, r+o_ 2
2.3
2
a 1,r+o_l
Some Properties of the atJ
Let us consider that
O~J r-l
=
linear component of S.S. Rows
=
quadratic component of S.S. Rows (adj. for linear)
=
(r - l)st degree component of S.S. Rows (adj. for
linear, quadratic, ••• , (r - 2)nd degree components)
19
Of1 = linear component of S08. Colso (adj. for Rows)
Of2
=
(8aj.
quadratic component of SoS. eals.
for Rows and
linear component of Cols.)
of,
0
-1
:=
(c -1 )st degree component of 8.8. Cols. (ad.j. for
Rows and li.near, quadr"atic, •
0
0'
(c - 2 )nd degree
components of eolso)
and
Yi1
=
Yi2
Yi,
linear component of 8. S. eols.
quadratic component of S. 8. Cols. (adj. for linear)
0-1
=
(c - l)st degree component of 8.8. Cols. (adj. for
li.near, quadratic,
0 •• ,
(c - 2)nd degree components)
Yf1
=
linear component of 8.8. Rows (adj. for Cols.)
Yf2
=
quadratic component of 8.8
0
Rows (adj. for Cols.
and linear component of Rows)
Yf,r-1 = (r - l)st degree component of 8 S. Rows (adj. for
0
Cols. and linear, quadrati.c, ••• , (r - 2) nd degree
components of Rows)
The subdivisions into linear, quadrati.c, etc., components is taken for
purposes of discussion only.
Obviously, from the definitions of C and
Q, any other subdivisions into individual components would serve equally
well.
20
NOW, 1 e t
... ,
US
+ _.'l.n th e
conSl.d er a 11 componen'JO
-, ~b-'1":"oe.I. . . cw V31.
r. 2
vJ: 2 ......
0
J
Oi,r~l) and then for the linear component of colllilli~s (adjusted
for rows),
0;1'
However, each of the components
ofa, ••. , Of,o-l
is
invariant with respect to the crder of aO.justment by all preceding
terms so that 0s2 a, ••• ,
2
os,
0
~1
are unchanged if the adjustment is
first by the linear component of columns (unadjusted) and then by rows
(adjusted for linear component of columns).
••• , 5f,0-1
Therefore, each of
ofa,
is orthogonal to the linear component of columns
2
(unadjusted) which is Yr;n'
Now,
and
Therefore, from the above argument,
or
2
Using a similar argQment for Yaa,
for i
= 1,
••• , c - 2.
o
Q
0
,
Y~-a, this can be generalized as
Thus, we can rewrite equation (2.2.21) as
21
O(r + i -1, c - i - I )
··
·
*
O(c - i - 1, r + i - 1)
O(c - i - 1, c - i - 1)
for i ::: 1, ••• , c - 1.
We will now proceed to show some aO_ditional relationships that
exist among the a i j •
~~
From equati.ons (2.2.10) we have
+ 2.,;2.,3 :::
;d"J:p
(2.3.8)
+ 1.;1;3
which can be rewritten as
~
'[
I
I
I
I
'£;1,£21 + • • • + .£a,r-1.£2, r-1 + 331.331 + ••• + .£3,C-1~,c-1
:::~
'[
I
I
3,a1SLa1 + ••• +,9,g,C-l.9:a,C-l
I
+:1'31~1
+ •••
I
]
~
]
+..9;3,r-1~,r-1 ~
or
Since ~1~1 ::: 1 and ~l~S ::: 0
(a,S
~ 2,1), if we pre-multiply by
~1 on both sides of equation (2.3.9) we obtain
or
2 2 2
1 ::: 8.:i.1 + ••• +
ail
+ ••• +
a
r + c -2, 1
(2.3.10)
and if we pre-multiply by ~~l and post-mu.ltiply by, say, ~2 on both
sides of equation (2.3.9) we obtain
22
or
(2.3.11)
By considering all vectors, ~1j, we can immediately generalize (2.3.10)
and (2.3.11) as
r+o -2
I
afj
(j
= 1
=
1, •.• , r + c - 2)
(2.3.12)
1 =1
and
r+o-2
I
1
=
o (j = 1,
a 1 j a 1 l3
1
••• , r + c - 2
f.
and 13
(2.3.13)
j)
By similar operations with the
~
j
we can show
r+o_2
I
j
afj
= 1, •.. ,
(i
=1
r + c -
2)
= 1
and r+ 0-2
I
j=.1.
a 1 j aOtj
::::
0
(i
1, r + c -2
=
(2.3.15)
and ex
f.
2.4.
Transformation of Bo
i)
From (2.2.10) we have
2
= Yl31
+
2
+ Y2,
0-1
and using (2.2.18), this becomes
6 C ::::
2.-*'At!* + ••• + §..*' Ac -1§.*
= 0*' [O~lA1J ~*
1= 1
(2.4.1)
23
If' we now express L1
R "" ~ 52 as
= 2-.* '[ _I(_r_-~l.,_r_-_l_)_l-_a(~r_-_l_,_C_-_l_)
1,
a(e -
=:
5*'K5*
r -
1)
a(c -
1,
c
J
2..*
-1)
(say)
and set
c-1
U
==
L:A1 -K
1=1
the null hypothesis can be stated as
Ho : 5* I U 5*
=:
a
This is the form of the null hypothesis that will be used throughout the
remainder of this paper.
If the alternative hypothesis is two-sided, we have
H1
:
5* I U 5*
1= a
The one-sided alternative that cor:responds to L1 > L1 is
R C
H1
:
6
* 'u
6* >
a
and similarly for L1 <L1
R C
2.5
Derivation of the Test Statistic
Let ¥. be a sample of size n compri.sed of the independent samples
¥.iJ each of size niJ from the rc cells of the arrayed population.
If
we let the within-cell density functions be N(~1J,cr2), then the likelihood function of the sample is:
24
!l
LC~,:;.§.,O'2)
2
= (2rTcr )
"2
exp
[= 2&~
(.2. - ,g,)' (.2.
=
!f) ]
(2.5.1)
or
L(};).§..,O'2)
:=
(2rrcr
2
r ~ exp{ ~ r2~ [(011 - £11 )2 + (~= ~)'~ - :!:a)
+ (~- & ) , (~- !s) + (§.4 - :f4 ) , (§.4 - !..4 )
+ (.2.5 - i6)' (.2.5 =.{5
The usual
st~aight-forward me~hod ~f
)J}
(2.5.2)
diffcrentia0ing lnL with respect
to each parameter i.n turn, equating each derivative to zero J and solving each of the result::"ng equations yield.s the necessa:-y maximum likeli,..
hooel estimates:
.2.
and 0'2 J in the un:r·e8~T:"cted parameter spa.ce.
placing the parameters in
(2.5.2)
Re-
with these estima+~es g:tves the likeli-
hood of the sample in the urrrestricteQ space:
n
L(O)
[~'s
w
]2
exp ( -
¥)
(2.5.3)
In order to find the m.l. estimators in the parameter space restricted by EO , it is necessary to find
iJ
;2, and A. that maximize
the function
(2.5.4)
where L'
=
lnL
and A. is a Lagrangian mllltupl.ier.
Since the
o's
a~e
all
linea~ly
independent, it follows that F will
"
b,.2.4
be a maximum with re spect te 2J., .2.4' and.2.5 when.§.l :=
= .{4 ,
,..
"
and (~ - 5) '(.§..5 -[6) "" Sw. Therefore, we will temporarily be con-
i.
cerned with maximizing F only with respect to
it is equivalent if we minimize:
~
and.
~.
To this end,
25
::: (2:* - gx·) , (2.* -
g*)
+- A (~X 'U~x. )
The derivative of F' w:tth respec t
::
:>J:i1'
~5*
20*
2
g-x
t ..)
(I + AU).§.*
Or, if we let w :::
rearrang~ng
:::
f
o'x- is
(2.5.6)
+ 2AUO'*
Equating the derivative t09.n (1' + 'C
are all zeros and
(2.5.5)
2 X 1) vector whose elements
-
yields
l.*.
(2.5.7)
(2.5.8)
(wI + U).§.* ::: w!:.* •
Thus, the m.l. estimator of
~
~*
:::
w(wI + U)
- 1
~*
(2.5.9)
'!...*.
Replac ing 2..* with ~* in (.§,* -
(~* - g*)'(~* - g*)
is
l* ) '(§.*
- lYe)
== [wg-X"(wI + U)-l -
we have
g,*.I] [w(wI + ur l ~* - g"*,]
:::g,*'[w(wI+ Ur 1 _ I][w(wI + Ur 1 - I] g*
::: f.*' (wI+Ur l [ wI-(wHU)][ wI-( wI+U)] (wI+Ur l
=g,*'(wI +- UrI
Thus~
replacing the paramete::'3..2,
u2
(wI + U)-lt.,*'
/£*
(2.5.10)
with their m.1. estimators in
equation (2.5.2) gives
1 [.
'( _
IJ. 'I
L ~) §.., u 2) ::: ( 2rru 2)- E:
2 exp{ - 2(j2
Sw + l*wl
+ U )-~.2(
u wI + U)-1. ~*]
A
(
•
J
(2.5.11)
26
from which we fl::J.d the m.l. esti'llator 'Jf' (Y2 tc be
(2.5.12)
Substitution of
functioL for
~2 for (Y2 l::.'l equation (2.5.12. )yi2:La.S the likelihood.
t:b~
sa.>nple in the rest:::'lr::ted pa:::'ar.letel" space:
L(w)
(2.5.13)
From equatlons (205.3) and (2.5.l3)J t:1.e li.k,,;li~lood rat::..c
~ [;,.+ g*
•
:'..8
fcnmd to be
E:
S
'(WH:)-l 1J 2 (wHU)-1
l*Y
- n
'2
(2.5.14)
]
Taking the large sampie theo!.'y result clue to Wilks (2,5) that - 21rJAH
is
distributed as X2 with degrees of freed,)m being the (Ufferecic,~ in
dimensionality of the parameter spaces 0 ar.;.d w-' we have as a test
statistic
_
,~
TH- n-U.l
1
[
/J.x-'( W -J.+H)-l
"T2(
+ 4f.
\.1
U VI 'T+1'T)~l
-'- ,-'
o*J
j!.
(2.5015)
Sw
which is approximately distrib'cted. as
xf.
A single degree of freedom
for X2 results from the w-space having di..meY1Sions one les8 than the
O-space due to the single constraint that is expressed as llQ.
27
In order to obtain the explicit value of the test statistic given
by (2.5.15) it is necessary to find the solution for the Lagrangian
A
multiplier w by substitution of 0* for 0* in (2.4.4), 1.~.,
~/(wI + U)-l U(wI + U)-l ~
=
0
(2.5.16)
which from (8.1.4.2) is seen to be, in general, a polynomial of degree
2(r +
C -
3) in w.
In section three, we will discuss two special cases
that result in second degree polynomials and in,section four we will
consider cases having polynomials of higher degree.
28
3.
3.1
TWO SPECIAL CASES OF THE GEl\lERAL RESULT
The Case with Equal Cell Proportions
If we let the cell proportions, p .. , be equal and subsequently
1J
obtain equal numbers of observations per cell, then from (2.3.1) and
(2.3.3) we see that:
=
2
Y2i
2
(3.1.1)
°3i
and
2
Y3i
2
°2i
=
so that (2.4.1) becomes
=
2-* '[_QoJ
=
.§.~3 •
0*
0
]
'01 I(c-l),c-l)
(3.1. 3 )
For (2.4.3) we now obtain
c-l
U = L: A.- K
.1 1
1=
= [-
.
I(r-l,r-l)
0
I I(c-l,c-l)
0
]
Therefore,
t?
= I{r
2,
+ c -
r + c -
2)
and
B = (WI + U) == [(W-l)II
o
.
0
(w+l)I
]
,
and
= (WI + U)-l
'=~~)II
0 .
o . (-1....)I
w+l
]
(3.187)
29
Now; we find that for the fo~egoing conditions, equation (2.5.16)
becomes
o
(3.1.8)
or
or
or by taking the square root of both sides of the equation
Rearranging; we have the two solutions for the Lagrangian mUltiplier:
(3.1.10 )
We will designate as w the solution having a plus sign in the numerl
ator and a minus sign in the denominator and the reverse as w •
2
Referring to equation (205.10), we see that the sum of squares to
be minimized is by application of (3.105) and (301.7):
30
(~* -
g*)' (~*
=
g*)
=
~'B-~2B-1g*
=g* 'B- 1 IB- 1g*
= g:.' (B- 1 )2gy,
g;g2
#..;).3
= TW-1)2 + (w~1)2
(W+1)2g;G2 + (W_1)2 £;;'3
=
(w 2_1)2
(3.1.11)
substituting w for win the numerator of (3.1011) gives:
1
8
(/2'£2)
(~)
(3.1.12)
And, substituting w for W in the denominator of (3.1.11) gives:
1
31
(3.1.13 )
Replacing the numerator and denominator of (3.1.11) with (3.1.12) and
(3.1.13) we have
(3.1.14)
Similarly, if w is used,
2
(3.1.15)
If we take the convention that (SSR)1/2 and (SSC)1/2 are the positive
square roots, then (3.1.14) will always give the minimum for ~/B-~2B-lg*.
Using (3.1.14) in (2.5.15) we find the test statistic for the case
with equal cell proportions to be
TH = n ln [ l_ + [(SSR)1/2 - (SSC )1/2 J2_ ]
2S
w
(3.1.16)
32
The Case wi.th r = c = 2
3.2
If the two systems of stratification to be compared each contain
two strata, then
c-l
L:
i=l
A. =A
·1
1.
a~la12 ]
=[a~l
a
a
12
12
,
= ~1~1
and
:I
=[~ ~J
K
k k'
=
(1, 0) .
where k'=
Therefore, from (2.4.3),
U::::::aa'-kk'
-1-1
--
Noting~.from (2.3.14), that
..
2
all
+
2
~12
=
1
and us;~ng (8.1'.1.2 )we have
From (8.1.4.2) we find, for r = c = 2,
=~
[wI - (U - p I)J
p*
1
2
(3.2.1)
33
where
P
l
=:
trace U ::: 0
and
thus,
Also, using
(3.2.4), we have
:::
:::
w2 1_2wU+U2
(w2=a2 )2
12
(w2 +a 2 )I- 2wU
12
(w2_a2 )2
12
To find the necessary solutions for w we substitute
(3.2.6)
into (2.5.16)~
~'(wI-U)(~l~{ =
lE
~')(wI-U)l*
=:
0
g'*"(WI-U)~l~{(wI-U)l*
=:
l*'(wI-U)! ls'(wI-U)g-)E>
[.§;{ (wI-U)g* ] 2
:::
[~' (wI-u)g* ] 2
:=
± k' (wI-u).i*
a' (wI-U).i*
-1
-
-
~
(3.2.3) and
34
or
which after expr:tnd:L:c.g and using (3.2.4)
=
wI
~
t:.)
siiT.plify beC;)rrL8~":
a12[a12R21 + (1-a11 )131 J
[ (I-all. )JL 21- a12i31 ]
as the solution usi.ng the negative signs
a,::;,&
(3.2.10)
as the solution using the positive signs in the numerator and
denominator of (3.2.8).
Using (3.2.5) and (3.2.7) in equatio:cl (2.5.10) gives:
R* 'B~VB-li* ::
-
-
a 2 f"l.' I (?,-l)'2;..x,
12-
-
(3.2.11)
si.lbstniitirtg w forw and.' simplifyirJ.g gi"ires:
1
[(1=a11 )f2l - a12f31J
2U:a11 )
2
35
=
2
=
[a12R21
~~:~~31J
[(1-alllf21 - a12f31J
2a
12
=
=
Replacing the parameters with observations in (2.2.18) we can write
t
2
21 =j*'AP;*
- -l-'-
t~l = (a1 J!21
+ a12f31)2
and
t
2
31
2
=
I*'A
i*
2-
t 31 = (a21f21 + a22(31)
2
36
From (2.3.13) and (2.3.15) we have the two equations
which when solved simultaneously give
(302.15 )
•
Substituting into (3.2.14) we have
Thus, (3.2.12) becomes
l*'B-ITiB-Ig*
= [t31
-131J [121
- t 21 J
2a
12
Similarly, using w for w in (3.2.11) gives the same result as (3.2.17)
2
but with positive signs.
Again i f we take the convention that the
square roots are positive, the minimum sum of squares is given by
(3.2.17) which may be written as:
!..* 'B':"YB-J.g*
=
[ (SSR .adj}1/2 _ (sse adj)1/2J[" (p~~")1/2~ {SS0)1/2J
=-----:..---.,.-------=::--.....- - - - - - = : : .
Therefore, when r=e=2, we have for the test statistic:
'H= n m[l+ [.(SSRadj l/2
-
(sse
ail" )1/2 J[. (SSR
2a S
12 w
)1/
2
- (sse )1/2]J
(3.2.19)
37
GtTIl\1ERAL SOL'GTlON FOR r > 2 ana. c ~2 J
4.
GR.n~ral
4.1
• • • ;1
.,.
Renarks
Ir, the preceding section we have shown that th2
LagraL.g~..an
illulti-
pIfer' J w., can be found by the solution of a simple quad.:cat.:c equation
and that the explicit algebraic form of the test statistic;;
'T
R» can be
gi.ven f;)r two special cases: for equal no. with an)' number of rQ1"s and
l.J
.
columns and. for r=c::2 with the no. not equal.
l.J
For ur"equal no
0'
1.J
when the problem size is increased to r=3 and
c=2 the polyLomial inw is of' degree 2(r+c-3)
size is increased to
2(r+c-3)
:=
r=c~
6; and so on.
= 4;
when the problem
the polynomial in w is of degree
We know that a general polynomial of fourth
degree is solvable in terms of rad.icalsj however, it does not appear
that the required cumbersome algebra is warranted in order' to obtain
a solution for one additional special case.
Especially since this
would contribute nothing toward a solution for the general case.
also
krJ.01oT
We
that the algebraic solution of the general polyno:rr';'.al cf
degree greater than four is impossi.bleo
Certain spee:i.al forms of'
higher degree polynomials ean be solved by direct fact.oring or other
means but the ca.se in quest.ion does not hold the promise of a. ready
solution due to the manner in which the polynomi.al coefficients are
functions of' both the obeervation values and the number of' observations
per cell, no
.0
l.J
We have J thez:'efore;l directed. the work toward obtaini.ng
a method of Ui.llllerical sc>luticm for those problems having r > 2 andc
••• J
r.
:=
Two somewha.t different routes leading ·to rm..."'llerical. e.olutions
are presented in. the next two sections.
These two dif'fe?' only in the
1. ,
38
point at which the algebraic development terminates and the numerical
solution is started.
4.2 Explicit
Algebraic Representation of the
IJ
I
-L_ -1.0
Equation . .t* B lJB
~*
== 0
For these problems of size up to and including r=c=4,
i.~oJ
poly-
nomials up to degree ten, the polynomial coefficients have been
derived algebraically.
.Numerical solution of the polynomial then leads
directly to a calculated value for AHo
In this section we will illus-
trate the derivation of the polynomial coefficients by using the
problem with r==c=3 as an
exa~pleo
The coefficients together with other
necessary quantities are given in append.Ix 8 2 for all combinations of
0
rows and columns up to r=e=4.
For r=c=3 we have
( a~l +
a~l
- 1 ) (alla 12 + a 2l a 22 ) (all a 13 + a 21a 23 )
2
2
(a
+ a
22 -1 )
12
U
:=
and from (8.1.1.2)
'*,
(a a
+ a a )
12 13
22 23
(2
+ a2 )
a
23
13
39
2 + 2
)
- ( all
a 21 - 1
o
o
o
o
o
o
o
o
It can be confirmed by inspection that the inverse of U is:
o
2
alla13a24
*
a
2 2
a
13 24
o
It is now necessary to find the sums of principal minors of U.
The sum of the one-rowed principal minors, Pl' is:
3 2
4 2
Pl = trace U = ~ a . + ~ a 2 . -2
j=l l J j=l J
or since a
14
4
=0
2
4
2
Pl = ~ a l · + ~ a 2 · - 2
j=l J j=l J
From (2.3.14), each of the first" two terms is unity; therefore, Pl = O.
40
The sum of the two-rowed principal minors of U is:
P2 = {[ (ail +
a~l
-
1)(a~2
+
a~2
-1) - (alla 12 + a 21a 22 )2J
2
2
2
2
2J
+ [ (all + a 21 -1)(a13 + a 23 ) - (a a 13 + a 21 a 23 )
ll
+ [(ail +
a~l
-
1)(a~4)
-
a~la~4J
which by rearrangement and use of equations (2.3.14) and (2.3.15)
becomes
Rather than proceed directly to find the sum of the three-rowed
principal minors of U, P3' we can save much algebraic manipulation by
making use of the relationship of the sums of principal minors of a
matrix and those of its inverse as given in appendix 8.1.3.
By inspec-
l
tion we see that the sum of the one-rowed principal minors of U- ,
is
p{ = trace
U-
l
=0
p{,
and from equation (8.1.3.3) the sum of the three-
rowed principal minors of U is
From the proof given in appendix 8.1.2 we have for the sum of the
four-rowed principal minors of U:
P4
= lui = ai3a~4
41
Summarizing J we have
p
1
P2
=0
:=
-
P3 = 0
2 2
P4 := a 13 a 24
In appendix 8.1.4 it is shown that the inverse cf B(4x4) can be
expressed as:
B-
1
=
MW
3
2
I - w (U- PI I ) + w(if- Plu+P2 I ) - (tf'-Plif+P2U-P3I)] •
(4.2.5)
Since in this case J PI
= P3 = 0
we have:
(It should be noted that P = P =0 does not hold in the other case
1
3
that yields 4x4 matrices U and B.
4x4 matrices but PI
:= -
2 and P
3
:=
When r =4 and c
2a2 2
and 1
a =
14
14
= 2, U and Bare
o.
See appendix 8.2.)
l
Using equation (4.2.6) we can expand B-luB- :
-L_-l ~TBT
( 1 )2{w6U - 2w
. 5 U2 + wU
4[ 3 + 2(U3 +p U) ]
B -UB
2
- 4W3[U(U3+P2U)] + w2[3if+P2I)(U3+P2U)]
If we express the inverse of U as given by equation (8.1.3.1)
U-
l
= (_~)4P4
=
-1
P4
[u3
[u3 -
Plif + P2U - P3 I]
+ P2U]
42
then rearranging we have
3
-1
U + P U = -p U
(4.2.8)
3
U
(4.2.9)
2
4
and
-1
= -P4U
- P2U
Substituting (4.2.8) and (4.2.9) into (4.2.7) gives:
-L_-l £ 1)2 { 6
5_2
41 ( - 1
]
B -UB
~TBT
w u - 2w U- + w L3 -P 4U - P2U) + 2P2U
2
3
2
l
l
- 4W [U(_P U- )] + w [(3U + P2 I)(-P4U- )]
4
(4.2.10)
Thus, from equations (4.2.10) and 4.2.4) we can write g*'B-luB-lg*
=0
as a polynomial in w:
[g*'ug*Jw
6
-
[2~,u2g*Jw5
+
[(a~3
+
a~3
+
a~4)l*'U~
-
3a~3a~4l*'u-lf*Jw4
2 a 2 [3"*
'U D*
(2
2
2 )/'* 'U-1J?*J
2 2 0* '1'*J
+ [ 4a13a24~
_ w3 - a 13
£ w2
24 ~ ~ - a 13 + a 23 + a 24 _
where
g*'U!..*
2
(£21 +
2
122 )
2 2 2
+ a 13 31 + (a23131 + a 24R32 )
R
- (al J!21 + a 12R22 )
2
- (a21'21 + a22f22 )
2
43
2)
R +132
1
{2
2 [( 2
-2~~2a 13 a 24
31
a a
13 24
+
2a13a24[(all~21
- (2
P21
2)J
+R22
+ a12R22)a24R31
+ (a21A?21 + a22l:22)a13R32
- (allR21 +
4.3
a12i22)a23~2
J }
An Example Illustrating Use of the Methods
Two systems of log grading (stratifying), each comprising three
grades (strata), were developed from different logical starting points.
System
"R" was developed with primary consideration given to size and
relative position of clear areas on the log surface.
system
"e",
In developing
emphasis was placed on size and character of visible de-
fects such as knots.
tures in common,
In addition, the two systems have certain fea-
~.~.,
each specifies the minimum log diameter for each
grade although these minima are not identical for the two systems.
A sampling study of white oak sp. logs was made in order to compare
the effectiveness of the two grading systems.
The methods discussed in
the previous section will be illustrated with the data from this stUdy1{
The pertinent information from this data is summarized in table 4.1.
The upper number in each cell of the table is the number of logs, n .. ,
lJ
and the lower number is the sum, Y. 0' of the observations, y .. k' on the
lJ
lJ
nl"Jo logs.
The observations, Yo Ok' express log value (i.e., total value
lJ
- -
of all boards produced from the log) on a per unit basis with one-thousand
board-feet as the unit.
In addition to the values given in table 4.1 we
1/ Data furnished by U. S. Forest Service, Forest Products
Laboratory, Madison, Wisconsin.
44
will need the pooled within-cell sum of' squares, S
w
the total sum of squares, ST
Using the computer
:=:
:=:
176,732.82, and
353,057.98.
progra~
described in appendiX 8.3, the fol1ow-
ing quantities were computed from the data appearing in table~4.1
all
= 0.48759129
a
~
a
a
12
21
a
0.0037259931
a
13
14
=0.19093343
==
0.87306405
:=:
0.0
_. -0.10884114
:=:
0.51739437
~1
:=:
338.70728
131
122
:=:
231. 73764
f.o2 '" 66.045057
22
Table 4.10
:=:
:=:
0.82704357
46.768679
Summary of' white oak sp. sawlog data
System C
Grade 1
Grade 1
Grade 2
49
88
7
144
5321.10
9047.85
650.05
15019000
299.
60
364
5
System
Grade 2
415.50
24560065
Grade 3
Totals
4444 035
29420.50
R
Grade 3
Totals
3
75
182
260
260080
4824 056
11036.85
16122.21
57
462
249
768
5997.40
38433.06
16131.25
60561. 71
45
t
t
::: 206.84622
21
::: 234.10209
22
Using the above quantities we have from equations (4.2.4),
~l. 450883
P2 P4
<-
0.52137355,
from equations (4.2.12), (4.2.13), and (4.2.14),
g*'ul*
= -70,835.812
g*'uF£* ~
110,918.71
g:'U-lg* "" -11,130.558 ,
and from equation (4.2.9),
If* 'u3g*
::;
-
-
-p
Jl* 'U-1i*
-
4-
~
p
R* 'U(l:'*
~-
::: - 45,253.084
Also, we find
!! 't'¥.o :::
174, 974. 21
Using the quantities given by (4.3.5), (4.3.6), and (4.3.8) to calculate the numerical values for the coefficients, the polynomial equation
(4.2.11) becomes
6 5 4
3
70,835.812w + 221,837.42w - 70,801.40w - 364,907.7Ow
2
+ 26,180.293w
+ 150,373.66w + 30,256.238 ::: 0
Numerical solution of this polynomial using the computer program described in
a~pendix
8.4 yields the following calculated real roots.
46
We must now determine which of these roots min:i.mizes the sum. of squares
f*B=~2B-l!.*
Note that since the left si(1e of equation (4 02.11) is
0
IBI20g*B-~-~~ we can obtain g*B-~~-ll* by s:i.mply increasing the
exponent of U by one i.n that equationo
TherefDre J by substituting the
numerical values from equations (4.3.5) through (4.3.8) into the modified equatior. (4 2 11) we have
0
0
6
5
4
3
+ 90 J 506.17Ow - I11 J 951.51w - .147,727.68w
2
- 56,1140538w + 60,512.476w + 47,563.306]/IB\2
[110,918.71w
From equation (8.1. 4. 7) we have
which upon substituting for the p. 's gives
~
IBI= w4
2
- 1.4580883w + 0.52137355
Using the four real roots of g*/B-luB-l~*
=0
g*/B-l~B=l~* = 328,166.22
=
14 J 784.071
= 162,307097
=
79,772.325
Thus, we have for the test statistic J
Til = n In [1 +
g*B~lu2p'*
]
w
= 768
In[l +
= 768
In(1.0836521)
= 61. 73
14,784.071 ]
176,732.82
we have:
for WI '
for w2
~
(4 03.13)
•
47
Since X(1,O.95)
=:
3.84, we conclude that there is s';ll'f'icient evidence
to reject the null hypothesise
From equations (4.303) and (4.3.4) we
have
SSR
and
~ +,2
~n~ + t 2
22
c..L
o"S'"v
-
=:
97,5890148
from which we find the following estimate:;:; of
within col:mnns:
2
s (within rows)
va:~iance
within rows and
ST - SSR
.. 3
- 241035
n2
ST - sse
8 (within columns) =:
U-3
=: 334 034
=:
Therefore, we conclude that system "R" is the better of the two systems
with respect to the reduction of variance within grades (strata)o
4.4 Numerical Solution for
of Size Greater than
~~oblems
r~=4
It is obvious that the methods presented in the two preceding
sections are suitable only for problems of modest sizeo ' Larger problems can be handled more easily if the numerical solution is started
directly with B-
1
as given by equation (4 0205), or in the general
equation (8.1.402)0
-1
In oruer to express B
as a polynomial in w with numerical matrix
coefficients we must find the numerical values for Pl", eoo.,.Pr+c-3 and
.2
,.r+c-3
for U, U-, ooe, U
Were it not for electronic computers, the
0
calculation of these quantities would be prohibitive in most cases.
However, every research computing center today has available programs
for matrix multiplication and solution of determinants for large
matrices.
The maximunl size matrix accommodated, of course, depends on
computer capacity and related factors.
Available programs could, in
most cases, be used for the necessary calculations with only slight
modification.
The amount of calculation can be consid.erably reduced by making
use of theorem (8.1.1) and the relationship given in (8.1.3).
Using
theorem (8.1.1) we have
from which it follows that
:::
and
and so on.
Thus, we need not find the (r+c_3)rd power of U but can
work with lesser powers of the partitions of U.
If r+c=3 is
even~
the
highest power of the partitions that is required is (r+c=3)/2 and if
r+c-3 is odd, the highest required power is (r+c-2)/2o
values for Pl' •
0 .,
p +
r c-
In finding the
3 if we use the relati.onship between the sums
of principal minors of U and U-
l
as given by equation (8.1.3.3) we can
work with determinants having about one-half as many rows and columns
as would otherwise be necessary.
For purposes of illustration, suppose that r=e=lO in which case
U would have dimensions (18 X 18) and we would need to find all powers
49
of U up to (r+c-3)
:=
17.
By using the approach as outlined above, we
could accomplish this by finding all powers up to (r+c-2)!2
andU22 , and the products UllU12 ;>
~lUl2'
titions each have dimensions (9 X9).
••• ;>
-1
where the par-
To find Pl' ••• , P17 we would first
find Pl' ••• , P9 directly from U then find Ufrom U •
~lUl2
= 9 of U1l
l
and derive p{ ••• ,
P~
Using equation (S.1.3.3), PIO' u., P
would be directly
17
I
I
available from P ' ••• , P •
S
l
Thus, the maximum size
solved is reduced from (17 X17) to (9 X9 ).
determinar~
to be
50
5.
EMPIRICAL SAMPLING EXPERIMENTS
In the development of a test statistic, we have used the result
from large-sample theory that, when He is true,
TH
=-2 ln A. has the
2
X distributionj however, in actual practice we usually will be working with relatively modest sized samples.
Therefore, empirical sam-
pling experiments have been made in order to study the behavior of
T
H
for some small sample sizes.
These empirical experiments included
a study of the power of the test as well as a study of the central
distribution.
Thirteen combinations of row and column dimensions and sample
sizes were tested.
These combinations are given in table 5.1.
each of these combinations the "trial values of T
step process using two computer programs.
H
For
were found by a two-
This process will be de-
scribed using the combination having r=c=2 and all n .. = 10 to
lJ
illustrate.
The steps performed by the first computer program are:
(a)
Input to program is 4000 cards each containing 72 random
digits (1.~., 10 four-digit random numbers plus an additional 32 random digits) from the RAND Tables (20).
(b)
Each four-digit random number transformed to a random normal
deviate as shown in table 5.2.
(c)
For the ten random normal deviates generated from a single
input card, the sum and sum of squares were calculated and punched on
an output card.
Each of the 4000 output cards also contained the last
12 random digits from the input card.
51
Table 5.1.
Rowand column dimensions and sample sizes for empirical
sampling experiments
p!::xperiment
number
1
Number of
rows and
columns
r=e=-2
Number of observations
per cell
Total
number of
observations
all n .. = 5
20
= 10
40
= 20
80
all Ii.. . = 5
30
= 10
60
= 20
120
J.J
2
3
4
r=3,?c=2
J.J
5
6
r=e=3
all n .. -
5
45
8
= 10
90
9
= 20
180·
7
10
r=e=2
lJ
nIl=20, n12 =10, n21 = 5, n22 =15
50
11
nIl=40, n12 =20, n21 =10, n22 =30
100
12
nIl=20, n12 = 5, n21 := 5, n22 :=20
50
13
nIl=40, n12 =10, n21 =10, n22 =40
100
.
52
Table 5.2.
Transformation of rectangular distribution of four-digit
random numbers to random normal deviates
a
Random
number
3
7
11
15
19
27
39
51
67
91
119
155
199
255
319
399
495
607
735
883
1055
1251
1467
Normal
deviate
-33
-32
-31
-30
-29
-28
-27
-26
-25
-24
-23
-22
-21
-20
-19
-18
-17
-16
-15
-14
-13
-12
-11
Random
number
1711
1975
2263·
2575
2911
3263
3631
4011
4403
4799
5199
5595
5987
6367
6735
7087
7423
7735
8023
8287
8531
8747
8943
Normal
deviate
-10
- 9
- 8
- 7
- 6
- 5
- 4
- 3
- 2
- 1
0
1
2
3
4
5
6
7
8
9
10
11
12
Random
number
Normal
deviate
9115
9263
9391
9503
9599
9679
9743
9799
9843
9879
9907
9931
9947
9959
9971
9979
9983
9987
9991
9995
9999
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
a A random number less than or equal to the number in the first
column yields the normal deviate in the second column.
53
The second computer program completed the following steps:
(a)
The desired values for ~ll' ~12' ~21' and ~22 were read into
the computer from parameter cards.
These values of the
~
.. were pre-
~J
calculated to give a required value for the variance ratio, V = cr 2/cr 2
R c
r
One computer run was made for each of the desired values of V.
r
four runs were made with V =1 (null case), V == 1.5, V
r
r
r
(b)
==
In all
2 , and V
r
in groups of four.
The
These cards were read into the computer
summa~
information contained in each group
of four cards was used, together with the parameters for the desired
V , to calculate a value of the test statistic.
To shorten the com-
r
putations, the form of the test statistic was changed as follows:
Pr {
[(SSR)l/:~
(SSC)1/2 J2
~
2
2 [1 -
e~ X12~1-21)
] }
== O!
w
Thus, the program calculated only
for each sample.
(c)
==
The input cards for the second program were the 4000 output
cards from the first program.
A deck of 1000 cards each containing one trial value of the
test statistic was the output for each computer run.
These cards were
sorted to arrange the trial values in ascending order,
the empirical distribution of the test statistic.
~.~.,
•
to give
3.
54
(d)
Between successive runs for V
I'
= Ip
1.5,
2~
deck was sorted on thTee co2.umns of the twelve rand.om
Ll each ca::'d.
3
the input card
Il'..unbers
contained
Thus, the empi:rical results between values of' V:r :for a
given size arx:ay and size sample are not independent.
This means that
the i!ldivi(lual poir.ts generated to const!'uct an empirical power curve
are correlated.
However, since new random deviates were generated for
each of the thirteen combinations given in table 5.1, their respective
power curves are independen:t.;.
The empi!'ical frequency distributions generated by the thirteen
s~~pling experj~ents
are shown in figures 5.1 through 5.5.
These dis-
tributions are arranged to show the number of observed values of
T
H
2
tha.t occurred in each ten=percentile range of a true X distribution.
'Using E (number of observations)
= 100
2
for each percentile group, X
for goodness-of-fit was calculated and is shown on each of the figures •
.A study of these empirical distributions indicates that for small
samples the 11pper tail of the
2
the X distribution.
T
H
distribution is larger than that of
However, when the number of observations per
cell is ten or recre, the discrepancy is small enough to be quite
tolerable,
10~0,
for a nominal type I error, a
=.
0010, the test may
have an actual a equal 0.11 or 0.12.
Since we have neither the central nor the nOrl=central small
sample distribution for
'Tn'
we cannot compare the empirical power
curves to the theoretical curves.
In fact, if the theoretical curves
were known, there would be no need for the empiri.cal sampling experiments o
However, it is inf'orrnative to compare the empil'ical power
curves for
'T H with
the power of an alternative test proced.uree
If
the total sampling ef'f'ort is divided so that n/2 observations are
55
160
140
120
r--
100
""--1--
----
~
80 ~
60 i
n=20
40
all n .. = 5
20
2
X = 27.24
~J
**
o
120
--
-
100
80
~r--
r---
--
l>a
= 40
n
60
40
all n
g.
20
~
X = 5.22 N.S.
0
S
CD
ij
= 10
2
120
100
80
n = 80
60
all n
40
01----1_--L._-L._..L..._J.----lL......-..L_--L._..J----I
10
20
30
40
50
60
70
80
90
100
Nominal percentile group
~.
',.
"
"
= 20
2
X = 10.36 N.S.
20
. . . .'.
.\...... ~ .
. : .~"
ij
Figure 5.1.
Empirical distributions for r=e=2
with e~ual cell sizes
56
1_-a.__
80
n ==
30
n ==
60
60
40
20
o
120
100
----
-
--'
~
-
f--
_r---
80
60
40
all n = 10
ij
20
?C == 14.18 N.S.
2
o
120
100
80
n == 120
60
40
all n .. = 20
~J
20
_ _4___l______l._....l
70 80 90 100
04----II.--.L.-..L--l--~_~
10
20
30
40
50
60
A
2
== 6.12 N°.S.
Nominal percentile group
Figure 5.2.
Empirical distributions for r=3 J c=2
with equal cell sizes
57
140
~
12 c
10{
e---
....~
8 01--
-
-
-
I---
n
=:;
45
n
:=
90
60
40
20
0
120
100
80
60
p"
0
l:l
\l)l
~
~7:
CD
40
all n, .:"~ 10
lJ
20
2
X = 10.,16 NoS.
0
J-;
~·:r.i
120
100
.--
"'-1---
I
I--
80
11 ,,~
180
60
8,11 n
40
. --
l,J
20
2
X :::: 4~ 88 1\1". S.
20
o
10
20
30
40
50
60
70
80
90
100
Nominal percentile group
Figure 5.3.
Empirical distributions for r=c=3
with e~lal cell sizes
58
n = 100
rill = 40
n
21
2
X
:=
== 10
n 12 = 10
n
22
== 40
10.90 N.S.
140
120
100
n == 50
80
n
60
n
40
ll
21
== 20
n
= 5
n
12
22
= 5
= 20
X 2 :::: 7.4 2 N.S
•
20
0
10
20
30
40
50
60
70
80
90
100
Nominal percentile group
Figure 5.4.
Empirical distributions for r=c=2
with unequal cell sizes-part one
59
140
12
n = 100
10'-1-----1
8
n
6
ll
= 40
n
12
= 20
n
= 10 n = 30
22
21
2
X = 10.90 N. S.
4
2
n
n
= 50
ll
= 20
n
12
n
=10
= 5
n =15
22
21
2
X = 22.42
**
10
20
30
40
50
60
70
80
90
100
Nominal percentile group
Figure 5.5.
Empirical distributions :for r=e=2
with unequal cell sizes~part two
60
taken to estimate average within-row variance and another independent
sample of n/2 observations is taken to estimate average within-column
variance, then the ratio of MS (within rows) to MS (within columns) is
distributed approximately as F(~ -
r,¥ - C)e
for F are much easier than those for
T
Since the calculations
, we might choose F as the test
H
statistic i f the two tests were about equal in power.
Power curves for the F..,test with type I error fixed at
Ol =
e05
over a range of V sufficient for comparative purposes were obtained
r
from tables given by Tang (22) and from nomographs by Fox
curves together with the empirical curves for
5.6 through 5.10e
T
H
(14). These
are show.n in figures
It is apparent upon inspection of these curves that
the gain in power of
over F is sufficiently large that in most
H
applications we would prefer T as the test statistic.
H
T
61
~------/
,
1.0
/
/
.9
/
' /
I
I
tJ;:l0
/
/
I
I
/.
/ /
I
.6
II
/
I
I
/
"I'"f
~
()
Q)
...
Q)
p:<
4-t
I
'/
I
.....
I
~
I
/
I
.4
,0
I
/
/
I
l>.
~
"I'"f
~
/ /
I
.5
0
..-f
I
I
I
/
I
I
I
.1
/
/
/
/
/
/
I
I ,
.2
"
/
"/ ./
,/
/1I'
.3
/
/
/
/
I
.7
W
'
/
f
/'
/
/
.8
~
/r /
, ~'
n. =
5
l.J
n. 0=10
l.J
n .= 20
l.J
o
------------
o
l
0.0 '-----L
1
---JI--
..J-
3
2
2/ 2
Vr=CYRCY C
Figure 5.6.
Empirical and theoretical power curves for
r=e=2 with equal cell.sizes
...._
62
,,
1.0
L:x---
.9
,
!(
,
...,
Q)
05
CH
,
of"f
~
.4
,0
e
.3
I
I
.2
/
I
I
/
I
I
I
I
I
I
I
I
I
I
I
I
r
I
Po!
/
/
/I
I
°I>t
~
/
/
i'I ,I
~
()
Q)
r-i
I
I
!I
.6
of"f
of"f
/
/
1I
0
,f
/
I,
p:j
I
III '
.7
~
/
,
.8
l:I:l
/
I
/
/
/
/
/
/
/
/
/
,!,
f
Il .. = 5
~J
n .. = 10 - - ~J
"
n .. = 20 - - - - - ~J
.1
o.ol----L.---------L.---------I..-----1 2 3
2
V r =crR/
Figure 5.7.
2
crc
Empirical and theoretical power curves
for r=3,c=2 withe~ual cell sizes
63
/~-/
1.0
t(
/
.9
I
.-
I
I /
/
I , !J
I
I ,
.8
I
I
/
II
.7
I
I ' I
II /
I I
III
I I
.6
I
I
.5
/
I
/
/
/
/
+iI
I I
I
I
.4
I II
I'
I
I
I
I
I
I
I'
I~
I'
I
n .. =
lJ
5
n .. = 10 - - - - - - -
lJ
n.
0=
lJ
20 - - - - - - -
o. 0 lI.--l~---------:!:2-------~3:!:------
Figure 5.8.
Empirical and theoretical power curves
for r=e=3 with equal cell sizes
64
1.0
.9
.8
.7
II?
tlO
1=1
.6
..-I
~
()
Q)
..,
~
CH
0
.5
~
~
'1"'1
r-I
or-!
.g
84
n
.c0
n
J.I
Pi
ll
12
= 20
= 10
5
21 =
n = 15
22
Yl
.3
n
.2
ll
= 40
n 12 = 20
n
n
.1
= 10
21
22
-------
= 30
o. 0 l...---Ll--------~2-----------,J3L...--------T2/ 2
Vr=O'RO'C
Figure 5.9.
Empirical and theoretical power curves for
r=e=2 with unequal cell sizes-part one
65
1.0
/
.9
/
.8
.7
:i
~
orf
.6
+:>
()
Q)
.,..,
Q)
fl:\
Cl-f
0
.5
~
+:>
orf
ri
"I"'l
~
.4
n
.0
0
n
~
n
.3
n
n
.2
n
n
n
11
12
21
22
11
12
21
22
= 20
=
5
=
5
= 20
= 40
= 10
= 10
------
= 40
.1
o. 0 l--~1--------~2L---------3:!:------"""'Vr =
Figure 5.10.
cr~/cr~
Empirical and theoretical power curves for
r=e=2 with unequal cell sizes-part two
66
6.
SUMMARY AJ.\ID CONCLUSIONS
The primary result obtained in this dissertation is the derivation
of a test statistic for comparing the average within-strata variances
for two systems of stratification of a single population.
The popula-
tion conceptionally is considered as a two-way array with rows defined
by the strata of one system and columns defined by the strata of the
other system.
'T
The general form of the test statistic is:
= n In[l +
H
g*
I
l
(wI+Ur
U2(wI+Url.e* ]
S
w
where n = total sample size
S
= sum of squares within cells of the two-way array
g*
= vector of linear functions of the observations (see section 2.2) •
w
U = a matrix that is a function of the hypothesis tested and
of the structure of the sample (see section 2.2 through 2.4)
w= a Lagrange mUltiplier
A direct algebraic solution for w is not possible in generalj however,
a solution is obtained under certain restrictions thus yielding explicit
algebraic solution for
'T
H
in two special cases.
When the population cell proportions are all equal and thQS all
n .. = n/re, then
lJ
= n In[l +
[(SSR)l/~~
(SSC)1/2 J2 ]
w
where
SSR = sum of squares between rows
SSC = sum of squares between columns
67
In the case with unequal cell proportions but with r=e=2,
lo~.,
two rows and two columns, the test statistic is shown to be:
]
where
SSR adj = sum of squares between rows, aO_justed for columns
sse adj = sum of squares between columns, adjusted for rows
a
,
12
,
= -21-'31
c q
= -31-;;:1'
c 0_
and the vectors
~21' ~31' ~l' ~31
are defined by the following
relations:
('£21.l)
2
= SSR
(:!.3il:)2 :: sse ad,j
<-'11)2
(
~3JJ
I
=
sse
)2 = SSR adJ'
When the cell proportions are unequal and
I'
> 2 (c=2, ••• , r),
the Lagrange multiplier, w, must be found by solution of an equation,
g*'(wI+U)-l U(WI+U)-lg* = 0, which is a polynomial of degree 2(r+c-3)
in w.
O l
For arrays up to size r=e=4, the coefficients of the w , w ,
2
2(r+c-3)
.
w , ••• , w·
have been derl.ved algebra.ically and are given in
the appendix.
Numerical solution of this polynomial is necessary to
obtain w to use in calculating
~H
for a given sample.
A computer
program to obtain this numerical solution, based on Bairstowis method
is given in the appendix.
This procedure for numerical solution is
illustrated with data from an array with three rows and three columns.
The problem illustrated i.s a comparison of two systems for grading
68
(stratifying)white oak Spa sawlogs.
For arrays of size greater than
r=c=4, the algebraic derivation of the polynomial coefficients becomes
unwieldy and it is recommended that the numerical solution be started
a step earlier.
The test statistic has been derived from the ratio of maximum
likelihoods, A.
:=:
tH~J'
by taking 'T H
sample distribution of 'T
H
Although the large
2
is known to be that of X , i t is necessary
:=:
-2 In A..
2
to obtain some indication as to how well the X distribution approximates that of 'T
H
for relatively small samples.
Empirical sampling
experiments that were performed indicate that the upper tail of the
'T
H
2
distribution is slightly larger than that of the X distribution.
When the number of observations per cell is ten or more, the actual
2
type I error may be Q' = 0.11 or 0.12 for a nominal Q' = 0.10 based on X •
2
Thus, we conclude that the X approximation to 'T
H
is sufficiently
close so that in application the critical region for 'T
H
can be taken
2
as 'T H ~ Xl, (l-Q')"
The empirical sampling was extended beyond the null case to construct
power curves for the test.
These empirical curves for 'T
with the theoretical power of F( ~ - r,
¥-c)
test of the null hypothesis being considered.
H
were compared
which could be used as a
This alternative test re-
quires subdivision of the total sample into two parts in order to obtain an independent estimate of the average within-strata variance for
each of the two systems of stratification.
The comparison of the power
of the two tests shows that 'THis greatly superior to F(
¥- r, ¥- c ) in
the ability to reject the null hypothesis when it is false.
69
7.
LIST OF REFERENCES
1.
Anderson, R. L. and T. A. Bancroft. 1952. Statistical Theory
in Research. McGraw-Hill Book Company, New York.
2.
Browne, E. T. 1958. Introduction to the Theory of Determ.inants
and Matrices. University of North Carolina Press, Chapel
Hill, N. C.
3.
Campbell, R. A. 1964. Forest service log grades for southern
pine. U. S. Forest Service, Southeastern Forest Experiment
Station (Asheville, N. C.) Research Paper SE-ll.
4.
Cochran, W. G. 1960. Comparison of methods for determining
stratum boundaries. Bulletin of the International Statistical
Institute 38(2):345-358.
5.
Dalenius, T. 1950. The problem of optimum stratification.
Skandinavisk Akuarietidskrif 33~203-2l3.
6.
Dalenius, T. 1952. The problem of optimum stratification in
a special type of design. Skandinavisk Aktuarietidskrif
35:61-70.
7.
Dalenius, T.
sampling.
8.
Dalenius, T. 1962. Recent advances in sample survey theory and
methods. Annals of Mathematical Statistics 33(2):325-349.
9.
Dalenius, T. and M. Gurney. 1951. The problem of optimum
stratification II. Skandinavisk Akuarietidskrif 34:133-148.
1953. The economics of one-stage stratified
Sankhya 12(4):351-356.
10.
Dalenius, T. and J. L. Hodges, Jr. 1957. The choice of
stratification points. Skandinavisk Akuarietidskrif
40:198-203.
11.
Dalenius, T. and J. L. Hodges, Jr. 1959. Minimum variance
stratification. Journal of the American Statistical
Association 54:88-101.
12.
Ekman, G. 1959. An approximation useful in univariate stratification. Annals of Mathematical Statistics 30:219-229.
13.
Feller, W. 1950. An Introduction to Probability Theory and Its
Applications. John Wiley, Inc., New York.
14.
Fox, M. 1956. Charts of the power of the F-test.
Mathematical Statistics 27:484-496.
Annals of
70,
15.
Grandage, A. 1958. Orthogonal coefficients for unequal
intervals. Biometrics 14(2):287-289.
16.
Hagood, M. J. and E. H. Bernert. 1945. Component indexes
as a basis for stratification in sampling. Journal of
the American Statistical Association 40:330-337.
17.
Newport, C. A., C. R. Lockard, and C. L. Vaughan. 1958.
Log and tree grading as a means of measuring quality.
U. S. Forest Service, Washington, D. C.
18.
Newport, C. A. and W. G. OIRegan. 1963. An analysis
technique for testing log grades. U. S. Forest Service,
Pacific Southwest Forest and Range Experiment Station
(Berkeley, Calif.) Research Paper PSW-P3.
19.
Petro, F. J. 1962. How to evaluate the quality of hardwood
logs for factory lumber. Canada Department of Forestry
(Ottawa) Technical Research Note No. 34.
20.
RrND Corporation. 1955. Million Random Digits with 100,000
Normal Deviates. Macmillan Company, New York.
21.
Robson, D. S. 1959. A simple method for constructing orthogonal polynomivl S W'l'i' the independent variable is unequally
spaced. Biometrics 15(2):187-191.
22.
Tang, P. S. 1938. The power function of the analysis of
variance tests with tables and illustrations of their use.
Statistical Research Memoirs (University College, London)
2: 126-157.
23.
Vaughan, C. L. 1958. Development of log and bolt grades for
hardwoods. U. S. Forest Service, Forest Products Laboratory
(Madison, Wisconsin) Research Paper No. TGUR-16.
24.
Watson, G. S. 1965. Equatorial distributions on a sphere.
Biometrika 52:193-201.
25.
Wilks, S. S. 1938. The large-sample distribution of the
likelihood ration for testing composite hypotheses. Annals
of Mathematical Statistics 9:60-62.
26.
Wishart, J. and T. Metakides. 1953.
fitting. Biometrika 40:361-369.
Orthogonal polynomial
•
71
8.
8.1
8.1.1
Theorem on
APPENDICES
Thec;rems and Derivatior:s
u2
I f U is the matrix defined by (2.4.3) and i f we define the
.partitio:J.s:
Ull(r-l,r-l)
(S.1.1.1)
U{2(c-l,r-l)
then
(8.1.1.2)
•
Proof:
.h
c th
and-ce~
the (a,~)t~ element in A~ is the product of the a th row
l
coll~
of A. , i.e.,
l
The expression enclosed by parentheses is given by equation (2.3.14)
to be unity;
( a,~ )
therefo~e,
th
2
element in Ai
~ aiaai~
which is seen to be the same as the (a,~)th element in A.• Therefore,
l
Ai is an id.empoter..t matrix, i. e.,
2
A.
l
•
~
(8.1.1.3)
A.
"-
The (a,~)th element in A.A. (i " j)' is:
l
J
a. a.,(a.la'C)+a. a. 2 (a. 2a' C)+••• +a. a.
~(a. r+ 2 a . C)
la l~ J
J~
la l
J J~
la l,L~C-C J,
cJ~
-l
= a.le'.1..,
a'C(a.la'
l
Jl
+ a.1. 2a'J 2 + '"
+ a
a
)
i,r+c-2 j,r+c-2
72
(2.3.15)
which from
A.A.
l
J
=0
is seen to be zeroj therefore,
(S.1.1.4)
(r+c-2,r+c-2)
Now, partition A. as follows:
l
A.l
(S.1.1.5)
=
. Then,
(S.1.1.6)
where
K is
Expanding
defined by (2.4.~).
if
gives:
2: [ I
u
= [
Ai -
KJ2
(i=l,
oe.
,c-l)
I Ai J2 + K2 - [ I Ai JK - 1{ I Ai J
which by using equations
(S.1.1.3), (S.1.1.4),
2
noting that K = K, becomes:
and
(S.1.1.6)
and also
73
=
which completes the proof.
8.1.2
Theorem on the Determinant of U
The determinant of U is:
(8.1.2.1)
",.Q)th element of U as:
write the ( ~,~
c-l
li
aiaai~ - e
aS =
i=l
Proof:
I
where a=l, ••• ,r+c-2 ;
e =1
= 0
for ot
~=l,
(8.1.2.2)
••• ,r+c-2 ; and
= ~ = 1, ••• ,r-l
otherwise.
(8.1.2.3)
and
Multiplying the athrowand Sth column in the determinant for all a and
~
gives:
lui
1
c-l
~
a
a
( a . a. - e)
= ~-:::--( C-l,ot c-l,S i~ lot lS
r+c-2 2
.TI
J=l
a
1 .
c-,
J
(8.1.2.5)
Now, the sum of the otth row, for ot ~ r-l, of the determinant on the
. right is:
r+c a 2
c-l
I[aC_l~aaC_l~~Iaiaai~J
~=l
- a
74
2
c-l,a
- i=l
r+c-2
= \L [a c-l~aa c-l~~Q(alaa l~Q+ ••• +a c-l~aa c-l~~Q)J
- a
2
c-l,a
S=l
r+c-2
=
L [ac-l,aa la (ac-l~~Qal~Q)+ ••• +ac-l,aa c-l,a (ac-l~~Qac-l,~Q)J
\
- a
S=l
r+c-2
r+c-2
= a c-l,aa la[ \L a c-l,~Qal~QJ+ ••• +ac-l~aa c-,a
2 [ \' a
L c-l,~Qac-2~~QJ
S~
S~
- a
2
c-l,a
=0
since from (2.3.14) and (2.3.15) the first c-2 terms in braces are
zero and the last term in braces is unity. Similarly, the sum of
the a
th
row, for a=r, ••• ,r+c-2 , is:
=a
2
c-l,a
Therefore, the last row of the determinant is:
2
o, ••• ~o,ac_l,l
2
2
' a c _l ,2 , ••• , a c-l,r+c-2
where the first r-l elements are zero.
If, in the same
way~
we sum columns then:
2
c-l~a
75
1
lui =----x
r+c-2 2
II a 1
j=l
c-,J
0
2
a c-l,l a c-l,r+c- 3ul,r+c- 3
a
u . .11. . . . . . . . . .
c-l,l
•
·•
··•
··
•
.•
...·•
..
•
.
o
.
2
o
a
a
•
2
c-l,r
c-l,r
··
a
2
c-l,r+c-3
r+c-2
I a~_l,C¥
2
a
2
o
a c-l,r+c- 3a c-l,lu r+c-3,1 ••• a c-l,r+c- 3ur+c-3,r+c-3
o ••• 0
0
•
c-l,r+c-3
c¥=r
(8.1.2.8)
or
• • • • fJ • • •
u
e ••••
•
·
1
lui = - - a
2
c-l,r+c ..2
····
·•••
·••
•
u
r+c-3,1
o •••
0
•••••• 0
a c-l,r
u
•
o
1,r+c-3
•
o
a
r+c-3,r+c-3
a
.
c-l,r
c-l,r+c-3
a c- 1 ,r+c- 3
00.
(8.1.2.9)
Repeating these operations using the a
CoO
2
0
,J
'000,
a lJo respectively
on the first (r+c-3), ••• ,(r-l) rows and columns gives:
76
O(r-l,c-l)
r+c-2
1
lui
\' a a
L lr c-l,O'
O!=:r
=---c-l 2
II a.~,r-1+.~
i=l
O(c-l,r-l)
r+c-2
a
I c-l,O' 10'
O'=r
a
0
•••
(8.1.2.10)
1
= ------c-l 2
II a.~,r-1+.~
i=l
Ull
I
0
(say).
~
And, by Laplace's theorem (2, p.20),
1
lui
=
----.IUlll.lu~21
(8.1.2.11)
c-l 2
II a.
1.
i=l ~,r- +~
Recalling from (2.3.6) that for i=1,000,c-2,
ai,r+l = ai,r+i+l = 000 = a i ,r+c_2 = 0
the second determinant above is:
2
aIr
a
a
lr 2r
r+l 2
I;a
O'=.t" 20'
a
a
lr 3r
r!;la a
0'=-'" 20' 30'
•••• 0
a
lr c-l,r
r~
. a a
..... O'=r
20' c-l,O'
•••• $
o
*
a
r+2
~ a
a
O'=.t" 3 0' c-l,O'
•
o
0
• o r+c..-2
2
].j a
O'=r c-l,O'
If we factor aIr from the first row and first column and a 2 ,r+l from
the second row and second column then:
77
1
a
(2r
)
a
2",r+l
a
2
(~r
a
a
a a
( 2r 3r, a
+1)
a 2,r+l
2,r+l
c-l,r
a a
) ••• ( 2r c-l....z.r + a
)
3 r+l
c-l,r+l
'
a 2,r+l
r+2
~ a
a
O:'=r 30:' c-l,O:'
r+2 2
~a
O:'=r 30:'
*
,r+c-2 2
~ a
O:'=r c-l,O:'
(8.1.2.13 )
a
Subtracting
••• , (a
2
(a r
X row 1) from row 2 and (a
3r
2,r+l
1
X row 1) from rows 3, ••
c- ,r
a
1
1~21
a
2,r+l
1
0
a 3 , r+l
.• ..
8
8
0
a.
c-l,r+l
1) ,
,c-l respectively gives:
2r
0
2 2
= a a
lr 2,r+l
8
x row
a'<..,r
:,
a
a 3 ,r+l
a
r+2
2
a3
0:'=r+1 0:'
•
~
8
8
.88
c-l,r
c-l,r+l
r+2
~ a
a
O:'=r+l 30:' c-l,O:'
···
r+2
r+c-2 2
~ a
a
••• ~ a 1
O:'=r+l c- ,0:'
O:'=r+l c-l,O:' 30:'
78
or
a
1
3 ,r+l
r+2
r; a
2 2
= a lr
a
2,r+l
a
•••
r+2
2
Q'=r+l 3Q'
c-l,r+l
r; a
a
Q'=r+l 3Q' c-l,Q'
*
r+c-2 2
!: a
Q'=r+l c-l,Q'
(8.1.2.14)
By repetition of this process we finally find
c-l 2
= i=l
IT a J.,r- l+J.
'
(8.1. 2.15)
o
Therefore, from (8.1.2.11) we have
(8.1.2.16)
From (8.1.1.2) we have
The matrix U has dimensions (r-l,r-l) ; therefore,
11
luFl=
(-1)r-llulll.lu221
lu.ul = (-1)r-llulll.lu221
lui. lui
and from (8.1.2.16)
= (-1)r-llulll.lu221
,
79
or
(8.1.2.17)
It now remains to evaluate IU221. From (8.1.1.1) and (2.3.6) we have
IU22 1=
c-l
c-l
~ a. a. -Ll.·. ~ 2 80 • a.
i=2 ~r l,L"'
i=c- lr ~,r+c~3
c-l 2
!; a.
i=l lr
a
a
c-l,r c-l,r+c-2
c~l
c-l 2
••• i~~ 2 80 ~,r+
•
1 80 ~,r+c•
3
Ea. '1
i~ ~,r-t'
a
··•
c-l 2
~ a.
3
i=e-2 ~,r+c-
*
a
c-l,r+l c-l,r+c-2
a
a
c-l,r+c-3 c-l,r+c-2
a
2
c-l,r+c-2
(8.1.2.18)
or, upon factoring
(a, 1 r+
c-,
c-
2 )/ (a 1
3) from the last row and
c- ,r,c-
last column
2
c-l,r+c-2 X
2
a c _l ,r+c_3
a
c-l
~ a. a.
i=2
~r ~,r+
c-l 2
~ a.
"
i~ ~,r+l
*
c-l
1 •••
~
i=e-2
a. a.
+
~r ~,r
c-
3
c-l
••• i=c~ 2 80 ~,r+
•
1 80 ~,r+c•
3
···
c-l
0
~ a~
3
i=e-2 ~,r+c-
a
a
a
c-l,r c-l,r+c-3
a
c-l,r+l c-l,r+c-3
··•
a
2
c-l,r+c-3
a
2
c-l,r+c-3
(8.1.2.19)
80
If we now subtract the last row, (c-l)·st , from the (c=2)-~d row,
subt rae t th e ( c- 1 ) st
"
COJ.:U1lL~
f rom t".he ( '~-c.)
,. ., \nd cO.Lurrm,
and +·h
v
en
multiply the l.ast row a:1.d last column by
c-l 2
L, a.
i=l ~r
lu22 I~'
2
a c-l,r+c~2
11(a c-1
J
r:-c=3
) we have:
c-l
L, a. a.
• •• a 0 a 2
3
i.=2 ~r 1.,r+l
c=""r c- ,9r+c-
a
c-l 2
~ a.
1
i=2 1.,r+
a c=l,r+l
••• a c- 2 ,r+l a c- 2 ,1-rC' 3
c-IJr
•
·
•a 2 •
o
c-2,r+c-3
*
1
(8.1. 2. 20)
Repeating the above operations with 1!(aC _ l ,r+C=4) , ••• , l!(aC_l,r)
being the fac-sors from the last row and last column for the
successive repetitions and S11btracting the last row and
from the (c-3 )rd., ••• , 1
st
coD~D
row and column for the successive
repetitions gives:
c-2
a
2
~a.l.ra.1.,r+l···ac- 2 ,ra c- 2 ,r+c- 3
i=2
o
c-2 2
~ a.
i=2 l.,r+l
o
c=1,r+c-2
.
..
·
• 2 •
a
o
o
1
••• a c - 2 , r+ 1 a C - 2 ,r-t·c = 3
c-2,r+c-3
o
o
(8.1. 2. 21)
Noting that the principal minor in the upper left is of the same
form as (8.1.2.19) , we see that continued. reductions must result in
a
=
222
a
a
c=1,r+c-2 c=2,r+c=3··· lr
c-l 2
II a.l.,r= l+1.
'
i=l
(8.1.2.22)
81
which, togetheT with (8.1.2.17) gives:
cc~pletes
This
3.1.:3
the proof.
Relationship Between the Sums of Pri:lcipal Yl:.rlC':':-'s
of a Matrix and those of its Inverse
A diTect result ()f the Cayley-Hamilton theoreni is
n~squa~e matrix~
of an
-1
... 1
[ r:.l
M -::::--- M
(_l)Dp
r.
where
p~
M , with rank
"'1-2
- PlM"
r~:
()n-2
+••• + -1
tha+~
the inverse
can be expanded as follows:
P
n-
..l.
2M ,
(S.l. 3.1)
is the sum of the i-rowed principal mi..nor's of M.
Rearranging
terms gives:
-1
::: - - - - [ p
(_l)n
n
l\R-l -p
.'1
n-l
T
- + P
n-
1I;:-2
2M +••• + ( --1. )n-lPILl
J
and multiplying both sides by ( M-1)n-2 gives:
-1
M
l
( M-l)n=l - Pn_l \'M• )n-2+.00 + (_1)n-lp1
=--- [ p
(_l)n
-'-J
.!.. "'"
n
(801..3.2)
which is seen to be of the same for.m as (8.1.3.1). TherefoTe, if we
I
Pn-i:::: Pi/p
where
n
P~-i is the S~~ of the (n-i)th-rowed principal minors of M- 1
0
82
8.1.4
Theorem on the Inverse of B
The inverse of the (r+c-2)-square matrix
+ wI
B := U
(8.1.4.1)
is
, 1 { Wn-1 I
B-.l..:=p*
n
-
~
oJ. (
••• "
-.L
W
n-2 CU-P -T·] + wn-3 CU2 -PlU+P2 I] +•••
l
)r-l n-rC_..r-l
W
U
•..r-2
...r-3
( . )r-l
]
-Pl U +P2 u
+••• + -1
Pr_1I +•••
] }
••• + ( -1 ) n-1C.~-1
lJ
-Pl .~-2
U
+p2U'~-3+••• + ()n-l
-1
Pn_1I.
(8.1.4.2)
where
P := sum of the r-rowed principal minors of U
I'
7
P::= determinant of B ,
n
and
n = r+c-2 •
Proof:
between p
I'
first it will be necessary to establish the relationship
and
p*= sum of the r-rowed principal minors of B •
I'
Any r-rowed princlpal minor of B can be represented as:
lB.J. I = I U.J.
I'
I'
+ wI
I
(801.4.3)
where rBi represents the (rxr) matrix of elements composed of the
intersection of the r rows and
I'
columns contained. in the
combination. Expanding the right side of (8.1.4 3) gives:
0
where
p .. = sum of the j-rowed principal minors of
J.J
U.
I' J..
.th
J..
83
rrherefore J the sum of the r-rowed. principal minors of B is
en
r
r
:E [w .:-
i=l
r-l
+••• +
n'lW'
k"l
(8.1.4.5)
1'-1
Now we will consider the coefficient of w
'" in p* •
r
crn
~ pc
= sum of the j-rowed principal minors of all C~
i=l lJ
r-rowecl principal m:i.nors of' U
e
Each of the r-rowed principal minors of U containE'. C~ j-rowed
,]
principal minors; thus, their sum (including all duplications) must
contain C~·Cn j-rowed principal minors of U.
J r
But, U contains only
C~ j-rowed principal mi.r-ors and by syrnrL.etry all must be duplicated
J
equally, therefore
CD
r
~
po
0
i=l lJ
=
(8.1.4.6)
and
"
P":::
r
Rearranging gives:
~
,..n-j
I.Jv
j=O n-r
r-j
w
p.
J
(8.1.4.7)
84
_f(!D_Crl=ICn.J.f'On<>2Cn_cn-3Cll+
+(=I)r-1CU-r+1CYl ) r
0.0 '''' r r- 1 l'v,r- 2 2 r- 3 3 000
1
1 W
r-~
In p
i
r
the coefficient of w p*
r-l
0
,
(i=I, ••• ,r), is:
n - r +i )
0.. +(_I)i-lCn-r+i-(i-l)C
1)
1
l0
1.=
("
1.0
= =(Cn-r+iCn-r+i_Cn-r+i-lCn-r+i+Cn-r+i-2Cn-r+i+
i
0
i-I
1
i-2
2
.00
000
+( -1 )i-lCn-r+i-(i=1 )C n - r +i +( -1 )iCn-r+i-iCn-r+i)
i-(i-l)
i-I
0
i
+(_I)iC~-r+i-ic~-r+i
But the expression inside the parentheses i.s zero (13, p047)j therefore,
i
the coefficient of w p* . i.n p
r-1.
is:
r
( -1) iC~-r+i
J.
(801.4.8)
Thus, the piS are expressed in terms of the p*i S
Po = P~ :::: 1
n
PI = P"~-CIWP~
n-l
n 2
P2 = P~-Cl wp!+C 2w P~
Pr
: P"r*_C1n -r+lWPr-l
* +Cn-r+2
2 * +
+(_I)r-lCn-lr-l
*+(_I)rCnWr *
2
w P -2 •••
r_lw PI
r Po
r
(801.4.9)
for r=O,.oo,n 0
If we now rearrange the terms within the braces of (801.4.2) as:
85
n-l
w
1PO
n~2D'
+wn-2 I P
~w
n-3Ip
+w
~W
1
···
Ii
0
' n-3_2
n-3 UP -rW
U-p
210
and substitute for P~ f~om equations (8.1.4.9) we have:
...
wn-1I(C~WOp~)
~ n-2 I (C n 1 *_Cn-1 0 *)_ n-2U(0.n 0 *)
w
1w Po 0 w P w
"Ow Po
l
n-3I(Cnw2 *_Cn-1 1 *+Cn-2 0 *)+ n-3U(CD 1 *_Cn-1 0 *)
+w
2 Po 1 w PlOw P2 w
lW Po 0 w Pl
+ ( -1 )
n-l 0 (n wn-l
n-l n-2
()n-l 1 0
)
w I. Cn- l
P*O-C n- 2w P~l·+ •• ·+ -1
COw p*n- 1 +•••
nO)
••• + ( -1 ) n-J_~-l(
-U-COw
I1)
(s. 1. 4.10)
If we sum coefficients of
P~
from the first parentheses i::1 rows
i, ••• ,n-l, from the second parentheses in rows 2, ••• ,n-l , etc.; and
sum coefficients of p! from the first parentheses in rows 2, ••• ,n-l ,
from the second parentheses :1.n rows 3, ••• ,n-l , etc.
for
p~
, ••• ,
P~-l
we have:
j
and similarly
86
n 1
w - I [C~-C~+ ••• +(-1)n-1C~_1]
~{
_wn - 2u [en_c n+••• +( _1)n-2Cn ]
1
n-2
°
+(_1)n-3W2~-3[C~_C~+C~]
+(_1)n-2w~-2[en_Cn]
.
°
-I- (-1 )n-1w°Ur:.-1[c n ]
°
1
}
wn - 21 [C~-1_c~-1+ ••• +(_1)n-2c~:~]
_wn - 3u [Cn-1_cn-1+••• +(_1)n-3Cn-l]
1
n-3
·•• °
(8.1. 4.11)
-I-(_1)n-4W2~-4[C~-1_C~-1+c~-lJ
+(_1)n-3w~~-3[Cn-1_en-1]
1
+ ( -1
···
·+·
.l:'n_2
'I',":f
{
+ p*n-1{
)n-2
Q~-2[
w u
°
}
n-1]
Co
W1I[C~-ei]
_wOU[C 2]
°
}
WOI[C~J
}
which may be rearranged as:
87
*J
( -l) u-lPol
n-I T [en _cn ±
±C~lJ
w ~
n-l n-2 ••• 1
n-2 [en _",n ±
±e1'l:t:1J
+w u n-2 ~D-3 •• vI
0
+w2"(J11•• 3[C n -C D+IJ
2
1
+wV=2[C~-lJ
+wOUn - 1 [lJ
+(=1)n-2
pl{
}
n 2
w-
1 [C~:~-C~:~± ... ±C~-~lJ
+wu - 3u [C n - 1 _c n - 1±••• ±cn-~lJ
n-3 n-4
1
(8.1.4.12)
+W2~-4[c~-1_e~-1+lJ
+w~..n-3[ en - 1 -lJ
1
+wOlfl - 2 [lJ
}
1
2
w I [c -lJ
1
+(_l)Op* {
n-l
+wOU [lJ
}
wOr [lJ
}
But, from (13, p.48), we have
Therefore, (8.1.4.12) becomes:
.
88
*[C n - l n-1 I +C u - l wn-2 U+
+C n - 1 ly-p-2+Cn-l ~p-1J
n-lw
n_2
••• 1 w u
0 W u
Po
:l<-[C n - 2wn-2 +C n - 2wn-3 +
,,,n-2wInI!·-2.......
n-2.WO..U.n-2
I
U ••• -'v
2
3
u
rv
J
1
o
.L
nn-
-p.,
•
+(-1)n-2p~_2[CiwlI+C;WoUJ
+(_l)n-lp~_l[cgwo!J
}
(8.1.4.14)
Substituting into the right side of (8.1.4.2) gives:
(8.1.4.15)
1
which from (8.1.3.1) is seen to be B- •
89
8.2
Matrices and Sums of Principal
I~nors
for Problem Sizes to r:::c:::4
For all values of the parameters rand c ,
r~c
, up to r=e:::4 the
.-1
sums of principal minors of U, the matrices U and Ii
polynomial
t"* JB-~"1l-lg*
are as follows.
"¥,.'
For r=e=2,
P1 = 0
u =
!!"'[ w2U +
2WP2 I -p2uJi~·
-
=0
For r=3, c=2,
P1
= -1
P2
= -a13
P3
--
2
a
2
13
2
(a -1)
11
U
=
*
a
a
11 12
2
(a12 -1)
a
a
11 13
a
a
12 13
a
2
13
,and the
90
o
-1
U
1
=-2-
a
2
-a
13
13
a
*
a
12 13
a
2
13
For r=e=3,
P3 = 0
P4
2 2
= a 13
a 24
2
2
(a11+a12 -1)
(a11a12+a21a22)
(ai2+a~2-1)
U
=
*
(a12a13+a22a23)
a 22 a 24
2
2
(a +a )
13 23
a
a
23 24
a
2 2
-a a 24
13
-1
U
1
=
a
2 2
a
13 24
2
24
0
2
a11a13a24
-a13a24(a11a23-a13a21)
2 2
-a a 24
13
2
a12a13a24
-a13a24(a12a23-a13a22)
a
*
2 2
a
13 24
0
a
2 2
a
13 24
2
3
1
6
5 2
g*'[w U _ 2w U -(P2U+3P4U-1) + 4w p 1 - w p (3U+P U- )
2
4
4
+2WP4(~+P2I)
+
p~U-1Jg* = 0
91
For r=4, c=2,
P1
= -2
P2
2
= l~a14
P3
2
= 2a14
P4
= -a14
2
2
(a11-1)
a
11 12
a
a
11 13
a
a
11 14
2
(a -1)
12
a
a
12 13
a
a
12 14
2
(a -1)
13
a
a
13 14
a
U ==
*
a
-1 1
U = -c:.0 a
14
2
-a
14
0
0
2
-a
14
0
*
-a
2
14
a
a
11 14
a
a
12 14
a
a
13 14
a
2
14
2
14
n '{w6U - 2w5[ U2 -p1U ] -w4[ p1U2 - (P 2
)
-lJ
~*
1 -P2 U~3P3I+3P4U
- 2w3[P3U-(P1P2+2P4)I+P1P4U-1J
+
w2[P3~-(P1P3+3P4)U+(3P1P4+P2P3)I-P2P4U-1J
+
2WP4[~-P1U+P2IJ
+
p~U-1
For r =4, c =3 ,
222
P2 == -(a14 + a 24 + a 25 )
P
3
=
222
(a
+ a
+ a )
24
25
14
}g*
=0
92
(a118.12+8.218.22)
(8.
22.
+8. -1)
12 22
(a11a13+a21a23)
(a11a14+a21a24) a 21 a 25
(a128.13+a22a23)
(a12a14+a22a24) a 22 a 25
(2
(a138.14+a23a24) a 23 a 25
2
1)
\a13+a23-~'
o
o
o
2
a12a14a25
2
a13a14a25
*
a
2 2
a
14 25
o
8
7
l*'{w U -2w [lf- P1U] +W6[3U3_4P1uF+(pi+2P2)U] -2W5[P3U-(2P4+P1P3)I
+(2P5+P1P4)U-1_P1P5U-2]
+W4[P3~-(3P4+P1P3)U1'(5P5+3P1P4+P2P3)I
~(3P1P5+P2P4)U-1+P2P5U-2] -2W3[-P4~+(2P5+P1P4)~-(2P1P5+P2P4)I
1
+P2P5U- ] +W2[-P4U3+(3P5+P1P4)if-(3P1P5+P2P4)U+(3P2P5+P3P4)I
-P3P5U=1]
+2w[P4P5U-l_p~U-2]
+
p~U-1
}i*
~
0
93
2
2
2
= a14(a2S+a3S+a36)+(a24a35-a25a34)
P
= 0
s
2
D.·1-l
1 J.
2
P4
3
D.· 1 a· 2
1 J. J.
3 2
D3.· -1
1 J. 2
u=
2
2
2
2
+a36(a24+a25)
3
:3
a
1:a·
1 J. 2 J.,3
3
3
I; a, 2 a . 4
1 J. J.
3
I;a· 2a. S
2 J. J.
a
a
32 36
3 2
I;a. 3-1
1 J.
3
I;a, 3 a , 4
1 J. J.
3
a
I;a.
2 J.3 J.. S
a
a
33 36
3 2
I;a'4
1 J.
3
I;a· a, S
2 J. 4 J.
a
a
34 36
3 2
I;a.
2 J. S
a
a
3S 36
3
1I;a'la':3
J. 1.
*
a a
1:a'
31 36
2 J.1 a.J. S
1 J. 1 a,J. 4 1:a'
a
2
36
where
ana.
~2 = a14a2Sa36
X
a11a2Sa36
-a36 (alla24-a14a21)
[all(a24a3S=a25a34)-a14(a21a3S-a2Sa31)J
a12a2Sa36
=a36(a12a24-a14a22)
[a12(a24a3S-a2Sa34)=a14(a22a3S-a2Sa32)J
a13a2Sa36
-a36(a13a24-a14a23)
[a13(a24a35-a2Sa34)-a14(a23a3S-a2Sa33)J
94
Cc~ter
8 3
0
P:::.·ogram for Transf'Jrmations
A program was written in FORTRAN rJ' for the IBM 1410 computer
to calculate the
(a)
f'ollowing~
The set of orthogonal vectors of linear coefficients de-
(8.3.1.)
.£31
c
-3,c-l
•
In the printed computer output these vectors are identified by the
heading:
LINEAR COEFFICIEN"l'S SET 1.
(b)
where,
A set of vectors analogous to (a) but not defined else-
2:.. ~. ,
Q*=
Q - =
2
3.21
(8.3.2)
Q
3
.9.:2,c-l
3.31
.9.3,r-l
These vectors are identified in the computer output by the heading:
LINEAR COEFFICIENTS SET 2.
95
(c)
The matrix having rows defined by (2.2.19) for i=l, ••• ,
c-l
!:l
=
~2
all
a
a
a
21
a
•••
12
a
22
1,r+c-2
(8.3.3)
2,r+e-2
•
a
-c-l
a
a
c-l,l
c-2,2
a
•• 0
e-l,r+e-2
This is identified in the output as MATRIX Ao
(Note: this should not
be confused with the matrix A as defined by (2.2.21).
o
~
(d)
!{
The vectors
= (t 21 ••• t 2 ,C_l).
g*
I
=
(121 ... L 2 ,r_l £'31
0
..
.13 ,C-l) and
These are identified as LINEAR FUNCTION Land
LINEAR FUNCTION T in the output.
It must be explained that the computer program does not calculate
the quantities in (a) through (d) exactly as defined.
section (2.2) we see then any vector ~S or
£as
Referring to
is of length n but
that it is composed of only rc possibly unique elements,; thus, for
computing purposes we consider
~S
and
£as
to be of length re.
In
addition, the program as presented here does not normalize the vectors,
1·~·, ~S ~S
~ 1 and ~S
£as
1= 1.
It was found to be desirable in
the early stages of the present study to have the exact values of the
coefficients at the expense of a small amount of hand calculating to
later produce the normalized quantities.
We felt that a program that
would calculate the exact coefficients for orthogonal contrasts when
the numbers are unequal would be useful for other purposes as well
96
and thus we have lef't it in that f'orm.
With some modest revisions the
program could be made to produce the normalized quantities.
There are a number of' papers in the literature (15~ 21, 26) that
give methods f'or calculating orthogonal polynomials; however, none of'
these is directly applicable to the problem at hand.
The method used
here is most nearly related to the technique that has come to be known
as "step-wise regression" or "step-wise reduction."
If' we consider, f'or example, the 3 X3 array as given in table
4.1 we have:
m
1
Xl
=
•
•
·
1·
0
o
o
·••
1
o
0
1
1
0
..
•
•
o
1
0
0
1
0
0
1
0
0
0
1
0
·.
1
··0•
0
1
0
0
1
0
1
0
0
0
1
0
0
1
0
0
1
0
0
0
(8.3.4 )
97
:.t
X{X1
=
n1 •
n2 •
n' l
n' 2
n10
0
n
n
n
n2 •
*
11
n
21
(8.3.5)
12
22
0
n' l
n' 2
768
=
144
364
57
462
144
0
49
88
364
5
299
57
0
*
(8.3.6)
462
and
y
X{l
.-
.. ,
Y
1"
Y2 0 ,
Y
'1'
Y
• 2'
=
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
1
0
0
1
0
0
1
0
0
0
1
0
0
1
0
0
1
0
!1
= Zl!l
(8.3.7)
where
Y11•
Y12 •
!1
=
Y ,
13
Y21 •
Y22 •
Y23 •
Y31 •
Y32 •
(e
Y33 •
(8,3.8)
98
In most computational sche:n:es, we woulcl augmer;.t the upper-triangular
portion of X!X, with the vector·Xl/~ and then proceed by the selected
1. -'-
method. to directly calculate sums of sQuares of the observations and
other desired Quantities from the data.
Quire not only the
Sl11l1S
For our purposes, we re-
of sQuares but also the individual vectors
of coefficients aS80ciated with single "degrees of freedom" of the
sums of sQuares; therefore, we augment the upper-triangular portion
of X{X1With the mat::ix Zl that is defined in eQuation (8.3.7) and
proceed with operations on this array.
We will represent the array
as:
IZ J
[ Upper triangular X{X l
a
=
...
a
a
a
l,r+c-l
a
2,r+c-l
r+c-l,r+c-l
a
a
l,r+c
l,rc+r+c-l
2,r+c
a
r+c-l,r+c
r+c-l,rc+r+c-l
(8.3.9)
Instead of the usual ffstep-wise reduction" calculation, b ..
lJ
ai+l,j+l - (al,i+lal,j+l)/all ' we will use b ij
al,i+lal,j+l
b
ll
= allai+l,j+l
_
to perform the reduction of (8 0309) to
b
12 ••• 1,r+c-2
b
b 22 ••• b
2,r+c-2
b
b
=
l,r+c-l
2,r+c-l
... b1,rc+r+c-2
o ••
b
2,rc+r+c-l
•
•
b
·
r+c-2,r+c-2
b
r+c-2Jlr+c-l 00 •
b
r+c-2Jlrc+r+c-2
(803010)
99
Similarly,\> we can compute clo "
J
:=:
bll- b-l+
.+~
1
~ ~'\>J ~
obtain a second reduction and so on.
- b'l~,\>--i+lbl_,JT~
0'" to
In th::ts way,\> 'by eliminating
the division, the exact values of the coefficients are developed
on the right side of the array.
One additional dodge is employed
in order to keep the size of the
n~~bers
as small as possible.
We
see that:
c.
°
lJ
= bllbi+l,j+l
= (alla 22
- bl,i+lbl,j+l
2
- a12)(allai+2,j+2 - a l ,i+2 a l,j+2)
- a 12a l ,\>j+2)
2
- a 12ai+2,\>j+2
- a 11a 2,i+2 a 2 ,j+2 + a 12a 2,i+2 a l,j+2 + a12a 2,j+2 a l,i+2 ] •
(8.3.11)
That is, all the elements in the array at any reduction are exactly
divisible by the (1,1) element in the reduction two steps previous.
By using Xl and Zl the procedure will yield the orthogonal
vectors of coefficients associated with the individual components of
the sums of squares for rows and columns (adjusted for rows) as given
by equati,ons (2.3.1).
If we let X be the rearrangement of Xl such
2
that the columns are ordered m, C ' C2 , R , R2 , and then define X~X2
1
l
and Z2 accordingly, the computational proceQQre will give the vectors
of coefficients associated with the individual components of the sums
of squares for columns and rows (adjusted for columns) as given by
equation (2.3.2).
100
Input to the computer program is in the following order:
(a)
Parameter card
Columns
1
3
6
Contents
-2
-5
-8
r+e-l
rc+r+c-l
2rc-:r-c
c
9 -10
blank
11 -80
(b)
Upper triangular portion of X{X •
l
Each element is a whole
number right justified in a five-digit field.
I
on cards serially by rows of XIX ,
1
••• , a l,r+c-l'. a 22'
(c)
... ,
Representation of Zl.
a
1.~.,
.
2,r+c-l'
.0 .
j
a r+c- 1 ,r+c-l •
One card for each element equal unity
in Zl except those elements of the first row.
Columns
Elements are entered
Card format is:
Contents
1 - 3
row number of element
4 - 6
column number of element
7 -80
blank
The column number required is that of the augmented array and not for
Zl alone,
~.~.,
elements of the first column of Zl are found in the
(r+c)th column of the augmented array.
(d)
Upper triangular portion of X~X2.
Card format is the same as
for (b).
(e)
Representation of Z20
(f)
Cell totals.
Card format is the same as for (c).
Each cell total is entered as a whole number
right justified'in a ten-digit field.
whole number for the analysis.)
rows into the cards.
(i.e., Y..
- -
l.J.
must be scaled to a
Cell totals are entered serially by
101
(g)
Number of observations per cell.
Each n .. is entered as a
~J
whole number right .justified in a five-digit field.
They are entered
serially by rows into the cards.
(h)
Indicator card.
Placed at the end of each problem set.
This card contains 00 or 01 in columns one and two for respectively
the last problem set a.nd any other problem set.
The program is as follows:
C
C
C
C
C
C
C
00001
00002
00003
00004
00101
00102
00103
00104
OOlOS
00106
OOOOS
00006
00007
OOOOS
OOOSO
PR¢GRAM F¢R Llr-J"EAR TRANSF¢RMATI¢NS
'r:rrIS PR¢G':R.AM CALCULATES
C* = LINEAR C¢E:F']'ICIENTS SET 1
Q;* = LINEAR C¢EFFICIENTS SET 2
MATRIX A
LINEAR FUNCTI¢NS L
LINEAR FUNCTI¢NS T
DIMENSI¢NX(11,4S),FL(10),FT(S)
F¢RMA.T(16FS.0)
F¢RMAT (I3 , I3 )
F¢RMA.T(SFIO.O)
F¢~1AT(I2,I3,I3,I2)
F¢RMA:TU/IIOX, 26HLINEAR C¢EFF'ICIENTS
SET, I2)
F¢RMAT(SH R¢W ,I2/(SF2S.0))
F¢lli~~(IIIIOX,SHMATRIXA)
F¢F3MAT eI//IOX, 19H LINEARFUNCTI¢NS
(SF2S. 0) )
F¢RMAT Ulllox, 19H LINEAR FUNCTI¢NS Til (5F2S. 0) )
F¢RMA:r ( I2 )
READ (1,4)M,MY,NU,K2
NT=l
D¢7I=1,M
D¢7J:.::l,MY
X(I,J)=O.O
READ ( 1, 1) ( (X ( I, J ) .' J =I , M) , I:.::1, M)
D¢SL=l,NU
READ(1,2)Nl,N2
X(Nl,N2 )=1. 0
J¢=M+l
D¢SOJ=J¢ ,MY
X(l,J)=1. 0
D=LO
D¢10Ml=2,M
K=Ml-l
D¢9I=Ml,M
D¢9J=I,M'f
LII
102
00009 X(I,J)=(X(K"K)*X(I,J)-X(K,J}*X(K, I) )/D
00010 D=X(K,K)
WRITE (3 ,101mI'
D¢12I=2.,M
N¢n¢W=I-l
00012 WRITE(3,l02)N¢R¢W,(X(I,J),J=J¢,MY)
IF(I~.GTol)o¢T¢14
00013
00014
OOOlS
OOOlS
00017
00018
00019
NI'=2
D¢13I:::2,M
WRlTE(4)(X(I,J),J=J¢,MY)
FEWIND4
a¢T¢S
D¢lSI:::2,M
WRITE(S)(X(I,J),J=J¢,MY)
REWINDS
READ(l,3)(X(M,J),J=J¢,MY)
READ(l,l)(X(M-l,J),J=J¢,MY)
IM=K2-1
JJYl'2 =M- 1
D¢lSI=l,IM
D¢16J=l,JM2
X(I,J)=O.o
D¢17I=2,K2
READ (s) (X(I,J) ,J=J¢ ,MY)
D¢18I=l,IM
FT(I)=O.O
D¢18J=J¢, MY
FT(I)=FT(I)+X(I+1,J)*X(M,J)
D¢1911=l,JM2
.
READ (4) (X(l,J) ,J=J¢ ,MY)
D¢1912=l,IM
D¢1913=J¢ ,MY
X(12,Ll)~(12,11)+X(l,13)*X(12+1,13)*X(M-l,13)
REWIND 4
REWINDS
WRITE(3,103)
D¢20I=l,IM
N¢R¢W=I
00020 WRITE(3,l02)N¢R¢W,(X(I,J),J=l,JM2)
D¢21I=l,..TM2
READ (4) (X(I,J) ,J=J¢ ,MY)
F1(I)=0.0
D¢21J=J¢,MY
00021 F1(I)~1(I)+X(I,J)*X(M,J)
REWIND 4
WRITE(3,l04)(F1(I),I=l,JM2)
WRITE (3 , lOS) (FT(I), 1=1, 1M)
READ(l,lOS)N¢PR¢B
IF(N¢PR¢B.EQ.l)Q¢T¢S
S'l'¢P
END
103
The fol towing te::,'ms used in the program are defined.
M = r+c-l
MY = rc+r+c-l
NU = 2:::::'c -r"'c
K2 = c
Nl = row m:unber for elements equal unity in ,Zl or Z2
N2
D
NT
= colum..YJ.
number for elements equal unity in Zl or Z2
= J. initially, then successively all'
=1
bll~
...
I
when program is operating on XIX and Zl
l
I
= 2 when program is operating on X2X2 and Z.c:.. .
N¢R¢W = row number in program output
FL
=
FT
= linear
N¢PR¢B
=
linear function of ob ser-V"at ions ,
f ij
function of observations, t ..
l.J
1 in all problem sets except the last,
=
0 in the last
problem set.
All other terms used are indices and are defined in the program.
An example of the program output using the data from table 4.1
is given in table
8.1.
The divisors required to normalize the
vectors in the two sets are found by calculating
r
c
2
~ ~ n.c ( )
i=l j =1 iJ QI~ ij
and are as follows:
Vector
Divisor
V69,009,40S ~ 8307.1901
104
Table 8.1.
Example of computer program output
LINEAR COEFFICIEl\1TS
SET 1
624.
-144.
-144.
O.
37440.
-52416.
-4637360.
-187200.
-157248.
51253384.
91350460.
384551092.
624.
-144.
-144.
O.
37440.
-52416.
-4637360.
-187200.
-157248.
-496913352.
-456816276.
-163615644.
711.
711.
711.
O.
O.
O.
920304.
-5636862.
-5636862.
9077244.
153211146.
-403613562.
-57.
-57.
-57.
14193.
14193.
14193.
5308182.
-1248984.
-1248984.
-26286204.
117847698.
-438977010.
-57.
-57.
-57.
-26334.
-26334.
-26334.
6372828.
-184338.
-184338.
266914428.
411048330.
-145776378.
22597632.
23003136.
420992053248.
295150946688.
-4849515474336.
0
25644176064.
404231406592.
115398573.
12068008308.
624
-144.
-144.
O.
37440.
-52416.
8990800.
13440960.
13470912.
-21059272.
19037804.
312238436.
ROW 1
0
ROW 2
ROW 3
ROW 4
SET 2
LIl'ilEAR COEFFICIENTS
ROW 1
ROW 2
ROW 3
ROW 4
MATRlX A
ROW 1
o.
ROW 2
817648128.
2458587364161264.
LINEAR FUNCTIONS L
281370576
38086166048492.
LINEAR FUNCTIONS T
105
£31
Y7,470,b03,984,885,760 ~
~32
V33 ,254,763,752,143,594,752
.9.21
Y31,124,736
86,432,076
~
5,766,694,400
5,578.9548
V265,742,266,482 ~ 515,501.95
(8.3.12)
The divisors for
not used.
~31
and
~32
are not given since these two vectors are
Using the nllillerical approximations in (8.3.12) as divisors
we directly adjust the remainder of the output to find
a
a
I
= 0.0037259931
12 = .£2~21
I
13
=:
'£31~1 = 0.87306405
I
0.0
a
14 = .£3~1
a
- -21-~ 2 = 0.19093343
21 -c
a
0.51739437
22 = £22-~2 =
a
=-0.10884114
~
23 =c
-31:-L2
a
I
= 0.82704357
24 = £3~22
:=
I
I
I
= 338.70728
= 231. 73764
46.768679
66.045057
= 206.84622
:: 234.10209
106
8.4
Compute~
Program to Obtain Polynomial Roots
The following program written in
FORTIUU~
II for the IBM 7072
computer was used to obtain the numerical approximations to the roots
given by equations (4.3.10)
C
MAIN PR¢GRAM
C¢E(1)=70835.812
C¢.E(2)=211837.42
C¢E(3 )=-70801. 400
C¢E(4)=-364907.70
C¢E(5 )=26180. 293
C¢E(6)=150373.66
C¢E(7 )=30256.238
CALL R¢¢TN(6,C¢E,R,CR)
PRDf.r 1, (C¢E(I),R(I),CR(I),I=1,7)
1 F¢RMAT(3E20.8)
Eli[)
C
SUBR¢UTINE R¢.¢.TN (N,C¢E,R,CR)
P¢LYN¢MIAL R¢¢T EXTRACTI¢N
C
C
¢:BTAI1'ifS REAL AND C¢MPLEX R¢¢TS ¢F A P¢LYN¢MIAL IN ¢NE
Ul\1KN¢.WN WITH REAL C¢EFFICIE:NTS. A VARIATI¢N ¢F BAIRST¢W S
METH¢D IS USED.
C
C
DIMENSI¢N C¢E (51),C¢EFN(51),B(51),C(50),R(50),CR(50)
ACe = •0000001
INDIC = INDIC
J=J
K == 1
Nl::.-:N+l
NP=~
6
5
2
9
8
7
NPl=N+l
FIN=O.
D¢ 6 I=l,Nl
C¢EFN(I)::C¢E (I)/C¢E (1)
IF(N-2) 22,23,5
IF (C¢E(2))9,2,9
BETAl ~ C¢EFN(N)/.OOl
BETA2 '" COEFN(N)!.OOl
a¢ T¢ 8
BETAl =e¢EFN(N)/C¢EFN(2)
BETA2 =COEFN(Nl)/COEFN(2)
D¢ 15 J==l,lOO
B(l)::C¢EFN(l)
e(l)=B(l)
B(2)= C¢EFN(2) + B(l)* BETAl
C(2)= B(2) + eel) * BETAl
107
D¢ 10 I:=' 3,NPl
B(r) == e¢EFN(I) + B(l-l)* BETj\~ -I- B(I-2)J,(- BETA2
10 e(I) :::: B(l) + C(I-l)* BETAl + C(I-2) .* BETA2
DELTA ,: C(NP-l) *e(NP-l) - e(NP-2) * a(NE)
DBETl :::: (C(NP1-3) * B(NP1) - e(NPl-2) * B(NP1-l) )/ DELTA
DBET2 == (C(NP1-l) * B(NP1-l) - e(NPl-2)
B(NP1)
DELTA
BETIP = BETAl
BET2P "" BETA2
BETAl :: BETAl + DBETl
BET.42 == BETA2 + DBET2
IF (ABSF(BETIP - BETA1) - ACC
(ABSF(BETAl)))11,11,15
11 IF (ABSF(BET2P - BETA2) - ACC
(ABSF(BETA2)))16,16,15
15 C¢l\1TINUE
ACe=ACC*lO.
IF(ACC-.l) 120,120,9
120 PRINT 12
12 P¢RMAT(87H PR¢GRA.M HAS ITERATED 100 TIMES.
TRY
lAGAIN lHTH II1EW TRIAL S¢LillI¢NS)
IF (:nnnc )1, :3 ~ 1
3 BETAl = -.2
BETA2 == 2.
INDIC = 1
a¢ T¢ 8
16 NP= NP-2
NP1==NPl-2
BETA1= -BETAl
BETA2= -BETA2
DO 4 I=l, NPl
4 COEFN(I) = B(l)
19 RAIl == BETAl
BETAl - 4.0
BETA2
IF (RAD) 20,21,21
20 R(K) = - BETAl72.0
R(K+l) = R(K)
CR(K) = SQRTF(-RAD) / 2.0
CR(K+l) = - CR(K)
K = K + 2
IF(FIN) 1,24,1
24 IF(NP-2) 22,23,9
21 R(K) == (-BETAl + SQRTF(RAD)) / 2.0
R(K+l) = -(BETAl + SQRTF(RAD))/ 2.0
. CR(K) = O.
CR(K+l) = O.
*
*
*
*
K = K + 2
25
22
23
1
IF(FIN) 1,25,1
IF (NP - 2}22,23,9
R(K) = - C¢EFN(2)
('~ T¢ 1
FIN = 1
BETAl = C¢EFN(2)
BETA2 == C¢EFN(3)
a¢ T¢ 19
RETURN
END
*
)1
108
The f'ollowing terms usee). in the program are a.ef'ined:
N
= degree
COE(I)
R
of' polynomial
= coef'f'icient
= real
of' term of i th degree (i=l, ••• , N+l)
part of' a root
CR = coefficient of the imaginary part of a root.
Program output for the example is:
I
COE
R
CR
1
0.70835812E 05
-0. 53806485E 00
O.OOOOOOOOE 00
2
0.21183742E 06
-0.26564956E 01
O.OOOOOOOOE 00
3
-0.70801400E 05
-0. 24406735E 00
O.OOOOOOOOE 00
4
-0.36490nOE 06
-0.13883567E 01
O.OOOOOOOOE 00
5
0.26180293E 05
0.91822165E 00
0.19682898E 00
6
0.15037366E 06
0.91822165E 00
-0.19682898E 00
7
0.30256238E 05