The Relative Efficiency of the
Approximate F-Tests Frequently
Encountered in Unbalanced Designs
Yea-Tsai Hsu
lnst. of Statistics Mimeo Series
4/=
1602
iv
TABLE OF CONTENTS
Page
LIST OF TABLES. .
1•
2.
3.
4.
...·....
1
·.....
·..........
THE ANALYS IS . . . .
3. 1 The Model . · · · · · · · · · · · · · ·
and Its Expected Value
3.2 The Sum of Squares
Type
II
Analysis.
The
3.3
·
3.4 The Type III Analysis ·· ·· ·· ·· · · ·· ·· ·· ·
3.5 The Unweighted Mean Analysis. · · · ·
·· ·· ··
THE TEST PROCEDURES . .
· . • 21
REVIEW OF LITERATURE . .
4.8
···
·
·
····
·
····
····
····
····
Introduction.
Procedure A
Procedure B
Procedure C
Procedure D
Procedure E
Procedure F
Procedure G
···
····
·
····
· · · ·
·
·
·· ··
· ·
· · ·· ·
····
·· ·· ·· ·
····
····
··· ·
·· ·
·
· ··
··
··
··
· ·
DISTRIBUTION OF APPROXIMATE F STATISTIC S
5.1
5.2
5.3
5.4
6.
·
INTRODUC TION .
4. 1
4.2
4·3
4.4
4.5
4.6
4.7
5.
· • vi
·
··
·
·
·
·
5
...
7
···
8
11
·
·
·
·
·
·
·
··
··
··
· ·
·
···
7
13
15
21
24
24
24
25
25
26
26
· . . • 27
Distribution of Quadratic Forms. . .
.
Distribution of The Ratio of Two
Independent Quadratic Forms . . . . . . . . . .
The Exact Distribution of Approximate F
Statistics. . . . . . . . . . . . . . . .
.
Computation of The Distribution Function . . . .
EVALUATION OF THE TEST PROCEDURES . .
27
36
41
44
· 56
6. 1
6.2
Ev al uat ion Pr 0 c es s. . . . . . . . . . . . . . . 56
Designs and Values of Variance Components
Investigated
58
6 • 3 Re suI t s . . . . . . . . . . . . . . . . . . . . 63
7.
8.
SUMMARY AND SUGGESTIONS FOR FUTURE RESEARCH.
7•1
Summary..
7.2
Suggestions for Future Research. . .
.
.
LIST OF REFERENC ES .
.
. .
. .
. .
.
.
. .
.
70
.
. .
. 70
. . . 72
· 74
•
v
9.
APPENDIX....................
76
e-
vi
e
LIST OF TABLES
Page
4. 1
6. 1
6.2
6.3
6.4
6.5
The Degrees of Freedom and Expected Values of Mean
Squares For Model (3.14).
.....···
The Cell Frequencies of Two-Way Designs · ·
Coefficients of The Expected Mean Squares ·
Ranks Of The Test Procedures. (VA=O) ·
·
Ranks Of The Test Procedures. (VA=.25) ·
··
Ranks Of The Test Procedures. (VA=5 ) · ·
···
····
····
····
····
··
··
··
23
60
62
64
65
66
9. 1
Power Functions In Percent For Design No. 1
9.2
Power Functions In Percent For Design No. 2
9.3
Power Functions In Percent For Design No. 3
··
79
9.4
Power Functions In Percent For Design No. 4
80
9.5
Power Functions In Percent For Design No. 5
9.6
Power Functions In Percent For Design No. 6
9.7
Power Functions In Percent For Design No. 7
9.8
Power Functions In Percent For Design No. 8
9·9
Power Functions In Percent For Design No. 9
··
····
····
····
····
····
··
··
· ·
·· ··
9.10 Power Functions In Percent For Design No. 10
9. 11 Power Functions In Percent For Design No. 11
9. 12 Power Functions In Percent For Design No. 12
9. 13 Power Functions In Percent For Design No. 13
77
78
81
82
83
84
85
86
87
88
89
9. 14 Power Functions In Percent For Design No. 14
90
9. 15 Power Functions In Percent For Design No. 15
91
1:
The
subject
of
INTRODUC TION
estimating
variance
components
in
unbalanced designs has generated considerable discussions in
the literature. However, little research has been done in
the area of testing variance components.
The main problems
concerning variance component testing in unbalanced designs
are:
1. The mean squares in the analysis of variance table,
with the exception of error mean square, do not have
chi-square distributions in general.
2. The
mean
squares
are
not
in
general
mutually
independent.
3. It is unlikely to find a pair of mean squares which
ha'le
the
same
expected
'lalues
under
the
null
hypotheses usually postulated.
Therefore, an exact F test is not usually available for the
unbalanced designs.
which
suggest
procedures
technique
There are, however, several procedures
approximate
are
based
on
F
tests.
the
The
synthesis
commonly
of
mean
used
square
introduced by Satterthwaite [1941 and 1946J.
In
these procedures, the nonindependence and nonchi-squareness
of the mean squares are ignored and a synthetic, approximate
F
test
is
square (MSN)
performed
and
a
by
constr~cting
jenominator
mean
a
numerator
square (MSD)
mean
using
2'
single or linear combinations of the mean squares from the .
analysis of variance table so that both MSN and MSD have the~
same expected values under the null hypothesis being tested.
A linear combination of mean squares is assumed to .have
approximately a chi-square distribution with its degrees of
freedom
computed
by
the
Satterthwaite's
formula.
An
approximate F statistic is then obtained as the ratio of MSN
to
MSD.
It
distribution
is
with
assumed
the
two
to
follow
computed
a.pproximately
degrees
of
an
F
freedom.
Numerous tests based on different 1 inear combinations of
mean squares from different types of analyses can be found
in this fashion.
A conventional F test involving the ratio of two mean
squares can be used as an al ternative to the synthetic F
test mentioned above, provided that their expected values
under the null hypothesis are only slightly different.
The
analysis
of
unweighted
means
alternative to the synthetic F test.
provides
another
The cell mean of each
subclass is considered as the only observation in the cell,
a.nd
the
balanced.
for
the
me,an squares
are
computed
as
if the
data
T. .
ere
The computations involved are very simple, and
two-way
crossed
designs,
there
is
no
synthesize mean squares because the appropriate
need
to
numerator
and denominator mean squares have the same expected values
under the null hypothesis.
e
3
The approximate F statistic does not usually follow the F
distribution
for
obvious
reasons.
Attempts
to
find
the
exact distribution of this statistic were successful only
under limited conditions.
The
purpose
of
this
research
generalized expression for
is
to
find
a
more
the exact distribution of the
approximate F statistic, to evaluate the performances of the
various approximate F tests under different conditions, and
to provide gUidelines for choice among these procedures.
We will confine our efforts to the random, unbalanced,
two-way crossed designs with no missing cell, and to avoid
complications, to the 1 inear combinations of mean squares
with positive coefficients.
Chapter 3
f~nctions
reviews the
sums of squares,
the
estimable
and the hypotheses associated with three different
types of analyses.
Chapter 4
test
describes
procedures
that
Satterthwai te f s
are
based
combinations of mean squares.
The
formula and
on
different
seven
linear
performances of these
test procedures are evaluated in Chapter 6.
Chapter 5 deals wi th the distributional aspects of the
approximate F statistic. First, we derive the distribution
of
a
quadratic
form
in
normal
variates.
Then,
the
distribution of the ratio of two independent quadratic forms
4·
is obtained.
Finally, the distribution of the approximate F '
~
statistic is derived.
The evaluation of the performances of the test procedures
is
treated
in Chapter
6.
Fifteen
two-way designs
wi th
different dimensions and cell frequencies are selected, nine
different combinations of values for the variance components
are used, a.nd the power functions of each test procedure is
computed at three different significance levels.
Chapter 7 provides
a
summary of this
suggestions for future research.
study and
some
5
2. REVIEW OF LITERATURE
When testing variance components in a random model with
unbalanced data, the exact F test in the form of a ratio of
two
independent
Nevertheless,
mean
we
squares
can always
does
find
a
not
usually
exist.
of two
linear
ratio
combinations of mean squares as an approximation to
exact
F statistic
technique.
This
using
the
synthesis
technique
was
first
Satterthwaite [1941 and 1946J,
who
showed
of
mean
square
introduced
that
the
a
by
linear
combination of mean squares can be approximated by a chisquare variate whose degrees of freedom is a function of the
expected values and degrees of freedom of the component mean
squares.
This
synthetic
or
technique
was later employed
approximate
F
statistic
in computing
by
synthesizing
numerator and denominator mean squares (see Searle r1971 ] ).
Howe and Myers [1970] suggested an approximate F test which
involves a single mean square in the numerator and a linear
combination of mean squares in the denominator. Gaylor and
Hopper [1969]
investigated
the
performance
of
the
Satterthwaite's formula when some of the coefficients in the
linear combination of mean squares are negative.
Krutchkoff [1968]
approximate
F
studied
tests
which
the
sizes
involve
and
addition
Hudson and
powers
as
well
of
as
subtraction of mean squares.
~ietjen
[1974J showed that if the expected values of two
6 '
mean squares from an unbalanced design are reasonably close4t
to
each
other
under
the
null
hypothesis,
then
the
conventional F test provided by the ratio of these two mean
squares may be more favorable than the synthetic F test.
The analysis of unweighted means was first introduced by
Yates [1934J for its simplicity in computation. Gosslee and
Lucas (1965J used it in testing hypothesis for two way fixed
model.
Hirotsu [1968]
and
Webster [1968]
extended
testing varince components in random model.
it
to
Their results
indicated that the approximate F statistic obtained in this
analysis follows the F distribution very closely.
All
of
procedures
the
F statistics
approximate
involve
ratios of
correlated
in
these
test
quadratic
forms
which are not distributed as chi-square variates in general.4It
Cochran [1 934 J showed that these quadratic forms, by way of
orthogonal
transformations,
combinations
of
can
independent
be
expressed
chi-square
random
as
linear
variates.
Box [1954J showed that the ratio of two quadratic forms can
be
transformed
into
a
ratio
combinations of chi-square
of
two
independent
random
and
linear
that
the
exact distribution of this ratio iz a finite series for some
limited cases.
It can be shown that the exact distribution
function of this ratio
distribution
functions
is an inf'ini te weighted sum of F
as
Pitman [1949J and Laha [19541.
suggested
by
Robbins
and
7
3. THE ANALYSIS
3.1
The MQdel
CQnsider the randQm mQdel:
where
Y is an nx1 vectQr Qf QbservatiQns,
XQ 1s an nx1 'vectQr Qf 1 's,
Xi (i=1 ,2, ... ,k) are nxmi knQwn matrices,
6 Q is an unknQwn cQnstant,
each
S·1 (i=1,2, ... ,k),
effact,
an nx1
an
mix1
vectQr
the
i th
randQm
uncQrrelated randQm
variates assumed tQ be distributed as N(O, ai2 1m.),
16i,6j (i~j) and e are independent Qf each Qther, and
e is
is
representing
Qf
vectQr Qf uncQrrelated
randQm variates
N(O, a 2 In).
e
N(XOS Q, V), where
assumed tQ be distributed as
Thus, Y is distributed as
V
a
This mQdel can alsQ be represented in matrix fQrm as:
''''here
8
and
a•
•
•
•
Bk
3.2
The Sum of Squares and Its Expected Value
The usual way of computing the" sum of squares in a random
model is to treat 6 as if it were fixed, and compute the
quadratic function of Y associated with testing a linear
function LS, where
The column dimension of L is the same as that of X in
equation (3.3), and it has a row :rank of nL'
Then the" sum
of squares corresponding to testing (H o : L6=0) is:
SSL • (Lb) 'CLeX'X)-L,)-l CLb)
T,yhere
and (X'X)- is a generalized inverse of X'X.
It is obvious that SSL is a quadratic form in Y:
SSL = Y'X(X'X)-L'(L(X'X)-L,)-lL(X'X)-X'Y
= Y' QY ,
(3.6)
9
and the matrix of quadratic form is:
Thus the expected value of SSL is:
The corresponding mean square, MSL, and its expected value
can then be
obtained by dividing SSL and
E(SSL)
by the
degrees of freedom nL.
Goodnight
and
Speed [1 978]
developed
the
following
theorem to compute the expected value of SSL when L6 is an
estimable function.
Theorem 3.2.1
If L is from the row space of X with full row
rank nL' and SSL is defined as in equation (3.7), then there
exists a matrix C • [Co!
ell ...
l~]
of the same dimensions
as L, such that
( a)
C =- (U,)-lL
(3.10)
(b)
where U is the
up~er
of L(X'X)-L',
and
triangle of
SSQ(C i )
~he
Cholesky Decomposition
is the sum
of squares of the
elements of the Ci submatrix.
The
proof
of
the
theorem
is
given
below
so
that
• 10
intermediate results may be used later:
If L is in the row space of X, then
L - L(X'X)-X'X
or
L1 - L(X'X)-X'Xi
Substituting equation (3.11) into equation (3.8), we have
(3.12)
This is a submatrix of L'(L(X'X)-L,)-lL.
If we fo rm the matr ix
and
perform
matrix,
the Cholesky Decomposition on the left hand
then
it
becomes
[Ulc]
where
U is
the
upper
triangular matrix such that
Thus
and
It follows from equation (3.12) that
X~QX. = C:C.
1.
1.
for i
1. 1.
= 1, 2,
and
•
I
I
~ ceo.
"'0 0 0 .... 0·
k
r
...
i=l
~SQ(~)
2
2
Iw~. cr. - ":lrcrQ
~
•
1.
--
... , k
f
11
As
can be seen from
inspection of equation (3.10),
a
direct consequence of this theorem is that if any submatrix
Li of L is zero, then the expected value of SSL will not
involve the i th effect.
3.3
The Type II Analysis
This Analysis is based on the type II estimable function
of SAS-GLM procedure (see SAS User's Guide, 1979 edition).
For the two-way classification,
the model
in equation
(3.1) can be rewritten as:
where
i=1,2, ... ,a
j=1,2, ... ,b,
k=1,2, ••. ,n ij
and
nij)O is the number of observations in the (i,j)th cell,
IJ
is a constant,
.,.. and eijk are row, column, interaction effects
1', 6·,
J
1J
and random error. They are independent random variates from
0.
'
norma1 popul a t lons
2
W1,
th zero means and
' ·cr 2 , JB'
2 JAB
2
var1ances
A
andJ e respectively.
The type II estimable function for row effect A is:
12
where
~
..
1.
n. j -
and
a
t n
i-1 ij
(see Searle [1971J p.304).
L2 can also be obtained from the associated rows of the
Forward Doolittle of X'X for the model which has been
rearranged so that all effects which do not contain A (i.e.
column effect B) are put before A.
then
rearranged
back
to
the
The columns of L 2 are
original
order
(see
Go odnight [1 980] ) .
The corresponding sum of squares
is equivalent to
RCAI U ,B), the reduction in total sum of
squares due to fitting A after
~
and B.
The type II estimable function :for the interaction effect
AB is:
for
.
W--
,:ll
I·
L:
i =- 1,2,
j
I
J
• 1,2,
... , a-l
. . . , b-1
(3.16)
13
where
Notice that
involves only the interaction effects. The
~ij
corresponding sum of squares
is equivalent to R(AB I U A, B), the reduction in total sum
J
of squares due to fitting AB after U, A, and B.
The matrices of quadratic form and the expected values of
SSA
and SSAB can be computed using equations (3.8)
2
and
(3.10).
3.4
The Type III Analysis
This analysis is based on the type III estimable function
of
SAS-GLM
It
procedure.
estimable function for
classification,
the
is
the
the balanced
type
III
same
design.
estimable
e ff e c t, A, is:
~
I
q, a-l
J
where
~.
~
=
~.
~
+ y.~. -
(~
as
y. )
a+
'a
the
type
II
For two-way
function
for
row
14
and
y.1.. - 1.b
Notice that
independent
'1 does not
of
the
b
E
i-1
y .•
l.J
involve the column effect and it is
frequencies,
cell
The
corresponding sum of squares
and its expected value can be computed using equations (3.7)
and (3. 10) .
The
type
III
estimable
function
for
the
interaction
effect, SSAB, is the same as that of type II analysis.
The error sum of squares, SSE, is:
Y' (I-X(X 'X)-X')Y
for both type II and type III analyses.
It can be shown that the type III estimable function for
A in equation (3.17) is orthogonal to the type II estimable
f~nction
for AB in equation (3.16).
SSA 2 and SSA 3 , the sums of squares for A in type II and
type III analyses, are usually different from each other.
However, it can be sho',m that if every cell in each row has
the same number of observations (the number may be different __
in different rows), then the hypothesis (H o : L2S=O) can be
15
reduced to (H o : L 6=0), and the two types of sum of squares
3
will be equal.
3.5
The
The Unweighted Mean Analysis
Unweighted
Mean
Analysis
was
Yates [1934] as a simple analysis.
first
introduced
by
The mean of each cell is
treated as the only observation in the cell, and a balanced
analysis
of variance
observation
per
Hi rutsu [1 968],
is
cell.
performed
Later
as
if there
Gosslee
and
and
Levy,
Webster [1 968],
was
one
Lucas [1965],
Narul a
and
Abrami [1975] extended its use to testing of hypothesis.
For the model in equation (3.3), we define the mean of
the (i,j)th cell as:
x 1.J
.. - Y.
j
1..
a
n ..
1.J Yi ·k
~
J
-. J
a
k-1 n ij
and the means of x ij as:
x.1..
-
x- ..
-
-
In
b
1:
j-1
b
~
j-1
addition,
let
x ij
x . ,.
b
-
x .
~
b
-
H be
a
x ij.
i-1 a ,
~
-1..
x.
i-1
-a
the
harmonic
1:
mean
of
all
cell
frequencies and
(3.18)
( see Se ar Ie [1 971 ] p. 366 ) .
16
The sums of squares for row and interaction effects are:
a
SSA
b t (x
-)2
u i-1 i. - x
• b
a
t
i-1
-2
x.
1..
_! ~
b i-1
- y r (Q1
and
(x .. -
l.J
x.. - a
l.J
such that
i.1. • - i . J. + i
b
-2
i: x.
j-1. J
- b
)2
a
i:
i-1
-2
-2
xi. + ab x
17
.J:.... 1
nil
nil
.J:....1
n
n
i2
·
i2
o
-
and 1 n
is an (n 1'jx1) vector of 1 '6;
ij
a 1 i JO)
°
2
(t
j-1 i-1 n ij
Y _ 1 bt
!rQ
2
a
- 1a
BB'
where
B"J.J =- OT?'
AJ.' J'
~J
r
1
oK' ,. j
.. J
LO
b
( y"J.] , ) 2
z:
1.:
n, ,
i=-l j-1
J.J
j
if
,. K
j".K
a
:II
,. eel
(3,21 )
, 18 '
where
C•
and
•
a:o1
DO'
(3.22)
where
All
A
12
D ,.
Alb
~1
A
ab
Notice that if we form the matrix
U =-
CD/AlalC]
and the matrix X in equation
so that Xc'
X1 '
X2 ,
and X3
is partitioned as:
corre~:l'pond
to the mean,
row,
19
column, and interaction effects, then U will be identical to
X if we change the values of all its non-zero elements to 1.
The expected values of SSA u and SSAB u can be computed
using equation (3.9). But, before we do that, we need to
compute
•
~ SSQ(X~A) - ~ SSQ(X~D)
(3.23)
Similarly,
,
~r(XiQABXi)
'1'
.. SSQ(XiC) -
b SSQ(XiA)
~ SSQ(X~B) + ~ SSQ(X~ D) .
-
It can be shown that
r
r
XoA .. b·1 a.
,
I
XoB .. a·l b
,
I
X0 C -
lab
,
ab
0
,
X.,A =I
X2B
X1A .. b·! a
,
X1B .. J ao.
a
I
~
,
X1C .. . 1: 1. 1.0
1.I
X D ..
..
,
.
'T'
..J
aO•
X1D - b'l a
,
a
X .A =- ':" 1+Ib
3
I
a'!b
X B
3
.
• w
1.-
la ® I b
(3.25)
· 20 .
where
,..
1b is a (bx1) vector
of 1 's,
I b is an identity matrix of order b, and
Ja,b is an (axb) matrix of 1 's.
Substituting
equation
(3.25)
into
equations
(3.23)
and
(3.9), we get
e
E(SSAu) • (a-l)nh cr; + (a-l) cria + (a-l)bcri
Similarly,
E(SSAB u ) • (a-l)(b-l)~ cr; + (a-l) (b-l) cr~
The expected mean squares can be obtained by dividing the
expected
sums
of
squares
by
their respective degrees
of
freedom.
The fact that? the expected :nean squares of A and AB are
identical when
O'.~
is equal to zero makes this analysis even
more attractive when the design is unbalanced.
e
'.
21
4. THE TEST PH OC EDURES
4.1
Introduction
In this Chapter, we will discuss seven test procedures
which will be evaluated by their performances on a selected
set of designs based on the model in equation (3.14).
null
( H 0:
hypothesis,
2
cr A =0 )
In
which
we
will
refer
to
hereafter,
The
is
•
each procedure,
the numerator
and
denominator mean
squares (MSN and MSD) are constructed usin~ one or two mean
squares in the form of
MS
where 0
and DFD)
<
= T·MS1
T ~ 1.
+ (1-T)·MS2
(4. 1 )
The corresponding degrees of freedom (DFN
are computed by the Satterthwaite's formula (see
Satterthwaite [1941 and 1946]).
Thus the degrees of freedom for the MS in equation (4.1) is:
f •
where f,
d1 ,
freedom
and
respectively.
d2 ,
the
and EMS,
expected
EMS1,
EMS2 are "the degrees of
values
Notice that EMS,
EMS1
of
MS,
MS1 ,
MS2,
and EMS2 are unknown
.'
22 '
quanti ties in practice; therefore, they will be replaced
bye
their observed values.
An approximate F statistic is then computed as
F' • MSN
f.mD'
This
statistic
is
assumed
to
have
approximately
distribution with DFN and DFD degrees of freedom.
an
F
Once the
statistics F', DFN, and DFD a.re known, we will follow the
usual
procedure.
rej ected
if
F'
That
is
is,
greater
the
than
null
hypothesis
W
... DFN,
DFD ,a.
at
will
a
be
given
significance level a..
The degrees of freedom and the expected values of mean
squares are shown in Table 4.1 wher.e K1, K2, K3, K4, and K5
are the sums of squares of different submatrices
Ci s
defined
in equation (3.10), and nh is the reciprocal of the harmonic
mean of the cell frequenc ies def:Lned in equation (3. 18) .
Only the mean squares needed for testing the null hypothesis
are shown here.
e
'.
23
Table 4.1 The Degrees of Freedom and Expected Values of Mean
Squares for model (3.14).
( a) The Type II Analysis
Mean Square
MSA 2
MSAB
D.F.
Expected Mean Square
a-1
K1
2
(jA
(a-1 ) (b-1 )
MSE
n-ab
2
2
+ K2 (JAB + (je
2
2
K3 (j AB
+ (je
2
°e
(b) The Type III Analysis
Mean Sauare
MSA
D.F.
a-1
3
MSAB
( a-1 ) (b-1 )
MSE
Expected Mean Square
2
2
2
K4 a A + K5 (JAB + 0a
2
2
K3 (jAB + (j e
2
n-ab
°e
( c) The Unweighted Mean Analysis
Mean Square
MSA u
MSAB u
D. F.
a-1
( a-1 ) (b-1 )
Expected Mean Square
2
2
b 0 A2 + °AB
+ nh °e
2
2
+ nh °e
°AB
· 24.
4.2
Procedure A
This procedure is based on the Type II Analysis.
( 1 ).
K2
Let T= y;
, S= T1 .
(2). If T>1 then
MSN=S·MSA 2 + (1-S)·MSE
MSD=MSAB,
DFN is given by (4.2)
DF D=(a-1 ) (b-1 ) ;
otherwise
MSN=MSA 2
MSD=T·MSAB + (1 -T) ·MSE,
DFN=(a-1)
DFD is given by (4.2).
4.;
Procedure B
This procedure involves the con.ventional F test based on
the mean squares from the Type II Analysis.
MSN=MSA 2
MSD=MSAB,
DFN= (a-1 )
DF D= (a-1 ) (b-1 ) .
4.4
Procedure C
This procedure is based on the Type III Analysis.
K5
(1 ). Let T= !C3
(2). If T>1 then
1
,S=:;;-.
'.
25
MSN=S •MSA 3 + (1 -S) •MSE
MSD=MSAB,
DFN is given by (4.2)
DF D= (a-1 ) (b-1 ) ;
otherwise
MSN=MSA 3
MSD=T' MSAB + (1 -T) . MSE~
DF N=(a-1 )
DF Dis g i v en by (4. 2) •
4.5
Procedure D
This procedure involves the conventional F test based on
the Mean squares from the Type III Analysis.
MSN=MSA 3
MSD=MSAB,
DFN= (a-1 )
DF D= (a-1 ) (b-1 ) .
4.6
This
procedure
analyses.
Procedure E
involves
both
type
II
and
type
III
MSN is constructed as a I inear combination of
MSA 2 and MSA . However, it is meaningless to do so unless
3
K2 is not equal to K5 and the value of K3 lies between that
of K2 and K5.
( ~)
I
•
(2 ) .
L t
e
T !C3-K5
. = K2-K; .
MSN=T'MSA 2 + (1-T)'MSA 3
DFN is given by (4.2)
MSD=MSAB,
DF D= (a-1 ) (b-1 ) .
26
4.7
Procedure F
This procedure is a modified version of procedure E.
MSN
is constructed the same way as it is done in procedure E.
But instead of computing DFN from equation (4.2), we simply
use the degrees of freedom for A.
(1 ) . Let T= K3-K5
K2-K5
(2) .
.
MSN=T·MSA 2 + (1 -T) . MSA 3
MSD=MSAB,
DFN= (a-1 )
DF D= (a-1 ) (b-1 ) •
4.8
Procedure G
This procedure is based on the Unweighted Mean Analysis.
MSN=MSA u
MSD=MSAB u '
DFN= (a-1 )
DF D= (a-1 ) (b-1 ) .
27
5. DISTRIBUTION OF APPROXIMATE F STATISTICS
In Chapter 4,
we
have discussed
the various ways· to
construct MSN and MSD using linear combinations of the mean
squares
from
three
types
of analyses.
Since
these
mean
squares are quadratic forms in Y, MSN and MSD, being linear
combinations of mean squares, are also quadratic forms in Y.
However,
in the unbalanced designs,
usually
independent,
distributions.
nor
do
MSN and MSD are not
they
have
chi-square
Therefore, the test statistic F', the ratio
of two correlated quadratic forms,
could not follow an F
distribution.
In this Chapter, we will show that the exact distribution
of F' is actually an infinite weighted sum ofF distribution
functions.
5.1
Distribution of Quadratic Forms
We know that the mean squares from the unbalanced designs
do
not,
in
general,
have
chi-square
distributions.
The
following theorems will show the exact distribution of the
individual or linear combinations of the mean squares.
Definition 5.1.1
If Y is distributed as
N(Xo~'
V), then the
quadratic form Y'AY is said to be translation invariant if
Y AY
I
a
(
Y-
Xo~
*) 'A
(Y-Xo~
*)
· 28 .
for any arbitrary
,
~*.
This is equivalent to the
restriction~
on A that XoAXo=O.
Theorem 5.1.1
any
If Y is distributed as
N(0 , V), and A is
real
sYmmetric matrix of rank r, then Y'AY is
r
distributed as W. 1:: Ai%i' where %i (i =- 1, 2
r) are
i-1
independent, central chi-square random variates each with 1
J
degree of freedom,
non-zero
•••
,
and Ai (i - 1, 2, ... , r) are the real
eigenvalues
of
the
matrix
product
VA
(see
Cochran [1934J and Ruben [1962J).
Proof:
This is achieved by the linear transformation
Y=LPX
where L is the lower triangular matrix such that
V=LL' ,
and P is the orthogonal matrix of the eigenvectors of L'AL
such that
P 'L 'ALP=D
·tlhere D is the diagonal matrix of eigenYalues of L'A:', or
eqUivalently of VA, and
A~S
1.
are the diagonal elements of D.
Thus Y'AY is distributed like
2
r
r
iii:ll
i=-l
Ai:t i
where Xi are
independent unit normal (N(O,1)) variates. This implies that
Y'AY
is
independent
freedom.
distributed
~hi-square
like
W =-
variates
r
!: A. zi
i-l J.
each
Nith
where
1
zi
degree
aree
of
29
Corollary 5.1 .1
If
N(XO~' V),
Y is distributed as
and A
is any real symmetric matrix of rank r such that Y'AY is
translation
r
t
invariant,
then
A.Z. where %i (i
1. 1.
-
Y'AY
is
... ,
1, 2,
distributed
as
are
independent,
central chi-square random variates each with
degree of
W •
i-1
r)
and A. (i • 1, 2, ... , r) are
freedom,
the
1.
real
non-zero
eigenvalues of AV.
Corollary 5.1.2
corollary
the matrix product AV in the above
If
has m distinct eigenvalues each with multiplicity
such that r 1 +r 2 +... +rm=r, then Y'AY is distributed as
tIl
!
A.x. where x ' are independent, central chi-square
J
. j-1 J J
random variates each with r j degrees of freedom.
r·
.J
W.
Theorem 5.1.2
mean
square
If Y is distributed as
from
the
distributed as W - t
i-1
type
II
or
A.Z. where
1. 1.
~l(Xo~'
type
%.
V),
III
then any
analysis
(i • 1, 2,
is
... , r) are
1.
independent, central chi-square random variates each with 1
deg!'ee
of
freedom,
and
Ai (i • 1, 2, ... , r)
are
the
eigenvalues of the matrix product VA for some matrix A.
Proof:
~nalysis
~nd
e-
Since any mean square from
~he
~ype
II or type III
is a quadratic form Y'QY with Q d.efined by (3.8),
it is associated with an estimable function L6 where L
is in the row space of X with row rank nT, all we need to
j.J
show is that Y'QY is translation invariant,
From equation (3.11), we have
i.e. ,X~AXo=O.
. 30 .
(5.1)
L(X'X)-X'X o =L 0
e
=0
since the
estimable functions
from
analysis do not involve the mean
type
lJ.
II
or
type
III
Combining equations
( 3 . 8) and (5. 1 ), we get
• X(X'X)-L'(L(X'X)-L,)-l L(X'X)-X'X
~ X(X'X)-L'(L(X'X)-L,)-l L
0
o
- 0
Thus
X~AXo=O.
mean square.
This is also true when MS is MSE, the error
Therefore, type II and III mean squares are
not only translation invariant, but are also distributed as
r
W - t A'Zi according to Corollary 5.1.1.
. 1 1.
1.~
~h eo rem
5. 1 • 3
If Y is d istr i buted as N(XoJJ, V), t hen anye
mean square from the unweighted mean analysis with r degrees
r
of
freedom
Zi (i
random
~
is
distributed
1, 2, ... , r) are
variates
as
independent,
degree
with
fiJ ~ • t
1.-1
central
of
Ai zi
where
chi-square
freedom,
and
Ai(i - 1, 2, ... , r) are non-zero eigenvalues of the matrix
product VQ for some matrix Q.
Proof:
Here,
~ean
squares of interest are MSA u
Recall that
=
.....L
a-~
Y' (0
·1
~nd
MSAB u '
31
and
MSAB u
1
• - ......- - - - SSABu
(a-l) (b-l)
•
(a-l~(b-l)
Y'(Q3 - Q1 - Q2 + Q4)Y
1
• (a-l) (b-l) Y'QAB Y
Q and Q4 are defined in equations (3. 1 9) ,.
3
(3.21) and (3.22), respectively.
We only have to
where Q1'
(3.20),
Q2'
show that Y'QAY and Y'QABY are translation invariant, i.e.,
,
,
XoQABXo=XoQAXo=O.
It
can be shown that
if Xo
vector of 1 's, then
Q1Xo
• Q2 Xo • Q3Xo • Q4Xo • D·
.
Alb
~l
where D is defined in equation (3.22). Thus
QAXo
a
(Ql - Q4)Xo
• 0
and
Q.~Xo
• (Q3 - Q1 - Q2 + Q4)Xo
• 0
is an· nx1
· 32 ,
Therefore, Y'QAY and Y'QABY are translation invariant and,
according
to
5.1.2,
Corollary
MSA u
and
MSAB u
distributed as linear combination of independent,
are
central
chi-square random variates.
Theorems 5.1.2 and 5.1.3 have shown that any mean square
in Table 4.1.1
is distributed as a linear combination of
independent, central chi-square random variates each with 1
Since the MSN and MSD constructed in
degree of freedom.
Chapter 4 are 1 inear combinations of these mean squares,
they
are
also
distributed
as
linear
combinations
of
independent, central chi-square random variates.
We
now
proceed
combinations
of
to
find
the
independent,
distributions
central
of
linear
chi-square
random
e
Robbins and Pitman [1949] used the characteristic
variates.
functions
of
chi-square
function
distribution
independent,
central
variates
of
to
linear
a
chi-square
random
show
that
combination
varia~es
is
the
of
an
infinite weighted sum of chi-square distribution functions.
A similar
result
was
by Laha [19541
obtained
using
the
distribution of Bessel functions.
Theorem
5.1. t1
combination
of
The
distribution
independent,
function
central
of
a
linear
chi-square
ra.ndom
variates with positive coefficients is a.n infinite weighted
sum of chi-square distribution functions.
e
33
Proof:
where
Let
X
J
(j - 1, 2,
... m) are
independent,
central
chi-
square random variates each with f i degrees of freedom such
m
that I: f. - M, and without loss of general i ty, we can
j -1 J
assume that
1 < Al
-
<A2
<
...
< Am
.
Let
. -2
Pj
f1
Since
xj
is
and
a
m
I:
j-l
..
Pj
chi-square
M
2
variate,
its
characteristic
function is:
<fI x. (t)
.. ( 1 - 2i t)
-p.
J
•
J
Thus the characteristic function of ~ is:
A
m
..
-0 .
II [A.(l - 2it:)
j-l
J
- (A. - 1)] 'J
J
34
1 k
-(p +1<.)
(1 • - ) j (1 • 2i t)
j
J]
Aj
Letting
and
we have
m
4>ulA. (c) =-
IT
j-1
-(p.+k..)
co
[
(1
~
a.. k
k.aO J, j
-
2it)
J
m
-
CD
•
~
K-O
~(1
- 2i:)
~
j a1
(p.+k.)
J J
J
J ]
CD
•
t
~(l
K-O
- Zit)
-(~ +
35
K)
Z
(5.5)
This is a linear combination of the characteristic functions
of chi-square variates whose degrees of freedom are M+2K
where K=O,1, ... , respectively. It follows that
CD
Pr(~ oS. w)· 1:
K-O
1\
~ FM+2K (w)
where FM+ 2K ( ) is the distribution function of a chi-square
variate with M+2K degrees of freedom. Letting V=AW, we have
If we let t=O in equations (5.2) an1 (5.5) , we get
!
a.. -
I-O
l\.
(5.8)
1
It can also be shown that
1:
k.=O
J
a.,k. = 1
J
J
tor j
=
1, 2,
...
I
~
This implies that the infinite sum in equation (5.7) is less
or equal to
combination
variates
is
Thus, the distribution
1.
of
an
independent,
infini te
central
weighted
f~nction
of a linear
chi-square
sum
of
distribution functions as given in equation (5.7).
random
chi-square
36
Distribution of The Ratio of Two Independent Quadratic
5·2
e
Forms
In
this
section,
we
are
interested
in
finding
the
distribution of the ratio of two independent quadratic forms
which do not have chi-square distributions. We will limit
our
discussions
to
quad ratic
forms
defini te and translation invariant.
that
are
non-negative
It has been shown, in
the previous section, that a quadratic form is distributed
as a linear combination of independent chi-square variates.
Therefore, the ratio of two independent quadratic forms can
be considered
as the ratio
of two linear combinations of
independent chi-square variates.
Theorem 5.2.1
linear
The distribution
combinations
of
f~nction
independent,
of the ratio of two
central
chi-square
e
random variates wi th positive coeffic ients is an infini te
weighted sum of F distribution functions.
Proof:
Let
x
u - z
-
-
A1X1 + A2XZ +
;1%1 + ;2%Z +
+ AmXm
+ ;nZn
where xi (i=1,2,o .. ,m) and Zj (j=1,2, ... ,n) are independent,
~entral
chi-square
random
variates
each
degrees of freedom, respectively, such that
with
Pl'
and
a·
.J
e
37
q. • N
J
'
•
and without loss of generality, we can assume that Ais and
•
~jS
are all greater than or equal to unity.
Then,
from
Theorem 5.1.4, the distribution functions of x and z are:
1:11
G(x) •
and
t
K-O
~FM+2K(X)
1:11
H(z) -
1:
L-O
0LFN+2L (z)
where a K and b L are defined as in equation (5.4).
Let
eW • u
Thus
We
TN'ill
first
derive
the
d istr ibution
of
w.
characteristic function is:
~w(t)
• fo=Jo=
e iwt dG(x) ·dH(z)
br
1:11
•
1:
L-O
N +L
22
~
!' (~ ....L)
f
0
1
N
l:ll
z
-it
1z
+L-1
e
-1 z
dz
Its
38
~ r(~ +K+it}
aD
• 1:
- K-O
t
r(~
r (~ +L-it)
bL
aD
L-O
+K)
r(~ +L)
Applying Fourier Transformation, we get the density function
of w as:
aD
-
1:
K,t-O
r (~
+K) • r (~ +L)
· di ~:e-iwt r(~ +K+it).r(~
+L-it)dt
Let
!N + L
- 'l.t
-
-h
,
Thus
~ +
it -
L + h
,
The integral in equation (5.9) can be written as:
1
- e
-w(~ +L)
i
N
-1 -
L+'
J-1'N - L'
I4l
J.e» e -wh
r c~ + K i- ~ +L+h) • r (-h) dh
-J.e»
-'(11
= e
2' +l..• )
eN
_CM+K+ N +L)
(1
+ e _rl1 ) !
2'
.
"(M
+ K •,!"TN . L) . .';1'
?
J. 2'
(5.9)
39
(see Whittaker and Watson [1927J p.289).
Substituting this into equation (5.9), we get
all
_w(li+ L)
- d~+ K +li + L)
W
axbL e
2
(l+e- ) 4
2
r(~+K+~+L)
few) - t - - - - - - - - - - - - - - - - - - - - - - - - - K,L-O
r(~ + K)· r(~ + L)
Since
e
W
-
u
and
dw _ 1
au
u
the distribution of u is:
f(u)·
t
K,L-O
where B( .,.) is the Beta function.
Let f 1=M+2K and f 2=N+2L,
then
The distribution function of u becomes
· 4.0'
ca
F(u) •
I:
K,t-O
~bL
l
f1
U
u
r -
1
du
f 1+f2
,.
r1 f2
2 B(r,
( l~..l)
r)
f1
Making the transformation u-r:;v, we have
2
f1
1 r v
(1:)
2
f 1+f 2
f
f2
GO
F(u) -
1:
K,t-Q
-
SXbL
ffiU
o
f
(l+lv)
2
CIt
I:
K,L-O
-
~bL • F f
~
1'~2
K,t-O
r -
1
dv
f1 f2
)
B(r' 2
f2
(r:- u)
1
CIt
1:
2
f1
~b L • FM+2K,N+2L
(N+2L u)
M+Zk
(5. 10)_
where F M+2K ,N+2L( ) is the F distribution function with M+2K
and N+2L degrees of freedom.
Thus the distribution function of the ratio of two linear
combinations
of
independent,
central
chi-square
random
variates with positive coefficients is an infinite weighted
sum of F distribution functions as given in equation (5.10).
This implies that the distribution function of the ratio of
~wo
independent,
non-negative
definite
~nd
translation
invariant quadratic forms is also an infinite weighted sum_
of F distribution functions.
41
5.3
The Exact Distribution of Approximate F Statistics
In Chapter 4, we have shown different ways to construct
the numerator and denominator mean squares,
MSN and MSD,
which are quadratic forms in Y. Let
MSN=Y'AY
and
MSD=Y'BY.
The approximate F statistic is the ratio
?' = MSN
MSD
From the discussions in Chapter 3, we know that both MSN and
MSD, being linear combinations of mean squares with positive
coefficients,
are
non-negative
invariant quadratic forms.
definite
and
translation
In an unbalanced design, MSN and
MSD are not usually independent, nor do they have chi-square
distributions.
Therefore,
Box [1 954]
d istribut ion.
function of F'
F'
showed
does
that
not
the
fo llow
an
F
d istr ibution
is a finite series if the mul tipl ic i ty of
every distinct eigenvalue of VA and VB is even, (V is the
variance-covariance matrix of Y).
satisfied
in
a
typical
This condition is seldom
unbalanced
design.
generalized distribution function is needed.
Gi v en any
'T
a1 ue f> 0,
A
more
. 42 '
Pr(F' .s. f) • Pr (~ .s. f]
Y'AY
• Pr [Y'BY
• Pr (Y' (A
- p,:, (Y'QY
Since
MSD
is
a
linear
oS f]
- f·
oS
B)Y .s. 0]
0]
(5.11)
combination of mean
squares
wi th
positive coefficients, we are sure that Y'BY is non-negative
d efini te;
otherwi se,
Furthermore,
since
equa"tion
MSN
( 5. 11 )
and
MSD
will
are
both
invariant, so is Y'QY. Applying Corollary
not
hold.
translation
5.1.2, equation
(5.11) can be written as:
Pr(F'
m+n
~
f)
• Pr [ !: A. z.1i=l
~
• Pr [
..
m
I..
i-1
m
!:
- Pr [i-1
n
!:
j-1
·,vhere
z..... (i =- 1
~hi-square
,'\ i ' (.:
" =-,-,
..
t
A.1- z.1-
A.1. z.1I"
1;'
J
y.
01.
~
-
n
!:
j-1
~ .:
J
y.
J
~
0]
-< 1]
(5.12)
J
2, ... , m + n)
~re
inde-pendent,
central
random variates each with Pi degrees of freedom,
I
?
-,
... , m
+ n)
eigenvalues of VQ such that
'3.re
the
non-zero,
distinct
43
Pl + P2 + ... + Pm • M,
Pm+l + Pm+2 + ... + Pm+n • N
and
I
I
'~J.
" • - Am+j'
Yj· Zm+j
for
j
• 1, 2, ..., n
Letting
f
t •
~1
,
for i - 1, 2, ... , m ,
and
for j - 1, 2, ... , n
equation (5.12) becomes
m
Pr(F'
~
f)
A E A.Z.
1. 1.
• Pre i-1
n
~
1]
... E ; jY j
j-l
~
m
E
A.Z.
1. 1.
• P-r [i=l
n
... ;jYj
';=1
..J
~
:
< ~]
- A
.
44
Applying theorem 5.2.1, we get
where F M+ 2K , N+2L ( ) .is the F distribution function wi th M+2K
and N+2L degrees of freedom, and aK' b L are defined as in
equation (5.4).
Thus
the
distribution
function
of F'
is
an
infinite
weighted sum of F distribution function given in equation
(5.15) .
Th is resul t
provides us wi th a powerful
tool to
evaluate the test procedures under different conditions.
5.4
Computation of The Distribution Function
It is obvious that we cannot compute the probability
equation (5.15)
because it is a double infinite sum.
i_
If
this infinite sum is replaced by a finite sum with K1 and L1
as the upper bound of K and L, respectively, then the
truncation error is:
1:'
w -
Pr F ~
('
f)
K1
•
L1
t
t
~bL· FM+2K,N+2L
K-O L-O
[(N+2L)~]
CM+zk)A
.
I ~
\~./.
16')
45
Since
!
K-O
~.
1 and
for K, L - 0, 1, 2, ... ,
equation (5.16) becomes
where
and
Equation
(5.17)
provides
the
upper
limit
of
the
truncation error due to replacing the double infinite sum in
e qua t ion (5. 15 ) wi t h a fin i t e sum . Th is e qua t ion ism 0 s t
helpful when we want to find the upper bounds of K and L.
46
If we choose K1 and L1 large enough such that the right
hand side of equation (5.17) is less than the predetermined
tt
upper
limit
of
truncation
error.
Then
equation
(5.15)
becomes
(5.18)
Notice that an F distribution function needs to be computed
for each term of the double sum.
consuming.
It
can
be
shown
This can be very time
that
by
making
the
transformation
x - 1
the
F
distribution
+ S-A
function
in
equation
(5.18)
can
e
be
replaced by an Incomplete Beta function, and this equation
becomes
where I x (.'.) is the Incomplete Beta function,
;
-1·
In addition,
M+2K and f
Z
~he
2 -
recurrence
N+2L
Z
for~ula
47
can
be
used
to
generate
most
of
the
Incomplete
Beta
functions needed in equation (5.19).
As for the coefficients a K and b L , they can be computed
using equations (5.3) and (5.4). Since the only difference
between
aK and
eigenvalues
bL
of
is
that
while
VQ
aK depends
bL
depends
on
on
the
positive
the
negative
eigenvalues of VQ (Q defined as in equation (5.11)), we will
only discuss the computation of aK here.
We recall in equation (5.5) that
•
t
K-O
~(l _ Zit)
m
•
-(~ + K)
2
QCl
[
IT
j-l
t a. k (1 - 2it:)
kj-O J, j
-(p.+k.)
J J ]
(5.20)
where
m
m
Kand
a K,
t
j-1
a· k
J, j
respectively.
M. t P
Z j-1 j
kj
are
defined
in equation
(5.4)
and
(5.3),
If we let x=(1-2it)-1, then equation (5.20)
becomes
QCl
t
K-O
a:{x
K
m
=-
II
j-l
QCl
(
!:
k.-O
~
a. ,
J
'~j
k.
x J)
(5.21)
..J
If the upper bound:;f K is chosen to be n, then equation
48
(5.21) becomes
n
t
XK
ax
K-O
m
II
- j-l
k.
n
( t
a
k.-O
J
j ,k
x J)
(5.22)
j
This is equivalent to multiplying m polynomials of degree n
a j ,k
is the k. th coefficient of the jth
J
j
po lyn om i al and aK is the Kth coefficient of the product
such
that
polynomial which is truncated to degree n.
When
n
in
equation
(5.22)
is
small,
the
following
algorithm can be used to compute the coefficients aK's.
Let
S be
a
(2x( n+1 ))
matrix,
C be
a
vector
of n+1
elements and I1, I2 be integers that are either 1 or 2 such
that the I1 th row of S contains the coeffic ients of the
product of the first j-1 polynomials, and the I2 th row of
stt
contains the coefficients of the product of the first
j
polynomials, and the vector C contains the coefficients of
the jth polynomial.
Algori thm 1:
step 1: Set I1 =1 , I2=2 and j= 1 .
step 2: Set S(I1 ,K)=aj,K
for each K=0,1, ... ,n.
step 3: Set j = j+1 .
step 4: Set C(K)=a. K
J,
for each K=0,1, ... ,n.
tt
49
K
e
step 5: Set S(I2,K)= t S(I1 ,K-i) ·C( i) for each K=O, 1, ... ,n .
i-O
step 6: Interchange 11 and 12.
step 7: If j=m then go to step 8; otherwise go to step 3.
step 8: Set aK=S(I1,K)
As
for each K=O,1, ..• ,n.
we can see from equation (5.3)
that the
following
recurrence relation can be used to compute aj,K:
for K - 1, 2, ...
·"here
-Po
a. 0 - A.
J,
J
J
The computing time for this algorithm is in the order of
O(n 2 ). This can be quite costly when n is large.
To find a
faster
that
algorithm,
we
acknowledge
the
fact
the
coefficients of the product of two polynomials are identical
to ":;he convolution of the two coefficient
original polynomials (see Aho,
Hopcroft and 1Jllman
p.255).
Le"t
p(X)
:II
',~ctors
q(x)
=
n
1:
j-O
of the
r- 1 9741.
50
be the two original polynomials, and let band c, both o f e
length
n+1,
be
the
coeffic ient
vector
of
p( x)
and
q( x)
respectively, then the elements of the coeffic ient vector of
the product polYnomial, d of length 2n+1,
can be computed
as:
K
d(K) •
b(j)"c(K-j)
t
j-O
for K - 0, 1, ... , 2n .
These are exactly the components of the convolution of the
two vectors band c, if we ignore d(Zn+1) which is zero.
Let B be the Fourier Transform of the vector b such that
B(K)
-
n
1: b (j) "~NKj
j-O
for K =- 0, 1,
... ,
(5.23)
n
e
21Ti
where
n
w =- e + i
and
i
is
the
square
root
of -1-
The
elements of b can be recovered from that of B by the Inverse
Fourier Transformation
1
n
r .
K-O
b (j) - n+l ,.
-1(.
B (K) • w • J
for j • 0, 1, ... , n .
Similarly, let C 'and D be the Fourier Transforms of c and d,
respectively.
We
observe
f~om
equation
(5.23)
that
coefficient of the Fourier Transform of b,
the polynomial p( x\
at the point x=w K.
B(K),
the
Kth
is ....
une value of
This implies that
e
'.
51
B(K) ·C(K),
x=w K,
the product of the values of p(x)
should be equal to
D(K),
the value
and q(x)
at
of the product
polynomial at the same point. The only problem is that the
product of two polynomials of degree n is a polynomial of
degree
2n.
This
can
be
solved
by
expanding
the
two
coeffic ient vectors, band c, to length 2 (n+1) and letting
their last n+1 elements be zeroes.
Theorem 5.4.1
two
vectors
(Convolution Theorem).
is
the
Inverse
Fourier
The convolution of
Transform
of
the
elementwise product of the Fourier Transforms of the two
"lectors.
Proof:
Let
b = [b(O), ... ,b(n),O, ... ,O]'
and
c = [c(O), ..• ,c(n) ,0, •.. ,0]'
be vectors of length 2(n+1), and let
d = [d(0),d(1), .. ~,d(2n+1)]'
be the convolution of band c such that
2n+l
d(K) •
!:
j=-O
b(j) oc(K-j)
for K =- 0, 1, ... , 2:1 . (5.24)
Notice that d(2n+1 )=0 and c(K)=O if K<O.
Let
B,C
and
D,
all
vectors
of length
2(n+1),
be
the
.'
Fourier Transforms of b,c and d, respectively. We need
prove that
D(L) .. B(L)-C(L)
for L" 0, 1, ... , 2n+1.
Applying Fourier Transform to equation (5.24), we get
D(L)"
2n+1 2n+1
t
t
b(j)_C(K_j)_wKL
K-O j-O
for L - 0, 1, ... , 2n+1.
Interchanging the order of summation and substituting s for
K-j yields
D(L) •
2n+1 2n+1- j
t
t
j-O
(
)
b(j)-c(s)-wL s+j
s--j
for L· 0, 1, ... , 2n+1.
Since b(j)=O if j>n, we can lower the upper limit of j to n,tt
and since c(s'=O if s<O or s>n and the upper limit of s is
at least n+1 regardless of how large
,j
is, we can replace
the lower and upper limits of s with 0 and n, respectively.
Thus
D(L)
n
=-
j-O
=-
s-O
B(L)-C(L)
for L" 0, 1, ... , 2n+1 .
We
have shown
~hat
the product of the Lth coefficient of
the Fourier 'rransforms of band c is ~he 1 th coefficient of t t
the Fourier Transform of d.
It follows that the convolution
53
of two
vectors
is
the
Inverse Fourier
Transform
of the
elementwise product of the Fourier Transforms of the two
vectors.
The importance of this theorem is that when multiplying
two
polynomials,
the
coefficient
vector
of
the
product
polynomial can be obtained by taking the Fourier Transforms
of the coefficient vectors of the two polynomials, obtaining
the elementwise
product of the two
transforms,
and
then
performing the Inverse Fourier Transformation on the product
vector.
The following algorithm is obtained by applying theorem
5.4.1 to the polynomial multiplications in equation (5.22).
It can be used to compute aK's when n is large.
Algorithm 2:
step
Set j=1.
step 2
Set S(k)=a.J , K
for each K=0,1, ... ,n,
and S(K )=0
for each K=n+1, ... , 2n+1 .
step 3
Obtain Fourier Transform of vector S in place.
step 4
Set,j=j+1 ..
step 5
Set C(K)=a,J,.K
for each K:0,1,.o o,n,
and C(K)=O
for each K=n+1, ... ,2n+1.
54
step 6
Obtain Four ier Transform of vector C in place-
step 7
Set S(K)=S(K)-C(K)
step 8
If j<m then go to step 4; otherwise go to step 9-
step 9
Perform Inverse Fourier Transformation on vector S
for each K=0,1,
, 2n+1 -
in place.
for each K=0,1, _._,n.
step 10: Set aK=S(K)
The Fast Fourier Transform algorithm (FFT) introduced by
Cooley and Tukey [1965J can be used to compute the Fourier
Transform
and
its
computing
time
for
inverse
FFT
is
in
in
the
above
the
order
algorithm.
of
The
O(nlogn).
Therefore, time efficiency can be realized when n is large.
Notice tha.t the above algori thm is an approximation
toe
the correct algorithm,
which can be obtained by deleting
step 9,
the following
and
inserting
three steps between
steps 7 and 8:
step 7.1: Perform Inverse Fourier Transformation on vector S
in place.
step 7.2: Set S(K)=O
for K=n+1, ... ,2n+1.
step 7.3: Obtain Fourier Transform of vector S in place.
What
this does
is to el im inate the
contr i butions of the
higher order coefficients of the product polynomials in thee
intermediate steps.
We know that if n is large enough and
55
the coeffic ients of the product polynomial is truncated at
the nth term,
the
above
accuracy.
the remainder is insignificant.
three
steps
can
be
ignored
without
Therefore,
loss
of
56
6. EVALUATION OF THE TEST PROCEDURES
6.1
The
purpose
of
Evaluation Process
this
chapter
is
to
evaluate
the
performances of the seven test procedures using different
designs and different values of variance components.
For a
given two-way crossed design, a given set of values of the
cri
variance components and the null hypothesis (H o :
=0), the
following steps will be taken to compute the probability of
rejecting
the
null
hypothesis
for
each
of
the
test
procedures:
1. Construct the X matrix as
in equation (3.2),
and
variance-covariance matrix V.
2. Construct the matrices of quadratic
f~rms
for the mean
squares in table 4.1.
3. Compute the coefficients of the variance components in
the expected mean squares using either equation ('3.10)
or
equation (3.18).
4. Depending
on
the
particular
investigated, construct the
combinations of the
m~trices
matri(~es
test
procedure
being
A and B using linear
of quadratic forms constructed
in step 2 such that MSN=Y'AY and MSD=Y'BY.
e
57
5. Compute the degrees of freedom DFN and DFD for MSN and
MSD, respectively.
6. For a given significance level
compute the critical
el,
value of the approximate F statistic
f=F DFN , DFD, el
7. Construct the matrix Q=A-f·B as in equation (5.11).
I
8.
Obtain
Ai (i=1, 2, ••. ,m),
the
m
distinct,
positive
I
eigenvalues of the VQ,
~j
and
(j=1 ,2, ••• ,n), the absolute
values of the n distinct, negative eigenvalues of VQ.
t
9. Divide the values of
value,
Xi (i=1, 2, .•. ,m) by their smallest
, and assign the resul ts to
equation (5.13).
,
~.
Divide the values of
value,
:.J
(j=1, 2, ... ,n) by their smallest
,and assign the resul ts to
equation (5. 14).
Ai (i=1 ,2, ... ,m) as in
Thus,
A.
1.
and
~.
J
~j
(j=1 ,2, ... ,n) as in
are all greater than or
equal to unity.
10. Given K1 and L1 , compute a K (K=0,1, ... ,K 1 ) based on the
values of
A.S and their mul tipl ic i ties, and compute
1.
I
and their
J
multiplicities using either of the two algorithms discussed
b L (L=0,1, ... ,L 1 )
based
on
the
values
of
~. S
in Section 5.4.
<
11. Compute
Pr (F!
~quivalently,
equation (5.19).
f)
using
equation
(5.18)
or,
58
12.
Compu~e
the probability of rejecting the null
hypothesis~
at the significance level a:
Pr ( F'
>
f)
=1
- Pr (F'
<
f).
A computer program has been written to
computations in the above twelve steps.
implement
the
The probability of
rejecting the null hypothesis, or the power function of the
test, is a measure of the performance of this test procedure
under the given conditions. It is compared against the power
functions
condi tions,
of
the
and
other
test
procedures
appropriately ranked.
under
the
same
Si nee we want the
power function to be as small as possible when the null
hypothesis is true and as large as possible when the null
hypothesis
is
false,
the
test
procedure
which
has
the
largest power function will have a rank of 7· if the nulle
hypothesis is true or a rank of 1 if the null hypothesis is
false.
The
probab il i ty in step
truncation error defined
12
was
computed
in equation (5.19)
so
that
its
is less than
10- 5 .
In a few cases, where the infinite series in equation
(5.8)
did
not converge
fast
enou.gh,
the
upper
limit of
truncation error has been relaxed to 10- 3 without disturbing
the ranks of the procedures.
6.2
Designs and Values of Variance Components Investigated
Fifteen two-way, unbalanced designs were selected for the
59
evaluation of the seven test procedures.
range from (3x3)
to (1 2x3).
Their dimensions
The cell frequencies of these
designs are I isted in table 6.1.
where the number in the
i th row and jth column of each design is the cell frequency
or
the number of observations
in
its
(i, j) th cell.
number
in parentheses on top of the design is
number
of
observations.
For
each
of
these
The
the total
designs,
the
following nine sets of values were assigned to the variance
components
222
O'A' O'.-\B and
O'e'
2
O'A
2
O'AB
0
.25
0
0
4
.25
.25
.25
.25
4
5
.25
5
5
4-
0'2
e
60
Table 6.1 The Cell Frequencies of Two-Way Designs
1- ( 1 5)
2. ( 1 7)
1 1 1
1 1 1
1 1 1
4 1 2
1 1 1
2 2 2
1 1 4
1 3 4
1 3 1
333
2 2 2
1 1 6
1 4 4
1 2 5
2 2 2
2 1
2 2 2
6. (24)
7 •. (24)
3.
8.
(20)
(28)
4.
9·
( 24)
(32)
5. (24)
10. (35)
1
1 1 1
3 3
1 1 4
1 2 4
4
4 1 1
1 3 1
1 2 3
1 1 3
2
3
1 1 2
4 1 2
1 1 3
1 5 1
4
1
1 1 3
1 1 2
1 1 5
1 2 3
2
1 2 3
2 2 1
5 2 1
411
5
e
1 1 2
11- (42)
1 2.
(42)
1 3. (48)
1 4.
(50)
1 5. (50)
2 1 2
4
2
1 1 2
1 1 2
3 1 1
1 1 3
3
4
1 1 3
1 4 1
1 2
3 1 2
3
2
1 2 1
1 2 2
1 1 1
1 1 1
3
2
1 1 1
1 3 1
1 2 4
1 5 2
4
1
1 1 4
2 1 1
1 1 1
3 1 1
3
1 2 1
1 2 1
2 1 1
1 2 1
2
3 1 1
1 2 1
1 3 1
1 1 4
3
2 1 1
1 1 1
1 1 1
321
2 1 4
2 1 1
2 1 4
1 4 3
1 1 1
4
1 4 1
1 1
e
61
Furthermore, for each given set of values of the variance
components,
the
probab il i ty that
the
null
hypothesis
is
rejected by each of the seven test procedures was computed
at three significance levels (.10, .05, .01).
The values of the coefficients of the variance components
in the expected mean squares shown in table 4.1 are listed
in table 6.2.
It was found in all fifteen designs that the
inequality
K2 2. K3 2. K5
held.
Therefore, the ratio of K2 to K3 was always greater
than or equal
procedure A.
to
unity,
and we only synthesized MSN in
On the other hand, the
~atio
al ways less than or equal to unity,
of K5 to K3 was
which means we only
synthesized MSD in procedure G.
The values of K2, K3 and K5 are all equal when the cell
frequencies within each row are the same, i.e.
n ij = n i
for i=1, 2, ... ,a.
;Nhen this condition is satisfied,
and
number
5,
all
procedures
~s
in the case of designs
except
procedure
G
~~e
identical.
iNe also noticed,
values
0
f
K3
in designs number 6 and 1 2,
and K5
are
equal
'Nhen
there
are
that the
only
columns in the design. In that case, procedures 0, D, E
F
~re
identical.
t'".,o10
~nd
62
Table 6.2 Coefficients of The Expected Mean Squares
K1
K2
K3
K4
K5
nh
1
4.8CCO
1.6000
1.6000
4.8000
1. 6000
0.6667
2
5. 0909
2.4242
1.3333
3.6828
1.2276
0 .. 8241
3
6. C417
2.3262
1. OS 77
I~.
7097
1. 5699
0.6759
4
5.2209
2.0303
1.5953
4.4313
1. 4771
0.6903
5
5.15eo
1.9167
1.9167
5.7500
1. 9167
0.5833
6.
4. 3916
2.6903
1.70 13
3.4026
1.7013
0.6033
7
4.4417
1.7247
1.35 d5
3.34B4
'42ti20
o. 79 44
8
5.2333
2.0093
1.61 20
4.4720
1.4909
0.6833
9
5.8e41
2.6201
1.6220
4.4121
1. 4707
0.6878
10
5. 3619
2.2472
1.5573
4.2:19
1.417]
0.7140_
11
4.9f15
1.9963
1.482b
4. 1106
1. 3702
0.7479
12
5. 1422
2.8981
2.2441
4.4882
2.2441
0.4792
13
4.6505
1.861 9
1.3943
3.9060
1. 3020
0.7833
14
4. d509
1.9209
1. 46 10
4.. C419
1. 3473
0.7639
15
4. CJ:; 4
1.5544
1.2405
3.5453
1.1018
0.8657
OES IGN
63
When two or more procedures are identical, the rank assigned
to each of them is the mean of the ranks that would have
been assigned to them had they not been identical.
6.3 Results
The
power
different
functions
designs
of
and
the
tests,
different
in
values
percent,
of
for
variance
components are listed, along with the other statistics, in
tables 9.1 through 9.15, where VA and VAB are the values of
cr 2
A
and
E(MSD),
2 , respectively, R is the ratio of E(MSN) to
AB
and P10, p05, P01 are the power functions of the
a
test at 10,5, and 1 percent significance levels.
Notice
that we have introduced a new variable R, the ratio of the
expected values of
MSN
to that of
Although it is not
MSD.
the expected value of F', it gives us a good indication of
how large the power function of the test will be.
null hypothesis is
procedures
except
true,
R
procedures
is
equal
Band
to
D.
hypo thesis is false, R will increase as
2 lncreases.
.
decrease as ,crAB
When the
unity
lihen
J'.i?
in
the
all
null
increases and
For a better view of the results, the ranks of the test
procedures are shown in tables 6.3 through 6.5 where VA and
VAB
are the values of
O'i
and
O'~, respectively.
see that these ranks are not random patterns.
1.ve can
~¥hile
the
power functions change wi th the designs, the ra.nks of the
test procedures remain very stable.
64
Table 6.3 Ranks Of The Test Procedures.
VilB
A
B
c
2
3.5
6
3.5
1
4
5
3.5
5
5
5
3.5
7
7
7
3.5
7
7
7
7
7
7
7
7
7
7
4.~
4.5
4.5
DESIGN
,
.3
5
6
0.25
I
8
9
10
11
12
13
14
,
15
2
J
4
5
6
1.00
7
ti
c
..
10
11
12
1.1
14
15
1
2
3
1+
5
4.00
6
7
~
9
10
11
12
1J
14
15
5
6
5
5
5
5
5
6
1
5
2
4.5
2
2
2
"]
3
7
7
4.5
7
7
7
7
7
7
7
7
7
7
1+.5
4.5
2'"I
~
2
1
2
.3
1
J
2
4.5
1
1
2
1
2
2
1
2
2
2
,
2
2
3
j
2
3
4.5
4.5
4
3~.5
3.5
2
6
6
3.5
3
3
3
3.5
2.5
3
3
3
3
3
3.5
3
3
2
7
5
4.5
.
5
5
5
5
4.5
5
5
5
14
~
-'
3.5
...
1
2
r.
3.5c
~
~.
6
1
1
.3.5
-
6
6
6
1
1
~
6 c:
J.
oJ
1
1
b
1
6
6
4.5
4.5
6
4.5
~
4.5
4.5
4. :;
4.5
,
4.5
1
1
1
1
5
7
7
7
7
7
7
7
G
5
5
:.>
7
7
F
4.5
7
7
4.5
"1
E
4
4
4
5
4.5
c'
5
4
5
4.5
4 ;)C·
7
aANK
C
1
3.5
2. 5
2
2
2
2
2
3.5
(VA=O)
1
1
1
1
1
4.5
~
1
1
4.5
4.5
2
1
2
1
1
4.5
1
1
1
6
6
6
6
6
6
6
4.. S
6
6
6
6
6
~.
5
4.5
6
6
6
b
(;)
1.1.5
6
6
6
7
~
oJ
ij
4
4
14
4
1
4
4
4
..1
-'
4
4
3
4 .~
r.:
1
3
3
3
.3
J
14.5
5
1
5
5
5
5
1.+
2
.3
2
4
1+
4.5
1
4
2
1+.5
6
4
4
4
4
3
:;
4
2
3
4.5
4
4
2
3
3
4
3
4.5
3
.3
4.5
3
4
4
1
4
4
2
4
J
3
I
65
Table 6.4 Ranks Of The Test Procedures.
VAD
DE31G~~
1
2
J
~
0.25
5
6
7
E
9
10
11
12
13
14
15
1
.2
3
4
5
1. 00
6
7
E
9
1C
11
B
3.5
7
5
6
3.5
7
6
b
7
6
7
7
7
1
1
1
1
1
1
3.5
Ii
7
3.5
6
6
7
6
6
7
4
S
7
6
3.5
7
5
6
J.5
7
7
7
9
10
11
7
7
,6
Co
e
12
13
1~
15
1
1
1
1
3",/5
I)
15
2
3
3. 5
3 .. 5
7
1
1
1
1
3.5
7
5
12
13
14
4.00
A
6
I
6
6
b
1
1
1
1
1
1
1
1
1
1
1
1
1
3.5
1
1
1
J.S
1
1
1
1
1
1
1
1
1
1
(VA=.25)
F
3.5
5
J
3
3.5
3.5
3
3
3
3 .
3
3 .. 5
3
3
3
3.5
S
4
3
3.5
J.S
3
3
3
J
3
3.5
J
3
.3
3.5
3
)
3
3.5
3.5
3
3
3
3
3
3.5
3
3
3
3.5
b
7
7
3.5
3.. 5
7
.
2
'')
.. ....l:
_
M
J.5
2
-..,
'J
7
2
6
7
Ao
6
3.5
6
7
6
3.5
0
7
7
3.5
J.5
7
7
2
7
7
3.5
6
7
6
-
0
7
7
3.5
3.5
6
7
6
'+
J.5
J.5
4
4
5
5
6
5
7
Ii
5
5
4
4
~
4
1.+
5
.,
2
4
."; c
.......
2
J.5
2
2 c:;
.
j:;
..,
~
2
2
2
2
J.5
.;:
~
_.2 c:
2
2 c:
.... 3.5
'1
7
J
~
.2
2
2
2
7
3.5
7
2
3.5
7
2
7
4
4
5
6
~
~
.J"' ... :)
J .. 5
4
3.5
3.5
6
0
c:
~
~2 ~
G
2
2
4
3
5
J.5
J.5
5
5
5
5
5
3.5
5
5
..
~
7
J
6
4
7
6
4
4
4
4
4
6
4
4
4
5
5
3.5
S
7
4
6
1+
4
5
3.5
3.5
7
Ii
5
5
5
3.5
4
4
4
4
4
5
4
t+
5
5
(,
4
5
5
66
Table 6.5 Ranks Of The Test Procedures.
VAB
DE~lG
1
2
3
4c;
....
(,
"}
0.25
8
S
10
11
B
c
3.5
7
6
6
3.5
3.5
3.5
7
7
7
7
7
7
1
1
1
3.5
1
1
1
1
1
1
7
15
7
1
1
1
1
3.5
3.5
1
2
3
4
~
~
....
6
7
0
c
J
10
11
12
13
14
1:5
7
7
7
6
7
1
1
1
3.5
7
7
7
.7
7
3.5
7
1
1
7
7
7
7
1
1
1
1
1
1
1
2
2
J
3.5
3. 5
J
3
3
3
3
3.5
J
RANK
o
E
F
3.5
3.5
J.5
3
4
4
4
7
7
3.5
3.5
6
6
6
6
2
2
3
3.5
3.5
J
3
3
3
J
3.5
J
J
2
3.5
-: c
.......
2
3.5
4
5
2
4
4
r::
6
3.5
3.5
5
7
7
J.5
J
3
3.5
3.5
3.5
':
c;
6
6
6
6
'7
:2
2
b
.......
2
6
2
6
6
6
2
b
2
2
- .....
,j
.., c:
.:.
..,
3.5
6
~
3.5
3.5
J
2
5
3.5
3.5
7
1
J.S
"
3
3
3.5
J.5
.3
3
3
J
J
3.5
.3
3
J.5
J
3.5
7
6
6
7
8
S
10
11
12
,J
14
15
7
7
7
7
7
7
7
7
7
1
1
1
1
1
1
1
1
1
...b
~
3
/
6
2
3.5
3. :>c;
6
6
6
2
3.5
~
Q
~
........
~
.2
6
6
2
.J ......
\)
c:
2
6
5
.
4
5
4
7
E
4
4
5
6
4
3.5
7
4
5
5
5
3.5
3.5
5
5
2 ...
~
oJ
5
5
3.5
5
5
5
:>
")
4-
5
J.5
3.5
5
5
5
2
.)
--
6 c;
4
4
2
7
6
5
5
5
......
:2
3.5
2
2 c:
b
4
~
<J
5
5
7
5
6
2
2
3
1
1
1
J.5
3.5
7
4
4
4
4
4
3
3
3.5
J
b
1
1
2
LJ.OO
A
12
13
14
1.00
N
(VA=5 )
5
3.5
5
5
5
4
4
4
4
l~
4
7
6
4
4
4
4
4
6
4
4
4
67
These ranks show that the performance of procedure B is the
best when the null hypothesis is false and the worst when
the
null
is
hy~othesis
true.
On
the
contrary,
performance of procedure D is just the opposite.
expected because,
value
when the
of R is always
null hypothesis
is
greater than or equal
the
This is
true,
the
to unity in
procedure B , and it is always less than or equal to unity
in procedure D.
A procedure with a large R value tends to
reject the null hypothesis more often than it should. This
can be verified by observing that
procedure
B
can
sometimes
be
the
twice
power function of
as
large
as
the
significance level even when the null hypothesis is true.
A
procedure with a small R value tends not to reject the null
hypothesis
as often as
performance
of
it should.
procedure
is
D
so
This
explains why the
good
when
the
null
hypothesis is true and so poor when it is false.
The performance of procedure A is the worst of the seven
procedures when the null hypothesis is false.
Its R value
is generally smaller than that of the other procedures, and
it is less likely to reject the null
null hypothesis is true,
"'lowe~rer.
improve
a~
~s
the ratio of
to
hy~othesis.
When the
its 'gerformance seems to
is increased.
This is
believed to be caused by the change of DFN, which decreases
9.S
the
ratio of
resul ts in
9.
statistic,
and
to
larger
makes
is increased.
~ri tical
~
.J..
+u
less
A
smaller DFN
'\Talue for the approximate F
likely for
procedure
A
to
· 68'
rej ect the null hypothesis.
Th is effect is ref! ected in the'
e
ranks of procedure A.
Procedure E seems to
hypothesis is false.
perform very well when the null
Unfortunately, it does poorly when the
null hypothesis is true.
To find out the reason, we compare
its performance with that of procedure F since both of them
have the same approximate F statistics.
The difference is
in the critical values for these approximate statistics.
In
procedure F, the DFN is the same as the degrees of freedom
for A effect.
and
is
In procedure E, the value of DFN is larger,
computed
using
the
Satterthwaite I s
formula.
This
resul ts in a smaller critical value for the approximate F
statistic,
and
makes
it more
I ikely
for
procedure
i. s
bel ieved
E to
reject the null hypothesis.
From
the
above
discussions,
it
procedures A, B, D and E produce poor results, and they are
not suitable for testing the null hypothesis.
Of
the
hypothesis
remaining
is
false,
three
procedures,
when
the
procedure C generally has a
power function than the other two.
AI~hough
are mostly less than one percent probability.
null
larger
the differences
lihen the null
hypothesis is true, procedure C performs well when the ratio
of
ak
to
a;
is small, but procedures F and G are better
when that ratio is large. Again, the differences in power
functions are mostly less than one percent probability.
~
69
Unlike procedure A, procedure C synthesizes the MSD using
linear combination of MSAB and MSE. Therefore, the computed
DFD is independent of
ai,
and remains the same whether the
null hypothesis is true or not.
It
is
worth mentioning
that
the
distribution of the
approximate F statistic obtained in procedure G is amazingly
closed to an F distribution.
The probabilities that this
statistic is greater than the deciles of the corresponding F
distribution have been computed for a few of the designs,
and they are all within one percent probability difference
from the actual probabilties.
70
7.
e
SUMMARY AND SUGGESTIONS FOR FUTURE RESEARCH
7.1 Summary
We have shown seven different procedures to construct an
alternative test of variance component when the exact test
1s
unavailable
as
Before choosing
in
the
the
right
case
of an
procedure,
unbalanced
it
is
design.
essential
to
understand the distribution of the approximate F statistic
computed in the procedure.
algorithm
to
compute
the
This research has provided an
distribution
function
of
the
approximate F statistic and the power function of the test,
i.e., the probability that the approximate F statistic ·is
greater than the critical value of the test when the values
of the variance components are given.
It can be used
e
to
judge the performances of the test procedures.
The resul ts clearly show that procedures A, B,
are inferior to the other three procedures.
D and E
When the null
hypothesis is false, procedure A produces an R value which
is generally smaller than that of the other procedures, and
it is the least powerful procedure"
Procedures Band Dare
equivalent to the conventional F test.
They are the only
two procedures in which the expected values of MSN and MSD
are :lot equal.
Our
resul ts
ind icate that
this
inequal i ty
cannot be ignored, and both procedures should be abandoned.
e
71
Procedure E synthesizes MSN using a I inear combination of
type
II
and
type
III
corresponding value
mean
of DFN,
squares
for
factor
computed from
A.
The
Satterthwaite's
formula, is larger than the degrees of freedom for factor A.
A larger value
approximate
committed
F
by
procedures
of DFN reduces the critical value
statistic,
this
should
and
worsens
procedure.
not
be
the
type
Therefore,
used
to
test
of the
I
error
these
four
the
variance
components.
Al though
the
performances
of
the
remaining
three
procedures are all acceptable, procedure C appears to be the
best
with
synthesizes
respect
the
to
the
denominator,
independent of the value
tested.
rejecting
its
performance.
degrees
of
the
null
hypothesis
G for
component
Since
freedom
it
are
of the "aria-nee component being
When the null hypothesis is false,
procedures F and
variance
overall
is
greater
the maj ori ty of
combinations
its chance of
than
the
studied.
that
designs
of
and
Therefore,
procedure C is recommended over procedures F and G.
Procedure F synthesizes the
numerator using the 1 inear
combination of MSA from both ty!'e II and type III analyses
with DFD equal to the degrees of freedom for factor A. Its
performance is close to that of procedure C and slightly
better than that of procedure G.
Procedure G, the test from the unweighted mean analysis,
is easy to compute. There is no need to compute the degrees
· 72 .
of freedom,
and the approximate F statistic follows an F
distribution very closely.
In addition to
the
comparison of the
procedures,
the
results also indicate that, while the power function of each
procedure
can
be
affected
by
changing
the
size
of the
design, the relative performances of the procedures appear
to be unaffected.
7.2 Suggestions for Future Research
The
idea
of
obtaining
approximate
F
statistics
by
constructing quadratic forms which are linear combinations
of mean squares can be extended to three-way or mul ti-way
designs.
Our results have shown that it is better to leave
the numerator alone and construct the denominator using a
linear combination of mean squares associated with the type
tt
III estimable functions.
For the two-way design, since procedure F performs very
well and
since the
inequal i ty K2 2.. K3 2.. K5 holds
in all
fifteen designs as shown in Table 6.2, it is suggested that
there might be a linear combination of type II and type III
estimable functions (L's) such that the corresponding mean
square has the same expected value as that of MSAB when the
null hypothesis is true, and that the ratio of these two
mean
squares
may
prove
to
be
a
better
approximate
F
statistic in testing the null hypothesis than any procedurett
examined here.
7;
Most of the estimates of variance components are based on
equating
quadratic
forms
to
their
expected
values
and
solving the system of linear equations. These estimates are
naturally quadratic forms.
The work presented in Chapter 5
may be modified to give the distribution of these" estimates
/
.
when the true values of the variance components are known.
74 .
8. LIST OF REFERENCES
Aha, A. V., J. E. Hopcroft and J. D. Ullman 1974. The Design
and Analysis of Computer Algori tms. Addison-Wesley,
Reading, Mass.
.
Box, G. E. P. 1954. Some theorems on quadratic forms applied
in the study of analysis of variance problems. I.
Effect of inequality of variance in the one-way
classification. Ann. of Math. Stat. 25:290-302.
Cochran, W. G. 1934. The distribution of quadratic forms in
normal system, wi th appl ications to the analysis of
variance. Proc. Cambridge Philos. Soc. 30:178-191.
Cooley, J. W. and J. W. Tukey 1965. An algorithm for the
machine calculation of complex Fourier series. Math.
Compo 19:297-301.
Gaylor, D. W. and F. N. Hop~er 1969. Estimating the degrees
of freedom for linear combinations of mean squares by
Satterthwaite's formular. Technometrics 11:691-706.
Goodnight, J. H. 1980. Tests of hypotheses in fixed effects
linear model. Comm. Stat.-Theor. ~eth. A9(2):167-180.
tt
Goodnight, J. H. and F. M. Speed 1980. Computing expected
mean squares. Biometrics 36: 123-125.
Gosslee, D. G. and H. L. Lucas 1965. Analysis of variance of
disproportional data when interaction is present.
Biometrics 21:115-133.
Hirotsu, C. 1968. An approximate test for the case of random
effect model in a two-way layout wi th uneo.ual cell
frequencis. Rep. Stat. Appl. Res. JUSE, 15: 13-26.
Bowe, R. B.
and R. H. Myers
1970.
An al"ternative
to
Satterthwaite's
test
involving
positive
linear
combinations of variance comnonents.
J. Am. Stat.
Assoc. 65:404-412.
Hudson, J. D. and R. G. Krutchkoff 1968. A Monte Carlo
investigation of the size and power of tests emnloying
Satterthwaite's synthetic mean squares. Biometrika
55:431-433.
Laha, R. G. 1954. On some properties of the Bessel function
distribution. Bul. Calcutta Math. Soc. 46:59-71.
tt
75
Levy, K. J.,
S. C. Narula and
P. F. Abrami
1975.
An
empirical comparison of the methods of least squares
and
unweighted
means
for
the
analysis
of
disproportionate
cell data.
Int.
Stat.
Rev.
43(3):335-338.
Robbins, H. and E. J. G. Pitman 1949. Application of method
of mixtures to quadratic forms in normal variates.
Ann. Math. Stat. 20:552-560.
Ruben, H.
1962.
Probability content of regions under
spherical normal distributions, IV: The distribution of
homogeneous and non-homogeneous quadratic functions of
normal variables. Ann. Math. Stat. 33:542-570.
SAS User's Guide, 1979 Edition. SAS Institute, Inc. Raleigh,
NC.
Satterthwaite, F. E.
1941.
Psychometrika 6:309-316.
Synthesis
of
variance.
Satterthwaite, F. E. 1946. An approximate distribution of
estimates of variance components. Biometrics Bulletin
2: 110-11 4.
Searle, S. R.
1971.
Linear
So ns, In c " New Yo r k •
Models.
John
wiley
and
\
'l'i et j en, G. L.
unbalanced
30:573-581.
1974.
Exact and
effects
random
webster, J. T.
1968.
An
Technometrics 10:596-604.
a1'1'r oximate
designs.
approximate
tests for
Biometrics
F-statistic.
Whittaker, E. T. ~nd G. N. Watson 1927. A Course of Modern
Analysis. Cambridge .University Press, London.
Yates, F. 1934. The analysis of Multi~le classifications
with uneaual numbers in the different classes. J. Am.
Stat. Assoc. 29:51-66.
.• 76
9. APPENDIX
77
Table 9.1 Power Functions In Percent For Design No.1.
VA
V,\ B
l' RCC
A-
0.25
E
C
1:
E
-E
G
A
E
0.00 1.0u
c
C
E
l'
G
A
E
4.00
c
C
~
2. 00
2. OJ
2.00
2.00
2.00
2. uO
2.00
2.00
2. 00
c
t
E
~.
\J \J
4.00
4. \.i 0
4.00
it.oo
4.00
4.00
4.00
4.00
4.00
~.c.;o
4.0 Q
4.00
~.OO
2.00
2.00
2. 00
~.CO
4. 00
4.00
~. u Ii
4.00
4.00
4.00
2. CO
2.00
2.CO
4.00
4.00
4.(;0
2.iJO
4.0C
ij. I,) 0
4.00
co
2. U\)
2. 00
2.00
2.CO
2. GO
2. uO
2.00
i.O
2. CO
A
2.0C
2.0U
2. CO
2.00
2. CO
~.
() U
£110
£105
EO 1
10.025
5.019
':.01 S
5. a 19
:. v 19
5.019
5. 01 S
5. 122
1.00 b
1.006
1.. OOb
1.000
1.000
1.000
1. JO 0
1.QuO
1. \,lOv
1.00:'>
10.025
10. a~5
10.Q25
10. a ~ 5
10.102
1.000
1.00v
1.iJOO
l.JOO
1.000
l.0JJ
1.000
1C .. l1t;
1J .. 116
10.11e
1C. 110
10.110
1';.11G
10.0Li9
1.000
1.00 J
1.0uO
1.0UJ
1.\'}Qu
1. QUO
1 • Gu ii
10.230
lC. ~ ~O
10.2J0
lv. ~::O
10.2JO
10. ~;0
10.iJi;o
1.857
1.857
1. ti5 7
1.,j57
1.057
j. 657
1. U13
21.32b 12.1t:6
lU.C~5
~1.3~8
21.328
.. 1 •
~1 • ..l":ci
~1.::~o
5. J 87
5.0 J 7
s. 17 ~
~.174
5.174
~. 17 4
5. 174
5.174
5. CO 5
1~.lo6
12. 1 u 6
i b
t:
12. 100
j~.lo6
21.049 11.981
1.462 16.369
1.46 :.2 1C.. 3t S
~.oo
1. lu2
1. 162
1. 16':
1. 162
1.1u.:
1. 1b2
1. 101
4u~
5.uJ7
5. Ob 7
.;"U E.
4.00
4.00
4. CO
4.00
4.00
I.J.OO
4. uo
1•
5. J8 7
:. J:J7
5. C6 7
1(j.J6~
1.462 16 • .169
1. u62 16. j 69
1.u62 le. ;1:S
1.4S0 lb. 1 btl
1.006
1.. 006
1.006
1.038
3.5
3.5
2.~50
.2.'J50
":.95J
2. 'j:; ()
2.897
7
J. 5
3. :)
3.5
J.5
3.5
3.5
3.5
3.5
J.5
3.5
3. 5
3.5
7
69.9SS
44.n:
~. ~C
G
2. CO
2.00
2.00
2.00
2. GO
2.00
2.00
A
2. CU
,\
E
C
I:
E
:
E
C
I:
E
E
Li
2.00
2.00
2.GO
2. IJ i)
2. CO
2. ao
2.eU
4. iJ iJ
4.00
4.
~v
4 .. 00
4.00
4.00
4.0u
4.00
4. uC
4. () 0
4.00
II. 'J
a
4.00
10.231
10. ;.;31
10.231
10. 2J 1
1 u. 231
tv •.d 1
10.0:';0
-:. 4d c
b. 486
t. L4d 6
ll.
480
c. ~o t
E9.~55
'+4.272
09.955 44.272
t,:).95:5
o~.95~
4~.272
~ 4.27 2
0<).511
~3.545
cS.~55
i.l4."7~
07.1150 5 S. j 1 5 2~.221
67. ~ec :':.315 2d.221
7 • tl U,; S:J.J15 2d.221
tJ
67.ticu :: ..:n: 26.221
07.0';0 55.315 2d.221
67.beU
i
1.942
2.003
4 .. 00 ld.143 79.~15
~. 00 1d. 1Li 3 7S.416
4.00 10.11+3 79.I.J10
4.0 U 16.1 .. 3 H. 4 16
4.uO 18. 143 79.410
4.00 la. 1 ~ J i5.41')
4.00 17.3bJ 7').110
C
3.5
...
2.003
2. CO
2.00
2.00
2.00
2.00
2.0u
E
J. S
3. 5
3 .. 5
J. :.
3.5
3.5
A
C
3.5
3.:)
a• ..;d 6
~.72J
12.i+55
1~. ~::
1••
2.003
2.0JJ
2.0C3
2.003
b. 20 2
1~.4_5
,
8.891
ci. b 91
8.09 1
8. U9 1
8.139 1
i::.IJ91
12.207
l...
' ijS5
r::
"1
2.950
2.950
2.00
4.00
1455
3.5
3.5
4.5
5
4.5
4.5
4.5
4.5
1
G
4.00
i~.
3.5
3.5
1.054
1.054
1.0S ..
1.054
1.054
1.054
1.00 1
4. () a
4.00
4.00
12... 35
3.5
3.5
I.J. 5
:+.5
... 5
4.5
4.3
4.5
2. au
4. () 0
RANK
1.0 2i
1.027
1.027
1. J2 7
1.027
1.J2 7
1.011
j.377
1.377
1.377
1.J77
1.377
1.377
1.30 7
E
C
C
E
E
:
I.J.O:J
.2.. 00
2.00
A
Ii
l.iJiJ
2.00
2. 00
2.0u
2. cu
E
5.00
2.CO
E
G
E
0 .. 2S
4.0C
G
E
4.00
2. CO
2.00
2.
C
E
E
E
O.2S 1.00
OFO
C
A
Q. ~5
DFN
:~.J15
2b.221
G7.o24 55. 13 J 27.7u3
.... 243 4j.1l87 JUu75
4• . :. 4..i 4~.t>;!,7 30.27:
4.211] ~3.;)d7 3U • .n 5
l.J. ~ ~~ J 43.6<37 ~C.27:
4.":43 i+3.t)d7 JJ • .::75
4. 243 :., J .. ti~1 2 ,.;. "" 7:
4 • .: 1 ~ !.I].6uS 3C.-JtJ;J
10.~'jll
lC.4'::0
1 'J • ,~ 'J Ij
1: • '+ '):J
, J .... 9 .~j
~C.4(j.·j
1 G. 177
3.5
"I
3. :i
3.5
J.5
3.5
3.. 5
J.::i
7
J. J
3.5
J. :s
3.5
3. 5
J. 5
7
· 78
Table 9.2 Power Functions In Percent For Design No.2.
VA
VAB
Hec
A
0.00
1.00
C
j)
f
F
G
A
Q
C
l)
E
P'
G
A
a
C
0
E
4.00
t'
G
I1
0.25
A
13
c.:
D
~
F
G
A
B
5. CJ 1.00
c.:
D
4.00
4.UC
4.52
4.00
4.. 00
4.00
4.00
1.525
2.159
1. 704
1.671
1. 7 14
1. 714
1.69<3
4.00
4.00
4.30
2.013
2. CO
2.CC
2. 00
2. 31
2.00
2.00
2. CO
2. 68
2. CO
2.00
J. 23
2.CO
2. ClO
~.oo
2.
~<j
2.(;0
2. 00
2.74 •
2.00
2.00
2.00
2.57
2.00
4;.
(j
J
~. 2U
2.00
2.00
2.uU
2. UU
l. co
2. CJ
2. 21
2. UO
2.00
2.00
2. S2
2. GO
2. VU
2. 11
2. CO
2. 00
2. UO
~.UO
4.00
4.00
4.1.10
4.0 v
4.. 00
4. 11
4.00
4. U0
4.00
4. uu
4.00
4. J 0
i:
2.09
2.00
2. l)Q
.2.00
2. Sb
4.00
4.00
.... 1 1
.... <) 0
4.00
G
2.00
i.l.
D
'i
2.GO
I:' 10
i?Ol
4.00
4. v u
4.JC
00
HANK
1~. ~2~
4.. eo s
b.437
0.958
d.915
':1.J74
9. 153
~. bO 1
4.29 :2
4.644
4. 4J 13
4.60 S
O.U1~
7
1
.2
0.~49
3
1.000
o.Scd
1.46U 15. B9
1. VO U 9.5CJ5
4. nJ
7.(,).:.5
4.57 "
4. 5:j 3
C.iJ·~l
1
7
1.0u0
1. \)00
l.0uO
9.653
8.689
3.917
~.:'::<jU
~.9c~
~. u7C
\i .. titi 1
17 • .1(;2
:.055
4. b87
=.26 J
4.. 959
4.9dt:
9.J~5
24.1.171
14.5 J 2
ld. 5~0
lS.3'17
19.015
19. ~69
10. OY 2
10. Bei 4
1C.t.tJ:
10.654
1.300 12.428
€'0926
1<j.~Cti
10.J'l>~
2~.J130
12. 734
1. 413 15.5cJ
b.174
1.349 14.7lj9 7.82 1
1.40d 15.714
8.57 C
1. 408 1:';. J 43 u. 17 5
b .. 2j:;
1. ~ 1 1 lS.::4J
1. 1 1 1
1. d90
1. 156
1.079
1. 1511
1. 150
1. 15S
,;.J~7
".01 J
19.940 10.U013
l.2 •.Bd
6.354
1 1 • I~ J 1
5.85">
1~.3'14
c. S" 1
12.()1'.J
tJ.1Ut+
c. 224
1".117
4. J G
7.000 ::~. 527
4.ul.1 12.377 71.:512
4. J U 9.2'JO ob.721
A
13
C
R
1. J3J
C.532
U.916
C.902
1.050
C• 'I J 0
0.901
1.. 041
G.952
0 .. 971
44.66S
o \J. 07 '"
~
4. '+ S t
6
q
5
j
2
6
4
5
C.715
1.847
c. '33 q
0.935
1 .100
0.995
G.996
1
7
5
2.u93
3.707
2 .. 067
~ .. 200
~ .. 5 .. d
7
1
2.J6b
2.434
.2
6
3
4
5
b
2
4
3
1. 441
3.039
1 .581
1.692
1.943
7
1
5
6
2
d 1J
J
C.:t14
2.359
1.. 242
1.212
1 • ... ll.1
1.2a9
1•2 ~ 1
1
1
3
1.738
j.
4.00 11.:0J 71.:1g ~S.~~2 31.-t57
4.0tJ 2'J.29o bl;. tHo 71.'J7~ 4-1.140
4. S2 15. 0')1.1 71.J~J Coc. JJt "'J.6u1
4.00 14. 791 76. ,; ~j9 l>5.5j5 .31.1. b 11
4 • .1J 13. i77 /7."::;1 'J/.hh ~C.jt.ti
4.uO 15.277 76.744 b 6. j 116 J9.515
4.uu l:+.joJ 10 • ... ~ i.l t.=c.JV(; j;.22~
.
G
1
3. bY 7
8.950
J.03
2. 53
2.00
2.0u
E
::'
4,.00
1. 000
ti.~7J
1.009 17. 154
1. 000 le. 1b9
9.405
0.9J3
1.000 10.~~5
1.000 9.912
1. ou 0
9. sa5
2. S8
A
!
4.00
4.00
4. 11
4.00
q. uO
4.. 00
4. vO
E
':
a
!
4. d08
4. ~o S
2.00
2.00
C
0
E
I
~.un
A
fl
C
G
I
O.<j~5
2.00
2.47
2. CO
2. au
f
I
4. 00
4.00
4. JO
4.00
4. uO
4.00
4.00
~.ao
A
,0.25 1.00
1. 000
1. 2v~
1.000
O. :idO
1.000
1.u00
1.00v
E
F
G
B
.
4. a a
4 .. 00
4.52
4.00
4.00
4.0\.l
4. 00
I)
G
0.25
4.20
2.00
.J
4.00
DEI:
IJ
C
0.25
DFH
1'J.22~
JJ.IJbq
.n . .; 79
4
6
2
J
4
7
1
2
0
J
4
5
7
1
2
o... ~j% Sl.6~7 2:+.147
i:5.'::E1 53.522 .2G.720
6:). ;~n ~2.ulu 25.555
6
J. L I 1 35.3d 2.::.7') 1 0.795
5. 7 (;8 51,6,,0 Je.015 111_ 76u
.... 110 .... J.~2~ Ju.2lc 1J .. JtU
).284
J.LJ41 ~l.'::H) .;u.v3'J
4. JIJi) 'I 2 • ,} ':1'1 d
liJ.J.:;d
. J6::
) .,
",
j . 7 1:4
.... u0o ... _ • ·L~ 1 _,j4 JO ...
.,;. J3 U
4. 10 'j 42.')qJ d.JiJc
7
1
2
d. d .. 7
'1.15d
9.15(;
9.22 J
65.672 52.d4<: 25.692
J
S
4
ti
.1
5
4
79
Table 9.3 Power Functions In Percent For Design No.3.
V.\
VAB
PROC
A
f
C
0.25
l:
E
Of
G
A
e
C
0.00 1.00
I:
::
E
G
.\
e
....
C
4.00
I:
if
~
.\
0
C
D
.i::
F
G
.\
B
C
D
F
G
;\
au
2.21
2.00
2.00
.2.00
J.9S
2.00
2.00
2.09
2.00
2.00
2. CO
3.9d
2.00
2.00
2.03
2.CO
2.00
ll. U a
Il .. OO
II.GO
4.00
4.00
4 .. 18
4. 00
4.00
1l.00
4.00
4.00
4.00
5.04
4.0 ()
".00
~. u 0
':+.00
4. vO
~.OO
4.5.2
4.UO
4.00
~.uQ
4. () 0
4.00
4.00
4. 18
4.00
4.00
4.00
4. () lJ
4.5"
4.00
2.00
2.00
4.00
4.00
2.02
4.00
4.00
4• 1 <.:
4.00
4• .) 0
4.00
4.00
G
;\
g
...
D
.
F'
G
2.UO
2. a ()
2.00
J. %
2.00
2.00
4.00
4. () C
i?0 1
4.993
5. Ii 7 <j
3. ':.118
4. ld 2
5. J7 5
4. 7:~ 1
1.009
1.164
4. a8 2
5. 1'~ J
6. 50 J
~.
6J q
4. q'j 2
5. B 1 1.4
:. 1J 2
4.957
0.~81
O.U13
1.• 146
0.932
0.973
1.0142
l.)iJ7
0.697
C.b7d
1. ,,03
n,UIK
~
1
2
6
3
l~
5
7
2
1
6
II
O.. 'J8U
3
276
1. hO
5.414
4.553
b.127
5.412
4.994
1.065
1.520
0.908
C.920
1.378
1.118
0.998
J
7
1. (32 '+ 21. 135 12.042
2. 1 11 :2 4• .} J 7 14. 3U 1
2.028
3.707
2.. ti21
2.571
J .. 590
2.959
5
1. ti40 21.i:l51
1.7SS 19.5~J
1.891 -'2.724
1. U9 1 ~ 1. =1.16
1. U10 ~0.Jn
~.
1:::. J 3 6
11.0')7
13.615
1'.27C
11.:j~:'
,.032
1.422
1.692
1. i+58
1. J 11
1.456
1. :+5 b
1.44-:;
f:. 79"}
1b.l~2
19.£.23 11.112
E
.. ();,..,)-, e.
H:. 7 1 'i
1... G02
7.910
1i.529 1O.J97
9.0;,n
10 .. 5J7
b. IJI" 7
16. G7:j
1.991
2.669
1.1'+3
1.1.401
1. 162
1.003
1. 155
1. 155
1. 16J
1,.:25
1b.Cd9
13.152
11.082
1J.ob2
12.8.ltJ
17.47<+
21.700
17.911
17 .031
1 H. 811
18. :.l11
4.vO 17. 20 J
4~liO
?05
1.042
4.00
4.()0
5.04
4.00
~. 0 J
4.00
2.03
2. 00
2. GO
2. 00
J.95
oog
9 .. 9jll
1.0ri
11. Hi 1
1 .. 000 0.:347
0.951
ti.E£:4
1. 000 10 • .!B4
1. 00 ()
Y.6U~
l.0u.> ~.UO':.l
1.
1.000 10. ij43
1.222 13.:£.5
1.00v 10.778
o. dbJ
9.0i~
1.000 11. J 45
1. 00 I) lG. tiC7
1.000 9.992
A
3
C
.."
P10
1.000 10. 192
1. 161.4 1'.5~2
1.000 9.609
~., LI ~ 1
O.d~~
1. viol 0 10.J'JU
1. JiHl lO.1Ul
1 • IJU Q Y.'JJ7
•(j
F
R
4.00
4.52
4. Oil
4.00
4. va
4.00
J.92
2.00
2.00
0
v
2.00
2.00
;)
..
~.OO
2.1 C
2.00
2.00
l.OO
Z.OU
C
1. Uo
2.CO
~
J:.
s.
2.31
2. 00
2.00
2.00
3.95
2.00
2.34
2.00
2.00
2. OJ
3.91
2. ao
2.00
E
0.25
11.00
4.0 ()
5.04
4.00
4.UO
A
C
<l.UO
2.68
2.Ou
2 .. 00
2.0u
J. au
2. 00
2.00
3.98
E
0.2S 1.1)0
OFO
E
l
(i
0.25
DFN
'J.442
11.7JS
10. ltd
~). 140
lu.127
10. 1:'; 7
9. 95 ()
1~.1':il
t
..
<;
1 '
wi 4 ....
8.75 U
£:.05J
5.719
7.60 '6.750
t;;.270
I .. 72lf>
1.739
2.506
2.057
1 • 914
1.J77
1.9b4
1.343
1.202
i .787
1. 457
1.JJ3
78.595 68.732 42.842
J 1. l;L 1 73.07'3 4J.Jbl
~1.';7'J 73.£<82 52.U2 1J
78.394 00.617 4".591
30.'.iCJ 7;;.. 4J 7 4t.GJ9
7'-).)':)9 70.037 45.:.:30
-J c, . v;- ('v ..,I E~. 1 -j ~ -. J .!.f 70
t S . :l S 1 : 2. 'N 7
70.200 5u. 57 'J
t ~ . j:; 1 ~7.'J~2
05 • .l~~ :,,~. ~9 6
lHj.~15 : 7. '4 lG
b7.:323 55.20 G
G7.~:42
~.:.v41
;;1:.547
32.038
J 2. 1 96
25 .. 988
31.517
~8.428
27.69~
4
1
6
5
2
1
3
7
2
4
6
5
1
4
7
2
3
6
5
1
3
7
2
,~
6
6
1
2
7
3
4
S
(i
1
2
7
3
q
5
3.8u1 41.jJS 2C .. 52S 11;.282
4.cluS 47.J93 34. it:': 13.070
4. ;: j:j
,>i 1 ] 1. Lu5 1 1. ~ 05
6
1
J
4. ~:;4 4 i.l • cIS
4.0'H 43.J30
4. 200 4 j_ c .... 1
c:.J
3.6;)7
4:.
'-to. (.J'.
~7. ~j'.J
j 2. ~J J
JJ.;:O~)
ji,;.uJ~
-).308
L:. .44d
10.711
lC.1Sa
7
2
:.+
• 80 •
Table 9.4 Power Functions In Percent For Design No.4.
VA
Vi\U
l?aOC
A
E
O.2S
C
J;
E
z
G
,
A
e
v.Ou
1.. () 0
14 .. 00
C
C
E
t
G
A
E
C
C
~
w
E
~
.\
0.25
E
C
C
E
,
G
A
0.25 1. CO
E
C
I:
E
:
I
G
!
A
E
4.00
C
I:
e
C
G
.
A
0.25
Ii
C
C
E:
r
1. VO
~. OC
r:
B
.c
C
I:
r::
E
G
6.00
6.00
6.00
1. vOO
1.. 070
1.0eO
£lOS
fOl
'1:~3~
9.229
14. 91 ~
5.7q
4.340
4.40 <;
5.343
0.989
1.180
0.706
0.:331
1. 13 2
O.ll92
0.9214
11
0.941
2
7
J
1
o. <J7'J 9. 1 11
1.0(;0 10.320
1.00v Y.llS]
1.. 000 9.705
b.OO
b. (,)0
o .. J7
b. J 0
6 .. 00
b.UO
6.00
ao 0 9.665
1. 16d 1~'0 c':i 2
1.000 :I.i3bO
....... 1.:;"
O. 95 ~
~. " .. v
1. GUll 1C.7bo
~.Sj2
1. UUU
1.000 9.912
J. 1 a
3.. 00
3.00
3. 00
4.91
UU
6.00
6.00
6.13
6.00
O.OC
6.. 00
0.00
'J.Q91
lj.6S14
1.000 10.297
o. ':Uti 9'0~63
1.000 11.013
1.000 10.C~7
1.000 9.'Ja9
14. 15
J.Gv
3.00
3. 00
ij.09
3. vO
3.CO
6.00
6. () ()
6470
6.00
1.73 J
2• .; 1 1
1.:309
1.771
1 • tl~2
l.d::'::
1.798
J.~O
J.• co
3. S6
J.OO
3.CO
3.00
~. 7d
J. 00
J .. uU
J.16
J. 00
3.00
3.. () 0
4. c lJ
3.1.10
J. GU
.... 15
J. Qu
J. CU
J.OO
4. 7~
J. JO
J.50
J. au
J. CO
3. JO
4. 7 'j
3.00
C
.00
6.7 C
b. OJ
Pl0
J.5b
J. au
3. ~o
J. (J U
14. dO
3.00
A
A
... 00
q_ 15
3.00
3.00
3. uU
14.65
3.00
J.OO
j.~Q
F
G
i
OPO
(;
E
S.OO
OFU
J. J
(J
3. 1 d
J. 0 ()
3.Uv
J.00
4. 7 a
J.Ov
3. ()Q
1.
4.625
14. 77 1
736
1:.53 C
l.J.
... ~ 50
4.
~o::
5.(81)
1.362
fJU 5
Q.d'15
1. 2S 1
0.975
oj.
P.AUK
2
I)
3
4
&
11. 0 '15
4. ~u 1
0.977
14
S
4.631
7. ou d
5.197
1.1.615
5.871
s. OJ 1
14. :.l9 1
0.U8B
1.481
l.u37
C.922
1. J 1b
1.016
0.997
2
7
S
1
b
14
14.243
U
14.1)0;')
13. 71 7
25. 'H7 16.026
~q.j4J 1 4. j~:;
24.251 14.2:37
3.d15
4 .. 706
J.717
3.517
4. b 13
3.. 750
3.7ijO
6.00
6.J 7
6. 00
6.1.10
6.00
').462
1.395 17.206
1.671 ~1.90S 1~.G21.1
1.~~7 18.253 10. 123
1. 3fj 1 16.9(;4
C; .. 312
1 • ~4J 1'J.2dO 11. 2b 2
~ • gi 1
1. 4 ~ J l i .. aso
1.4~'+ 17.:.141
:1 .. :J, I
2.207
3.181
2.323
2.167
2.946
2.348
2 •.d6
5.00
6.13
6.0C
b.OO
1.. 139 12.0d3
1.. '+ 13 17.CJS
1. 160 1.3.2U1
1.J:J~ 11.9:<6
1. bo 14.0US
1. bo 1~.(;42
1. 16 U 12 .. 827
1.259
2.078
1. ~94
1.317
1. dSb
1.440
1.42J
6
39 ... 40 tiJ .. 52d 04. )70
g 1• .: 5 g eo:.~ciO oS.SciJ
~0.444 d~. 172 66.621
6
1
J
7
2
d~.b:;il
JJ.-.l12 :;5 • .227
5
tl.'.ivj 7G.:j11
ou.577 43.5·+]
7
1
J
6
2
5.vO
5. 'JO
6.00
6.00
b.vO
o. () 0
6.UU
b.OO
.-
0.
,,,\
U
"I
v
b. 0 ()
6.7iJ
6. 00
b.OO
6. () 0
lJ.uo
o. O~
1. JOu
1.
~Jb
2J.9a~
27.76
2- .... 738
c: - ,
~
~
....
_
1E;.'Jdi
~J
t,
15.60J
1'1.7j'J
17.181
16.81d
17 ... 4.:17.442
10 .. ':IS..;
: •.
~
0.15 d
s. 227
b.952
L.. 1:.J 6
7. -, b 0
E. 71 3
b. b 7 (]
:: g. It':J 5 uJ.524 u'.l.724
JO.ooo b~.:',:j~ o9.J69
'JU.0;2 J4.263 65 .. 'JuS
6. VO 11. nb 02.7::0 7 ~.
J \)
u S(;.S2U
b.37
6.00
0.00
6. U 0
0.00
'J4~ -.iQ .. 9u7 71.~J1J7 .. d. () 04
9.492 79.~:6 G'). 112 j 44 .. 422
9 .• ~tJ ~ J1.J.ll 7..:..7.30 ~~.600
9.602 60.1S2 IC.614 a5.825
b.uO
3.77')
0.1 J
6.0 U
4.),01
J. ::IJJ
o. () 0
9.
9.074 :.W. 2~d
~ ..
772
6.00
4. 1 1'.)
tJ.UO
1~.
u. () 0
4. 11 v
1') d
7\:;.725 4S.i.I9Y
1. 1J2 37.517 15.361
9. 14 1 '15. 7 • 1 21 .. UU5
s.
1 JO, I.~ 1. j 7 G H • ..:a I
...
, • ..; .. :.J ";~.7u~
1c.270
S.ut;u
:OJ.JU7
19.9b6
.... '1 C
j
~; j )
v. ) .. , 17.256
":5"j ... J. vI:) 17... 00
~.,
(
~.,
3
1
3
7
2
~
5
6
1
3
7
2
5
4
1
3
7
2
4
5
~
5
4
7
1
J
()
2
.
S
81
Table 9.5 Power Functions In Percent For Design No.5.
VA
VAS
niCe.;
A
a
0.25
C
0
t
f
G
A
a
0.00 1. 00
C
C
0
E
C
D
f
G
i\
B
C
D
i::
r
G
A
Ll
l.:
I:
w
"
10.40
10.4b7
10.467
10.467
10. q €I
10.407
10. C 12
3. OU
3.00
J.CO
3.0u
3.00
3.00
3.00
3.00
3.00
J.CO
J.OO
3.00
J. Qu
3.00
3.. 00
J. OJ
3.00
3.00
3.00
J.OO
.l.UV
3.CO
J ..
CO
3. uo
J. (J 0
3.00
J.OO
3. 00
••a
J. CO
..3. iJ 0
J.OO
3.00
L:
C
i::
E
4.JJ
1. vOJ
1.000
1.UOO
1.000
1. JOO
1.000
1.000
r
...
1. ao
1 u. 2 ~4
11l.254
1\;.254
10.09d
J. CO
3.00
3.00
E
F
:
5.00
1.000
1. uvO
1.000
1.uuu
1.000
1. JQIJ
1.000
G
C
"
!
6.00
D
a
0.25
b.IlO
6.00
b.OO
G
4.00
3.00
3. 00
3. 00
3.00
3.00
3.00
3.00
r'
I
1.000 10.0El
10.0bl
1. ()U 0 10.061
1.01.10 10.0ul
1.000 10.061
1.000 10.vbl
1.00 i.l lO.31.j6
A
.~
1.00
3.00
3.0u
3.00
6.00
b.OO
6. uO
b.OO
6.00
6.00
6.. 00
3.00
J. CO
3.00
B
o• .25
J.OO
J.OO
3.00
J.OO
E
B
0.25
Oft
b.OO
6. uO
6.00
b. 00
U
f
G
4.00
DFN
3. GO
J.CO
3. UO
G
3.0U
j\
J.Ou
E
<.:
C
...'.'
if
"~
J. UO
J. C;J
J.I)U
J.(;(j
J. JO
J. 00
6.00
6.00
6.00
6.00
6.JO
6.. 00
6.00
6.00
6. J 0
6.00
6.0 iJ
6 4 00
6. JO
I
6.00
u.oo
6. u()
6.00
6.00
6.00
6. oj J
6. J I)
b.OO
6. (J 0
5.00
P05
['01
RA NK
5. O~ 7
5.047
5. Ol~ 7
5. 04 7
5. v4 7
~. 04 7
1.016
1.016
1 .016
1.016
1.010
1.. Ii 1(j
1.095
3.5
3.5
3.5
3.5
3.5
3.5
~.
~.
19 C
198
1 'j ;;
1~j
5. 198
5. 19 (3
5.076
5:
1.068
1.068
1.068
1.068
1.06U
1.0uo
1.026
5.366
5.3bo
5.36 C
5. Jt.6
5.366
5. Jub
1.127
1.127
1.127
1.. 127
1.127
1.127
1.003
4.5
Ll. 5
4.5
It.56S
10. 56 j
27. leu H. 56 S
27.10d 10.509
27. led 16.56S
27 .. 1,)0 1&.5&';
:':0.291 l:.9bu
4.082
4.002
... 662
4.00.2
3.5
3.5
3 .. 5
J.5
4.btl2
345
7
19.C77 11;. uJ 2
194077 10.JJ"2.
l~.C77 hi. oj 2
l~.O77 10.332
19 .. Ci7 1u. 332.
19.077 10.832
lG.i;C7 i;j ....,C
2.b~1
2~ b91
~. 6-:11
3.5
3.. 5
3.5
3.5
3.5
3.5
1
1;.4 b
U.419
Li.41S
1 • £. 1 J
1.613
1.. 61)
1.613
1. u i J
i • G1 )
1.441
1.UOO
10.~=4
lu .. 254
10.~:4
1.972
1. n;.
1. '..J 7"21. J7 2.
1. '::IV 0
4~ 3
1.4~.l
1.
1. l+9J
1.493
1.4Yj
1.l+93
1... 7ij
6.00 20.4':i7
6.0u 20 • .. 37
6.vu 2iJ.4J7
6.00 2iJ. 4j.,
91.427
S 1.. '4 ~ 7
91.427
S L 427
u.uu
5. uO S
1. '.J72 27. leg
13. ·.19
1':: • ;4 1 ':i
12.91:;
0.0 J
c;
1. ')"U. ;'7.1uf.J
1.1U6
1. 16u
1. lu6
1. 166
1. iot>
1. 1'')0
1. 1b 4
6.~0
=.272
lj.~19
7.172
7.172
7. 172
7.172
i. 17'::
i. 17"
c.. 7 j
6
0+.682
4.456
24691
2~691
2.691
2.510
~ lij 70.008
6 c. 41 E ie.GOd
;)0.410 iv.OOa
::,1(;.
"i:.41c }1~.OlJ8
b .0lJ 2v.!.tJl ~ll • ,.27 00 .... ~ '.J 7J.jJci
6.00 20.43 7 <; 1. (j~7 (j C. l!l c: 7C.OOd
0.00 1:J.\)01 91.000 Q~.d4:) ud.J07
6.Ll~
b.d(;
6.0",
6. UJ
6.vu
b.Ou
6.00
b. 0 J
6. ,j 0
;).,,0
b. J 'J
b.0e
b. UJ
t,).
UG
1,j.JS7
lU.857
10. Jj 7
10. ~57
1 U. d5 7
lv.ciS7
;0.474
l+.J17
4. J 17
4. J 11
'f. J 1 7
4. J 17
1I4 J 17
4.2.73
31.524 7..:.S5d
01.:~4 7:". 5S::
d 1.5 ",4 n.SSJ
cl 1. 5:.: 4 L .• 55e
81.524 7i..SSd
d I.S::t 7:':.~:i1:;
(n.~~4 72.'!lJ
:<0.U12
-!ci.d12
:';:'~. d 12
~U.d12
:.+ti •.'3 12
4<:.012
~u.
109
... 9 1 'I 1. u j J lJ.S45
'I • 'i 1 ~l.t)la 11:.545
4. ) 1 .. l.;..oJJ , o. :,:. S
~
• S·
4. '-)
,I • :;
') • 1
1 41.1)3'.)
1 41.uJ0
1 4 1.
0
i) 1 ,:
41.,471
llj.545
1d.~"5
IC.:J '.:J
11.<300
1
Ll.S
:t. 5
4.5
I•• ~
4.5
'+.5
1
4.5
4 .:Jr.:
4.5
1
3.5
3.5
3.5
3.5
3.5
3.5
3.5
7
3 • :JI"
3
r.:
.~
3.5
3 .. 5
J. 5
3.5
-,
3.5
J.~
J. 5
J .. 5
3. 5
3.5
"I
.
3.5
J ~
J.5
J.5
J. 5
J .:)
7
~
82
Table 9.6 Power Functions In Percent For Design No.6.
VA
'lAD
DIN
DEC
4.00
4.00
G
7.. 02
4.00
4. 00
4. (;0
4.00
4. 00
4.. 00
A
5.32
l:5CC
A
jj
c
0.25
j)
.f
:'
,
I
E
F
G
4. vI,}
4.00
4. GO
A
13
C
D
4. 40
4.00
4.00
4.00
4. 00
4.00
4.0u
4.00
4.00
4.00
4 .. 00
4. Ii 0
4.00
4.00
4.0 J
4.00
.....
4.00
4.0 \)
G
5.7B
4 .. 00
4.00
~_ 00
4.00
:.t '" r"
4. 00
A
5.01
C
D
G
4.00
4 .. (;0
4. UO
... 00
4.. 00
4.00
4.00
4. uO
4.;) 0
4.00
A
13
4. J 1
4.00
D
.... cc
C
D
1.00
4. all
E
F
G
A
d
C
0.25
I
I
J)
E
f
a
I(). 2S
1.00
~
I
1-,
F
C
14.00
E
f
G
Ii
i3
C
D
1::
f
G
0.25
~
;
~
,
13
1.00
5. 00
J
I
I
4 00
1 •
.
~.OO
4. OJ]
~. 00
~.GO
4.0:)
~.oa
4 .. 00
4~UO
4.\)0
4.00
4 .. (10
4.00
4 .. CO
4.00
4.(;0
4.. 00
PO 1
4. citi4
C.. 974
1. JO 1
O.. ~42
b ..
Ji~
2
4.3':1 i
Ii. 397
4.391
4.397
4. tU 5
l.\)Ol} '). ~ ~ 2
1.300 14.77'0
1.000
9.dC6
1 • ~O U 9.800
4.57 S
7.609
4.07 J
4. ti7)
1. 000
1.000
9 .. 800
4. U7]
4.416
1. 00 (j
~ .. d 15
10 .. 295
10.054
10.0YIi
10 .. 1194
10.094
~. %9
9.t:U6
1. UOO
l.~07
9.C56
4.
un
4
"1:;<:
...
J
RA NK
O. d ~~ 2
0.042
(). 842
0 .. 956
C.593
1.570
C.~bb
0.906
1.:.90b
0.'.106
C.847
O.~G5
4.. UO
4. Ui)
A
4.
:3
4.(;0
4. UO
11~
-:
4.0u
;~
4.00
4. GO
4. () iJ
4.CO
4. J ()
4.00
i•• 00
4.00
4.00
I~. 0 ()
:~.O()
!~. () ~
4 .. (jO
'L ,it)
4.(;\)
ii. :) U
1
7
4.5
4.5
4.5
2.1~4
2. 12 ..
2 .. 145
3.5
3.5
6
1.257
1.. 77 3
1.. J 1S
1.. J 15
1. J 1S
1. J 15
1.. J 12
13.3S5
€. b4~
21.107 11 .. 559
14 .. 638
7 .. 634
1 '~. t)]U
1. oJ 4
1... 63d
7 .. 534
1~.,)J.g
7. oj ~
14. :33
7.503
1.. 404
2.5d5
1.. 612
1.. U12
1.612
1.0 1.2
1.59d
7
1
3.5
3.5
3.5
J. 5
Ii
1. Jd:J
1. \)4 U
1. 109
1.. 10 'J
1.1<)9
1.. 109
1. 1011
10. ~uti
HI.5d2
4.. 9U i
9.764
7
11.1:t2
1 1. 7 fl 2
11. 7 ~2
11.6C3
5:1:;;0
0 .. 902
2.0 53
1.230
1.230
1.230
1.2 jO
1 .. 20J
11.,7fl~
c
'J9 C
5.9':1 0
5.':HJ
5. ci ':I II
·).1~4
i. 1
lJ
4.JO
4.GO
4.5
4.5
4.5
4.5
1
2.036
3.2J5
4.5tJO
1J.742
10.57')
1 2. ~iJ6
!1.00 12.9jI.J
4.. 'J U 12. ·130
J0
2
7
L ul)7 17.5r:1
9.359
1.9 1. . . 2~. 1db 13 .. 078
1.. 597 13.427
9 .• d 1 ~
1.597 1(3.1.l~1 9.81~
1.597 1~.1.j~7 ':i.Ule
1.:.i97 1\).~27 9.313
1.530 10.3 j j
':i. 775
1.000
1 • 00 iJ
1.0UO
1.000
1.Il0J
4. J lJ
4.lJO
4. J 0
4 ..
2.5
5
1.708
1.017
1.017
1.017
'.017
0.995
18
4.00
...
6
7
2.5
2.5
2.5
8.3 0
5.063
5.063
~. 063
5.06 J
<t. 0 U
G
i:'
G
4.(JO
1.000 9.797
1. 17 J L! .. 424
1.. 000
'J.Ct5
1. 000
9~Oo5
1.00 (J
~.Ob5
1. 000 9.005
9. ] 49
1. 00 iJ
P05
t:1.t:72 r:~.112 35.697
tl'J.S07 ;)0. 73·~ 51 .. 135
cS • .iSu 74 .. 32<': 42.008
u:J.2')O 74.320 42.003
C.S.2':1'; 7~.0 32G 42 • .1Jo
~.oo 12.93b tiS. :':-::1{) 7<.+.320 ... .2.vG8
4. U J 1 2. 11 'J :.J j . 1 17 ]4.J4C 41.577
4. 21
4.00
4. \)u
4.00
4.0U
4.00
4. 0 \J
J
I
4.00
4. Oil
4.00
~
1
:
4 .. 00
4. i) 0
4.00
4 .. UO
4. UO
4.00
4.00
4 .. 00
4.00
a
0.00
4. U\)
4 .. UO
4. \J 0
1!1Q
R
i.
6 ~ 1 \J t:.'J72
<.+~ ..
75 t
1l:.d96
9 ... ')5 "77.)..; .. 04. ~4 '. J L 209
7• .2') d 70.2ud ~4.j7L 23.349
7 • .!9 ti 7').2ud 54.d72 2J.]lI9
7.2'Jd n. ~ljl3 ~ fl. 07;: 23.)49
7.29d 7,) • .: J J :J4.G72 2J.J49
7. 2. J 7 iJ. ~~'1 :4.td~ 2J.JdJ
2.77')
4 ..
J20
3.. ldJ
bJ
J. l1.iJ
j . 1j J
j.
J. 17 ~
, .,
J~
•
1.
40.
~ I)
52
4J .. .." I1
"u. ;1... 11
;~ \)
.. J. 17
'j
.
5.717
'1.6JC
JS.J7J 11.324
2:.91 1 7."~J
25.)11
7. 1.5 J
'::. 'J 1 1 7.453
7. ,15J
..: 5. '} i i
7.J1d
2:.7~c
4.5
2
7
1
~~ ~
1
3.5
J .• 5
3.5
3.. 5
6
7
1
3.5
3.5
3.5
J~
6
5
7
1
3 .. 5
3. 'j
J.5
3.5
6
7
1
J.S
J.5
J •S
3. ')
6
!
.
83
•
Table 9.7 Power Functions In Percent For Design No.7.
V.\
VAD
DPN
OFO
5.50
8.00
8.00
8.71
B. 00
8.00
8. () 0
8.00
4.81
4.00
4. 00
4.00
5. e6
4.00
8.00
8.00
G
4.CO
A
PROC
P05
EO 1
9.342
1.06d 11. j52
1.000 9.360
O. 9ti6
~. ~ 12
1.000 10.390
1.000 ~. 4e 4
1.000 9.7'-13
q. 93 5
5.707
4.48 q
4.465
5.30 d
4.025
,~. t.l26
0.996
1.194
0.786
0.840
1.139
C.ddS
9.050
... 778
ti.OO
8.0 Q
0.00
8.00
1.000
1. 155
1.0CO
9 • ..:..:.)
1. 000 1u.JOu
1.000 y.oC ..
1.000 9.931
::1.694
4.1314
4. <J .. 5
4.28
4.00
4. 00
4. 00
5.99
4.00
4.00
8.0C
8.00
8.15
8.00
1.000 9.401
1.228 14. ~ 34
1. \JOO 10.270
o. ~5J 9. j4 3
1. 000 11 .. 052
9.9<;;0
1.001l
1. 000 9.991
4.577
'7.437
5.179
4.655
5. (i3 5
5.00 ':i
~. 99 3
E
5.56
q.OO
~5.420
tt.OO
15.39 1
10.J76
15.744
14.9"15
17.295
1 :4 474
15.465
349
5.480
4.301
i+.v53
Ii
1
C
8.00
8.00
8.71
ti.00
8.0C
4.254
4.272
4
1 J. 181+
13.845
10. S 1 9
10.16 :
12.151
10.002
1
i •
2.. 471
3.000
2.004
2.464
3.314
.,2.028
"...,-,
6
A
f
C
0.25
C
E
-E
...
"
,\
E
0.00
~
1.00
C
E
~
E
(;
4.00
C
E
E
G
A
I.:
O.~5
'4.
00
4.00
4.00
5.70
4.00
q. CO
... 00
1:3.~
C
~.OO
8.00
8.00
...
,.f
5.75
4. 00
4.1,)0
,\
4.81
4.00
4.00
4.00
5.24
4. 00
4.00
8.00
8. () 0
4.28
4.00
4. 00
4.00
5.90
4.00
8.CO
8. 00
8.15
8.00
8.00
8. V0
~
~
E
C
D
0.25 1.00
a
e
:
c;
d. U0
!;l.O 0
8.00
8. 00
8.40
8.00
8.00
P10
1. 000
O.%d
1~. %2
9.~qq
c.717
4.891
'1.1:01
1.653
1. a97
1.72d
1.704
1.737
1. 737
1.718
20.1::0
~5. 1il7
27.5':+9
25. EG 7
25.723
1.371
1.620
1.421
1.376
1.41'::1
1. 419
1 ... 1 d
10.218
~ J. : ~ 5
1:i.JJO
18.194
2 0.512
18.5i9
lSI. hO
1. 1 J 6
1.400
1. 157
,. 102
1. 153
1. is 3
1. 1S I)
12.:.+0:0
6.35 ;)
18.1 So 10.027
13.643
7.240
12. 117 J
c. 527
~9.4111
.~.
• w_
~
J.J
I
f
A
E
l:
C
..
4.01l
:;'
f
"
~
A
E
C
C
E:
F
0.25
I
S.OO
1.00
4.00
4.00
S.80
4.00
,\
c
4 .. 81
4.00
..
I.:
~.OJ
.E
~
4.00
5.81
4.00
4.00
A
E
4.26
4.00
;;
4.00
..:
I
5.56
4.00
4.GO
I:
I
co
I..i
lo'
4.00
4.
E
:
-;
:t.(;()
5. oS
OJ
4.00
1.+.
a.oo
H. :i4;)
13. ~ b
13 • .::96
ci.11~
c.991
Ih9d:t
O.'JJU
0.939
1.41ti
0.930
0.'101
1.258
C.. '.J67
0.980
0.d69
1.Sti2
1.04]
0.930
1.329
1.011
0.997
~.
5.240
_.~.J
..
1. J 15
2.345
1.597
1 .416
1.990
1.536
1.51 d
RANK
~
2
1
0
J
q
2
7
4
1
6
J
5
1
7
5
2
6
4
3
3
7
2
5
1
3
7
2
5
1+
7
1
3
6
2
~
5
14.05d 93.392
11.646 ::5. ~ 11
15.57J 'h.S07
15.350 g~.JJO
8.~0 1S.I'd 9 ... 7':; ..
8. J IJ 15. 743 94.~21
j . J G 15.Jo': :J~. '';09
76. '~ 10
-;I.'J5E SC.276
'joJ.9uJ 7Cl.7l.i9
-)C.091 76.473
'J 1... j 7 7'].761
'1C.516 77 .248
7
1
3
6
2
7:...o61~
5
E. 0 0 0.417 05.330
ti.00 10. 5 7 ~ =I;.~:~Y
2.~ 0
'1.42'1 (37.~17
8. 00
9. 120 :6. J~ti
8.00 9. J 7 i4 e7 • .:iJd
8. () 0
Y.374 E7 • Gi,; ':
fj. I) 0
9 .. 359 87.()~1
77.7'.3 :Jb.02J
ti2.S9J I) J .263
dv_'+Jl bu.2~1
70.606 57.451
J 1. JIB 02.10]
7'J.672 56.566
7'J.7i5 5J.614
7
1
J
6
2
8.0U
tie J 0
d.71
S. 00
13.00
8.00
c.15
J. 71 J 50.712
o. J 0
... \;7 0 oJ,JuS
4. J 7U t 1.:, 77
l.l. 0
a
d. U U
8.00
4.67':J
4.1 j j
3. ')44
o7.2~Y
02.:;0:,)
oJ. c -l 1
4. 12 Y u2.:.J16
3J.9~1
;~.ld~
32 1
4.c1\J2
j.
9.01:;
7. 41 .2
1.60 b
0.70 '+
'J. Jv 1
1.312
".0']7
:i.QUO
J. 022
7 • .127
'l.uJl
4.29 t3
4
5
4
7
1
J
6
2
5
'.+
84
Table 9.8 Power Functions In Percent For Design No.8.
VA
VAB
EliCC
A
a
C
0
0.25
·1:
G
A
a
A
C
D
E
F
G
A
J
C
0
E
F
G
A
8
C
D
E
F
G
0.25 1. UO
A
3
1••
c
00
D
E
f
G
A
1 0•
B
C
25
I
1.00
5.00
4.
I
\J
U
I
8. a()
8.00
0.50
8.00
Ii. 0 I)
0
.E
F
(j
A
3
L
D
E
8.00
4.946
5. &29
4.430
4. Jb 4
5.5t.l5
4. b 14
4.765
1.205
0.756
0.iJ14
L2 j 'J
O.U84
0.919
4. SJ 7
6.747
4.:J10
4. t+d·J
4.707
7.418
5.2JE
4.")07
u.152
5.032
4.991
d.OO
4.22
8. 00
8.00
tI.17
8. (j 0
8.00
8.00
8. 00
1.. 000
\J.b12
1.213 14.169
1.000 10. :57
o. ~35 9.072
1.000 11.37U
1.000 10.028
1.000 9.Sb'.!
~o
ti. a 0
8.00
tI. ~ 3
8.00
d. (J 0
4. 00
d.OO
1.74iJ
2.003
1.ti15
1.775
1.829
1. ()2 9
1.1:104
~6 .. ug5 lL3J3
J 1.0 ... 5 20. 123
~d.~49 17.)6 t
Zb.714 16. 150
::0. 113 1':i.463
27.73:1 lw. '1i.u
';.7.5C3 10.d41
1.. 402
1.653
1. 4~9
1. J82
1.445
1.445
1. l.J46
1a. 6 10 10.5'+2
24.230 14.379
~().06o 11.437
1tl.~(;b 10.no
~1.61S 13.06 C
19.61':> 11.121
1S.01.:5 11. 1<+1
1. 141
1.389
1. 161
1.085
1. 156
1. 150
1. 161.1
1-;'. iC6
6.550
18.200 10.055
13.E;:7
7.306
12.20....i
0.372
15.013
b.S06
13. JaQ
7.062
13 • .: 7 S 7.iJ32
4.CO
4.00
4.00
6.74
4.00
4.00
4.72
4.CO
it. 00
4.00
b.
;". co
4.47
4.00
4. 00
4.0v
6.57
4.GO
4.00
4. 19
4.00
4.00
!.t.GO
6. 71
4.CO
4.GO
... 07
4.00
4. () 0
4.00
b. S2
... OJ
4. U 0
4.07
4.0U
4. (] (J
4.CO
6. 54
OJ
4. 00
A
i3
C
4.u6
i:"
G
d.OO
1.000 9. aE 1
1.071 11.420
1.000 9. :: 15
O.97d
9.045
1. 000 1C~ 64d
1.000 9.444
1.000 'J.6Y9
8.00
:1.
E
8.00
0. 00
l' 0 1
4.CO
F
G
j)
d.~3
i?05
P 10
1. 000
~.74ti
1. 1S2 1J.OJO
1.QOO
9.~4J
0.954
9. 109
1.000 1 1. 1 14
1.000 9.dl3
1.000 9.910
4.00
B
0.25
4.61
G
F
8.00
a.oo
l:
0
4.00
5.38
ope
4.CO
4.00
4.00
b.40
4.00
4.00
4.00
4.00
4. (;0
6.59
C
1. UO
0.00
lJ Ei~
4. U0
4.0u
4. uO
6. Sci
4.00
4. 00
8.00
ti. 0 a
8.00
8.• 50
6.0u
8. U0
8.00
d. 0 0
8. UI)
8.00
8. 17
8.00
li. 00
8.00
8.1)0
5.00 15. ';63 :'4.7::'7
8.00 1'3.721 %. Out..
8.93 17.232 ~5.~c6
8.00 'b.']19 9LL ~JO
B. C J 17~57J ::;:>.764
d.CG 17.573 95.303
8. u G 17.072 95.~:1
5.
4~
C
4. ti8(i
~ .• ~2 S
C.. 998
aJHj K
5
1
2
1
I)
I.3
(j.9S3
1.429
C.,):!.7
0.073
1. J 54
0.971
2
"I
G.901
1. S85
1.058
0.697
1.430
1.018
C.9<J7
2
4.,,25
6
1
G.ns
6.2~J
4.';)]9
494
'.1.';07
:t.tiJ4
~ ..
4
1
6
J
5
7
5
1
6
4
3
J
7
2
1+
~.1CJ7
5
2 .. 567
6
1
3
7
2
5
J. d 79
:':.U32
2.:' 1'3
3.712
2. i76
2.7ul+
1.3 b U
2.365
1.635
1. J 79
:L 150
1.557
1.531
';; I. J 1 ~ 77.)90
clJ.l:;2
g2.5~6 cio::.1JU
) 1.50 4 79 • .JG5
n. 05 ':: 83.215
'L;.06 1 iO.J74
<JJ.J21
~
6
1
3
7
2
4
5
7
1
J
6
2
II
i~;.7GJ
5
00
'i.037 2f:.:::'c it:. '-'25 57.H2
8.00 11.,.,J d':J.~4) JJ.'1S7 bS.49L
8.5 U '1.97d bd.~~'j c2.:J44 <; 2, • (j'I'J
8.lJO 9. :;10 b7.210 JlJ.U18 5':1.212
d. () Q
9.'jU2 8':1.11" :,i';. 2 Li 03.JOl
d. () 0 9.)02 UB.,)u2 d 1.162 LO.(bd
'). 'J 1 1 6d.\.':17 1j1.2~4 00.971
8. vO
J
Sl.b-:E
tj.
G. u 0
8.00
tie 1 7
6.0a
~.
tl.
J
(J
uQ
U.JO
J.oH
'0.
S(Lt~2
.. c. 1 'J
n.7 07.717 :'S.23
:+. 2. 12 63.5]j :: ~. 70
J.')3:.> bu. tJ:~~
:•• 122 e4 .. ~~fJ
,~. 1"':_
62.:"':vJ
4 .;JJ oJ.~~j
4
4/. '12
: J • ~) 1
'. ') • :. 1
:':J.JJ
1.')18
9. f) ,>3
5.'jJ2
q
tj"Jt: 6
j • \J (~
S\~ t~
q • jj~ S
'••
1
J
6
:.2
5
4
7
1
J
6
2
5
~
I
~
"
.
85
"
Table 9.9 Power Functions In Percent For Design No.9.
VA
'.fAil
PROC
1\
f!
C
0.25
C
E
.f
G
A
E
C
0.00 1.00
C
E
0.828
2.0d6
2
7
t3.00
6.00
d.. () 0
4.275
5. I.+il 3
4.719
".li70
e.til ')
,. 18 I
0.n3
0.953
&
u.oa
0.. '142 1J.767
1.000 10.4<)2
'1.572
1. OOJ
1.000 'J.UJU
J
5
0.637
2.456
1.046
1
7
5
8.oJ
1.. JuG
'J.797
~.
79b
1.340
1.940
1.lt4'J
1. Jb3
1 ... J '1
'.439
1....... 1+
c
J:
<;'
~
r
I.i
I
A
E
C
,.
.
C;
~
G
A
i:
C
i:
"
!:o
i
Ii
;\
i:
(;
C
...
t'
F
G
A
E
C
...
S
r
I
4.415
'1.J77
0.00
a. () 0
8.b3
A
l3
I
9.132
5.01
4.00
4.00
4.00
f
G
I
17.I~G3
Cl.uU
9.IUJ
1.. 640
2.221
1. dUb
1. 7Sd
1.d19
1.61'1
1. aeo
c
1+.00
4. o'J 2
4.2d2
4.540
Il.014
d.OO
9.19
8.00
E
U
4.00
5.&5
4.00
4 .. CO
d.OO
J. 0 0
d.OO
8.uo
d.. ~O
6.00
8.00
5 ..
oj
lI.OO
4.00
2.130
:.353
2.. 771
.2.390
J.297
3
6
2.73 ,
2.662
~
4
7
1
3
6
2
5
4
7
1
17.c:~7
'.1.:;52
2.0 .. 017 12.20 J
1S.270 HJ.8JE
1').ouo 1'.0ill
11. ~d 1
n. :+25
u. i)O
0 52.2J9
7J.lJ4d
o~. 17:(, 63.191)
79.62.3 Sl'i.574
d2.4uO \)3.039
tiu.9d7 t:C.:'+07
:.I1.1tJU Qu.;)90
4.15
4.1)0
9.333
17.207
30.0::1 Hl. 577
1'J.90J 11. Jj .2
4.5~2
4.672
J,).~25
!:i.GO
... 12
4. 0 ()
:.+.lJO
7
1
3
&
2
5
4
'JJ.JOd
<j(j.cil0
17 .. 1JJ :J:' • :.i3 0
d. 0 l) 1 b. E; t,J':l ';4.uJ6
8.Lie 17.J7t! ')5 .... 10
tl. () a 17.370 ':)5.283
4.10
4.00
4. UJ
4.00
5.54
4.00
4. C J
24.~2t!
1~.J2:l
I)"
1j.91~
7.. Jid
C.CJO~
'111.52 t; d:.7'Jl
'j2.b24 >32.449
91.351 7c.968
.) 2.724 d':.2:-+J
'12."1 de. 2 07
iJ.t:iJ 10.995 ):;.014 '.:1.6J9 7<J.5rJ2
..., • .) ,j
<j,. , 9
tl.JiJ
6.LlI)
3.
l) I)
e. v (~
8. () v
8.lJ a
8.00
8.00
8) I
d. J J
._~
iJ.vu
b. 00
J,. (j 'J
22.~3·J
7.')2,J
12. 'j 6 J
'3.92 ':1
9.350
9. Tl::.,
'J.77J
3. o j 7
J4. ,2'34
75.
114
'.i 1 • 7 <; I (; 6. t. t)'~
Qd.\)UI)
C6.<i]S
c:i).7L5
07.'374
jd.J5d
'+ 1. i.J~ 7 1J.156
7j.1!<) C ,. 11 J 1 j~.j43
4. '::'0; oJ.044 :iv.717 ~~.%4
J .. J 0:) 3).9~7 '16.65 i.: 22 • '14 d
'1.U7. uJ.-j74 :'>'.'.)7'.) 2.7.127
~~ j 7 ~ b2.:: 1 1.l8. 'j 1': .21~.1211
J.4~C+
5.~~')
:+ • .;JO
1
007
7.7 J2
4.763
4.224
5. 5v 1
l~.
7.02J
Ij.UO
l!
25.070 14.855
J5.d% ~,j.55:
.::.l.u55 17.124
~ti.l~cJ 15.609
2d. 'J ..W 18. 1<30
:;. 7. ';:j 1 16.S~2
27.451 1b. 63 9
, 3. 1uJ
13. J
U. iJ U
I.
2
1.271
C.974
0.994
13.1301
11.7c9
1. 121
1.729
1. 1b J
1.066
1.1Sit
1. 154
1. 1& 0
2
1
6
3
0.849
1.. 0 423456J
1.621
1.3u9
1.91 b
, .4 9J
1.527
tLJO
8.00
8.22
d.OO
1).00
4. 00
4.00
5.56
4.00
4.00
o. as 3
S. 52 '.
14.380
7.341
6. '15
4.3 S
4.00
4.00
4. 00
5. tiS
4.00
4.00
4. JIJ
;
9. 71~u
lJ.'j6"
1.000
0.97 J
1.000
1.00U
1.uOO
3.00
C
1.00
5.o~
i
5.b8
4.00
~. 00
4.00
5.4d
4.00
A
E
5. \10
1.000
1. 3d 1
5.44
q. 00
4.00
4.00
4.83
7.0c,L
4.0a8
1 J. _.: u
a.G:'.!
d.S19
J.~72
1.000 8.S5J
1.53 J ~O. ~47 11.033
,~ 000 10.329
5.212
4.315
d.740
0.919
5. 71 1
1.000 10.015
4.875
1.000 9.798
4. 9d 4
1.. 000 9.980
f
G
I
8.00
~.l+2
4.00
4.00
9.~l!~
RArlK
0.%8
'4 5~ 4
0.1>4d
0.710
0.982
U.7d6
0.835
tie
8.00
d.. 00
8.22
8.00
8.00
8.00
8 .. 00
1:
0.;';5
0
1.°9
1. 1 d
tl05
4.44
4.0J
4.00
4.00
5.90
4.00
4.00
B
4.00
8.00
00
9.19
8.00
0.00
U.OO
8.00
ro 1
1:10
A
C
0.25 1.00
7.29
q.OO
4. 00
4.00
B
.... 00
4.00
a
0.25
OFD
r
l.i
4.00
OFN
.ij~~)2
\)j. 199
50. os 1
':1~.920
6
3
4
7
1
~
.......
:~
S
,
7
J
b
2
~
5
7
1
J
I>
2
S
'~
86
Table 9.10 Power Functions In Percent For Design No. 10.
VA
VAD
;FCC
A
I:i
\.:
D
E
'0.25
F
G
A
C
C
D
1.00
0.00
f
f'
G
A
a
C
D
f
F
G
4.00
A
&
C
D
E
F
G
0.25
A
B
C
0.25 1.00
0
E
F
Ii
A
B
C
4.0v
D
E
F'
G
I
i
5. 00 1. UO
c
10.00
10.00
10.78
10. C0
10.00
1 c. 0 \J
10.00
1.JOO 9.i.l~2
1.270 10.Uti7
1.0JIJ 9.':64
a.7~J
0.'145
1• .:l00 1u. d 8 LI
1. 000
9.0 '4J
1.000 9.dtS
5.45
5.00
5.00
5. CO
7.7B
10.00
10.00
10.27
10.00
10.00
1.00 J
9.018
4.290
1.Jd2 1 d. 576 10.118
1.000 10. ,H36
5.255
0.923 d.65'9
4.270
1.0iJO 11. , 1n
t. J2J
1.00U 9.877
4.932
9.So).
1. VOO
4.9l:io
5. CO
lC.OO
5.00 10.00
as
D
;;
f
G
24 2
4.3d4
:+.u42
4.612
d.027
i.l. U7 C
~. 2lJ 1
:. 761
4. 7b 4
4.d:.12
]
14
0 .. dS')
2
1
6
C.961
5
C.774
2.2%
1.070
2
7
O.~jS
0.839
12 • .; 13
c. 22 E
23.710 13.7 J 6
14.32/
7.702
1L.O"J8
&.317
15. ;;'f:7
0.057
lJ.o'+l
7.~19
1j.d~4
7. J2 2
1.':51
3.508
1.749
1. j 73
2.207
1.60 u
1.021
~2
5. CO
5.
co
57.2c.ic
':id.j~o
c; .•
~5
4.010
J.(}I·!9
3. 1:; J
t :;7.374
~1.,45d
b:'. 'U '; c7.(;",J
)1.0714 7lL 24')
~
c .. Ju J 7J.55u
db. 397 0').519
r; b.:; 0 1 'li.S/I))
d7.537 71.J7,.
ci. 1,0 C /1.075
3.570 04.1,1 :1.1Q2
S. 0 1) 0 7"1.41'0 l)b.74J
", ) ...
"'
.... 1Lid 78.4.:0 c~C.4_
J.JuJ 00. Vj,+ ~.::...04Y
4. ,)7 'J 7 I. I': 1 '; 'j • 'J:j t
t4.J7J (j'L J~2 J ( ) . b 1 iJ
1:l 1 U-J.'JIO ,:7.70 ...
....
5.049
6. 961
5.50Q
5.552
0;'7.741 -::6.056 ::'i.~l'JO
97.J':+4 ') 5. 29 1 07.273
S7 • ~ t: 'j , t; • ~ 'j i. ~~.J~\i
'17.0<24 '15.7J6 >:i:.L2J7
'j 7.';·j ~ ; : . ~ Ij v :~1.oJ"
1 O~ 00
u• .:65 ~J.~b;;
11).;)0 1 j . 75 J ':l4.752
C). 7'):' S ~.; .; L
1J. 7 a
10. \) 0 S..~5') 'Jl.717
10. (j U 9.679 'J J. " 1<j
1 C. G(.)
'J.u7J 92.:;':~
10.0(,)
9. 7'.+7 <:;'.::t6
5.. :.45 10.. 00
5. 00 10.00
S. O,) 10.27
:i.":'; 1 0. iJ 0
7. 5u 10. J U
5 .. ~u 11J.uv
10.iJO
~. au
~
CJ7.uu:
1
6
1. 12d
1. S6 7
1. 159
1.070
1. 154
1 • 15 ...
1. 15-9
10.00
10.00
11..144
1 C. 0 0
5
5.6)5
3.5'J<:
5.45
5.00
5.00
5. C I)
7.7J
5.CO
5. CO
d.05
5. C(J
5.00
5. CO
3
6
2.737
5. S 76
3.205
2.699
10. 00
10.00
10.27
10.00
10. G(.)
10.00
10. Uu
I~
1.392
0.990
C.9'JS
S.76L4
11::.587
7
C.91'-J
u.& is
1.2n
15.410 10.901
29.989 1U.694
~ 1.... 1'4
1~". 4() 5
19. 1 uQ 10. 8J 8
~~.7(;5 13. dO 4
20.713 1 1. 878
20.Sci L .• O~ 1
'9.8C6
1
.;.Ul0
1.'J2i.l
18.S7~
~
2
6
1.36J
1.79 ..
1.4 i W
1. J 61
1.43 ..
, .434
1.437
~'i.:Jld
'(;\'/1\
1.09U
o. g 11
6. 42 10. 00
5.00 10.00
5.00 10.78
5.00 10.00
7.50 10. U0
5.CO 1 0.00
5.00 10.00
5. () u
C
~.
,9.1<;4 lb.3Je;
37.2.:6 2<.1.780
30.5";7 1';.211
2d.51::i 17. SO 5
32.171 ~ 1. GO =
F
G
"
• •~ £
C.7'J7
0.734
1.669
2.0d9
1. 785
1.740
1.799
1.799
1.777
5. CO
5.0u
5.00
7. 3:J
:5.00
5.00
l:;
D
4.
b.b
4.252
4.0'15
10.00
1 c. 00
11. 1+ 4
10.1,) 0
10.00
1 C. 00
10. ~o
8.
5. 00
::). CO
7. -.2
!J
4.0u
6. 42
5.00
5. (j 0
s. UO
7. 54
5. co
5.00
6.
!J
'?y g y. ,CJ<~q
1.000
5.783
1 • 1:.! 1+ 12.8:.!')
1.000 9.0c6
O.lJ7~
0.042
1. QOu 10. ~~v
1.JOO 9.1.:.0
9,.5:2
1. OUO
A
..
Ii.
~Ol
10.00
1 C.OO
11.44
1 C. 00
10.00
1 c. a 0
13. 00
G
I)
[105
P1il
8.05
5 .. co
5.00
5. CC
7. 22
5. CO
5.00
14.37J
21J.'421
10. on
16.277
i. J:) lU.uQ 1 b , ; i 0
s. CO 1 0.00 lb. no
5. JJ lU.OO 16. ~ ... 7
A
3
C
0.25
R
2t.1';5
3
I~
1
J
..,7
...
:)
q
7
2
5
4
7
1
3
6
...
5
...
7
1
3
I
b
M
~
'j
7
1
J
0
2
5
II
I
...
7
.. 1.lJ41)
J. 2J 'j
1
j
J
3::.':; 1':1
31.1)0
32.:)] ...
2
5
L'.J. i) 1b
I
6
1
J
t
...
II
·
87
,
..
Table 9.11 Power Functions In Percent For Design No. 11.
'I A
VAu
I?ROC
A
E
C
0.25
C
E
f
G
A
e
c
0.00 1.. 00
C
E
,..~
'"
A.
E
C
4 .. 00
C
E
E
i.i
A
E
C
C
0.25
E
f
Ii
A
E
C
C
E
E
0.25 1.00
ti
A
E
C
4.00
C
E
r
G
A
E
C
0.25
C
E
F
G
A
13
C
D
E
F
G
5. 00 1.00
,\
13
C
;J.UiJ!
I
,
I
0
E
'l
G
DFN
a
DFD
R
FlO
£105
PO 1
RANK
2
1
6
3
4
10.. 9
"I. 0
7 .. CO
7.00
10.23
1. CO
7. co
1 ~.OO
14.00
15.68
14.. 00
14 .. 00
14.00
14.00
1.000 9.871
1.09 .. 12.60 9
1.000 9.301
0.919 13.d~7
1.000 10.042
1.000 ~. ~dJ
1.000 9 .. 7J9
4.958
0-.. 51 7
4.44 J
4.219
5.552
4.490
It. 792
1.006
1.ijl0
0.78J
0.766
1. ~09
(J .. 839
0.n4
8.67
7.00
7. CO
7.00
10.. 00
7. uo
7.00
14.00
14.00
14 .. 93
14.01)
14. uo
14.00
1.000 9.6So
1.. ~o 1 15.634
1.000 9.9dO
d. e jJ
O.. S::'S
1.000 11.227
1.000 9.7~1
1.000 9.918
4.711
e. 55 1
4.961
4.30 S
6.018
4.82J
4.935
0.935
1.955
0.966
o.132 7
1.. 395
C.. ~s 1
0.976
5
1
6
3
4
7.. 55
7.. 00
7.. 00
7.00
10.89
7. 00
7.GO
14.00
14.00
1 ~ .. 33
14.00
14.0a
14.00
14.00
1.. COO 9.401
~. sli 2
1.. 296 lb. 444 10.195
1.000 10.443
5.305
0.935 8.731
4.303
1.000 11 .. 576
6. ~98
1.000 9.977
:.005
4.991
1.000
9.989
0.&52
2.4lJ
1. 101
C.dsO
1.510
1 .015
2
7
5
1
6
4
8.83
7.00
7. CO
7.00
10.33
7.0U
7.CO
14.0C
14.00
15.68
14. uO
14.00
14.00
1 IJ.OO
1.072
1. 9~9
1.765
1.. 71.9
1 .. 778
1.77'13
1.752
32.197 21 .. 022
"I I. 4eo 20 .. 5134
35.159 23.030
jJ.Olj~ ~ 1. 231
37. ~~2 25.599
.34 • .5 1 ~2.41:
34.082 22.131
G. 7~6
lC.718
7.698
(:.799
7
1
3
6
2
d.17
7.00
7. CO
7. UU
10.56
7.00
7. CO
14.00
14.00
14.93
14. li 0
14.00
14.00
14.00
1.371 21.7i.l1
1.7'J1 ';~.4'::2
1.434 ~3. 953
1.369 ~1.5::5
1.429 25.832
1.429 23.3G2
1.429 23.~11
L:. 662
: U. 002
14.32 a
12.574
16.. 283
1J.7lJ2
13.d61
3 • .358
6.6'12
4.012
5.17U
2.794
3.810
7. '+0
7.00
7.CO
1.IJO
I U. 03
7. vv
7. GO
14.00
14.00
14.33
14.00
1<+.00
14.00
14.00
1. U3
1.475
1. 159
1.083
1. 1~ 4
1.. 154
1. 158
7.000
14.560
8.288
15.1 n
12. S 7'-)
1:.826
9. u 17
16.5l.;-"
14.55 I
7. dJ C
14.o3j
7.852
1.. <+92
3.961
1.. 966
1.53Q
2.604
1.808
1.78t!
l~.GO
7. 1:J 14.00
7. vO 14. vU
7.(;() 15.08
7.0U 14.00
1 U. 44 14.0 U
7.00 14. U0
7. 0 () 14.v0
13.441
~4.559
9~5~2
7. J 58
7.295
3.366
14.44.l 99.1% 98. 3u 1 94.615
19.193
16.3(;9
15.575
1 0 .. :>5~
16.552
io.U32
':is.:SJ 9~.ldC 57.198
99.419 :38.,,81 <j 6.439
99.313 '.Jti.651 S5.5:J1
~9.4():l ';9.01c3 'Jo.!.lu3
55.3';':1 ~c.t:05 96.U08
99.J30 96. cd ~ 9S.od1
7. 18
7 wCu
7. OJ
7. (; 0
10.
7. (; 0
7.0U
14.00
0 .• 42 1 Su.1C2
14.00 11.1')9 9d.lJ~ (I
14.93
:i. b 7 I 97.~:~
14.00 9.. 234 ~6.741
1 4. 00
':l.506 S7.!l')J
14.00 9. ~do 97.120
14. uO
9. 5 (,1" >;7.1"d
7. 1...
7.00
7. J;)
7. () U
1v. So
7. CO
7. OJ
14. U0
14.JO
14. J 3
J4.QO
14. iJ 0
1 .... 00
1 t•• 00
4"
0.997
'i2.90S 81.5&u
%. J25
tJ9.4~9
'J:.U12 :.31:.735
123 d4.299
(J e • 1 1 <J
u5.54!
':i 11.702 :35.060
'j L+.
9~.5S7
'J~.725
3. (;:J 6 74.'::31.1 63.41 i: 3b.229
d5.176 7b.{32<)
ou.125 7~ • .22 1
77. 1 do tJtJ.4~7
c1w14b 72.1:.t"
4. e70 79.1T1 oo.J20
4~ 15 '.J 7,).'1"J u <}. u'i 1
.... d76
.... 1 I !
3 ~ 'i 01
4. J 10
5 £•• 077
4(;.J00
41.8:31
4S.d6,)
.. 4. JJ j
45.4QS
~
.,2
3
4
5
6
1
3
7
2
5
q
6
1
J
7
2
S
4
7
1
J
6
2
4
S
7
1
J
b
2
5
4
7
1
3
b
2
5
4
88
4
Table 9.12 Power Functions In Percent For Design No. 12.
VA
VAl!
2RUC
A
f
0.25
C
C
E
°E
G
A
e
0.00
l.iJO
C
J:
E
E
G
4.00
j\
E
C
C
E
E
G
A
B
o. ;::5
c
C
E
F
4.937
t:.2H
4.!l39
(I. ()j 5
4.039
~89
O.9iJ7
1.297
0.1393
U '08'1.3
0.d')3
C.U9J
0.tl79
7.00
7.00
7. 00
7.. 00
7.00
7. (j 0
7.00
1.000
1. ~U:2
1.000
1.UUO
1.()(j0
'.000
1.. JOO
9.705
14. 1cu
10.119
10.119
10. 11 1}
10 .. 1 '9
9.851
7914
7 .. 3d 1
:;.084
0.938
1. :, 02
1. 0 26
5.00 '+
'.02u
7.33
7. 00
7.00
7.. 00
7 .. 00
7.. 00
7.00
7.00
7. vO
7 .. 00
7. 00
7 .. 00
7.. 00
7.00
1.iJOv
1.262
l.00J
'.000
1.000
1. 000
1. 000
9 .. ~25
15 .. 4~7
10.424
10• .:j ~ l6
10 .. 42~
4. &5t.1
d.40
7.00
7.00
7.00
7.0')
7.. 00
7.00
7.00
7 .. 00
7. ij 0
7.00
7.00
7.00
1.63 a 25 ... 33 14.782
7.00
7.00
7.00
7.00
16.915
11.763
5. :JJ 9
18. 6.J 1 10.095
6. 77 1
12.912
E.. 77 ,
1~.':i12
~o
7. J a
7.00
7. CO
7. 00
7. C()
7. 00
7.011
7.00
7.00
7 .. 00
7.00
7.uu
7.0'J
7.00
1. 1J 0
1.3'11
1. 112
1. 112
1.. , 12
1. 11 .l
1. 112
7.15
7. v..;
7. CO
7. vv
7. i~U
7.00
7. CiJ
7.0U
7. Ju
7.00
7.. \) 0
7. C J
7. ~ 0
7 41J 0
13 • 754
17.57:)
15.376
15. J 76
b.37u
15. J7f.J
, 't. i 1
7. 1 4
7. iJO
7. GO
7.00
7.00
7.00
7. 00
7.0 IJ
1. \i 0
7.00
7.0U
7.;)0
7. ";0
7.00
7.137
'J • 127
7. j I J
7. 'J 17
7. J 17
7.')17
7. i(J J
7 ~ 11
7. 00
'l.0\)
7. 0 oj
7.00
7. iJ J
"7. i) 0
7.00
7. J 0
7.110
7.v0
7.JO
7.00
7.00
)
__
A
S
C
t
A
e
c
,,-I:
...
E
G
i\
C
C
C
E
f
G
A
E
C
C
E
E
li
7.uo
7.00
4.
4.
5.
JJ!~
~ ..
ud 4
4.897
d. iJ7 2
5.301
5.301
5.301
5 • .30 1
4.989
10.4~'~
9.983
~
7 .. 00
C
t
E
4.bJ'i
1. 'j 28 .31.9"15 15.042
1.719 2b.6d~ 1 ~. 06 ~
1.71) ~6. OCS 1:.'S6Z
, • 7 1 ~ 26.G:35 '5.(;02
-.
1. 71 ':t ..:b.cc:_
1~.u62
l.bdo ~5.~~~ 15.070
1.3 'J 7
1.5'.JU
1. J40
1. J ij 6
1. 340
1. J4b
1. JJ U
F
G
4.00
8.07
7.00
7 .. 00
7.00
7. ell
7. Oil
7.0U
1.000 9.886
1. 105 12. 190
1.000 9.47)
9.1.1 7 J
1. Ju 0
l.001J 9.473
1.000 9.47 )
1.000 9.401
7.80
7.00
7. GO
7. JO
7. La
7.00
7. vO
E
5.00 1.. 00
7.00
7.00
7.00
7.00
7.00
7.00
7. (JO
L=O 1
A
G
0.25
9. ij 9
7.00
7. CO
7.00
7. CO
7.00
7. GO
£105
&'10
7.
:
4.00
DPD
U
e
0.25 1.00
DFN
1
.,.
:d.S~d
'd.J9J
1S.CC;U
10. v'l0
ld. aso
17.ti:i9
~.'J51
'j. jb 4
1~.912
lJ.77 1
C. 77 1
12.912
12.41'J
b. J] (3
1 .. 026
0.96')
Sci.o~o
'Jt. ':iJC
91.'Jj') "~.53J
'37.'j:';; ') 5.55 J
,-, ,j'
97.95'1 1'- .):J
S7.'J:'J J s. 55 J
__
r
"'17,
'97.03'J :"J..).
:..
.J •
3.778
5.533
4.060
4.Ci>J
4.0!; 0
4.060
J. d
jJ
2.0 JJ
J.~IO
~. 320
2.326
2.3,,6
2 .. 325
..2. 179
1.231
2.270
1.463
' .. 4013
1 .llb d
1 • ~6 U
1. J 41
a7.~b5
8J.519
d.l.S1'J
J J. ) 1 'J
dJ.51 J
::..:: .06 ~~
D8.204 7 '-j .. ,~ () It 51.250
':i ~. 'J 7 lJ ct. :12 J 03.2:+0
J'J.9'1(" :i 1. ';! 0 1 50. J 15
8'3.;<:'0 Jl.9u1 5u.J15
.,9.9:10 01.'1\)1 :)0 .. J 15
d'j. ':ho '" 1.)\) 1 5(,.,)15
90.027 81.7')7 55.JJO
37.852
:0.'113
~ .2. j,; J
~~.juJ
~
3.5
3.5
3.5
3.5
1
1
7
4.5
4.5
4.5
4.5
2
0.905
1
7
'.715
' .. 094 4.5
1.094 4.5
'.094 4.5
1.. 094 '16.5
0.967
2
')7.521 1):".602 dO.541
qt'"
'.
."
t)
SJ.297
3.ijJ<J 66.u40
j . ,." ) 57.'.:J7-4
J,.~4~ 57.5/ :1
j • .; 'I ') ":;7 • .;7 ..
J. ~ 4) 57.5'] 0.1
3 • ..:3 j 'j7.~4f.J
~
9. OU ':)
n.717
'i.951
9.'j51
'i.951
1.021)
RANK
l~.
220
.~2.553
i7.h7
17. 1 :1 -,
c+.:.. _Hi 3 17.1u7
17. 1 i') 1
.~ " .. Jd j
.. i • 'J ') ') 1 b .. .; 1 Q
,
7
3.5
3.5
3.5
3 .5
I)
7
1
3.5
3.5
3.5
3.5
6
7
1
3.5
3.5
J .. 5
3.5
ti
7
1
3.5
3.5
J.5
3,.5
b
7
1
3. 'j
3.5
J.5
3.5
6
,
7
J.5
3.5
J.5
3.5
1>
'#.
•
1
89
..
Table 9.13 Power Functions In Percent For Design No. 13·
V 1\
V AS
E1iCC
G
11. 1 b 1 u. VO
9. UO 1 u. 00
~.oo 19.06
9.00 1 U. 00
13. 1b lU.OO
9. CO 18.00
9.00 18.00
~.ti7b
1.000
1. 195 10.412
9.scl:l
1. aou
o. ~b 1 d.U~u
1.0UO 11.~"'1
1.000 9.719
1.000
':I. ~ ~ 1
A
9.72 18. 00
C
D
e
A
Il
C
D
1.u 0
r-
F
S
C
0
4.00
c:
F
G
,
A
U
o.~s
D
J:.
P
G
t
A
0
C
0.25 1.00
,
0
E
F
G
It
5
C
4.00
lJ
t.
f
G
A
13
C
D
0.25
i
1:'
G
A
J
5.00
C
0
E
1.00
f
G
A
i:l
C
J
~.IJO
E
r'
I
P 10
Ii
1.000 9.897
1.0d7 12.780
1.000
Y.25a
0.9dJ d.U86
1. uoo 10.6Sb
1.00u 'J.J 0':1
1.000 9.761
-f
G
0.00
DH
13.211 1 e. 00
9.CO 9. JO
9.00 19. d g
9.00 18.00
1~. 75 1 d. 00
9. 00 1 U. 00
9. 00 1d. 00
A
D
0.25
DEN
G
.
£'05
ij. 792
6.675
4.494
... 2~5
5.Stl2
ij. 504
4. ao s
1.010
1.447
C.tlO7
0.774
1.220
0.839
c.no
4. 7Y:
0.942
2.093
D.nll
Ii. 951
4.957
4. J 12
c.023
4.812
4.937
4.571
1.000 9.4C3
1.284 1~ .4n 10.940
5.27 'j
1.UOiJ 10. 180
4.284
0.944 d.6d4
1.UOI) 11.579
c. 294
1.000 9.94~ 4.38 J
4.991
1.000 S.~e9
O.J~6
1 .3 9 'J
O.:)4b
5
7
2
1
6
3
4
~
<.
7
5
1
6
,
oJ
C.'J76
II
C.8S7
2. bd 1
1.094
0.843
1.513
1.0U7
0.997
2
7
7
1
3
9.00
9.00
9. CO
13. 51
9.00
9.00
18.00
1l.i• .3 9
18.00
1 tie 0 a
18.00
18.00
11. 35
9.00
Y.OO
9.00
12. 36
'J.. 00
9.00
1 U. (j a
1':l.. 1j1j
1 d.O 0
1u. 00
18.00
t d.. 0 0
d.l13
1.640 35.~oj'j ~ 3. 676
1. ~4 ~ 45.J92 J~. 13 J 1,. J 03
s •2 ll~
1.7 J 7 3d.:~(' ~:.d9C
iJ.2Jd
1.707 Jo.jJ4 24.104
1.747 l.iO.S7·~ ~b. i 1 4 11.361
8.9a2
1.747 37.9Ycl .; 5. 309
e.77~
1.726 .17.475 ~ 4.99 II
10. 53
9.00
~. 00
9. CO
13. 12
9.00
9. QO
HI.OO
18. GO
lY.Ub
18.00
1 d. 00
18.00
10.00
1.36 ..
1.681
1... 2 4
1.309
1.421
1.421
1. 42 1
d.GCS 14. Ou 2
35.530 23.391
~ 5.. S i 0 15.040
2J.5.34 14.;)13
25.325 15.280
;.~. 4::: 1 1':.J'J2
..... ]2
:+
t. 132
14.C~1
7.JdC
2b.bjIJ 16.148
6.718
15. U23
1J./HJ7
7.206
17.3.11 10.13':
15.2u'J o.2SU
U.30(;
15.~~4
1 .0 C9
4. b27
2.10ti
1.647
2.aJ4
1. '140
1.:134
6
1
3
7
2
5
9.25 16. u U 13. tjl2 S9.7E7 99.4U4 97.S60
'1.0 U 1 ::i .. Ii 0 1 d. J 2 ';j 9'i.S')7 9Y.776 9').09b
7
i
3
llJ.(j(J
9.63 Hi. JO
1 6.. 0 U
1 ti. J 9
li3.00
1".00
9.CO 1d.(1)
9.00 18. u 0
9.00
9. VO
9. CO
13. 44
1.401
1. 157
1. un
1. 15J
1.bJ
1. 157
2d.I.:Ca
t7.97t
3. cHi?
8.002
4.b42
3.'324
5.966
4.395
5
1
6
3
4
0
2
4
5
7
1
3
6
2
5
4
9.00 19.09 15. 7 J4 9~. 844 :;S.60S
9.00 18.liO 15 ... 65 99.012 95.597
12. ~ 'J 1~.li0 lS.'~j7 'j>:;.t;t5 'J 4; .. 71S
9.CU 1ti.:JO 1 s. ~J 7 9<].03:.1 99.o4u
9.0J 18.1,.)1) 15.517 ~ ~ . t 1 "i So; .006
96.753
98" .. 45
-,c.JS2
9d.611
'j (,) • i~ 7(}
d. 27 J 96.';;J H.7JO
9.2J lti. J ;)
9. GO 18.UU 10.'J07 'J 'J • 2. ./ '1 98.542
9. OJ 1 ~L J tl
'~. ~u ... 5t.'j~J Si. ;j':.io
97.44')
'J.oo 1d.iJO 9. 1 1u 9d.u:Jd
c ,", %1 152
lJ. 02 1 t). I) 0 <J ... 13 ",j.y';O
97.7'-1 b
9. (,;0 lti.uJ
9. ~ 13 9U.8od
"
'j7.7iJC
.01CC:l
':J. 4 I 1 C
9.00 ld.OO
n .. JJ3
95.115
'JJ.376
7
1
3
" 11 • L 84
2
~
9. 19
<).1.:0
'1. 0 J
9. CO
13. 1 J
Y.ca
J.OO
10. J a
1 ti. I) 0
1J. J \)
1 d. D J
1 01
iJ :j
lu1,li)
1 d. 0
v
3.047
~.
u..: 0
i;~.C41
72.
4U J
LJO.7JlJ 31+.5&2
b.,;l'; 7o.htJ
1 <.I 0
3.91 J t~ 4. ~ 4~ j 75.S65
'• • 063 c7.4,)'1 tiO.4 .. 3
'+. Jo 3 d~.')7U n .67 ]
io. I Jb tt.b17 78.57 f:
'4.
')2.0:15
6
2
4
5
6
92.7713
9::.d50
4
t.676
7
l>.
1 2 (,
S
1
7.:+35
J
0,
IJ.76d
u
L 14 i 4
..;.'}92
:;. ~ 2J
<-
5
,
.~
90
•
Table 9.14 Power Functions In Percent For Design No. 14.
VA
VAG
PllOC
.\
E
c.:
0.25
C
E
-1'
G
d.
,...E
\). 00
...
1.00
C
E
l'
G
Ii
E
C
4.00
C
E
F
G
·
n
I;
C
c
0.25
E
;:
G
A
E
l.:
0.25 1.00
!
C
E
F
i.i
I
A
E
4.01l
C
C
::
f
\.i
0.25
A
E
c;
J:
~
·
<:
~
A
5.00
1.00
-
C
C
:::
·
G
,\
c
I
I
I,
I.
4.00
C
.,.C
w
r
.3
DFt~
DPD
13.04
9.00
9.eo
~. 00
1 J. 2 5
9.00
18.00
16.uO
20.22
1 U. iJ \}
1 S. 0 0
16. GO
1 d.OO
11. 02 18. 00
'3. ou 18.00
9.00 19.2 J
':i.OU 1 tie 00
14.00 18.00
9.vll 18. 00
9. CO 18.00
~.oo
flO
Ii
P05
fO 1
1.000 9.925
1. \JUb 12.i7b
1.000 9.Ju9
o. ':i7Y d.7'U.
1.000 10.9V
1. aoo
~. J 1U
1.000 9.7ti9
4. 99l~
6.b7 ':i
4.496
4. 2u J
s. 76 8
4.512
i4.833
1.1l1tl
1.451
0.804
1.000 9.816
1. 1':J a 1 b. j t: 6
1.000 1C.04~
O. ';54
d.6 .. ~
1.vOO 11.537
1.0UO 1;;.780
1.000 9.9Jl
4. :J86
8. ')5 7
5.013
4.241
6. 2b 2
4.657
4. 91+ 5
0.376
2.116
0.923
2.730
1.135
0.834
1. 0 ~ 1
1.041
0.997
U.762
1.29 J
C.84J
0.'139
0.<)88
C. d 12
1.49U
0.9&2
0.9UO
9.67
9.00
9.00
9.00
14.37
9.0U
9. 00
18. (J 0
1 b. 0 0
18.41+
18.00
18.00
1~. 00
18. a 0
1.000 9.ci7')
4.772
1.27 J 15.362 10.935
1.JOO 10.54J
5.396
0.934 8.:":2
4.216
1. ao 0 11.944
6.595
1.000 10.075
5.076
1.000 9.991
4.992
11.20
9. uJ
9. 00
9.00
13.60
9.00
9.00
18.00
1 ti. 00
(3.59 J
37.012 24.568
46. ~ 17 22.')J4 lJ.41l0
J'1.~19
20.702
':1.680
0.470
.J7.01S 24.534
4;t;.20 I 30.027 12.20&
';1;.E!J5 ~ c. os:
'].247
1.740 33.0;;;0 25. 1173
9.035
20.22
1U. 00
18.00
1b. 00
18. 00
1.67 J
1.974
1. 751.1
1. 719
I. 7b 9
1.769
18.0 C
1ti.OO
19.2 J
1B. 00
18• .10
9.00 18.00
9. CO 1 d. 00
1.37 J 24.G91
1. bti] .is.o73
1.430 2o.3bO
1. Jb<4 23.440
1.427 ~J. 720
1.427 ~:.c35
1.425 .i.S.043
~.
~.
1. 1J ...
1.451
1. 158
1. au 1
1. 15 j
I. 153
1. 157
10.42
9. Oil
9.00
9.00
1 J. 95
59 1 a.oo
OU 1u. ,)"0
9.CO lU.OO
9. UO 18.00
1.... 30 18.00
~.OU
1u. U 0
9.GO 1 B.00
9.22
':1'
• "v,
9. CO
'1. 00
1 J. 02
'J. (1)
cJ. 0 J
18.00
1 U. U0
20.22
10.00
1 d. v 0
18.00
1 d. u u
14.1+1 L;
18. U0
1 u. ()')
1 U1·. 4
1d. v U
, d. d J
1 J. J Ij
18.uO
12
1
6
3
4
3
7
5
1
&
2
4
2
7
5
1
6
4
J
6
1
J
7
2
l~
5
6
1
J
.1.930
7
2
~.'i'J4
5
1.723
4.694
2. 183
1 .031
J.OI9
1.'199
6
1
3
7
2
4
5
1... 450 j') • 7 lq) :;~.5Jj '11.3.151
16. tiS 1 S~.S~5 9'3.792 SY.159
1u.11d ~9.tl:'d J9.&99 ~ti.!367
15.7!32 'i ';.52 J '19. 62 C Sd.5JO
IO.Jd2 ~'J.=~\j
::Y.75i 99. U7 i.
c4" .... .; c:_ ..') ']9.677 '1r..71d
lb. Ju 2
l~. 7')4 :19.<3':0 )\].030 'JJ.5()4
7
1
3
6.382
4.523
l'i.J 7S
7.065
26.4: 8 lb. 120
16. UJ2
a. btJ 2
1J. ::: 75
7. 101
17.7'j4 10.5:> 2
IS. J ~ 7
c.302
1:.i.H8 8. J2:':
1.'.)39
.~~
.J
d. 4tJ 5 90. "~l~ io. ;;07
;I. ~ 1 1 8. \J I)
9.0J 1&. U0 11. Vip,) gr::. 2~ci St.Su2
~. (,0 1 <J.'<; j
9. u 1 Q <)8. '"JQ 7 97. '-nJ
9. 1 bb Sc.b~9 .:.,7.45J
9. ao lb. J 0
13. us 18.00 9.53 J 9'J.:JOJ 'jtl.2S 1
'J. 53 J '::i d. :; '17 97. d0~;
':J. JO Hl. JO
9.0v 18.00 'J. 504 98.'JlJu 'J 7. UJ 0
'j. i 8
'3. VU
':14 C:J
!J. vJ
1 J~ 9-;
9. vJ
':l.CO
".iJd)
8 .. 165
4.!JOl
-d. S7S
1 b. 1b 5
1..1.974
113.668
1=.54C
15.539
ilANK
J.lJU,+
b 17
.... 1t> j
~.
J.:J\;O
4. vb d
... ·Jo,J
<I. 14 'j
2.1':-12
v. ::,; 4
(J.b..;:)
4.JSJ
7.u:.ll
5. :.. G J
1+
6..,
4
5
'JO.4UJ
7
qJ.o2~
)
~:;.'9J
':12.078
'J .~. () '4
I
6
2
n.950
9J.042
5
72.094 It':l.290
~4.J2;~
o:.'J2l:i
7
1
J
6
2
5
7d.bj~
7:;. 2 '1 1
dJ.7(JS
51.7~d
") ... '"' c
: .. ., I ~'".)
(I
1.782
77.l125 55.,)011
u./l'j IS.lld Ju.J71
4
'+
,
91
Table 9.15 Power Functions In Percent For Design No. 15.
VA
'lAB
0.25
H<;c;
1. 00
22. 00
22 .. UO
22.37
22.00
22.00
22.00
1.000
1. 211
1. 000
0.961
1.000
1.000
1.000
13. J3
11. JO
11.00
11.\;J
15. Jq
11. CO
11.00
,a.ao
22.00
1.615 .37 .... :7 ~=.Jgl <;.066
1 .. U3 0 4S.4<rJ 32.254 13.034
1.6J4 4;; .. ~C9 ~ 7. 204
S.'.J9J
1.bb~ ..18.5~5 l5.~" 1
~. 229
1.691 42.630 JO.2tH 12.371
1. b 91 )Si.644 ';0. i9 1 9.709
1.672. ':9.CU5 ~t. J5J
':J.S71
C
D
E
F
C
l:i
E
f
G
A
13
C
D
.E
F
G
12. 57
1 1 • 00
11. 00
11. 00
IS.56
11 • ot)
11. 0 (J
~2
.. 00
23.65
22.00
~2.00
22.00
22.00
22.00
~2.00
22.97
~2.00
22.00
22.UO
22.00
23.65
~2.0U
22.00
22 .. 00
2~. 00
22. 00
~2.QC
22.97
22 .. 00
22.. 00
22.00
22.00
I
A
B
C
0
E
F
G
A
0.';5
i3
C
D
:
F
"
"
A
5.
~o
1.. 00
3
C
0
E
1"
G
i~ • i)
I
I
0
(1. 86 tj
11.77
11. CO
11.00
11.; CO
15.91
11.0 C
11.00
A
4.00
:.002
b. 20 tj
4 .. 652
HANK
22.00
C
0
E
t'
G
a
I
l'Ol
13. 1~
11. CO
11.00
1 1 • vO
15. &0
11. CO
11. OU
G
0.25 1. 00
P05
,\
A
O.2:i
22.00
I? 10
14. 'l a
11. CO
11.00
11.00
15. 24
11.00
11.00
B
4 .. 00
R
A
Il
C
D
E
f
G
11
o. U(J
DE':I
A
i3
C
0
E
F
G
11.6 d
11. CO
11. 00
11.00
b. ",0
22.00
22.00
22.37
22.uo
2..:. 0 (J
~ 1 • : v 22.00
11. liJ 22.00
11. 2b 22. a 0
11 • OJ ";2.00
11. J 0 2J. 05
11. CO 22.00
15. '17 22.0 J
11. GJ 22.00
i 1. J J :c:: 2.0 'J
11. 25 22. () Q
11.0;) :.: 2.
11~ GO ~ 2.
(j
'j
a
7
11. Du 22.00
15. '''9 22. OJ
11 • Gu ..:2. uC
l:.()u 22. u u
I I. 20 22. UJ
11. 0 U 2~.OiJ
i 1. 0 J .: 2. J 1
11. GO 22.,,0
1S. '; ') 22 .. J 0
11
t
'·,1
... \,j \J
~2.0u
1 1. (;:) 22.00
1.000 9.9~O
1.0bO 12.135
1. 000
S.55')
O.9t39 9.202
1.00u 10. ~ 12
1.UOO ') .. ~oa
1.000 ~.914
1 .0 1 ~
1.34
O.JJ4
1.2 tiO
0.UU2
C.974
4.4& 4
5.74 a
~.
tl4 4
4.931
1.000 S.d44
1.1LtO 15. 143
1. 000 10. GiN
v. n4
9. lJ'.>
1.00u 11. J;O
1.iH)(j 'J.U,W
1.000
9.570
4.920
8.179
4.99 E
4.495
Lt. 882
4.97t
0.07 J
1.421
0.967
C':I91
9.bC;~
4.c03
9.974
5.26 CJ
4.485
~. 33 4
5.050
4.99 €
0.93·9
:':.442
1.094
0.893
1.532
1.029
0.999
1.359
17.U'JJ
10. ;6(,
9.liJJ
11.1::40
10.041
9.9~5
~5.~~4
0.986
1.1191
C.CJ92
6. UU v
15.344
4.ll77
U.019
5.094
l~. ii95
6.569
4.895
1.590 J5.17"/ ';0.100
1.406 ~7.2~U 1(; .. cl27
1.J09 25.J80 15. 398
1. ~04 2~.501 lS.179
1.404 2b.7U9 10.'418
1.402 26.77':; It.• 414
1!L S:;S
c. 05 4
25.477 1 ~. 425
16 • .:: Ll9
s. 073
11+.483
7.849
10.C1J 1iL 616
S.714
15 .. 1375
•• 154 15. e (; 1
8.677
13.291 9".S~J
lb. 461 ~9.')o';
14. cd4 SS.S~:
14.519 ':19.937
14.82 ~ (j'3. '~~ ...
14.U2;J 9').9-.5
1 '~.
'-i '~4
';".~.;7
d. 1U7 <:9.': CJ
99.05')
'-J. 12 S C;~.~~~
0 .. 600 ~2· ~:+-l
9. Co ~ 'j '"j. ;; c..:
9 .. Uti .. CJ9.50~
'J.UL+v SS.5";1
lQ.i:+o
2.232
1.846
2.)99
2. 10 1
2 .. 056
99.~ld
39.142
')').:JYS
:;'::.459
-]S.572
5-::.359
\.. 6 .. ~51 r;'1 .6\l7
99.267 '17.256
(; co. :1 b S ":6.)';13
':10.822 )'3.81J
:.i g. 145 S7.J29
'Jd.9J9 3.:.149
'Jd.942 ')6.167
4 .• ::;95
c7.~~6 7::J.d55 5c.J09
93.'::0;;: Jo.J53 72.-570
j. )jij
4. Jjd
4.uJ(j
J9. '. J ') J2.1:.1 I
S1.~lj H. J5 S
'W...)?1 (.d.J67
J. 701
'I. J'JS
~.(,;bJ
SG.7~~
9(;.;: .:J
-j~.5'1
,'J4.513
t;6.Jb~
ti.:.
i.J
J:32
...J
..'.1. iJ2
..,.
::J .... I
~
7
5
1
6
2
4
2
7
5
1
6
4
3
7
1
3
6
2
4
5
7
1
3
6
6
1
'·H .. 87Q
S9.~5~
3
1.35 &
4.464
99.360
\;.;,.o~3
2
4
2
4
5
S;:i.U1C
1
.;S.072
')9. d5 5
9~.']1
1
6
sel
~,
L 135
1.JBO
1. 155
1.. 109
1.. 152
1 • 1J 2
~.
3
v
I
5. i.J b ')
3
7
2
4
5
7
1
3
."45
6
7
1
3
6
2
5
4
7
1
3
6
2
5
4