.
ON JUSTIFYING THE INTUITIVE BASIS FOR
THE METHOD OF WEIGHTED RANKINGS
by
Ibrahim A. Salama and Dana Quade
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1411
August 1982
I.
INTRODUCTION
Let X" be the observation on the j-th of
lJ
of
n
m treatments in the i-th
complete blocks. and consider testing the hypothesis of no treatment
effects. specifically
X'l •... • X,
l
lm
are interchangeable for each
i
(By definition random variables are interchangeabZe i f their joint distribution
function is invariant under permutations.)
The alternatives under consideration
are fairly general; however. two specific examples are:
(1) Additive treatment effects
H :
l
There exist quantities Ll ••.•• L (treatment effects); not all
m
equal to zero. such that for i=l, .•. ,n, X
- LI"·,,X
- T
m
il
im
are interchangeable.
or
(2) Ordered treatment effects
at least one strict inequality.
Standard nonparametric procedures for attacking this problem are based on
within-block rankings:
for example, the tests of Friedman (1937) and Brown
and Mood (1951) for H ; and Lyerly (1952), Page (1963) and Jonckheere (1954)
l
for H .
2
The only assumption they require is that the blocks be independent,
i. e. ,
(I)
The random vectors X, = (X'l" •• ,X, )'. for i=l, •.• ,n
~l
l
lm
{ (the blocks), are mutually independent.
However. to simplify the exposition it is convenient also to assume
(II)
P{X., = Xl' J' i} = 0
l]
so that with probability 1 there
for
~vill
j" j
I
be no ties within blocks.
But suppose
we make the additional assumption of additive block effects, as is common in
the usual "parametric" approach to this problem:
-2There exist quantities
(III)
13.,
... ,(3n
1
such that the random vectors
(block effects)
(Xl' 1 -
13"1 ... , X,1m
13.)'
1
-
are all identically distributed.
Then comparisons of observations are possible between blocks as well as
within, and procedures which use only within-block comparisons waste
informa.tion.
A method of weighted within-block rankings, which generalizes the
standard method based on unweighted rankings, has been introduced by Quade
(1972, 1979).
The idea behind this method is that blocks in which the
treatments are more distinct are more likely to reflect any underlying true
ordering; hence these blocks, which may be referred to as more
should receive greater weight in the analysis.
credihZe~
(in practice, credibility
is measured by apparent variability; but note that by Assumption III the
t~ue
variability is the same in all blocks.)
To determine the weight for the i-th block, use some location-free
statistic
D. = D(X. , ... ,X.)
111
1 m
which measures the credibility of the block
with respect to treatment ordering, and let
D'l, ..• ,D.
n
Q
i
be the rank of
D.
1
among
Again for simplicity of exposition, make the (unessential) assumption
(IV)
P{D.
1
= Do,}
= 0 for
1
i
#
i'
so that there will be no ties in the ranking of the blocks.
Let
o -<
b < ••• <b
1- - n
be a fixed set of block scores; and weight the i-th block proportionally to
Then in testing against the ordered alternative one may use the weighted
average external rank correZation
W=
where
C.
1
nI InI
b
i=l Qi
C.
1
i=l
b,
1
is the correlation between the alternative ranking and the ranking
observed within the i-th block.
Against the general alternative one may use
-3the weighted average internal rank correlation
C=
C.. ,
where
i
II
.J.. I
1.r1.
b
b c .,/«Eb.)2 - Eb~)
1.
1.
Q1.• Q1.., i 1.
is the correlation between the rankings observed in blocks
1.1.
., •
and
1.
The purpose of this paper is to examine the notion on which these weighted
rank correlation coefficients are based, which we may call the credibility
hypothesis.
II.
FORMULATION OF THE PROBLEM
Let
X1, ... ,X
~
~n
F(X1, ... ,X )
m
be
n
independent observations from
of variability defined on
m
R .
function" on
m
R .
be a distribution function defined on
m
R .
Then
F(X).
Let
Let
D be a measure
D may be regarded as an "ordering
Note that, for simplicity of exposition, we assume
P{D(X)
= D(Y)} = 0
for all
X and
Y
E
m
R
Using this notation, we may say
x
< Y if and only if
Let the random variables
order of magnitude, as
and call
Let
~(i)
G
D(X) < D(Y) ,
D(X1), ... ,D(X)
~
~n
D(~(l»<... <D(~(n»'
X and
Y
E
m
R
be arranged in ascending
Then we may write
~(l)<"'<~(n)'
the i-th ordered vector (i=l, ... ,n) with respect to D.
be a "score" transformation defined on
m
R .
An example of
G
which we will keep in mind throughout the discussion is:
S
m
is the set of permutations of the integers {l, ..• ,m} and
R.
1.
~.,
1
... ,X}.
m
is the rank of
t
1.
We note that it is not necessary to restrict attention to the
linear rank case; the discussion may be extended to
where
X. among
is a vector of scores.
Now,
the
G(X)
credibility
= (tl, •.. ,tm)
hypothesis
may be examined by studying some of the statistical properties of the random
-4-
G(~(i»' or in our case
variables
GR(~(i»' i=l, ••• ,n.
Again we narrow
discussion to the case of testing against ordered alternatives.
Statements
may be transferred to the general case with obvious modifications.
One
way of confirming the credibility hypothesis would be to show that
where
00
is
is the true ordering of the
{l, ... ,m}).
ill
treatments (which we will assume
This can be done by studying
This is the criterion which will be used in this paper, although we mention
another possible criterion:
In the next section we give an example illustrating these concepts in a
special case.
THE CASE OF TWO TREATMENTS AND TWO BLOCKS
III.
Consider the situation where we have two treatments
i~o
blocks.
X and
Y, and
The raw data may be represented as shown on the left below;
the ranks are as shown on the right.
X
I
II
Y
r-;-
~.
X
Y-
I'
R(l),l
Y
Z
II'
R(2),1
1
Y
R(l).2 I
R~l
j
Note by this notation that the range of the observations in the first
block
I'
is less than the range of the second block
Let
]J
and
Ae-Ay
> 0
x > 0
11'.
-5Assume that
X and
Yare independent.
Let
Our object is to find an expression for
D(X,Y)
!X-Y\
E[R(.)
.]
1 ,J
and hence the
expected correlation with the true ordering for each block.
let
and
To do that,
Z = Xl - Y ' so that we have
1
AZ
A]J
e
]J + A
z <
°
A]J
-llz ,
e
~~
z >
°
fZ(z)
Similarly, by letting
are independent.
by
W = X2 - Y2 ' we have
fW(w) = fZ(w) , and
Thus the joint density function of
Z
and
Z and W
W is given
g(z,w), as shown on the following graph.
w
All)2 AZ -llw
[ A + 11 e e
All )2 -llz e -llW
-- e
[A + ]J
z
z
A]J ) 2 AZ AW
[ /\1 + 11 e e
w
To finish the calculations, we need to evaluate the probabilities of
the 8 regions
as shown below.
AI" .. ,A
8
determined by the lines
w=O, z=O, w=z, and W=-Z,
w
z
z
w
-6The respective probabilities PI"",P
for simplicity of presentation.
have been written on the graph,
S
That Pl=P ' PZ=P , P =P 7 , and P 4 =P S is not
3
S
6
a special feature of the example, but follows from the fact that Z and Ware
independent and identically distributed.
=1/2.
In particular, P +P 2+P 3+P 4=P{!zl2.lwl}
l
Now,
P{R(X)=l, jZI2.lwl} + 2P{R(X)=2, IZI2.lwl}
p{!zl2.!wj}
(P +P ) + Z(Pl+P 4)
Z 3
1/2
since R(X)=l (or 2) if and only if Z, or equivalently W, is negative (or
positive); or in general
In the example,
3
2
1 + A + A ~ + 2A~
2
(~+A)3
A similar argument shows that
E[R
]
(2),1
=1 +
3
2
A + 3A ~
(~+A)3
3, for i=l,2 we have the following expectations:
x
y
I
I'
I1
I~
2
2
3
A + A ~ + 2AjJ
+
3
(jJ+A)
!rI
II'
2
3
A + A jJ +
2 3
(jJ+A)
ni I
I
1
3
1 +
A
2
+ 3A jJ
3
(jJ+A)
3 +jJ-A
~+A
A3 + 3A2~
2 3
(jJ+A)
jJ-A
3 -~+A
I
I
I
3
6
~
-7To evaluate E[corr(R(i)'OO)] we take
4It
example if A<V·
Then corr(R(l)
,°0)
°0
= [1,2], which is true in the
= 1 if R(l),l = 1 and -1 if R(l),l = 2.
Thus
P{R(X)=21Izl~-'WI}
P{R(X)=lllzljwl} -
P{R(X)=l, IZI.~lwl} - P{R(X):-~21Izl~IWI}
p{ lz 15..lwl}
P2 + P3 - PI - P4
P + P + P + P
2
4
3
l
and in the example this is
2
2
(V-A) (V +A )
(V+A)
3
In a similar manner we find that
P + P - P - P
4
3
l
2
P + P + P + P
2
4
l
3
2
2
(V-A)(V +4VA+A )
(V+A)
3
and hence clearly E[Corr(R(2)'00)J > E[Corr(R(l)'OO)] if A<V.
IV.
A CONJECTURE
Based on these results, we state the following conjecture for the ordered
alternative case:
There exists a treatment j 0' 1 5.. j 0 5.. m, such that
E[R
Cl ) ,J.] 5.. E[R(k) ,].]
for all
i > k
and
j -< J.
O
E[R
Cl ) ,J.] 2'- E[R(k) ,J.J
for all
i >k
and
.
j -> JO
and
Hence
E[Corr(R(i)'OO)] ~ E[Corr(R(k)'OO)]
for all
i > k
This conjecture is examined in the next section, using a simulation experiment.
-8V.
AN EXPERIMENT TO EVALUATE THE CREDIBILITY HYPOTHESIS
I t was decided to evaluate E[R("
1.)
,J for i=l, .•. ,n and j=l, .•. ,m for
,J
experiments with up to n=lO blocks and m=5 treatments.
simulated 500 times.
Each experiment was
The measure of credibility used \vas the block range,
and the distributions used for X,. were as follows:
1.J
(Appendix 1)
Normal with mean j (and unit variance)
(Appendix II)
Uniform on the interval (j, j+5)
(Appendix III)
Exponential with mean j
(Appendix IV)
Cauchy with median j (and unit scale parameter)
Each Appendix contains 36 tables (j=2,3,4,5; n=2, ... ,10) in the same
format, showing E[R(.)
1.
,J in the i-th row and j-th cloumn.
,J
For example) the
first table is as follows:
1
2
1. 332
1.155
1.668
1. 845
This shows the expected ranks in an experiment with two blocks and two treatments, for the case where in each block (row) the observation on the first
treatment (column) is normally distributed with mean 1 and variance 1, while
the observation on the second treatment is normal with mean 2 and variance L
In particular, (say») the expected rank of the second treatment in the block
with larger range is estimated to be 1.845.
Estimates of the expected ranks
i.n a block chosen at random can be obtained as the column averages.
The results of this Monte Carlo experiment clearly confirm the conjecture
of Section IV, and hence the credibility hypothesis, for the normal, uniform,
and exponential data:
within each table the expectations decrease monotonically
in the left-hand columns and increase monotonically in the right-hand columns,
except for very slight discrepancies ascribable to sampling error.
(The
magnitude of these discrepancies can be appreciated on inspecting the middle
columns for the 3-treatment and 5-treatment experiments, where by symmetry
the true values must be 2 and 3, respectively; or by noting also that within
4It
-9each row the true expectations must be symmetrically placed.)
For the Cauchy data, however, the results are clearly contrary to the
conjecture and hypothesis.
The within-column trends are not clear for this
case, although there is some suggestion that the expectations are most spread
out in the middle rows:
i.e., that the blocks with middling range are more
credible than those with extremely large or small ranges.
These results are consistent with Monte Carlo results reported by Silva
and Quade (1980) for the general alternative, and by Salama and Quade (1981)
for the ordered alternative, in which simulations indicated weighted ranking
procedures to be preferred over unweighted for normal and uniform data.
These
previous papers did not consider exponential or Cauchy errors; instead, they
looked at Laplace errors, but with rather inconclusive results.
VI.
THE CASE OF LINEAR WEIGHTS
In defining the weighted average internal correlation, suppose Zinear
weights are used:
that is, suppose
b . = Q
i
Ql
for
i=l, •.• ,n
Then
3
C = II Q.Q.,C .. ,/(n -n)(3n+2)
'J"
l
l
II
12
lTl
and a little algebra shows that
E[C]
Now,
n
1
+ I{D <D } + I I{D.<D }
2
l
i=3
l
l
and
n
1 + I{D <D } + I I{D.<D }
l 2
2
j=3
J
-10-
multiplying,
n
1 + I{D <D } + E I{D.<D }
2
1 2
j=3
J
n
+ I{D 2 <D 1 } + I{D 2 <D l }I{D l <D2 } + I{D2<Dl}j:3I{Dj<DZ}
n
n
n
n
+ E I{Di<D } + I{D <D2 }.E I{Di<D1 } + E I{D.<D 1 } E I{D.<D2 }
l
1
1=3
1=3
1=3
1
j=3
J
n
2 +
+
n
E I{D.<D } + E I{D.<D2 }
i=3
1 l
j=3
J
n
n
n
n
E I{D.<D <D } + i=E I{D 1.<D.1<D_?} + E
E I{D.<D l }I{D.<D 2 }
j=3
J 2 1
i=3 j=3
1
J
3
Hence
+ 2Cn-2)E(C12I{D3<D2<Dl})
+ (Cn-2)E(C12I{D3<Dl}ItD3<Dz}]
+ Cn-2)(n-3)E[C12ItD3<Dl}ItD4<Dz}]]
Consider the situation where there are n=4 blocks.
Let p. for i=l, ... ,24
1
denote the 24 possible permutations of (1,2,3,4) and let Q be the observed
ranking of the 4 blocks with respect to credibility; then PrtQ=P.} = 1/24 for
1
-11-
i=l, ... ,n (and this is true regardless of whether the hypothesis of interchangeability of treatments within blocks holds or not).
Then
24
i:1E[C12I{D3<Dl}!Q=Pi]P{Q=Pi}
Let E .. = expected correlation between the i-th and j-th leqst credible
1J
blocks among 4,
1~i<j~4.
We then find
Letting
We have
2~ + (n-2) E
+
13
12
i n- 2)
6
E
+
14
iE- 2 )
3
+ (n+2)(n-2)
(n+l) (n-2)
12
E24 +
6
E34
E
23
-12I{D <D }I{D <D } I{D <D }I{D <D } E[e ]
3 2
12
4 -23 1
- -3- -l - - -
i
P.
I{D <D }
3 1
I{D <D <D }
3 2 1
1
1234
0
0
0
0
E
l2
2
1243
0
0
0
0
E
12
3
1324
0
0
0
0
E
13
I,
1342
0
0
a
a
ED
5
1423
0
a
0
a
E
14
6
1432
a
0
0
0
E
l4
7
213 Lf
0
0
0
0
E
l2
8
211+3
0
0
0
0
E
12
9
2314
1
0
1
0
E23
10
2341
0
0
0
0
E
23
11
2413
1
0
1
1
E
24
12
2431
0
0
0
0
E2/+
13
3124
1
0
0
0
E
13
14
3142
0
0
0
0
E
l3
15
3214
1
1
1
0
E
23
16
3241
0
0
0
0
E2 .")
17
3412
1
a
1
1
E
34
18
3 /,21
1
a
1
1
E
34
19
4123
1
0
0
0
E
14
20
4132
1
0
0
0
E
14
21
4213
1
1
1
0
E
24
22
1:231
1
0
0
1
E24
23
4312
1
1
1
1
E
34
24
4321
1
1
1
1
E
34
12
4
8
6
:1.
e
e
-13Thus
24
(n-2)
2(n-2)
(n+l)(3n+Z) ~ + (n+l)(3n+Z) E13 + (n+l)(3n+Z) E14
E[G]
4(n-Z)
(n+2) (n-Z)
Z(n-Z)
+ (n+l)(3n+2) EZ3 + (n+l) (3n+Z) EZ4 + 3n+2 E34 '
or E[e] =
31
EZ4 +
3Z E34
1
+ 0(;).
Some special cases are as follows:
n
E(C]
2
~
3
1
44 {24~ + E13 + ZE 14 + 4E 23 + 5E Z4 + 8E 34 }
3; {lZ~ + E13 + 2E 14 + 4E Z3 + 6E Z4 + 10E 34 }
1
3~ {8~ + E13 + ZE 14 + 4E 23 + 7E Z4 + l2E~4}
4
5
3} {6~ + E13 + ZE 14 + 4E 23 + 8E Z4 + 14E 34 }
1
184 {24~ + 5E
+ 10E
+ ZOE 23 + 45E Z4 + 80E }
13
14
34
1
39 {4~ + E13 + 2E 14 + 4E 23 + 10E Z4 + l8E 34 }
6
7
e
8
Note that E[C] depends only on the expected correlations in a sample of 4
blocks:
thus, to find E[C] for any n by simulation one needs only to simulate
in sets of 4 blocks at a time, if linear block weights are being used.
VII,
SOME RELATED PROBLEMS FOR RESEARCH
Recalling the notation introduced in Section II, we may ask:
the statistical properties of the random variable(s)
m
G is an appropriate transDormation defined on R ?
G(~(i»'
what are
i=l, ..••n. where
We give some examples to
illustrate the idea and to open some specific questions for investigation.
A.
To attach a meaning to the functions D and G we consider the following
examples:
(A.l):
Let X. = (X.,Y.), i=l, •.• ,n, be
~1
1
1
a distribution function F(X,Y).
n
independent observations from
Let D(G) be the projection map on the first
2
(second) coordinate of R , that is
D(X,Y) = P1(x,Y)
X
-14and
G(X,Y)
Then G(X(,»
1
~
P2(X,Y)
~
Y
= Y[.1,n ] ' is the concomitant of the i-th order statistic, as
defined by David (1973).
(A. 2) :
Let X.
= (x'l'''''x, ),
-111m
F<,~).
from a distribution function
m
on R (D(X , ... ,x )
m
l
Range(xl,···,x
-
»·
1
2
.
m
J
m
.
Let G(!) :: G(x , •.• ,x )
m
l
(P,OG)(X(,», where P
--
independent observations
Let D be a measure of variability defined
1.
rank of xj among xl"'" x
J
n
E(x.-x) 1m-lor D(X}, .. "x ) :: max(x,) - m1n(x.)
=
m
i=l, •.• ,n, be
=
J
=
(R ,.,. ,R ), where R is the
j
l
m
RandoIll variables of the form
G(~( i»
and
is the projection map on the j-th coordinate of RIll, have
j
appeared naturally in studying weighted rankings analysis.
(A. 3):
In the previous examples we gave
t~vo
meaningful choices of D"
For other choices of G we may consider
Gl(Xl,···,Xm)
x,
G (x ,···,x )
2 l
m
X(j)' the j-th order statistic of (xl, .•. ,x ),
m
G (x ,···,x )
3 l
m
R., the rank of x. among (xI, •.• ,x ),
GLf (xl'
..• ,xm)
J
J
m
(xl"" ,x ), the identity map, etc.
m
\<lhile meaningful examples of choices of D are hard to obtain, examples
of choices of G are numerous and they lead to interesting statistical and
mathematical problems.
G.
We now consider a specific example.
Let
x,y,a~O
otherwise.
Let
~l
= (Xl,Y l ) and
~2
= (X2 ,Y 2 ) be two independent observations from the
distribution function F(X,Y).
Let D(X,Y)
= IX-Y! (the range), and G(X,Y)
(Rl ,R2 ) (again, R (R ) is the rank of X(Y) among (X,Y».
l 2
shown (based on Section III) that
Then it can be
-15-
and
Noting that (PlbG)(~(i»
+
E«P2oG)(~(i»)'
Out of this simple example we may pose the following
i=1,2.
(P2oG)(~(i»
= 3, i=1,2, we can easily obtain
questions.
1.
With D defined above, what is the distribution function of the
random variables P,] (X(
,», i,j = 1,2?
l '
2.
Let Gl(X,Y) = min(X,Y), G (X,Y) = max(X,Y) and D as defined earlier.
2
What is the distribution fWlction of the random variables G,(X '»'
J - Cl
i,j = 1,2? We would also want to investigate the previously mentioned
points for n>3.
C.
Let X, = (x'l" •. 'x, ), i=l, •.• ,n be
~l
l
lm
a distribution function FC!).
n
independent observations from
m
Let D be an order function defined on R , and
consider the corresponding ordered random vectors
~(i)'
i=l, •.• ,n.
Let x(i,j)
be the j-th order statistic within the i-th ordered vector of observations,
namely
~(i).
Then for all i=l, •.• ,n we have
Define
Then, a natural question presents itself:
of XCi,j)' i=l, •.. ,n and j=l, ..• ,m?
What is the distribution function
The sequence {xC . . )} may be called a doubly
l,]
ordered sequence, and the element x(, ') may be called a double-order statistic.
l,]
At this point we note that problems in the theory of order statistics may be
formulated as special cases of this setting by appropriate choices of the functions
D and G discussed earlier.
-16NOTE:
Section VI of this paper is enti.rely due to Quade,
The mathematical
work and the initial drafts of the remaining sections are due to Salama, wi.th
Quade having edi.ted them into final form.
ACKNOWLEDGMENT
The work of the first author was sponsored by grant
5-T32~ES07018-·05
from the National Institute of Environmental Health Sciences.
e
-17BIBLIOGRAPHY
Brown, G.W., and Mood, A.M. (1951)
On
Median Tests for Linear Hypotheses,
Proceedings of the Second Berkeley Symposium on Mathematical Statistics
and Probability~ Berkeley: University of California Press, 159-166.
David, H.A. (1973) Concomitants of order statistics.
Statistical Institute 45~ 295-300.
Bulletin of the International
Friedman, Milton (1937) The Use of Ranks to Avoid the Assumption of Normality
Implicit in the Analysis of Variance, Journal of the American Statistical
Association~ 32, 675-701.
Jonckheere, A.R. (1954) A test of significance for the relation between m
rankings and k ranked categories. British Journal of Statistical
Psychology ?~ 93-100.
Lyerly, S.B. (1952) The average Spearman rank correlation coefficient.
Psychometrika l?~ 421-428.
Page, E.B. (1963) Ordered hypotheses for multiple treatments: a significance
test for linear ranks. Journal of the American Statistical Association
58~ 216-230.
Quade, Dana (1972) Analyzing randomized blocks by weighted rankings.
SW 18/72, Mathematical Center Amsterdam.
Report
Quade, Dana (1979) Using weighted rankings in the analysis of complete blocks
with additive block effects. Journal of the American Statistical Association
?4~ 680-683.
Salama, I.A •. and Quade, Dana (1981) Using weighted rankings to test against
ordered alternatives in complete blocks. Communications in Statistics:
Theory and Methods AlO:
385-399.
Silva, Claudio and Quade, Dana (1980). Evaluation of weighted rankings using
expected significance level. Communications in Statistics: Theory and
Methods A9:
1087-1096.
-18APPENDIX I:
m
1
:2
1.332
1.155
1.668
1.845
1
1 .. 406
1.344
1.269
1.198
1.132
1.078
1.594
1<.656
1.731
1.802
1.868
1.922
2
:3
4
5
6
0::
Normal Data
2 treatments
1
1 .. I-,~03
2
:5
4
le369
5
b
7
1~316
1.221
1.205
l.125
1. 061
1
1.460
2
3
1.404
1.375
4
1.307
1.226
1.197
1.168
1.129
1.103
1.065
5
6
7
8
9
10
1,,597
,L631
1.,684
1 .. 1"19
1.79:;
1,,875
1.939
1.540
1 .. 596
1.62:5
1 .. 693
1.774
1.803
1.832
1.871
1.897
1.935
-19-
m = 3 treatments
1
2
1.421
1.223
2.00:
2.007
2.575
2.771
2.017
2,,018
2.033
2.013
1.982
7
1.604
1.416
1.339
1.253
1.237
1.186
1.159
1
1.595
1
2
3
1.+
5
6
1
2
3
1
2
3
4
..
1 9 464
1.265
1.216
1.527
1.341
1.224
1.178
1.987
2.02:
1.981
1.976
1.979
2.028
2.009
3
4
5
2. 1.97
2 '. 679
2.748
2.813
2
I.Lt61
3
1.387
1.306
1.239
1.229
1.189
1.139
4
5
6
7
8
1.528
1.381
1.278
1.209
1.165
1.985
1. 991
2.019
2.039
2.009
2.487
2.628
2.703
2.751
2.826
1
2
3
4
5
8
9
2
:3
4
5
6
2.781
2.05Lf
2.,760
1.983
2.859
2.001
2.003
1.993
2.003
2.012
1.999
2.007
2.002
2. Lt05
2.537
2.620
~.691
2.749
2.773
~
.80Q·
2.859
\:".
6
7
1
2~628
2" 73 Lf
2.549
2.710
2.803
e
1
2
2 .. 379
2,,566
1 0 529
1,,417
1 0 317
1.259
1.211
1.163
2.011
1.991
1.985
1.991
1. 993
1.990
2.460
2.591
2.698
2.750
2.796
2.847
1
2
3
1.630
1.463
1.396
1.333
1 .. 296
1.241
1.202
1.169
1.142
1.599
1.455
1.393
4
1.3Lt2
5
1.307
1.269
1.213
1.218
1.183
1.153
6
7
8
9
10
1.983
1.983
1 0 969
1.991
2.387
1.995
2.011
2 .. 020
2 .. 001
2.55 L}
2" 63~i
2 .. 677
~. 2:~ .. ,20
2.76 1+
2 .. 787
2.811
2,,857
2.016
2.024
1.994
1.983
2.014
1.9,91
2.017
1.974
1.989
1.994
2 .. 385
2.521
2.613
2.675
'2.679
2 .. 739
2.770
2.808
2.828
2.853
1.98!f
1
1.506
lc430
c:
"
.,
.I.
2
l
2
3
1.409
1.273
1.'+32
1.328
1.245
2,,880
? c 92'+
2.125
2 0 066
2.099
2 .. O!i 2
2.049
3,,585
3 0 736
2~936
30535
3 0 693
2.913
3~793
2 .. 933
2~Q3e
2.9,82
~;c929
3,,68;?-
5
2~955
h
1~227
2~937
3",707
3,,741
7
1e173
2,,056
2.06C
2.,095
2 e 082
?921
3$817
1
1.552
2
1$427
3
1>373
1.330
1.298
1.262
1.220
1.191
2.828
2.868
2.926
2.904
2.964
2.967
2 .. 933
2.910
3",+'+9
3.586
3.635
:3. 6 9L~
3.698
3.717
3.769
A
.'
.::I
Lj,
~~
,~-
3.609
1~.371
7
:2.889
2.926
:~ 0937
. 2.930
3<578
lG332
lQ27U
5
2.114
2.057
2.068
2.059
3e485
2'~92~
e
L}
(,
1.471
1.356
1.290
1.226
2~t:,'Y7
05 7
Ci
3
I{
1.
2
2 1.-::, :~
2
2.170
2.118
2.,065
2.071
2.03'3
2.053
2.077
2.095
3~803
3.525
3.660
s
3fJ704
.t:-
3 0 784
rt
I'"l
(1)
1
2
3
4
5
1.'+64
1.392
1.299
1.21+1
1.214
2.134
2.073
2.057
2 .. 113
2 .. 065
2.899
2,,900
2.962
2.905
2.94·0
1
2
3
'+
5
3$502
3$635
3,,681
3$741
30781
(;
7
B
q
..
1
2
3
Q
5
(;
1" 55:2
1 • 37i~
1.333
1,,306
1.26.3
1 .. 200
e
2,,142
2 .. 087
2.825
2 .. 930
2 .. 084
2 .. 1398
2 .. 927
2 .. 050
2. .,;060
2,,086
~l"
94 5
~!~906
3 o !}80
30608
3~68l{
3~,
5
807
l" 58'+
1.1+69
1.369
1 .. 369
1,,333
2.182
~,,822
2e;11Q,
2 .. 905
2 e 917
08~;,
2~896
2., 05'S;
2.928
1~307
2~OGJ>
2,,927
209B5
2Q95,O
?c943
::',,925
2~.O77
2
('J
6
7
16271
2(;;03S
B
1 .. 21+3
2c:~053,
'9
10
•
2.135
2 .. 097
2.087
2.051
2.052
2,,021.1
2,,068
2,,067
2.067
~
.. 907
209l.!3
2.954
.- F",sr67
?,,934
2¢·916
2 .. 929
3.422
39558
3.623
3.644
3.667
3.710
3.7'+7
3,,774
llJ
rt
~
:::
rt
Ul
3~802
,~!
1
2
3
4
3,,716
30731
2,,871
2.882
1.571
1.'+62
1.382
1.360
1 .. 326
1 .. 298
1.250
1.242
1.200
e
•
102113
2~G4'6
::1.181
2. 07-:'
3.411
3,511
3,,636
3e650
3 .. 679
3.704
:~Lo704
3.153
3,,787
3,,815
"
e
I
N
0
I
-21-
m
=5
treatments
1
2
1.399
1.256
2.073
2.112
3.026
3.007
1
1.422
1. 341
1.257
2.102
2.040
2.085
3.013
:l\.007
3.000
3 .. 905
3.926
3.910
4.558
4.685
4.147
1.457
1.360
1.306
1.218
2.124
2.080
2.055
2.154
2.970
3.010
3.054
2.983
3.92lt
3.920
3.899
3.878
4.524
4.630
4.686
4.766
1.488
1.353
1.328
1.2R8
1 .1'=)6
2.088
2.104
2.073
2.092
2.143
3.014
3.026
3.026
2.991
2.994
3.903
3.908
3.910
3.90lt
3.892
4.506
4.609
4.66lt
It.726
4.776
1.507
1.377
1.355
1.316
1.265
1.227
2.129
2.125
2.080
2.084
2.071
2.108
2.983
2.950
2 .. 978
3.015
3.036
2 .. 994
3.865
3 .. 937
3c957
3.896
3.884
30895
4.515
4.611
4.629
4.690
4.743
4 0 776
1.507
1.399
1.369
1. 322
1.2')9
1.266
1.197
2.114
2.09=
2.090
2.095
2.064
2.100
2.150
3.067
3.815
3.934
3.937
3.927
3.939
3.886
3.859
4.497
4.568
4.639
4.649
4.686
It.736
4.792
2
3
:+
1
2
3
4
~"
)
e
1
2
3
4
5
1
2
3
4
5
f,
1
2
3
4
5
6
7
~.003
'.965
'3.007
?l.O12
3.011
3.001
-22~
1
2
3
4
5
6
7
8
2.986
2 .. 996
"lii.OO8
~ .. 996
2.977
1.298
1.250
1.198
2.118
2.120
2.067
2.065
2.073
2.042
2.108
2.143
1.503
1.432
1.402
1.370
1.336
1.308
1 .. 283
1.246
1.201
2.122
2.108
2.0811
2.072
2.097
2.069
2 .. 091{
2.100
2.113
3.040
2.998
1 9 529
1.440
1.390
1.366
1.347
1. 311
1,,291
1.273
1.235
1.183
2.118
2.082
2.094
2.0B8
2.082
2.070
2.107
2.070
1.523
1 .. 406
1.380
1.355
1.:3~3
~
.. OO7
2.984
?9B8
3 .. 882
3.9143.931
3. 89L~
3.935
3.930
3.899
3.869
4. 1+92
4 .. 564
4,,614
4.690
4.682
4,,723
4.759
4.801
3.86Lj.
3.909
3.896
3.91/.f
3 .. 922
3.922
3.923
3.894
30862
4.471
4.554
4,.. 620
4.648
4.652
4,,108
4.698
4.764
4.814
3.896
3,,91tt
3.928
4 .. 465
3~940
4,,612
4,,669
'-J
1
2
3
4
5
6
7
8
9
~.998
2.996
2.993
2.99lf
:!.OO2
2.995
~ .. 010
10
:I.
2
3
4
5
6
7
B
9
10
2.0ee
2.13~
2.992
3 .. 009
3 .. 006
2.995
3.014
3.014
?992
7\.023
3.013
:'1,,004
3 .. 889
3.946
3.898
3,,896
3.90'+
3.874
4,,55~j
4,,582
4.659
tt.712
4.737
4.760
4.804
e
-23APPENDIX II:
m
Uniform Data
= 2 treatments
1
4-
1~363
5
1.314
1.236
1.123
1.547
1.592
1 0 653
1.637
1.6B6
1.764
1.871
1.442
1.443
1.384
1.362
1.326
1.304
1.226
1.096
1.558
1 G 557
1..616
1.638
1.674
1.696
1 .. 774
1..901.J
1.463
1.430
1.420
1.391
1.330
1.300
1.274
1.196
1.091
1 .. 537
1.570
1.580
1.609
1.670
1.700
1.726
1.80Lf
1.909
1.457
1.440
1.421
1.363
1.326
1.336
1.312
1.256
1.208
1.079
1.543
1 .. 560
1.579
1.637
1.674
1.,664
1.688
1.741f
1.792
1.921
2
3
1
2
1
2
3
1.409
1.238
1.591
1.762
1.410
1.351
1.209
1.590
1.649
1.791
6
7
1
2
3
4
5
6
7
8
e
1
2
:3
4
1.432
1.353
1.273
1.192
1.568
.1.647
1.727
1.808
1
2
3
4
1
2
3
4
5
1.451
1.392
1.380
1.303
1.155
1.549
1.608
1,,620
1.697
1.845
1
2
3
1.468
1.401
1.365
1.317
1.275
1.129
1.532
1.599
1.635
1.683
1.725
1.871
4
5
6
1.453
1.Lf.08
1.347
5
(,
7
B
9
1
2
3
Ii
5
(,
7
B
9
10
-24-·
m
= 3
treatments
1
2
3
4
5
1
;2
6
7
1.645
1.344
1
1
2
3
1.717
1.495
1.299
1. 983
2.013
2.021
2.301
2.492
2.679
2
3
4
5
6
7
8
1
2
3
4
1.749
1.595
1.417
1.229
1.
1,,801
;'
1~633
3
1.515
1.380
1.231
4
5
1.968
2.009
2.005
2.029
1.960
2.012
2.003
1.962
2.013
2.283
2.396
20577
2.741
2 .. 233
2.355
2.481
2.658
20156
4
5
6
1.793
1.636
1.534
1.425
1.332
1.218
2.033
2.035
2 .. 048
2.019
2.012
1.979
2.173
2.329
2.418
2.555
2.656
2.003
1. 981
1.987
L995
L,984
1..986
2.016
1.979
1.817
1.692
2.023
1.969
1.967
2.009
2" 0 O~~
1.985
2.030
2.000
1. 68L~
1.550
1.450
1.366
1.249
1.179
2.191
2.334
2.387
20505
::'.6:1.2
:20681
2.B29
,;)
c_
~
<!~:;;;;, (-:.~
Lf.. J
,/
~~~33~j
2 .. 3 11-9
2@L(ql
2,,546
2 ~ 6'+9
2.721
2,,823-
e
1
2
3
4
5
6
7
8
9
1
1
:2
.3
1.827
1.679
1.618
1.511
1. lt 02
1.303
1.192
~
34
5
6
7
8
9
10
1.841
1.725
1 .. 675
1.565
1.489
1.440
1.319
1.259
1.177
1.986
2.011
2,,023
2.003
1.986
2.003
2.006
2.027
1.981
1.865
1.736
1.709
1.634
1.543
1.486
1.412
1.307
1.259
1.173
1.973
1.997
1.972
2.014
2.020
1c972
1.983
2.015
1.975
2.001
2.173
2.26~)
2~303
2~1+33
.- 2-", 525
2,,557
206'75
2,,71. '+
2~B41
2.162
2,,267
2.319
2.352
~"t}37
2.542
2.605
2.679
2 .. 765
2~825e
1
2
1
2
.,
A'
l.l.
5,
f:.,
1.731
1.440
1.940
1.726
1.586
1.488
1.429'
1.287
1
2
If
_1.945
1.759
1.680
1.575
5
1.lf92
(;,
1.421
1.265
3
2.212
2.107
2.345
2.227
2.133
2.139
2 .. 076
2.131
:>.766
2e,sT?
2.650
2.783
3.290
3,,515
3.061+
3 .. 263
~0Bb7
3,,413
.. 855
1.904
2.872
3&511
3.590
~
3<1709
7
1
2
:3
4
5
6
7
8
'3
10
2.017
1.832
1.717
1.629
1,,578
1.523
1.4841 .. 441
1.342
1,,227
2.377
2.275
2.17.3
2.11€
2.122
2.087
2.133
~.6l5
3~O62
?c762
3.203
~
3.352
3.442
3.530
" 7 SIJ,
2.864
~.854
?.891
2.858
3.744
2<.920
2.990
3.159
3.252
3.368
3 .. 426
3.477
3.514
3 9 585
3 .. 654
2<.0868
3.788
2.41lf
~.578
2.338
2,,267
2.,210
2.126
2 .. 113
2.075
2.059
2,,082
2.670
?,,763
2.792
2~116
.3.600
?e870
? .. 886
?.925
'-~9lL}
-
-26-
m = 5 treatments
2.366
2.220
3.000
2.992
3.655
3.778
1.782
1.590
1.413
2.391
2.224
2.229
3.016
3.004
2~986
3.585
3.792
3.789
1.832
1.707
1.560
1.385
2.397
2.217
2.170
2.18:
3.025
:2.998
2.996
2.993
3.580
3.761
3.798
3.800
1.872
1.730
1.594
1.511
1.331
2.449
3.006
2.994
3.522
4 .. 150
2.23lt
3,,734
L},,:3
2.221
2.170
2.220
~.963
3.C19
3.024
3.839
3.826
3.789
4 .. 383
4 .. 475
4.636
1.864
1.734
1.613
1.573
1.470
1.326
2.486
2.304
2.220
2.174
2.129
2.220
3.540
30727
3 .. 786
3.846
3.883
3.799
4.109
4.249
4.365
4.410
Lt.510
4.678
1
2
1
2
3
1
2
3
4
1
2
3
..
5
1
2
3
4
5
6
3.001
2.986
:1.016
2.997
~.OO8
2.977
4 .. 226
4" 3-B"9
4"Sl3lf
4.166
4.317
4.47lf
4.636
08
e
-27-
1
:2
3
4
5
6
7
8
9
10
1.935
1.024
1.716
1.668
1.657
1.599
1.519
1.!j.94
1.398
1.221
2.56:
2.368
2.297
2.206
2.162
2.150
2.161
2.144
2.146
2.274
28982
~.OO5
3.GOO
3.030
2.980
~.O16
3.045
2.985
~.O16
~.OOO
3.440
3.620
3.726
3.788
3.809
3.851
3.825
3.829
3.838
3.763
4.077
4.183
4 8 261
'h307
'i. 393
4.384
4.450
4.548
4.602
4.742
APPENDIX III:
m
1
2
1.400
1.260
=
Exponential Data
2 treatments
1.600
1.71fO
1.
2
25
4
1
2
3
1.431
10349
1.231
1
4
1.456
1.390
1.308
1.191
1.'·~'-5
1~376
1~503
1.559
1~585
1 ~ 62 1t
10 69 5
3." TI'Ll
1.879
5
6
7
1.305
1
:2
3
1 .. 1+87
:1" ~):J. 3
1,.458
1 .. 403
1.346
1.339
1.278
1.217
1.122
1 .. 542
le226
10121
1.569
1.651
1.769
4
2
3
1.497
1.441
1.541f
5
1 .. 610
1.692
1.809
G
7
.9
1. ':5'37
1." 65 1f
1~6hl
:L,722
1.783
1.878
e
1
2
:3
4
5
J.
"
2
3
4
5
(.,
1.477
1.409
1.328
1.252
1 .. 154
1~~79
1.430
1.373
1.322
1.217
1.148
1.-523
1
1.591
1.672
2
3
4
5
6
7
8
9
1.,7lf8
1.846
1'.521
1.570
1.627
1.678
1.783
1.852
1,,488
1.448
1.428
1.408
1.352
1.321
1.260
1,,215
l.U.!
1..512
1.55,2
1 .. 572
1~592
1 v bile
1,,679
1.140
1,,785
1,,889
1
1~494
1~,,506
2
3
IG451
1.421
1.428
1.371
1.350
1.289
1.232
1.207
1.102
10 5 49
4
5
6
7
8
9
10
1. 579
1.572
lo629
1.650
L 711
1.768
1.793
1.89€
e
-29-
e
m
=
3 treatments
1
1
2
1.671
1.474
2.257
2.461
2.071
2.065
2
3
4
-5
6
7
1
2
:1
1.721
1.615
1.435
2,,219
2.325
2.528
2.061
2. 061
2 8 037
0
1
2
.3
4
5
6
7
1
2
3
4
1.811
1 .. 668
1.508
1.393
2.057
2.069
2.067
2.035
~
2.132
2.263
2.425
. 2.573
8
1
3
4
5
1
2
3
4
5
(;
1.827
1.700
1,,5[\0
1.471
1.385
2.064
2.059
2.101
2 .. 109
2.018
2.109
2.241
2.319
2.421
2.597
1.833
1 .. 724
1.631
1_527
1.431
1.373
2.049
2.070
2.09:
2.118
2.206
2.274
20377
2.521
2.621
2009~
2.-049
2.005
2.063
2.125
2.100
2.084
2@O99
2.083
2.173
2.267
2.346
28373
2,,071
2~505
2.6 Lt'l
2.001
1.849
1.724
1.674
1.621
1.563
1.495
1.403
1.351
2.046
2.083
2.104
2.090
2.089
2.045
2.0-86
1,,987
1 .. 847
1.823
1.723
1.650
1.507
1.451
1.387
1.348
2.029
2.035
2;089
2.084
2.081
2.100
2.076
2.059
1 .. 984
1.904
1.761
1.693
1.634
1. (; 0 11.553
1.487
1.434
1.363
1.326
1.994
2.079
2.080
2.083
2.100
2.059
2.119
2.072
2.089
2.012
2.105
2.193
2.222
2.289
2.31+9
2.459
:?511
2 .. 663
~-
e
1
2
1.853
1.701
1.633
1.570
1 .. 528
1.424
1.351
2
3
4
5
6
7
8
9
1
2
3
4
5
(;
7
8
9
10
1.5e2
2.124
2.142
2,,188
2.266
2.337
-'
~ ,,:593
2.473
2.553
29668
2 .. 102
~.160
2_227
2.283
2.299
2.388
2.394
2,.494
:?548
2.662
1
2~O~32
2
1 0 963
1 .. 799
1.733
1.686
1.577
1 .. 570
3
I.t
5
:1
;2
1 .. 906
1.669
2.LfS€
2.367
2<''760
2,,799
2 9 875
3 0 163
b
1
1
2
3
:L
2
3
1.975
1.766
1.618
2.476
2.l.t03
2.311
2.689
2.798
2 .. 845
2e>860
3.032
3.224
4
5
6
7
8
2.114
1.986
1.866
1.771
1.658
1.631
1.623
1.584
2 o Sl4
~1
h.~C
2~,478
~~725
2~f{t}7
2~776
~ ~ "-...
'./
.-
2o~12
?,,815
2<353
2.322
2 .. 221
2<060
2.500
2.471
2.485
2.469
2.483
2.396
2.309
2.199
~.64-9
2 .. 885
2 0 830
2.692
:'..735
2.729
2.810
~.830
~.810
2.797
2~745
2 .. 833
2 .. 977
3c039
3 .. 1°0
3.215
3 .. 378
2.735
2.850
2.913
3.030
3.048
3.142
3.257
3.419
s
II
1
2
2\
4
2.030
1.826
1.660
1.593
2.493
2.451
2.417
2.26€
2.689
2,,772
2 .. 791{
2.846
2.787
2.950
3.128
3.292
~
rt
1
2
3
4
5
6
7
1
2
3
4
5
1
2
3
4
5
6
2.037
1.856
1.709
1.635
1.591
2.090
1.904
1.781
1.694
1.628
1.594-
e
2.491
2.477
2.436
2.367
2.249
2.521
2.487
2Q1+7~
2.436
2.,357
2.,219
2$667
2.801
2.817
2.805
2.823
2,,668
2.804
2e865
3,,037
3.192
3,,336
2.725
2.769
2*720
2<>883
2e974
?.,815
3",05t~
2.(;>33
2 <01:: 26
3",182
3,,360
B
9
1
2
3
4
5
6
1
-8
9
1.
2.160
1.989
1.898
1.833
1.724
1.669
1.641
1 .. 585
1.573
2.515
2.503
2 .. 502
2.471
2.430
2.337
2.308
2.293
2.161
2.208
2.010
1.920
1.875
1.740
1.722
2 0 472
2.516
2.514
2.482
2..,1.+7 0
2.614
2.683
2.694
2.,761
2~ln8
2.839
1~641
2.412
1.623
1.603
1.614
2 3cr,""j'
2,,841
2d386
2 0 267
2 0 153
?
0
2.639
2.678
~.704
c·
2.760
2.820
? .. B-L+ 2
2.84L,j·
2.839
2.8t~4
?~826
2" 8l.!·5
g
7 8L~
2.685
2.829
2.095
2.935
3.025
3.150
3.206
3.281
3.422
2.705
2.790
2.071
2e-882
2.963
3.020
3.105
3.144
3.284
3.448
!'i
(D
~
rt
S
(D
::i
rt
!Jl
e
I
w
0
I
-31-
m = 5 treatments
1
2
1
2
3
1
2
3
4
1
2
3
4
5
1
2
3
4
5
6
1.846
2.736
2.570
:3.131
2.127
1.922
1.823
2.7e4
2.653
2.513
3.166
3,,199
3.095
3 .. 397
3.526
3.589
3.526
3 .. 700
3.980
2.165
1.950
1.823
1.770
2.891
2.783
2.652
2.51l.!
3.121
3.213
3.138
3.359
3.425
3.567
3.622
3.464
3.629
3.820
4.016
2.260
2.007
1.910
1.828
1.783
2.876
2.794
2.664
2.586
2.488
3.116
"3.211
3.208
3.108
3.318
3.431
3 .. 482
3.572
3.562
3. l t30
3.557
3.736
3.905
2 .. 276
2.038
1.904
1.891
1.826
1.807
2.923
2.873
2.743
2.640
2.587
2.'471
":5.153
3.204
3.270
3.347
3.448
3.495
3.540
3.616
3.378
3.538
3 .. 669
3.728
3.938
4.079
~.084
3.192
-
~.o-78
~.056
~.235
3.247
:'i.108
3.027
4.112
-32-
e
1
2
3
If
5
£;
7
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
9
2.298
2.055
1.964
1.8B6
1.839
1.7R6
1.775
3.356
3.520
.3.581
3.716
3 Q 843
3.935
It" 124
3.350
3.255
3.194
3.120
3.136
3.091
:';.020
3.250
3.359
3.453
3.485
3.574
3.565
3.582
3.615
3.154
3.302
3.140
3.198
3.240
3.194
t .. 226
3.348
3.426
3.508
3.540
3.519
:3 .18 L}
3i!'588
~.O95
3,,677
3 .. 620
3.1'16
~.161
2.350
2.122
1.964
1.953
J..877
1.630
1.815
1.7f,4
2.917
2.858
2.772
2.731
2.629
2.603
2.521
2.426
3.1.34
2.328
2.133
2.918
2.883
2.817
2.767
2.676
2.633
2.517
2.464
2.418
2.064
1.902
1.868
1.el+O
1.020
1.790
1.806
1.7£0
7
3.277
3,,360
3. lf84
39566
3,,560
30582
3.619
3.030
9
10
(,
2.779
2.6E.l8
2.582
3e143
3.230
3.192
3.144
2.453
B
4
5
2.83~
2.536
2.372
2.156
2.035
1.959
1.904
1.889
1.849
1.827
1
2
3
2.926
1.7£34
2.929
2.905
2.838
2.797
2.746
2.680
2.580
2.586
2.458
2.454
~.198
3.023
3.144
3.186
3.233
"3.207
3.356
3.467
3.456
3. Lr8G
3.553
3.576
3.615
3.576
3.188
~.169
3.205
3.205
3.098
3.082
~.O16
3.30tt
3.1+62
3.555
:5 .. 636
3.800
3.1367
3.992
4.1.74
3.297
3.497
3.494
3.583
3.724
3.781
3.891
3.974
Ii· 131.1
Ij
3.322
3.444
3.564
3~58a
3.726
3.740
3.813
3.914
4.062
1.f.194
e
-33-
APPENDIX IV:
Cauchy Data
m ::: 2 treatments
•
1
2
.3
1.418
1·352
1.jtj2
1.olf8
1.295
lo"U~
Ii
1.6Uj
:Lbje
7
10 305
1.317
1.362
J..436
1
2
1.398
1.bU2
1 .. 374
l.b~b
.3
4
5
1.317
1.6tl6
6
7
1.312
l~6Utl
1,,375
lo6~:J
8
1.Lt08
1.~92
5
6
1
2
1.350
1.368
1
1.369
2
3
1.326
1.355
1.65U
1.6j2
1.6..11
1.6.,4
1.0,+5
l..6';j~
1.~64
1.294
1. • "' U6
1.30lt
1.6'%
-34-
m
= 3
treatments
1
1
2
10585
1@b50
1.';l'::i1:3
2.417
~.UU7
2.339
2
3
q
5
6
:::
';(
..-
1.565
1,,580
1,,673
1.'3e'3
2.4/j·5
1.':)':16
2.42'+
~.u~o
2.30'7
1.5;;9
;2
16533
1 .. 573
1 .. 715
3
4
1.,:)':J8
1.';;':J6
2.433
2.471
~.Uj1
2.389
2.U21
2.25'+
;')
1Q575
1.535
1.915
1,,54"/
1.'375
1.976
2 e Ut!6
q.
1,11;46
';)
10713
~.uu:
2.451
2.459
2. tH8
:~.37e
2.261
l.'j~l
2. 1+6'::J
2.39'::1
1.786
~.UUO
1
2
1.639
1.519
1.503
1.501
1.550
1.606
1.689
1.769
1.':)<'}3
3
l.'j'jj
2.U11
2.Ull
2 •.349
2.211+
2" .)r:J-;
2 0 4"(0
2 Q Lj·B6
1. ';i'.n
2.507
1.'j~6
2. 1HA
1.':)H7
2,,407
2 .. U25
2.286
1.':)';;1
2.240
e
1
1
2
~.4(1~
2.48'J
•
q
~o40:C
1.':)17
1. ':H.l'j
5
6
1
8
:t
l.'j/jj
l.'j'~
1.6U.
1.658
-;
1
1.615
1.539
1.533
1.533
2
.3
4
5
6
7
8
9
1.615
1.540
1.50n
1.50p
1.538
lQ559
1.652
le685
1.821
2.U1.3
1."85
2.UU8
1.':)';;4
2.U04
2.U54
2.37-5
2. £+ 7~i
2.4C4
2.506
~.U17
2.'+5£3
2.387
2. '57 j '
~ .. 29/j
l.'j')~
2e183
1.':)/5
,
1~600
~~ut!C:!
2.37U
1
1 .. 643
~.Ol'j
2 .. 331
2
1.513
'.U15
~ .'+7~
1.525
~9U14
~:J
J,.501
~.Uj9
2.46U
2
3
~eU~~
I{
1.563
1.661
1.727
~QUC:!5
~.41~
1.'9tJa
2.518
J..';;':Jl
2.348
t!.27b
1.503
1.494
1.528
1.582
1.603
1.673
1.758
1.795
2. /,;.61
2.47':)
:J
,()
1.<-J':J7
4
5
6
7
8
9
10
1."'32
2 .. t:·80
1.'9':J8
2~42(]
1.'366
1.<-Jb.3
2
e
42t:l
1.'j'j8
2.364
2.244-
~.U19
2.186
e
•
e
1
2
1.717
1.858
2
.3
q.
c.t::!Ul
2.7f:'.2
c.2~U
2 .. 731
3 .. 298
3.16U
5
6
7
1
2
1
2
3
1.701
1.766
1.927
2.243
2 .. 736
2 .. 738
2.2'J~
2 .. 706
2.2~6
3.306
3 •.?51
3.071
•
~
3
q.
5
6
7
U
1e760
10658
1.646
1.7'fO
1.7g0
1.905
2.060
1.723
1.653
1.656
1.720
1.739
1.849
1.911
2·025
~O~U~
~.lt)~
".247
2.78U
2.79 '+
2.761
oc.~2b2
2.7':J<J
".276
2 .. j,j-'
2,,727
" • ~)'5 4
2.67U
2.63j
".l'.HJ
2.B?7
2.787
2.806
£.22~
c..7S1
2.264
2.7442.735
2.724
2.751
£.1/7
2 .. 1~5
2.t::!~8
2.27~
oc..2b2
3.251
3.365
3.344
3.238
3.206
3.087
2.972
e
3.272
3.364
3.339
3.266
3.252
3.157
3.0 136
2.'361
s
II
J.
2
3
L,I
1.712
1.680
2.1':1U
£.223
2.791
2.79?-
1oi323
2.2~O
'l .. 727
1.963
2.2H2
2.692
3.306
3.304
3.199
3 .. 063
.prt
r-j
ro
1
2
.3
q.
5
6
1
'2
3
If
5
1.745
1.683
1.726
1.855
2.009
c.'2Uj
2.1tjl
'2.773
2.792
c.c21
e..74e
~.iC':1'j
2.721
2.674-
2.317
3.278
3.344
3.304
3.125
2.999
7
S
9
1
2
1
2
3
4
5
t
1.735
1.632
1.719
1.751
1.869
2.042
c.~11
2.813
:3" 2'.0
3. 3 f t 0
3.'328
3.256
2.226
2.801
2.1"'-'
2.241
"'.75:'
2.217
2.'2bt$
2.714-
~.745
"t
o
67'-}
.3 .. 139
3,,011
1.714
1.671
1.679
1.645
1.722
1.787
1.846
1.946
2.061
8
9
1.796
1.683
1.652
1.665
1.664
1.769
1.836
1.869
10942
10
2~O39
3
4
5
6
7
2.25U
2.1-f8
2.1714
2.2~6
2.215
2.coU
2.313
~",j46
2.320
c.'2U3
c.l'j5
~o2U4
2.2j;:'
2.':::]'?
2.21~
2.800
2.823
2.801
2.752
2.756
2.77U
2.705?-.681
2.685
3.236
3.327
3.3 1t5
3.3'+6
3.306
3.193
'3.136
3.026
2.933
2.784
2.767
2.79';:)
2.773
2.774
2.733
2.711
3.216
3.354
3.347
3.327
3.344
3.?85
3.179
3.126
3.041
2.912
2.2 US
e.,,'2CJq
2",72U
2 • .jue;
2,,708
~.~111
2.677
p;
rt
S
ro
::J
rt
UJ
I
w
lJl
I
-36-
m
1
,.
'j
1
2
;;
1
2
.3
If
1
2
3
4
:>
=
5 treatments
2.9?U
1.827
1.995
~.976
1.82!~
~.IfUO
1.890
2.145
r-.1f~6
2 .. 99(J
tl.002
'.~69
2.973
1.794
1.839
2.024
2.163
'.j~l
2.992
'.3~1
.1.015
'.3~9
.~.O34
'.~7U
2.991
1.816
1.762
~.31'
3.001
?.95B
~.jt$l
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