•
\
Ji,
9'
",.,
e
SOME SIMULTANEOUS INFERENCE MEDIAN TESTS
by
Lester Ware Preston, Jr.
Institute of Statistics
Mimeograph Series No. 694
August 1970
•
v
TABLE OF CONTENTS
Page
LIST OF TABLES .
• viii
LIST OF FIGURES
1.
xiv
INTRODUCTION
1. 1
1.2
1
General
Nonparametric Background
1. 2.1
1. 2. 2
1. 2.3
1. 2.4
Defini tions
Scope
AssLJmptions
C0nstructioh Methods for Nonparametric Tests
1.2.4.1
•
1.2.4.2
1.2.4.3
.,
1.3
1.4
Sign Tests
Component
2
3
3
3
6
7
10
10
10
11
11
12
Definition of the Proposed Simultaneous Inference
Median Tes ts
... .
. . •
12
A SIMULTANEOUS INFERENCE MEDIAN TEST FOR COMPARING ALL
PAIRS OF TREA'lMENTS
2.1
2.2
2.3
2.4
•
Simultaneous Inference
Simultaneous !nference
Randomization Tests
Simultaneous Inference
Simultaneous Inference
Simultaneous Inference
1
2
Signed-Rank Tests
Rank-Sum Tests
Median Tests
1.4.3
1. 4. 4
1. 4. 5
2.
Tests Based on the Binomial
Distribution . . . . .
Tests Based on the Method of
Randomization
..
...
Tests Based on the Hypergeometric
Distributions
Simultaneous Inference Background.
Review of Existing Nonparametric Simultaneous
Inference Procedures
....
....
1.4.1
1.4.2
1.5
1
1
Data
Description.
Assumptions
Test Procedures
2.4.1
2.4.2
2.4.3
One-tailed Test
Two-tailed Test
An Example
13
13
13
14
15
15
16
17
vi
It
TABLE OF CONTENTS (continued)
Page
2.5
Test Derivatiqn . .
2.5.1
2.5.2
2.5.3
2.6
3.
..
32
38
45
Test Procedures
Power . . . . .
45
48
Data
Description
Assumptions.
Test Procedures.
3.4.1
3.4.2
3.4.3
3.5
3.6
3. 7
56
57
57
58
One-tailed Test
Two_tailed Test
An Example
Test Derivation.
Power.
Discussion
56
58
59
60
64
.
66
70
,
4.
LIST OF REFERENCES
71
5.
APPENDICES
74
5.1
The Null Distribution of m"
Pairs of k Treatments: One-tailed Critical Region.
The Null Distribution of m.. ! when Comparing All
74
5.2
Pairs of k Treatments: Two_tailed Critical Region.
Regression Coefficient Estimates for the Inverse
Quadratic Power Response Surface when Comparing
All Pairs of Treatments . . . . .
.
.
..
Approximate Stochastic Differences among Treatment
Populations when Comparing All Pairs of
Treatments at Given Power Levels . . .
... .
78
5.3
•
5.4
..
Derivation .
An Example
A SIMULTANEOUS INFERENCE MEDIAN TEST FOR COMPARING
TREATMENTS VERSUS A CONTROL
3.1
3.2
3.3
3.4
22
28
30
32
Discussion
2.7.1
2.7.2
•
Rationale
Exact Null Distributions
Approximate Null Distributions
Power .
2.6.1
2.6.2
2. 7
22
~~
I
when Comparing All
~~
82
95
•
vii
TABLE OF CONTENTS (continued)
Page
5.5
5.6
5.7
5.8
5.9
..
The Null Distribution of m when Comparing
Oi
Treatments Versus a Control: One-tailed
Critical Region.
....
. .
The Null Distribution of m when Comparing
Oi
Treatments Versus a Control: Two-tailed
Critical Region
....
....
.
Regression Coefficient Estimates for the Inverse
Quadratic Power Response Surface when Comparing
Treatments Versus a Control . . . .
..
..
Approximate Stochastic Differences among Treatment
Populations when Comparing Treatments Versus a
Control at Given Power Levels
Computer Programs . . .
5.9.1
5.9.2
..
•
5.9.3
5.9.4
116
120
124
137
158
Enumeration of the Exact Distributions of
m..
i
and mO'~
.
~~
158
Generation of the Approximate Distributions
of m..
, and mO'~ . . .
..
.
~~
162
Generation of the Approximate Powers
Estimation of the Parameters for the
Inverse Quadratic Power Surfaces . . . . . .
167
172
.
viii
LIST OF TABLES
Page
2.1
Anti-inflammatory effect data (decreased "paw volume")
18
2.2
Comparison of exact and "nonparametric" Monte Carlo null
distribution probabilities
3L
Combinations of k, n, distributions and experiment-wise
error rates for which the power was investigated and the
number of observation orderings generated for each
33
Comparison of exact and Monte Carlo (non-parametric and
parametric) estimates of experiment-wise error rates
for k = 3 treatments and n = 4 observations per
treatment. . . . .
.
. . . .
36
Absolute deviations between the "all pairs" estimated
inverse quadratic power surface and the Monte Carlo
generated powers for selected k, n, test type and
error rates
... .
.. .
39
2.6
Comparison of power for different sample sizes
51
2.7
Power as a function of S2
52
2.8
Comparison of power for different distributions.
53
2.9
Comparison of power for different populations numbers
54
3.1
Pain threshold data (millivolts)
61
3.2
Comparison of exact and "nonparametric" Monte Carlo
null distribution probabilities.
65
Combinations of k, n, distributions and experiment-wise
error rates for which the power was investigated and the
number of observation orderings generated for each
67
Comparison of exact and Monte Carlo (non-parametric and
parametric) estimates of experiment-wise error rates for
k = 3 treatments and n = 4 observations per treatment.
68
Absolute deviations between the "treatments versus control"
estimated inverse quadratic power surface and the
Monte Carlo generated powers for selected k, n, test type
and error rates . . . . . . . . .
.
69
2.3
2.4
2.5
a
3.3
3.4
3.5
5.1
..
..
. ...
The null distribution of m.. , when comparing all pairs of
~~
k = 3 treatments with n units per treatment: one-tailed
critical region with an experiment-wise error rate
75
LIST OF TABLES (continued)
Page
5.2
The null distribution of m.. , when comparing all pairs of
11
k = 4 treatments with n units pet treatment: one_tailed
critical region with an experiment-wise error rate
5.3
76
The null distribution of m.. , when comparing all pairs of
11
k = 5 treatments with n units per treatment: one-tailed
critical region with an experiment-wise error rate
5.4
5.5
~.6
5".7
5.8
The null distribution of m.. , when comparing all pairs of
11
k = 3 t~eatments with n units per treatment: two-tailed
critical region with an experiment-wise error rate
79
The null distribution of m , when comparing all pairs of
ii
k = 4 treatments with n units per treatment: two_tailed
critical region with an experiment-wise error rate
80
The null distribution of m , when comparing all pairs of
ii
k = 5 treatments with n units per treatment: two_tailed
critical region with an experiment-wise error rate
81
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n = 6 units per
treatment when comparing all pairs of k = 3 normal
treatment populations
...
..
.
..
82
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n
10 units per
treatment when comparing all pairs of k = 3 normal
treatment populations.
...
..
. .
0
5.9
77
•
•
•
••
83
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n = 14 units per
treatment when comparing all pairs of k = 3 normal
treatment populations . . . . . . . . .
....
84
5.10 Regression coefficient estimates for the inverse quadratic
power response surface in the case of n = 6 units per
treatment when comparing all pairs of k = 4 normal
treatment populations . . • . . . . .
. . . .
85
5.11 Regression coefficient estimates for the inverse quadratic
power response surface in the case of n = 10 units per
treatment when comparing all pairs of k = 4 normal
treatment populations
...••.•.
..
86
x
LIST OF TABLES (continued)
Page
5.12
5.13
5.14
.
5.15
5.16
5.17
5.18
5.19
5.20
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n = 6 units per
treatment when comparing all pairs of k = 5 normal
treatment populations . . . .
87
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n = 10 units per
treatment when comparing all pairs of k = 5 normal
treatment populations . . . .
89
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n
6 units per
treatment when comparing all pairs of k = 3 normal
treatment populations . . . .
91
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n = 6 units per
treatment when comparing all pairs of k = 3 normal
treatment populations
92
coefficient estimates for the inverse quadratic
power response surface in the case of n = 6 units per
treatment when comparing all pairs of k = 3 exponential
treatment populations . . . .
93
Regression coefficient esti.mates for the inverse quadratic
power response surface in the case of n = 6 units per
treatment when comparing all pairs of k = 3 uniform
treatment populations . . . . .
94
Approximate stochastic differences among k = 3 normal
populations in the case of n = 6 units per treatment
when comparing all pairs of treatments at given power
levels . . . . . .
. . . .
95
Approximate stochastic differences among k = 3 normal
populations in the case of n = 10 units per treatment
when comparing all pairs of treatments at given power
levels . . . .
. . . . . .
. . . .
96
Approximate stochastic differences among k = 3 normal
populations in the case of n = 14 units per treatment
when comparing all pairs of treatments at given power
levels . . . .
. . . . . . . . . . . .
97
=
Regr~ssion
•
xi
LIST OF TABLES (continued)
Page
5.21
5.22
5.23
.
5.24
5.25
5.26
5.27
5.28
5.29
5.30
Approximate stochastic differences among k = 4 normal
populations in the case of n = 6 units per treatment
when comparing all pairs of treatments at given power
levels
...
. ....
98
Approximate stochastic differences among k
4 normal
populations in the case of n = 10 units per treatment
when comparing all pairs of treatments at given power
levels
..
. ..
.
. . . .
101
Approximate stochastic differences among k
5 normal
populations in the case of n = 6 units per treatment
when comparing all pairs of treatments at given power
levels . . . .
. . . . . .
.
..
104
Approximate stochastic differences among k = 5 normal
populations in the case of n = 10 units per treatment
when comparing all pairs of treatments at given power
levels
...
. . . . . ..
..
.
no
The null distribution of m when comparing two treatments
Oi
versus a control with n units per treatment: one_tailed
critical region with an experiment_wise error rate
.
117
The null distribution of m when comparing three treatments
Oi
versus a control with n units per treatment: one-tailed
critical region with an experiment_wise error rate..
.
118
The null distribution of m when comparing four treatments
Oi
versus a control with n units per treatment: one_tailed
critical region with an experiment-wise error rate.
.
119
The null distribution of m when comparing two treatments
Oi
versus a control with n units per treatment: two-tailed
..
critical region with an experiment-wise error rate
121
The null distribution of m when comparing three treatments
Oi
versus a control with n units per treatment: two_tailed
critical" region with an experiment-wise" error rate .
.
122
The null distribution of m when comparing four treatments
Oi
versus a control with n units per treatment: two_tailed
critical region with an experiment-wise error rate. . .
123
..
xii
LIST OF TABLES (continued)
Page
5.31
5.32
5.33
5.34
.
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n ~ 6 units per
treatment when comparing two normal treatment populations
versus a control . . . . . . .
. . . . . . . . ..
124
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n ~ 10 units per
treatment when comparing two normal treatment populations
versus a control . . . . • . .
. . . . . . . . ..
125
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n ~ 14 units per
treatment when comparing two normal treatment populations
versus a control . . . . . . .
. . . . . . . . ..
126
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n
6 units per
treatment when comparing three normal treatment populations versus a control
..... ...
127
Regression coefficient estimates for the inverse ql,1adratic
power response surface in the case of n
10 units per
treatment when comparing three normal treatment popula_
tions versus a control
.
128
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n ~ 6 units per
treatment when comparing four normal treatment populations
versus a control . . . . . . .
. . . . . . . ..
129
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n~ 10 units per
treatment when comparing four normal treatment populations
versus a control . . . . . . .
. . . . . . . ..
131
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n ~ 6 units per
treatment when comparing two normal treatment populations
versus a control . . . . . . .
. . . . . . . ..
133
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n = 6 units per
treatment when comparing two normal treatment populations
versus a control . . . . . . . . . . . . . . . . . . . . .
134
~
....
e
5.35
~
....
5.36
5.37
5.38
5.39
.
.
...
xiii
LIST OF TABLES (continued)
Page
5.40
5.41
5.42
..
5.43
5.44
5.45
5.46
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n = 6 units per
treatment when comparing two exponential treatment
populations versus a control .
. • . • . • . .
135
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n = 6 units per
treatment when comparing two uniform treatment populations versus a control . . . . •
136
Approximate stochastic differences among k
3 normal
populations in the case of n = 6 units per treatment
when comparing two treatments versus a control at given
power levels .
. . • . . .
137
Approximate stochastic differences among k = 3 normal
populations in the case of n = 10 units per treatment
when comparing two treatments versus a control at given
power levels .
. . . . . .
138
Approximate stochastic differences among k = 3 normal
populations in the case of n = 14 units per treatment
when comparing two treatments versus a control at given
power levels .
. . . . . .
. . . .
139
Approximate stochastic differences among k = 4 normal
populations in the case of n = 6 units per treatment
when comparing three treatments versus a control at
given power levels • . . . . . .
5.48
e
to
140
Approximate stochastic differences among k = 4 normal
populations in the case of n = 10 units per treatment
when comparing three treatments versus a control at
given power levels . . . . . .
143
Approximate stochastic differences among k = 5 normal
populations in the case of n = 6 units per treatment
when comparing four treatments versus a control at
given power levels . . , • . .
• . . .
146
Approximate stochastic differences among k = 5 normal
populations in the case of n = 10 units per treatment
when comparing four treatments versus a control at
given power levels . . . . . . . . . . • • . . . . • .
152
0
5.47
o
•
•
•
•
xiv
LIST OF FIGURES
Page
2.1
Power versus stochastic difference between populations
i=l and i'=3: one-tailed all pairs median test
(S2 = 0.500)
2.2
..•••...•..••.•.....
41
Power versus stochastic difference between populations
i=l and i'=3: one-tailed all pairs median test
(S2 = O. 700)
2.4
40
Power versus stochastic difference between populations
i=l and i'=3: one-tailed all pairs median test
(S2 =0.600)
2.3
.......••..•.•.•..••
...••...
0
•
•
•
•
•
•
0
•
•
•
•
42
Power versus stochastic difference between populations
i=l and i'=3: one-tailed all pairs median test
(S 2 = O. 800)
.....•....•..•
0
•
•
•
•
•
43
•
1.
INTRODUCTION
1.1 General
:1:'J.'il.-,If;iiJ
n
Similar to many other aX'llNi$ of lnllthern.mtical and appHed
tics) the &(imesis of ne:!.the:r nonparamett'ic nor
statis~;lcal
theory
andte~hniques
.$ imul taneOU$
statis~
inforence
Typically, also,
ia i.dentifiable.
both of these areas have experienced a pattern of somewhat m¢teoric
growth that has been follQwed by auetained and widespread developmental
activity.
1. R. Savage pheas the fftrue beginning" of nonparametric
atattstics in 1936 [29J) while R. G,
Mill~r
dates the great
spu~t
of
interest in and research on simultaneous inference proc$dures as the
"late fot!:ies· t [20 J.
these two areas of
.
e
In 1959) Steel (.3Q,31]) c,Otnbin1ng facets of
introduced the first two non.
stat1st~cal inferenc~J
parame,tric sim.ultaneous
infer~pce
pr\:lcedures..
This paper) too, is concerned with thi$ combined area and con.
dders,
sp~if1c~Uy)
two s:i,.multanaous inference extensions otthe
nonpar~etric median t~sts as originally proposed by Westenberg [37J
and Mood [21J and .eubsequently extended by Brown and
~od
(31.
In the t'etl\aincler of the Introduction) the l'equisite nonparartietdc
and slmu.ltaneous inference backgrounds are presented and
parQmetrie
the
stm~ltaneous
l!I.pec~f1e
tnference developments are
sp~~ific
revt~wed.
non.
Finally)
testB proposed in this p4per are def1ned.
1.2 Nonparametric Background
d . U
,_
ill .
J
• " ~
I
1.2.1 Deiinitions
While Kendall and Stuart [14J make a rather rigid distinction
'between the term~ t!nonp~rametr1c'iaXld t1dbtribution... freeJ It many othel:"
2
authors
[.:..~:
2, 33, 34] acknowledge this difference but adopt the
convention of using the terms
d~~ree ~s
license, which to a large
wholly acceptable
paper.
inter~qangeably.
de£in~tiQn
Both terms will be
a consequence of the lack of a
for either term, will be followed in this
u~ed
requ~re
procedures that do not
This semantical
to refer to those statistical inference
complete
~pecification of
the sampled
population and thus remain valid for a wide class of parent distributions.
1. 2. 2
Scope
The scope of the
extensive:
nonp~rametri~
statistical inference field is
Walsh's [34, 35, 3&] three-volume Handbook of Nonparametric
Statistics, covering development$ pri9r to 1958 and presented in part
in a highly condenseq style, consists of almost 2,000 pages.
paper is limited to tqe
considera~ion
location-type hypotheses
fo~
This
ot non-sequential testing of
two Of more
~nivariate
sampleS.
Specif-
ically, the simultanepus inference proGedures proposed in this paper
are extensions of
a nonparametric
that the medians of k (>2)
procl;ldure that tests the hypothesis
s~mpled PQPu~ations
are identic?! against
the alternative that at least one of these medians is different.
1.2.3
Assumptions
While, by
def~nition,
minimal population
a~sumption~
are required
for dis.tribution..,f:ree procec)u17es, it is assumed for many of these tes,ts
that all sampled variates
~re continuous~y
therefore, the samples will yield
•
distributed.
nO tied observations.
Theoretically,
The validity
of this continuity assumption, however, does not preclude ties that may
3
result from the inherently
discre~e
tively imprecise) values.
While various methods for the "handling of
nature of all measured
(~.!.
rela-
ties" have been proposed [e.g.
........ 2, pp. 48..54], the method adopted
depends largely on the type of nonparametric procedure being used.
For
the median tests, only those ties that affect the combined sample
median are critical, and such
tie~
can most readily be resolved by
randomly assigning to equal observations the ranks they would have had
if not tied
1.2.4
..
(~.!
if
differ~n~
very slightly).
Construction Methods for Wonparametric Tests
1.2.4.1
Tests Based pn the Binomial Distribution.
Perhaps the
f
simplest of the
nonparamet~ic
te$ts are the ones that are based on the
binomial distribution and are generally known as sign tests.
The
. "median difference" version of the sign 'test appears to have been the
first nonpararnetric test ever used [1] and was discussed extensively by
Dixon and Mood [4] in 1946.
This test is based on the signs of the
differences between paired values and is a test of the null hypothesis
that the median of a continuously distributed population of differencescores is zero.
Despite its limitation to paired values and its rela-
tive inefficiency, the sign test has strong intuitive appeal, is simple,
and has been widely
tabu1at~d (~.!. r~quires
only tables of the
binomial distribution); henc;e, it Qas been widely used.
1..2.4.2
Tes ts Based on the Method of Randomization.
randomization, or
pe~utat~pnJ
was
an approach to the constructipn qf
statistical hypotheses.
o~iginal1y
t~sts
The method of
proposed by Fisher [9] as
of certain nonparametric
Fisher's method is predicated on the fact that
Co
e
4
any ordering of the sample values from a continuous distribution function has the same probability.
For example, in the case when the null
hypothesis, assumptions, and test conditions together imply that all
of several samples have been drawn from identical populations, these
populations may be considered as a single common population yielding a
pooled sample whose "component samples" are merely arbitrary labels for
groupings of observations.
Each of the different possible random
assignments (say, R) was, prior to sampling, equally likely to be the
obtained sample if the null hypothesis of identical populations is
true, but unequally likely to be the obtained sample if the null
hypothesis is false.
By choosing a test statistic which is sensitive
to the alternative hypothesis and calculating its value for each of the
R different possible random assignments, one obtains a set of R equally
likely values of the
te~t
statistic under the null hypothesis.
The
test's rejection region is simply the r most extreme of these values,
each of which is exactly
as
likely as any other value when the null
hypothesis is true, but which become unequally probable when the
alternative hypothesis is true.
If the test statistic for the actually
obtained sample falls within the rejection region, the null hypothesis
is rejected at the r/R level of significance.
Fisher proposed specific tlio-sample permutation tests for both the
matched and independent sample case.
more generally [25] and
extend~d
Pitman investigated these tests
them to the multi-sample problem for
both matched and independent samples [26].
These tests by Fisher and Pitman are examples of the method of
randomization applied to the original observations and as such are
#I
5
•
referred to as "component" or "observation" permutation tests.
They
are highly efficient but most impractical because the sample space for
the test statistic varies from one application of the test to the next,
thus making it impossible to provide generally applicable tables for
the null distribution of the test statistic.
This problem has been
alleviated, with relatively little loss of efficiency, by simply
replacing the original observations by their ranks and thus permitting
the preparation of requisite tables.
Many of the most widely used
distribution-free tests belong to this so-called "rank_randomization"
test category.
Among the more common "rank-randomized" tests for location are the
following:
(1) Signed_Rank Test (Wilcoxon)' [38J _ a test for the two-sample
matched pair problem.
(2) Rank-Sum Test (Wilcoxon-Mann-Whitney) [18, 38] -·a test for
the two-sample, independent sample problem.
(3) Friedman's Test [10] _ a rank-sum type multi_sample(k > 2)
test which is appropriate for the randomized
complete-block
.
.
design.
(4) Kruskal-Wallis Test [15] _ a rank-sum type multi-sample
(k > 2) test which is appropriate for the completely
randomized design.
Mention should be made of the so-called "normal· scores tests" in
•
which a second transformation is introduced.
This transformation re-
places the ranks by the expected values of the corresponding "normal
order statistics" before applying the method of randomization.
Other
6
transforma'tions that have been used to "replace the ranks before
utilizing the method of randomization include "inverse normal scores"
and "'random normal scores."
Several versions of these tests are
appropriate for the two_sample location problem} and rather extensive
investigations
properties.
1.2.4.3
[~.!.
12) 19] indicate that they have many desirable
Their use) however} has not been widespread.
Tests Based on the Hypergeometric Distributions.
Fisher's exact method for analyzing 2 x 2 contingency tables with fixed
marginal totals is} in fact, a special case of tests that are based on
his method of randomization and was proposed by him as a test of
independence between two populations whose characteristic of interest
is expressed in a dichotomized form.
As is well known} this analysis
is based on the hypergeometric distribution.
In the nonparametric Westenberg-Mood Median 'Test [21, 37] for
the independent two-sample location problem the data are cast into the
form of a 2 x 2 contingency table with all marginal totals fixed.
This
structuring of the data is accomplished by reducing the data to the
number of observations in each sample that are above or below the
median of the combined sample.
The probability associated with a
particular sample configuration can be calculated from the hypergeometric distribution; it can be determined from specialized tables
[~.!.
8] or from Lieberman and Owen's extensive hypergeometric tables
[17]; or, if the "expected" cell frequencies are large enough} it can
be approximated by the classical chi-square test for 2 x 2 contingency
tables
0
7
An extension of this two-sample median test for the independent
k-sample location problem was proposed by Brown and Mood [3].
~asey
a 2
In this
k contingency table is formed on the basis of the number
H
of observations in each sample that &re above or below the pooled sample
grand median.
Under the null hypothesis of equal medians for the k-
populations y this 2 x k table has a multivariate hypergeometric distribution.
While theoretically the significance level for a given table
can be obtained by summing the probabilities for all tables as extreme
or more
80
(~.!.
with the same row and column totals), it 1s usually
difficulty or impossible, to define appropriately the more extreme
configurations.
In any casey these calculations become increasingly
laborious as ttte number of sampled popul.ations end/or the individual
sample sizes increase.
The usual chi_square test for 2 x k contingency
tables, however, prOVides a fairly good approximation of the 81g01f1cemce level.
1.3
Simultaneous Inference Background
It seems certain that few experimenters today would deny their
r~peated
involvement in experiments that necessitate that they make
elli1lultanoous statistical inferences about k
means.
0r
Hance, a statistieal
proeed~re
(;> 2)
specific treattnent
that only permits one to reject
not reject the overall hypothesiS is not adequate.
To fill thiS
need, a plethora of so_called multiple comparisons or simultaneous
infer$nce techniques have been proposed, but the selection of the "most
....
appropFiate" one in any g1vensituation involves philosophical and
pra~tic
.
as well as theoretical considerations .
8
It is reasonable to consider that there are three principal types
of situations in which an experimenter may desire to make simultaneous
statistical inference
statements~
(2)
!
!
priori
(3)
!
posteriori multiple comparisons (frequently non-orthogonal).
(1)
priori orthogonal multiple comparisons;
non_ortho~onal multiple
Implicit in the definition of both of the
comparisons;
.~
priori statement types is
that the experimenter has one or more specific hypotheses that he
wishes to test before he conducts his experiment.
"simultaneous inference" procedures
j
The term
however, commonly is applied only
to those procedures that are appropriate for making the a posteriori,
or "data snooping," types of multiple comparisons.
No specific con-
sideration is given in this thesis j therefore j to the first two situations.
Undoubtedly the most involved and
fundament~l
factor to be con-
sidered in the selection of a multiple comparisons procedure is the
choice of an appropriately defined family error rate j which is the
simultaneous inference analogue of the Type I error.
The family error rate [saYj Er(6)] is defined j as
Er(6)
=
follows~
N (6)
w
N (6)
where
N (6)
= number of statements in the family
Nw (6) = number of incorrect statements in the family •
•
9
The choice of the appropriate famiLy of statements (e.g.
- - an individual
."
.oJ
comparison, a hypothesis, an experiment, etc.) is largely a subjective
matter for the experimenter.
The family error rate is a random variable whose distribution
depends upon the procedure utilized in selecting
~he
family of state_
ments and the underlying probability structure.
Two of the criteria
that can be used to characterize this distribution are "the probability
of a nonzero family error rate" and the "expected family error rate."
A number of simultaneous inference procedures, including the ones
proposed in this thesis, have been designed to control the so-called
experiment_wise error rate, which is defined as the probability of
declaring falsely at least one statement in the family of statements
that constitute all the statements made about a particular experiment.
Depending on the purpose and nature of the experiment, the
a posteriori simultaneous statements that are frequently of interest to
the experimenter concern a comparison of all possible pairs of treatment means
(~.
"all pairs,") or a comparison of each of two or more
"new" treatment means with a single control (or standard) treatment.·
mean (viz. "treatments versus control").
The development of nonparametric simultaneous inference procedures
has been predicated on two general approaches.
In one approach the k
samples are combined.pairwise and the maximum value of the usual nonparametric test statistic for two samples is utilized.
This develop-
mental approach leads to the so-called "Maximum Two-Sample Statistic"
type of nonparametric simultaneous inference procedure.
In the second
approach, all of the observations from the k samples are considered
10
simultaneously in obtaining the appropriate nonparametric multiple
comparisonss ta tis tic.
1.4
Review of Existing NOllparametric Simultaneous Inference Procedures
1. 4. 1
Simultaneous Inference Sign 'res ts
The "treatment versus control" simulta,neous inference sign test,
the first nonparametric multiple comparisons procedure, was introduced
by Steel [31] and was subsequently extended by Rhyne [27, 28J.
test uses the "Minimum Two-Sample Statistic ff approach
[!.. .=..
This
complement
of the "Maximum Two-Sample Statistic" approach] and has an experimentwise error rate.
The "all pairs" version of this test was introduced by Douglas
[5J, working with Steel, and was first discussed in 1961 by Steel
Nemenyi [23].
l
and
Subsequent theses by Nemenyi [24J and Rhyne [27J
extended the earlier results.
1. 4.2
Simultaneous Inference Component Randomization Tests
Miller [20] briefly
discus~es
an "all pairs" simultaneous
inference extension of Pitman's [25J two-sample component type permutation test and mentions the existence of a corresponding "treatments
versus control" version.
1
'.
.
Steel, R. G. D. (Decel1\ber, 1961). Multiple comparison sign
tests. Paper given at annual meeting of the American Statistical
Association, New York. Department of Experimental Statistics,
North Carolina State University at Raleigh.
11
1.4.3
Simultaneous Inference Signed-Rank Tests
Nemenyi [24] proposed multiple comparisons signed-rank tests for
both the "all pairs" and the "treatments versus control" simultaneous
inference problem.
These tests are of the "Maximum Two-Sample
Statistic" type and have an experiment-wise error rate.
Miller [20] points out that
N~enyi's
However,
statistic is not distribution-
free and suggests some approaches for making it asymptotically
distribution-free.
1.4.4
Simultaneous Inference Rank-Sum Tests
In 1959, Steel [30] introduced the first simultaneous inference
technique based on ranking the observations in the form of a "treat_
ments versus control" rank...sum test that is of the "Maximum Two-Sample
Statistic" type and has an experiment-wise error rate.
The next year,
Steel [22] and Dwass [7] proposed independently the equivalent test for
the "all pairs" problem.
Nemenyi [24] proposed exact and approximate simultaneous inference
extensions of the Kruskal-Wallis (one-way classification) and the
friedman (two-way classification) nonparametric analysis of variance
procedures and provided limited tables of the requisite statistics for
both the "all pairs" and "treatment versus control" problem.
More
extensive tables of the approximated statistics have been published by
Wilcoxon and Wilcox [39].
Dunn [6], using the Bonferroni inequality,
proposed a procedure that is similar to Nemenyi's for the one-way
classification problem.
12
1.4.5
Stmultaneous Inference Median Tests
Nemenyi [24] proposed both "treatments versus control" and "all
pairs" multiple comparisons extensions to the independent multiplesample, Brown_Mood Median Test [3].
In these extensions, Nemenyi
predicates his test statistic on the overall combined sample.
There appears to have been no proposal for simultaneous inference
extensions of median tests that utilize the "Maximum Two-Sample
Statistic" approach
(!..'=..
no simultaneous inference extension of the
Westenberg-Mood Test [21, 37]).
1.5
Definition of the Proposed Simultaneous Inference Median Tests
In Chapter 2 of this thesis; the author proposes a simultaneous
inferen~e
median test procedure, one- and two-tailed, that is
appropriate for testing "all pairs" of k treatments for data from a
completely random design.
The "Maximum Two-Sample Statistic" approach
is used and, as such, this test can be considered to be an extension of
the
two~sample
Westenberg_Mood Median Test [21, 37].
Tables of the
distribution of the test statistic are provided for a limited number of
treatments
(~.
k ; 3, 4, 5) and sample sizes
[~.
n
= 3(1)15J.
The small sample power of this procedure for various parametric alter.
natives; as determined by Monte Carlo techniques, is also tabulated for
selected values of k and n.
Chapter 3 presents the corresponding "treatments versus control"
.
simultaneous inference version; provides tables of the distribution of
the test statistic, and reports on the small sample power findings for'
various alternatives.
13
A SIMULTANEOUS INFERENCE MEDIAN TEST FOR COMPARING
ALL PAIRS OF TREATMENTS
20
The test proposed in this chapter is a simultaneous inference
analogue of the nonparametric k-sample Brown-Mooq Median Test [3J.
However" since it is based on the "Maximum Two-Sample Statistic"
approach" it is actually an extension of the two-sample median test of
Westenberg [37J and Mood [21].
This proposed test has an experiment-
wise error rate and is applicable for simultaneously testing the
significance of the individual differences among all possible pairs of
k treatments for data from a completely random design.
2.1
Each observation" X.. (i
~J
= I"
Data
00"
k > 2; j
= 1,
•.. " n)" repre-
sents a univariate measurement" of ordinal or interval level, on the
j-th experimental unit in the i-th treatment group.
either a continuous or discrete variable.
This value may be
While there are no inherent
limitations on the number of observations in each treatment group nor
on the number of treatment groups" only the equal sample size case
(n l
=
(k
= 3"
'0'
= nk = n
~
15) for three" four and five treatment groups
4" 5) has been considered herein.
2.2
Description
These tests provide for making simultaneous statistical inferences
k
about the (2)
= k(k-l)/2
pairs of differences among the k treatment
groups with an experiment-wise error rate.
A test may be either one-
tailed or two_tailed" but both one-tailed and two-tailed alternatives
cannot. be considered at the same time
,"
pairs).
(~.!.
for different treatment
14
Interest is focused on the detection of differences in location as
characterized by the medians of the treatment populationso
The null
hypothes1s~
is to be tested against the one-tailed alternative
hypothesis~
where the inequality must hold for at least one of the treatment pairs
and $\(X t ) is the median of the i=th treatment population (1 ;:: 1,
000'
k; i < i ').
One may also test against the two_tailed alternative hypothesisg
where the inequality must hold for at least one of the treatment pairs
(i < i ')
0
203
The Xij (i ;:: 1,
000'
Assumptions
k. > 2; j ;::
1, ... , n) are assumed to be k.
independent samples, each consisting of n independent observations.
The X.. 's are assumed to have a distribution function F 4 (x).
~J
.
~
While
the underlying distribution function [say, F~ (x)] may be discrete, the
Fi(x) can effectively be considered to be continuous due to the
~act
that the Fi(x) is are assumed to include probability effects which are
.
."
in.tl:'oduced to handle any tieso
15
204
20401
(1)
Test Procedures
One-tailed Test
Using the "Tables of the Null Distribution of m ' When Comparing
U
All Pairs of k Treatments:
One-tailed Critical Region" in Section
501, select for the appropriate values of k and n, the critical
value, m*, which most closely corresponds to the desired
experiment-wise error rateo
(2)
Designate the treatment samples by the integers i = 1,
000'
k
in such a manner that the associated alternative hypothesis is
expressed in a monotonically non-increasing order
[£o~o
where the inequality must hold for at least one of the treatment
pairs (i
(3)
< il)]o
k
Form the (2) combined treatment samples, each of size 2n and
consisting of the n variate values from the i=th treatment sample
and the n variate values from the i'=th treatment sample (i
= 1,
000' k; i < i')o
(4)
Form the
[lX(~ ]~_statistic vector: ~. = [m12m130 0° m1km23 00om2k
ooomii,ooomk_l,k] where each vector element, mii "
is a count of
the number of the i=th treatment variate values that are greater
than the median of the combined i-th and i'=th treatment samp1eo
(5)
Compare each of the
(~2 elements, m..
" with the previously
11
selected critical value, m*, and make a significance statement for
each of the comparisons, depending on whether the corresponding
16
m..
is equal to or greater than m* or not.
i
1.1.
That is y reject the
(~) comparisons
null hypothesis of equal medians for each of the
for which the corresponding m.
'f'
1.1.
(~.
m
ii
, ~ m*); otherwise, do not.
Two_tailed Test
2.4.2
( 1)
=..
is equal to or greater than m*
Using.the "Tables of the Null Distribution of m.. , When Comparing
1.1.
All Pairs of Treatments:
Two-tailed Critical Region" i.n Section
5. 2 9 s e lec t for the appropria te val ues of k and n.? the upper
critical value, m*' and its associated lower critical value,
u
*
~y
which most closely corresponds to the desired experiment-wise
error rate.
(2)
Arbitrarily designate the treatment samples by the integers
i
(3)
= l,
... , k.
Form the (~) combined treatment samples) each of size 2n and
consisting of the n variate values from the i-th treatment sample
and the n variate values from the ii_th treatment samples (i
k; i
< i') .
..• m.. i .•. mk 1 k J where each vector element, m..
1.1.
= 1,
-
,
1.1.
I'
is a count of
the number of the i-th treatment variate values that are greater
than the median of the combined i-tp and i'-th treatment sample.
(5)
Compare each of the (k') elements, m.. I' with the previously
1.1.
2
selected upper critical value, m*, and its associated lower
u
critical value, ~, and make a significance statement for each of
the comparisons, depending on whether the correspondingm .. i is
1.1.
17
equal to or greater than m:, equal to or less than~, or not.
That is, reject the null hypothesis of equal medians for each of
k
the (2 ) comparisons for which the cQrresponding m..
11
equal to or greater than the upper critical value
I
(~.!.
or is equal to or less than the lower critical value
m ,
ii
2.4.3
*
~ ~);
is either
m ,
ii
~m~)
(~.!.
otherwise, do not.
An Example
In an experiment to test the relative effectiveness of three new
chemical compounds as anti-inflammatory agents, 30 rats were treated
with standardized inoculations of yeast cells to create an inflammatory
process in the inoculated paw.
After a 24 hour incubation period, the
rats were randomly divided into three treatment groups of ten rats
each.
Each of these groups was then dosed with one of the three new
compounds.
The anti-inflammatory effect was measured in terms of a
decreased "paw volume" during the first 24 hours of treatment.
The
data for this experiment are shown in Table 2.1.
For illustrative purposes, assume initially that cost considerations dictated an interest in the one_tailed alternative hypothesis
(say) :
Median (A-15)
~
Median (A-IS)
~
Median (A-27)
[at least one directed inequality holds]
Using the one-tailed procedure outlined in Section 2.4.1, we have the
.
.
following:
a
The data have been rearranged within each treatment group to
facilitate inspection.
1.
From Table 5.• 1, for k
=3
treatments and n
per treatment, a critical value, m*
= 7,
= 10
experimental units
was selected.
This
critical value corresponds to an experiment-wise error rate of
0.212, which was considered to be acceptable at the early stage
of investigation.
2.
Consonant with the alternative hypothesis, the treatments were
designated as follows:
.
19
e
Treatment No. (i, i')
Chemical Compound
1
2
3
3.
The (3)
2
pairwise samp 1es were formed as follows:
COMBINED SAMPLE
COMBINED SAMPLE
COMBINED SAMPLE
1 and 2
1 and 3
2 and 3
(A-15 and A-18)
(A-15 and A-27)
(A-18 and A-27)
Treat. No.
Data
Treat. No.
Data
Treat. No.
Data
1
1
1
1
1
2
2
1
1
2
3.19
3.15
3.08
2.88
2.81
2.79
2.75
2.75
2.68
2.68
*
*
*
*
*
*
*
*
*
1
1
1
1
1
1
1
1
3
1
3.19
3.15
3.08
2.88
2.81
2.75
2.68
2.60
2.49
2.47
*
*
*
2
2
2
3
3
2
3
2
3
2
2.79
2.75
2.68
2.49
2.45
2.38
2.37
2.36
2.28
2.25
*
*
*
*
*
It
=3
A-15
A-18
A-27
*
*
Median----------1
1
2
2
1
2
2
2
2
2
(Note:
4.
.
*
*
*
-----------Median
2.60
2.47
2.38
2.36
2.31
2.25
2.21
2.20
2.07
1. 91
3
3
1
3
3
3
3
3
3
3
2.45
2.37
2.31
2.28
2.20
2.16
2.01
1. 89
1. 85
1.84
2
2
3
3
2
3
2
3
3
3
2.21
2.20
2.20
2.16
2.07
2.01
1. 91
1. 89
1. 85
1.84
It is only necessary to tabulate observations that are
larger than the median of each combined sample. All
observations are shown above for illustrative purposes.)
The m - statistic vector was determined to be:
"
•
20
where m = 7 is the count of the number of the first treatment
12
(£.!.
A-1S) variate values (marked with an asterisk, above) that
are greater than the median of the combined first and second
(£.~.
A_1S) treatment sample; m = 9 and m = 6 are the corresponding
13
23
counts for the first and third
(£.~.
A_27) and the second and third
combined samples, respectively.
S.
These m _ statistic vector elements were then compared individually
with ,the selected critical value (viz. m* = 7) and the following
significance statements were made:
a.
Since m = m*
12
(£.~.
7 = 7), the
"0:
(A_1S) was rejected; hence, the H :
l
Median (A.1S) ='Median
Median (A.1S) > Median
(A_1S) was accepted.
•
b•
Since m > m*
13
(£. ~.
9 > 7), the
"0:
(A_27) was rejected; hence, the H :
l
Median (A.1S) = Median
Median (A-1S) > Median
(A. 27) was accepted.
c.
Since m < m*
23
(£,~,
6 < 7), the
%:
Median (A_1S) = Median
(A.27) was not rejected,
These three significance statements are made jointly with an error
rate (one. tailed) of 0.212.
Assume now, for purposes of illustration, the two-tailed alternative
hypothesis (say):
Median (A_1S)
FMedian
(A.1S)
FMedian
(A_27)
[at least one inequality holds]
•
Using the two_tailed procedure outlined in Section 2.4.2, we have the
following:
21
1.
From Table 5.4) for k
=3
treatments and n
= 10
experimental units
per treatment) the following associated critical values were
selected:
m*
u
= n-2 = 10-2 = 8
*L = 2
m
These critical values correspond to an experiment-wise error rate
of 0.056.
2.
For brevity the same treatment designations were used as for the
one-tailed example) but this designation can be considered to be
arbitrary instead of ordered.
=3
3.
The (3)
4.
The m-statistic vector was determined to be:
5.
On the basis of comparison of the individual vector elements) miil )
2
combined treatment samples were formed.
with the selected two_tailed critical values (viz. m*u
= 8') retL = 2) )
the following significance statements were made:
a.
b.
c.
He:
He:
Median (A_l5)
= Median (A-18) ) not rej ected.
Median (A-15)
= Median
(A_27) )
rejected; hence) the
Hl :
Median (A_l5)
FMedian
(A-27) )
accepted.
H :
O
Median (A-l8)
= Median (A-2 7) ) not rejected.
These three significance statements are made jointly with an error
",
rate (two_tailed) of 0.056.
..
22
2.5
2~5.l
Test Derivation
Rationale
In the sense that the test procedure ol,ltlined in SE;!ction 2.4 makes
use only of information about the magnitude of the observations with
respect to the pairwise
r;;~multaneous
combi,ned treatment sample medians in making
inferences a1;>out the significance of the individual
differences among
pairs of treatments} it is appropr:Late
al~ pos~ible
to cpnsider that; it is a IIMaximwn Two Sample
Statisti~1I
tYPEJ
s~multa-
neous inference extension of the two-sample median test of Westenberg
[37J anq Mood [21J.
pro~edure
differs
From tQe
stan~point
substantial~y
of derivation} however} this
from the usual two.sample median test.
Whereas the test of significance in the
Westenberg~Mood median
test is based on the hYPergeometric distribution} no explicit use of
this distribution is made in the proposed multiple comparisons proce.
dure.
For simplification} each element} m..
~~
I}
-
of the m.statistics
vector ir;; merely a count of the number of the i.th treatment
~ians
that are larger than the median of the combined
sample (i
= l)
... } kj i < i
l
).
}
and
i'~th
Under the test conditions of fixed
marginal totals} there is} of course} a
between each element} m.. i
u.
i~th
ob~erva.
one~to~one
correspondence
and the hypergeometrically derived con-
figurational probability of the associated 2 x 2 contingency
tab~e
as
used in the two.sample median test.
To illustrate the basic rationale of both the Westenbers.Mood
median test and this proposed multiple comparisons procedure as well as
the relationship between them} consider the following two.sample
23
problem with n
ordered
<'!:.=.,
=4
observations per treatment,
Consider that the
decreadng) observations have been replaced by their
treatment designations <.!:!-. i
= 1"
2):
For the Westenberg-Mood median test the appropriate 2 x 2
contingency table is formed and Fisher's exact test" which is based
on the hypergeometric distribution, is used to test the null hypothesis, as follows:
Treatment
Totals
#1
#2
> Median
3
1
4
< Median
1
3
4
4
4
8
Totals
+
=
16
Jrf
+
1
irf
=
17
70
=
0.243
one_tailed
(alternative)
If we now c,onsider that these observations constitute a part of
a larger experiment (~o!-. k > 2), then in accordance with the procedure
outlined in Section 2.4, the first element of the
[lX(~)J ~-statistic
vector is obtained by counting the number of the first treatment
variate values that are greater than the median of the combined first
and second treatment sample
(~''=''
m12
= 3).
24
' (~.!. m 1=:0, n) are
U
It is obvious that extreme values of m
U
asapc!ated with statistically significant differences between the i-th
and i'-th treatments.
For simultaneous inference purposes, however,
the appropriate test statistic is, of course, the [lX
(~)]~-statistic
vector, as characterized by its maximal and/or minimal element(s)
(1. e. depending on the alternative hypotheses).
.... -
Hence, it is necessary
to derive its null distribution.
The derivation of the null distribution is predicated on a
randpmization approach.
Under the assumptions and null hypothesis, any
ordering of the kn observations
(~.~.
where there are n observations on
each of k treatment groups) has the same probability,
(n!) k
(kn)
~
Corresponding to each of these possible orderings, there exists an
m_statistic vector.
Its null distribution is obtained by calculating
its value, as characterized by its maximal and/or minimal element(s),
m "
ii
for each of these observation orderings.
For concreteness, consider the derivation of the null distribution
of the
m~statistic
of k
3 treatments:
~
(1)
vector for the case of n
3 observations on each
Under the asaumptions and null hypothesis,
(kn) !
(n!) k
=
--
I
=
are
=3
x 3
=9
observations.
can assume (n + 1) values
Since each vector element, m "
ii
(1. e. m..
th~re
1680
equally likely orderings of the kn
(2)
=:
0 1 1, ... n) and each vector is comprised of
l.1
(k)
(~) elements, there are (n + 1) 2
"possible" m-statistic vectors.
(3)
= (3
+ 1) 2
= 43 = 64
In this example only 42 of
25
these m_statistic vectors are "feasible" (.=.o!,o there is no ordering of the observations that corresponds to m
..
[0, 1, OJ)
(3)
= [m12m13m23] =
0
For purposes of derivation, it is convenient to further classify
these 42 "feasible" m-statistic vectors into 14 "Vector Types,"
on the basis of the element values only
(~o'='o
independent of
position of each element in the vector), and to tabulate them
accordingly:
Vector
Type
Number
Column Integers
3
2
1
Total Number of Observation
Orderings Corresponding to
this Vector Type
o
+
1
1
+
+
3
216
+
63
216
4
+
5
+
6
+
+
7
+
+
+
36
864
+
72
1
8
+
9
+
10
+
+
11
+
+
12
+
+
13
+
+
14
+
+
+
4
36
18
+
63
18
+
+
72
Total
1680
26
A
"+" under a particular "Column Integer" (£..:.: 0, 1, 2, 3)
implies that at least one element of that vector has the value of
that integer.
A "_" implies that none of the elements of that
vector has the value of that "Column Integer."
For example, the
36 vectors of "Vector Type Number 5" consists of elements, mii! ,
that have values of 0 or 2 only, specifically, as follows:
m-statistic Vectors
(Vector Type Number 5)
Number of Observation Orderings
Corresponding to This Vector Type
[0
0
2]
9
[0
2
OJ
0
[0
2
2J
9
[2
0
0]
9
[2
0
2]
0
[2
2
OJ
9
Total
36
(4) The one-tailed null distribution of the m-statistic vector can then
be derived as follows:
..
27
e
m_s ta tis tic Vector
Elements
.
At least one m.. ,=3
u.
At least one m.. ,=2}
~~
but no m
ii
Vector
Types
Number of
Observation
Orderings
Cumulative
Number of
Observation
Orderings
Probability
8} 9}
10} ll} 12
l3} 14
212
212
0.126
4} 5}
6} 7
1188
1400
0.833
2} 3
279
1669
0.999
1
1680
1.000
Cumulative
One",tai.l~9
, =3
At least one m.. I =1}
~~
but no m.. 1=2 or 3
~~
...
At least one m..
~~
I=O}
but no m.. ,=1 or 2
~~
.
e
1
or 3
Total observation orderings:
1680
(5) The two_tailed null distribution of the m_statistic vector can then
be derived as follows:
Cumulative
Number of
Observation
Orderings
m_statistic Vector
Elements
Vector
Types
Number of
Observation
Orderings
At least one m.. 1=0
1}3}5}7}
8}9} la}
1l}12}13}
14
384
384
0.229
1296
1680
1.000
~~
or 3
Cumu1a.tive
Two-tailed
Probability
At least one m.. ,=1
~~
,.
e
or 2, but no
m.. i =0 or 3
~~
2,4}6
Total observation orderings:
1680
28
e.
. 'Tha exact .and approximate' one- and two-tailed null distributions of
.. t;he .m-statistic}
a~cha;aete:rlzed'
by itsmaximaland!orminimal
....
.
.".
....
.
-
."
.
el~m61nt.(s).;
.m
H
'
.
,}. ate.disqussed. in Sections 2.6.2 and 2.6.3;:
<respec'tively} and' ther~'Sulting tables are presented in Sections
S.land 5.2 for k :: 3} 4.15 treatments with, n=.3(1)15 observations
for each treatmeqt.
2.5.2
Exact Null Distributions
i
tJ€
The direct
th~
-
p~ssible
n~ber
treatment
.-
e.
to the qerivation of the null di$triput;ion of
m.statistic vector is obviovs1y predicated on the total enwnefation
of all
the
appro~ch
orderings of
of treacments
(!..!..
~he op~eTVation$
(!.!.
for each combination of
k) and the number of
I'll;' :j.. '" l, "') k).
ob$e~ations
per
:r:t is equally obvious that total
enwnell'ation is not practicable except for very small values of k and
·I'lt'·
.Even. thenumber.of the null· di-stribution tabulations per
!.!.. is
_ca.sive unless the values of the nils are restricted; this thesis is,
thus}. li,m.ited to the cases .where there are an equal number of observa_
tiel'l.l
per treat\1lent (!: .!:l'll '"
This
tQ~al
approach was
enumeration of all pOSSible observation orderinss
empl~Y8d,.mak~ng
use of the computer program.
de~~ribed
and
listed in Sectien 5.9.1, to generate the m-statistic null distribution
for the following cases:
29
e
N~ber
n
-.
k
~
3
2
:)
3
4
2
3
4
4
5
~~
90
1,680
34,650
2,520
369,600
113,400
:3
:2
total enumeration
of Observati9n
Orderings
appro~~~
r~qui~ed
values of n because of the excessive computer time
~Quld r~quire
n
::r;
the generation of
756~
4: 63,063,000;. k ::; .5, n :;:: 3:
inp1ude
t:h~
for example, the k
observ~tion ordering~.
generate the
£o~ la~~~r
is not feasible, however,
759 observation
168,168,000.
T
to
3,
p ~5 ~ase
k
o~d~r~ngaJ
~
4,
~pprQaehes
Othel' direct
gene;t:'ation of only the "extreIIJe mU' value" vec:rf;:9r types..
use of recursive schemes ?nd/or th~ use of combinatorial ana1Y$i~.
Each of these
a~proa~hes, howev~r, ~s
an adequate number of
o~seryation
dependent on the
o;t:'perings
(~.!.
availabili~y
of
for various valu.es of
~ ~nd n) and/or associat~d .....
m-st~tistic•
The six distributions fenerated by computer
3, 4;. k :;::·4, n
= 2,
(Vi~.
k
3, n ::; 2,
~
th~Ei~
3; k :;: 5, n =Z) provideq little 8UPPQJ'lt fQr
otqe~
<i.!.
~el'Js,
to derive the following "extreme I\lU I value" vector
~hort-cut)
direct approaches,
It was
pQssib~e,
neverthe-
type~
f<?r
the CAse of k, ::; 3 treatment groups;
Vector Types
(m-statistic Vector Elements)
,
At least one mii' ::; n (onetailed)
At least one m.. ,
u,
(two-taUed)
•
= 0 0+
n
Number of Observiitlo11
Ordering
I
3(3n)
n
; I.
2(2 n)
n
6(3n) _ 2(2\)
n
n
I.;
30
2.5.3
Approximate Null Distributions
All of the remaining tables of the null distribution for the
~-statistic
vector (viz. k = 3, 4, 5; n = 2 (1) 15; Sections 5.1 and
5.2), as characterized by its maximal and/or minimal element(s), m "
ii
have been generated by a "nonparametric" Monte Carlo procedure.
The
requisite program description and listing comprise Section 5.9.2.
The number of observation orderings that were randomly generated
by the computer for various values of k and n are as follows:
n
2
3
4
5
6
•
7
e
8:
9
10
11
12
13
14
15
k
=3
Exact
Exact
Exact
51, 000
10, 000
10,000
10 ,000
10 ,000
11,000
10,000
10 ,000
5,000
10 ,000
5,000
k
=4
Exact
Exact
10,000
10,000
10, 000
10,000
10,000
10,000
10 ,000
5,000
5,000
5,000
5,000
5,000
k
=5
Exact
10 ,000
10 ,000
10 ,000
10 ,000
10 ,000
10 ,000
10 ,000
5,000
5,000
5,000
5,000
5,000
5,000
Since the distribution of the observation orderings with respect
to the "vector types" is symmetric for the one-tailed case, the upper
and lower tail results were pooled in deriving the one-tailed tabulations.
The validity of this "nonparametric" Monte Carlo procedure is
indicated in Table 2.2.
~
e
,;
~
e
'e
Table 2.2
Comparison of exact and "nonparametricl! Monte Carlo null distribution .probabilities
k=3'treatments
One-tailed
Critical Values
4
3
2
1
Exact
0.039
0.498
0.974
1,000
a
LOOO
Monte Carlo
0.037
0.494
0.973
1.000
1.000
Difference
+0.002
+0.004
+0.001
0.000
0.000
a
Exact
0.214
0,958
LOOO
1,000
Monte Carlo
0.313
0.959
1,000
1.000
n = 4 observations/treatment
Two_tailed
Critical Values
0; 4
1; 3
2'
Monte Carlo
0.072
0.759
1.000
Difference
+0.002
+0. 007
0.000
n = 3 observations/treatment
Two-tailed
Critical Values Exact Monte Carlo
Difference
+0.002
0.357
0.355
0; 3
1,000
1,000
-0 .001
1; 2
0.000
0.000
Difference
+0.002
0.000
k=4.treatments
. ~. One- tai led
Critical Values
3
2
1
:
k=' 5treatmen ts
One-tailed
}
Exact
0.074
0.766
1,000
:
:
n = 2 observations/treatment
Two-tailed
Critical Values
Exact
Monte Carlo
Difference
Critical Values
Exact
Monte Carlo
2
0,687
0.688
-0.001
0; 2
0,873
O. 870
=to ,003
1
1.000
1.000
1.000
0.000
0.000
1·J
1.000
1.000
0.000
a
LOOO
Difference
w
~
32
2.6
2.6.1
Power
Derivation
The small sample power of this proposed "all pairs!l simultaneous
inference median test was estimated, using a Monte Carlo approach, for
a limited number of normal population cases with equal variance (viz.
k
= 3,
n
= 6,
10, 14; k = 4, n
= 6,
10; k
= 5,
was also estimated for four additional cases
n
=
(a1l~
6, 10).
The power
k = 3 .' n
= 6)
in
order to investigate the effect of normal population variances, equal
and unequal, as well as different distributions (viz. uniform and expo_·
nentia1).
Table 2.3 outlines the details for these eleven combinations
of k, n and distributions.
For the purpose of this investigation, the term "power" has been
used in the sense of the probability of rejecting the null hypothesis
of equal medians for at least one of the
(~) multiple comparisons,
given that some two or more of the population medians are not equal.
To facilitate a comparison of powers for different population distribution~
differences in location among the populations have been tabulated
in terms of "stochastic differences."
The first step in this investigation of power consisted of the
random generation of 1,000 sets of kn observations (i.e. n observations
on each of k treatments) for a selected point in the k_dimensiona1
space which consists of the means of each of the k treatment popu1ations.
For each of the 1,000 sets of data, the
~-statistic
vector was
computed in accordance with the proposed multiple comparisons procedure,
and then the results for the 1,000 sets were used to generate the
33
Table 2.3
Combinations of k, n, distributions and experiment-wise
error rates for which the power was investigated and the
number of observation orderings generated for each
Experiment-wise
Error Rates
k
•
..
.
e
n
Distributions
2
Normal (cr. =1,1,1)
~
Observation
Orderings
Generated
One-tailed
Two-tailed
.003
.102
.006
.186
72,000
10
3
6
2.
3
10
Normal
(cr.=l,l~l)
.002
.030
.004
.056
21,000
3.
3
14
2
Nonna1 (cr.=l,l,l)
.
.009
.074
,018
.137
17,000
4.
4
6
2
Normal (cr. =1,1, I, 1)
.007
.179
.013
.303
26,000
5.
4
10
2
Normal (cr. =1,1,1,1)
.003
.060
.005
.108
39,000
6.
5
6
Normal (cr.2 =1,1,1,1,1)
.009
.255
.018
7.
5
10
2
Nonna1 (cr.=l,l,l,l,l)
.005
.091
.010
.161
52,000
8.
3
6
2
Normal (cr.=1,3,8)
.003
.102
.006
.186
18,000
9.
3
6
2
Normal (cr.=2,2,2)
.003
0102
.006
.186
19,000
10.
3
6
EKponentia 1
.003
.102
.006
.186
18,000
11.
3
6
Uniform
.003
.102
.006
.186
18,000
2
~
.
~
~
~
~
~
~
~
27,000
34
.
distribution of the
~-statistic
tremalelement.(s), . m..
i'
~~
vector, as
characteri~ed
by its ex-
The requisite computer program is described
and listed in Section 5.9,3,
For example, 1,000 sets of data were generated and the m_statistic
vector computed for the following case (k :;:: 3, n :;:: 6):
to
Treatment No. 1:
2
Normal (1-1 :;:: 0,0; 0"1
1
:;::
1.0)
Treatment No, 2:
Normal (I-L
Z
:;::
2
LO; 0'2
:;::
1.0)
Treatment No. 3:
Normal (1-1
3
:;::
2,0;
:=;:
1.0)
Z
CY
3
On the basis of the 1,000 sets of data, the distripution of the m_
statistic vector was determined to be as follows:
35
The power} corresponding to a specified experiment-wise error
rate} is thus the "cumulative probability" of the occurrence of an
m_-statistic vector whose extremal element(s)} m..
~1
i}
is equal to or
more extreme than the critical value m* associated with that specified
experiment-wise error rate.
For example} in the k
=3
and n
= 6 case}
the critical values corresponding to the one-tailed experiment_wise
error rates of 00003 and 0.102 are m*
unde:r the null hypothl';'.sis) and
m~'(
=5
6
=
(~o~.
(_L~,
~
at least one m..
1.1
=6
at least one m..
i
u.
under the n\ll1 hypothesis)} respectively (see Table 5.1).
6)
?
or 59
Thus J from
the results given above for this case} the approximate powers corresponding to these one-tailed experiment-wise error rates
(vi~.
0,003
and 0.102) are 0.292 and 0,854} respectively.
While no direct verification of the validity of the power estimates is possible} it is of interest to note that the "experiment_wise
error rates" generated by the "parametric" Monte Carlo "power" program
(viz. 5.9.3) for various distributions
<.~,:~.
by "setting ll the means of
the k treatment populations equal) correspond closely to the "exact"
ones obtained by enumeration
(~.
5.9.1) as well as the "approximate"
ones generated by the "nonparametric" Monte Carlo "null distribution"
program
(~.
5.9.2).
Results for the k
=3
and n
=4
case are shown
in Table 2.4.
For each of the eleven combinations of k} n J and distributional
types investigated} this power approximation procedure was repeated for
various combinations of the location parameters in such a way as to
ensure coverage of the "power_space."
.
n
= 6}
normal distribution
(rri
= I)
For example} in the k
= 3}
I} 1) case} 72 different computer
..
e
-
II
Table 204
-
,.
Comparison of exact and Monte Carlo (non-parametric and parametric) estimates of
experiment_wise error rates for k :=: 3 treatments and n :=: 4 observations per
treatment
One_tailed
Method
-
Distribution
Observation
Orderings
Generated
Two-tailed
(At least one m..
n
1.1.
:=:
_.
4
4 or 3
4 or 0
Exact (Enumeration)
None
(34,650)
0039
0498
.074
Monte Carlo (Non-Parametric)
None
10,000
0037
0494
.072
Monte Carlo (Parametric)
Normal
1~000
0040
.502
0074
Monte Carlo (Parametric)
Uniform
1 1 000
.036
0493
.068
Monte Carlo (Parametric)
Exponential
IJOOO
,036
0493
0068
~)
W
0\
37
runs were made in each of which 1,000 sets of data were generated and
the corresponding distribution of the m-statistic vector enumerated.
In these 72 computer runs, the variance of the normal distributions was
2
the same (viz.
~.
~
= I) 1, 1) for all treatments, but the differences
between the means of the three distributions varied over a range of
from 0.2 to 2.0 units.
The power data were then summarized by fitting to them a secondorder inverse po1ynomi.a1 response function [22].
This general function
is defined by
where x ,x , ..• x represents the levels of k independent variables and
l 2
k
y is the response.
re~Ronse
In contrast to ordinary second-order polynomial
surface models they are generally non-negative, bounded, and
have a second_order form which has no builLin symmetry.
properties are consisten.t
~
These
pri0E!: with those of a lipowerll response
surface.
Consonant with the spirit of distributi.on-free statistics as well
as in an attempt to make meaningful comparisons of powers for different
treatment distributions, the independent variables in the inverse
polynomial power response function are the flstochastic differences II
between the i
i < i l ).
..
=
1 treatment and the il-th treatment (i
=
I) ... , k;
For the purposes of this thesis) the stochastic difference
between (say) treatment population i
~
1 and treatment population
i' = 2 is defined as the probability that a randomly selected member
38
of treatment population ii
=2
will be larger than a randomly selected
member of treatment population i
= 1.
Thus, for example, the inverse
polynomial power response model for the k = 3 cases is as follows:
where
8
2
= stochastic
difference between population i
= land
ii - 2
S3 = stochastic difference between population i = 1 and it = 3 .
The regression coefficients in this model as well as those in the
k
=4
and k
=5
cases were estimated by means of the computer program
listed in Section 5.9.4, which is based on a weighted least-squares
criterion proposed by NeIder [22].
The estimated regression coeffi-
cients are shown in the tables in Section 5.3.
Table 2.5 shows the
maximum, mean and minimum absolute residuals for the principal cases.
The tables in Section 5.4 indicate the !istochastic differences!i
among the treatment populations that correspond to given power levels
over the range 0.50 (0.10) 0.90 and can, thus, be used in estimating
the !irequiredll sample si2:es.
in the next section.
This usage is illustrated by an example
Figures 2.1, 2.2, 2.3 and 2.4 graphically
duplicate these tables for two selected cases (viz. one-tailed; error
rate of approximately 10 percent; k
2.6.2
= 3)
n
=6
and 14).
An Example
Assume that an investigator plans to conduct an experiment to test
the pairwise
null hypothesis that three different formulations of a
39
Table 205
One-Tailed
One-Tailed
Two-Tailed
Two-Tailed
.003
.102
.006
.186
.341
.855
.343
.483
.055
.047
.053
.051
.013
.018
.013
.018
.000
.000
.000
.001
10
10
10
10
. Normal
Normal
Normal
Normal
One-Tailed
One-Tailed
Two-Tailed
Two-Tailed
.002
.030
0004
.056
.351
.839
.351
.519
.066
.073
.066
.OT?
.026
0035
0025
.033
.005
.010
.006
.007
14
14
14
14
Normal
Normal
Normal
Normal
One-Tailed
One-Tailed
Two-Tailed
Two-Tailed
.009
.074
.018
.137
.837
.917
.841
.528
0095
.071
0091
0072
.034
.026
.032
.025
.010
.000
.005
.000
6
6
6
6
Normal
Normal
Normal
Normal
One-Tailed
One-Tailed
Two_Tailed
Two-Tailed
.007
.179
.013
.303
.372
.719
.374
.707
.033
.036
.035
.060
.013
.014
.012
,023
.000
.003
.001
,003
10
One-Tailed
One-Tailed
Two-Tailed
Two,~Tailed
.003
.060
.005
.108
.565
. '787
.565
. 788
0278
.099
0279
0098
.055
.022
0056
0022
.000
.001
,000
0000
3
3
3
3
6
6
6
6
3
3
3
4
4
4
4
.
Normal
Normal
Normal
Normal
Distributions
.3
Estimat~ Absolute Deviatio n
Max Mean
Min
Power
Error
Rate
n
.3
~
Test Types
k
3
3
3
e
Absolute deviations between the "all pairs" estimated
inverse quadratic power surface and the Monte Carlo
generated powers for selected k) n, test type and error
rates
4
4
4
4
10
Normal
Normal
Normal
Normal
5
5
5
6
6
6
Normal
Normal
Normal
One-Tailed
One-Tailed
Two-Tailed
.009
.255
.018
,385
,990
.365
.069
.030
.049
.017
.009
.015
.003
.001
.000
5
5
5
5
10
10
Normal
Normal
Normal
Normal
One-Tailed
One-Tailed
Two~, Tailed
Two-Tailed
.005
.091
.010
.. 161
.989
,805
.391
.810
0127
.114
.176
.111
.046
.032
.046
.030
,001
.004
.000
.002
10
10
10
10
aAll variances equal to one (~.~. ~~ ~ 1) all i)o
b
Estimated power corresponding to maximum absolute deviation.
40
II
I
0.90
0.80
0.70
0.60
p:::
0.50
ILl
~
Pol
0.40
e
0.30
I~
II:
k=3; n= 6; Error Rate = 0.102
k=3; n=14; Error Rate = 0.074
0.20
0.10
0.500
...
Figure 2.1
0.700
0.800
o.
00
Power versus stochastic difference between populations
i=l and i'=3: one-tailed all pairs median test
(8 = 0.500)
2
41
II
I
0.80
0.70
0.60
0.50
f:j
:s:
0
p..,
0.40
e
0.30
I:
II:
k=3; n= 6; Error Rate = 0.102
k=3; n=14; Error Rate = 00074
0.20
0.10
0.600
Figure 2.2
00650
0.700
00750
0.800
0.850
0.900
Power versus stochastic difference between populations
i=l and i!=3: one-tailed all pairs median test
(8 == 0.600)
2
42
0.90
I
0.80
0.70
0.60
0.50
0.40
I:
0.30
II:
k=3; n= 6; Error Rate = 0.102
k=3; n=14; Error Rate = 0.074
0.20
0.10
0.700
Figure 2.3
0.750
0.800
0.850
0.900
Power versus stochastic dtfference between populations
i=l and i'=3: one_tailed all pairs median test
(8 = 0.700)
2
43
II
I
/
0.40
I:
0.30
II:
k=3; n= 6; Error Rate = 0.102
k=3; n=14; Error Rate = 0.074
0.20
0.10
0.800
Figure 2.4
0.825
0.850
0.875
0.900
Power versus stochastic difference between populations
i=l and i ' =3: one-tailed all pairs median test
(8 =0.800)
2
44
"
sustained release antihistamine (say:
A J B and C) are effective (as
measured by a standarized intradermal histamine challenge procedure)
for an equal length of time after oral dosing.
Because of manufactur-
ing costs J he is interested in the one_tailed alternative hypothesis:
Median [Effective Time (A)]
~
Median [Effective Time (B)] <
Median [Effective Time (C)]
[At least one directed inequality holds].
SpecificallYJ he wants to have a 0.80 power of detecting differences
between the pairwise
populations if the median effective time for
formulation B is stochastically longer than that for formulation A by
0.70 and the median effective time for formulation C is stochastically
longer than that for formulation A by 0.90.
That is) the investigator
wants to design an experiment whose sample size will ensure that he
will have an 80 percent probability
(£.~.
= .80)
power
of rejecting the
null hypothesis when in a conceptual population of experimental results
the probability that a randomly chosen effective time for formulation B
has a 70 percent probability of being longer than a randomly chosen
effective time for formulation A (i.e. stochastic difference of 0.70)
and that this corresponding probability for formulations A and C is
90 percent
(!..~.
stochastic difference of 0.90).
Expecting non-
normalitYJ the investigator plans to use the proposed one-tailed "all
pairs ll median test with an error rate
(!..~.
llsignificance level") of
approximately 0.10.
Referring to Table 5.18 (k
= 3)
n
= 6)
for a one-tailed test) an
error rate of 0.102) a power of 0.80 and an 8
2
(£.~.
formulation A
.
45
versus B) stochastic difference of 0.70, he finds that the S3
(~.~.
formulation A versus C) stochastic difference must be 0.898.
Since
these values are consonant with the investigator's desires) he elects
to use a sample size of n = 6,
If) however, the 8
3
stochastic dif_
ference indicated in the table (other requirements assumed to be the
same) had been greater than 0.90) it would have been necessary for the
investigator to use a larger sample size in accordance with the
corresponding information from Tables 5.19 (k
(k
=
3) n
=
14),
2,7
..
2,7.1
= 3) n = 10) and/or 5.20
Discussion
Test Procedures
There is little point in making detailed comparisons between this
proposed "all pairs" simultaneous inference median procedure and the
corresponding normal univariate techniques
(~,~.
Tukeyi s Studentized
Range, Scheffeis F Projection, Duncan's Multiple Range):
in common
with nonparametric techniques in general, it is appropriate to consider
applying this proposed test whenever conditions dictate that the
tabulated significance level is essentially as desired, regardless of
the underlying population distribution} or whenever there is evidence
of sufficient non-normality to jeopardize the optimality of the normal
theory techniques __ even though they may be reasonably robust.
It is
appropriate, however) to compare several aspects of this proposed
median test with three non_parametric procedures that have been proposed for making "all pairs" simultaneous inferences about data from
..
a completely random design.
Although it appears that none of these
46
non-parametric alternatives has been used extensively (at least in the
biomedical literature), the most widely known of these simultaneous
inference procedures include the rank.sum test of Steel [32] and Dwass
[7], Nemenyi's [24J and Dunn's [6J extension of the Kruskal_Wallis test
and Nemenyii s [24] median test.
From a practical standpoint, the
s~ultaneous
inference median
test proposed in this chapter should require about the same amount of
t~e
and effort as any of these three nonparametric alternatives.
common with the
Steel-~ass
In
and the Nemenyi median multiple comparisons
procedures, however, the use of this proposed median test is limited
because tables of critical values are not available for the unequal
sample size case.
The use of the Kruskal_Wallis_type simultaneous
inference procedure is not restricted by this
equal sample size case, only very
l~ited
l~itationo
Even for the
tables of exact critical
valqes are available for any of these four nonparametric simultaneous
inference procedures, but asymptotic results have been derived for the
three
existin~
ones.
However, Gabriel and Lachenbruch [11] have
recently reported, on the basis of a Monte Carlo study, that the
Bteel~Dwas8
k-sample statistic is increasingly biased for a large
number of experimental groups and, hence, questioned whether one may
use the multiple comparisons version of this technique even approxi_
mately except for relatively few quite large samples.
It is possible
that extended critical tables could be approximated for this proposed
simultaneous inference median test by the use of a Bonferroni X2
statistic, but this development has not been investigated in this
thesis.
47
From a conceptual standpoint, both the proposed median and the
Steel-Dwass simultaneous inference procedures possess a quality that,
intuitively, makes them preferable to the Kruskal-Wallis-type and to
Nemenyi's median procedures.
In the latter two tests all of the
observations from the k samples are considered simultaneously in
obtaining the appropriate nonparametric multiple comparisons statistic.
As a consequence, the outcome of the comparison between population i
and population i' in the Kruskal-Wallis-type and Ne.menyi median
simultaneous inference procedures depends upon the observations from
the remaining k-2 populations.
Hence, for the same set of sample
values from populations i and i', the comparison of i vs i' can be
significant in one experiment and not significant in another.
This
inherently undesirable situation does not occur with the Steel-Dwass
procedure nor the proposed median procedure because they are based on
the so-called "Maximum Two-Sample Statistic" approach in which the k
samples are combined pairwise in obtaining the appropriate nonparametric multiple comparisons statistic.
Neither does it occur with
the normal theory simultaneous inference procedures.
A closely related disadvantage of both the Kruskal-Wallis-type and
the Nemenyi median simultaneous inference procedures is that they do
not provide confidence intervals for the differences in the location of
population pairs without undue labor because each comparison depends on
all observations and thus the confidence interval for each
e. ~
e~
~
has
to be based on the projection of a multivariate region for all the
parametric differences.
As Miller states [20, p. l69J:
tion of the multivariate region is virtually impossible."
"The calculaOn the other
48
hand, relatively easy numerical and graphical shortcuts exist for
obtaining the confidence intervals for the Steel-Dwass multiple
comparisons procedure [20J.
Similar shortcut procedures might also be
developed for the proposed median test.
2.7.2
Power
In discussing the power of normal theory univariate simultaneous
inference techniques, Miller [20J points out that the overall power is
readily available from existing tables only for the Scheff~ F Projection, and that the power functions for the studentized range and the
studentized maximum modulus are unknown.
limited (viz. k
= 3;
Nemenyi [24J made very
n = 4) Monte Carlo power performance comparisons
among the Steel_Dwass, the
Kruskal~Wallis-type and
range multiple comparisons procedures.
the studentized
Otherwise, essentially nothing
is known about the overall power function for any of the three non_
parametric alternatives to this proposed simultaneous inference median
test.
Even if one resorts to Monte Carlo estimation in order to obviate
the insuperable difficulties of the direct calculation of the power of
simultaneous inference procedures, certain problems remain.
First,
there is the basic question of what inferential errors should be
utilized in defining the power of the procedure.
For example, when
(say) populations 1 and 2 are identical while population 3 is moved to
the right, a test in a given experiment may "accept" the alternative
hypothesis that a difference actually exists between samples land 2.
Nemenyi [24, po 3l.4J refers to this rejection of the null hypothesis
.
49
50
whenever the critical values correspond to significance levels of
"practical" importance (say, p < .10), essentially all of the power of
the two-tailed test is associated only with one tail of the test
statistic's distribution.
Table 2.5 is abstracted from the tables in Section 5.4 and is
intended as an illustration that, as expected, the power of this proposed median test increases with increasing sEmple size.
Table 2.6 illustrates another property of the power characteristics of this test:
as the location of population 2 moves from that of
population 1 to that of population 3, the power is reduced to a minimum
at approximately the mid-point on the stochastic difference scale.
An abstract of the power findings
different populations.
is shown in Table 2.7 for five
The type test (one-tailed), significance level
(0.102), number of populations (k
same for each distribution.
= 3)
and sample size (n
= 6)
are the
It will be noted from the table that the
three "normal" populations differ with respect to their variances
2
2
2
0"1 = 0"2 = 0'2
2
2
2
3'J 0"3
0"1 = 1·J 0"2
(i. e. first column:
l'J second
third column:
8) •
column~
2
2
0"1 = 0"2
=
2
0"3 = 2 J.
For the specific case studied,
the power for this proposed test is relatively stable for the normal
and exponential distributions.
There is evidence, however, that it
lacks robustness for the uniform distribution.
With the exception of the special use of this proposed median
test as a test of "slippage," there appears to be no other basis for
making a comparison of the powers for comparable tests with different
numbers of populations
..
is shown in Table 2.8.
(£.=..
k).
This "slippage" test power comparison
It will be noted that, in general, the power
51
Table 2.6
Comparison of power for different sample sizes
a
SAMPLE SIZE:
n=6
n=lO
n=14
ERROR RATE:
.003
.002
.009
STOCHASTIC
DIFFERENCES
..
-
_
.
...
S2
S3
.50
.50
.005
.008
.028
.50
.60
.0lO
.016
.061
.50
.70
.025
.038
.161
.50
.80
.082
.132
.598
.50
.90
.469
.697
1.000
.60
.60
.012
.018
.068
.60
.70
.027
.042
.166
.60
.80
.080
.130
.466
.60
.90
.288
.484
.732
.70
.70
.030
.045
.179
.70
.80
.084
.134
.457
.70
.90
.259
.447
.680
.80
.80
.090
.143
.509
.80
.90
.275
.471
.758
.90
.90
.330
.551
1.000
aOne_tailed.
k=3:
2
2
2
[1: N(j,JIp (J1 =1) J. 2: N(j,JI2 J (J2 =1) J. 3: N(j,JI3 J (J3=1)].
52
"
Table 2.7
Power as a function of 8
aOne_tailed.
k=3:
2
a
•
e
53
Table 2.8
Comparison of power for different distributions
STOCHASTIC
DIFFERENCES
S2
S3
A
e
Normal
b
POPULATION DISTRIBUTIONS
c
d
e
Exponentia1
Norma1
Norma1
.50
.112
.104
.124
.106
.096
.50
.60
.197
.184
.208
.185
.166
.50
.70
.350
.330
.349
.327
.292
.50
.80
.614
.587
.570
.570
.517
.50
.90
.948
.907
.846
.869
.840
.60
.60
.206
.196
.241
.203
.171
.60
.70
.348
.337
.391
.344
.290
.60
.80
.570
.564
.608
.565
.489
.60
.90
.829
.833
.847
.815
.767
.70
.70
.358
.349
.432
.367
.294
.70
.80
.566
.565
.647
.582
.482
.70
.90
.804
.817
.864
.815
.748
.80
.80
.586
.581
.690
.615
.489
.80
.90
.827
.831
.890
.848
.757
.90
.90
.886
.869
.926
.910
. 788
.547
.537
.576
.541
.479
aOne_tailed.
Er ror
Rate
:
0.102.
k=3.
2
2
2·
[1: N(I-Lp 0"1 =1) }. 2: N(1-L 2 , 0"2 =1) }. 3: N(1-L3 , 0"3=1)] .
2
2
2
c
[1: N(1-L , 0"1 =2) }. 2: N(1-L , 0"2 =2) }. 3: N(1-L , 0"3=2)] .
2
1
3
2
2
2
d
[1: N(1-L1' 0"1 =1) }. 2: N(1-L 2 , 0" 2:=;3); 4:N(1-L 3 , 0"3:=;8)] .
b
e[l:EXP (Q'1); 2 :EXP (0!2) ; 3:EXP (QJ3)]'
At
Uniform
.50
MEANS:
e
a
f
[1: U(O!l'
O!
1+1); 2: U(QJ2' 0!2 +1) ; 3: U(QJ3' 0!3 +1)].
n=6.
f
54
Table 2,9
Comparison of power for different populations n~bersa
NO. OF POPULATIONS:
k=3
k=4
k =5
ERROR RATE:
.003
.007
.012
STOCHASTIC
DIFFERENCES
S2' ••• , Sk_l
...
..
e
It
...
e
Sk
.50
.50
.005
.009
.012
.50
.60
.010
.018
.024
.50
.70
.025
.042
.056
.50
.80
.082
~
132
.164
,50
.90
,469
.499
.530
.60
.60
.012
.021
,028
.60
.70
.027
.044
.053
.60
.80
.080
.107
.109
.60
.90
.288
.269
.220
• 70
.70
.030
.048
.063
.70
.80
.084
.109
.121
.70
.90
.259
.259
.242
.80
.80
.090
.120
.153
.80
.90
.275
.295
.331
.90
.90
.330
.376
.533
.138
.157
.176
MEANS:
aOne_tailed.
2
(tT =1, all i).
i
n=6.
All populations are Normal with equal vqriance
55
inc~eases
slightly with an increasing number of populations, but that
this increase also corresponds with an increase in the
erro:t :.ra tea •
..
~ssociated
56
3.
A SIMULTANEOUS INFERENCE MEDIAN TEST FOR
COMPARING TREATMENTS VERSUS A CONTROL
The test proposed in this chapter closely parallels the one
proposed in Chapter 2:
it is a multiple comparisons analogue of the
k-samp1e Brown-Mood Median Test [3]j it is based on the t1Maximum TwoSample Statistic tl approach and hence is actually an extension of the
two-sample median tests of Westenberg [37] and Mood [21]j it has an
experiment-wise error ratej it is appropriate for data from a
completely random design.
However} this proposed test is appropriate
for making simultaneous inferences in those situations where the
...
experimenter1s interest is focused on the differences between a socalled control (or standard) treatment and (k_1) other (presumably)
experimental) treatments.
3.1
Each observation} X.. (i
~J
= O)
Data
• e •
J
k-1 > 2 j j = 1} '''} n)
represents a univariate measurement} of ordinal or interval level} on
the j-th experimental unit in the i-th treatment group.
The "i,
= 0"
treatment is assumed to be a "control" (or standard) treatment and the
remaining (k-1) treatments are assumed to be "experimental" treatments.
The measured value may be either a continuous or discrete variable.
While there are no inherent limitations on the number of observations
in the treatment group nor on the number of treatment groups} only the
equal sample size case (nO
..
five treatment groups (k
=
= 3,
= nk _ 1 = n S
15) for three} four and
4) 5) has been considered herein.
..
57
3.2
Description
These tests provide for making simultaneous statistical inferences
about the (k-l) differences between each of the (k-l) "experimental"
treatments and a single "control" treatment with an experiment-wise
error rate.
A test may be either one-tailed or two-tailed, but both
one_tailed and two-tailed alternatives cannot
be considered at the
same time (2;..::.. for different treatment comparisons).
Interest is focused on the detection of differences in location as
characterized by the medians of the treatment populations.
The null
hypothesis:
is to be tested against the one-tailed alternative hypothesis:
where an inequality must hold for at least one of the (k-l) comparisons, El, (X,) is the median of the i-th "experimental" treatment popula"2
tion (i
~
= 1,
... , k-l) and El\(X ) is the median of the single "control"
O
treatment population.
One may also test against the two-tailed alternative hypothesis:
where the inequality must hold for at least one of the treatment pairs.
3.3
The X., (i = 0,
~J
k-l
Assumptions
~
2j j
= 1,
... , n) are assumed to be k
independent samples, each consisting of n independent observations.
58
The X..
I
~J
S
are assumed to have a distribution function F. (x).
~
While
the underlying distribution function [say, F~(X)] may be discrete, the
Fi(x) can effectively be considered to be continuous due to the fact
that the F.(x) 's are assumed to include probability effects which are
~
introduced to handle any ties.
3.4
3.4.1
Test Procedures
One-tailed Test
(1)
Using the "Tables of the Null Distribution of mOi When
Comparing (k-l) Treatments Versus a Control:
One.tailed
Critical Region" in Section 5.5, select for the appropriate
values of k and n, the critical value, m* , which most clearly
corresponds to the desired experiment-wise error rate.
(2)
Designate the "control" treatment sample by the integer i
=a
and the "experimental" treatment samples, arbitrarily, by the
integers i
(3)
= 1,
... , k-l.
Form the (k-l) combined treatment
sampl~s,
each of size 2n
and consisting of the n variate values from the "control"
treatment sample and the n variate values from one of the
(k-l) "experimental" treatment samples.
(4)
Form the [i x (k-l)J m_statistic vector m
= [mal'"
mOi
m , k_l J where each vector element, mOi ' is obtained as
O
follows:
(a)
If HI:
O) > 6J:! (Xi)' mOi is a count of the number of
e~ (X
the "control" treatment variate values that are greater
than the median of the combined "control" and the i.;.th
"experimental" treatment samples.
59
(b)
e~(XO)
If Hl :
<
e~(Xi))
mOi is a count of the number
of the ffcontrol" treatment variate values that are less
than the median of the combined ffcontrol" and the i-th
"experimental" treatment samples.
(5)
Compare each of the (k-l) elements) mOi) with the previously
selected critical value) m* ) and make a significance state_
ment for each of the comparisons) depending on whether the
corresponding m is equal to or greater thanm* or not.
Oi
That is) reject the null hypothesis) of equal medians for
each of the (k-l) comparisons for which the corresponding m
Oi
is equal to or greater than m* (~.~. m ~ m*)) otherwise) do
Oi
not.
3.4.2
Two-tailed Test
(1)
Using the "Tables of the Null Distribution of m When
Oi
Comparing (k-1) Treatments Versus a Control:
Two-tailed
Critical Region" in Section 5.6) select for the appropriate
values of k and n) the upper critical values) m*) and its
u
associated lower critical value)
m;:)
which most closely
approximates the desired experiment-wise error rate.
(2)
Designate the "control ff treatment sample by the integer i
=a
and the ffexperimentalff treatment samples) arbitrarily) by the
integers i
(3)
= 1) ... ) k_l.
Form the (k-l) combined treatment samples) each of size 2n
and consisting of the n variate values from the ffcontrolff
treatment sample and the n variate values from one of the
(k-l) "experimental ff treatment samples.
60
(4)
Form the [1 X (k_l)] ~-statistic vector ~
= [mOl'"
mOi
mO, k-l] where each vector element, mOi ' is a count of the
number of the "control" treatment variate values that are
greater than the median of the combined control and i_th
"experimental" treatment sample.
(5)
Compare each of the (k-l) elements, m ' with the previously
Oi
selected upper critical value m*, and its associated lower
u
critical value, ~, and make a significance statement for
each of the comparisons, depending on whether the corresponding m
is equal to or greater than m:, equal to or less than
Oi
m~ or not.
That is reject the null hypothesis of equal
medians for each of the (k-l) comparisons for which the
correspondingm
is either equal to or greater than the
Oi
upper critical value (1. e. mO' ~ m"l'r) or is equal to or less
-
-
than the lower critical value
J,.
u
c,!:.=..
m ~ mi; otherwise} do
Oi
not.
3.4,3
An Example
During the early human testing of a chemical series of three
potential analgesics, an experiment was conducted to compare each of
the new drugs with a placebo.
Twenty-four human subjects were selected
from a previously "screened" pool and were assigned, randomly, to the
four treatment groups
(.!.. .=.
six subjects per group).
The "analgesic"
response to the treatment was measured in terms of each subject1s
"pain threshold" as indicated by his ability to tolerate graded levels
of electric shock under standardized conditions.
experiment are shown in Table 3.1.
The data fo·r this
61
Table 3.1
Pain threshold data (millivolts)a
_IIExperimental"
" Control"
Placebo
(i ::: 0)
...
(i
= 1)
3lC
3lB
3lA
(i
= 2)
(i
= 3)
47
52
50
55
47
55
52
57
49
59
53
58
51
60
55
58
53
60
57
61
54
63
58
62
a
The data have been rearranged within each treatment group to
facilitate inspection.
The experimenter was interested, obviously, in the one-tailed
alternative that the median "pain threshold" for each of the three
"experimental" drugs was higher than the "pain threshold" for the
placebo, i. e. :
1.
Median (Placebo) < Median (3lA).
2.
Median (Placebo) < Median (3lB).
3.
Median (Placebo) < Median (3lC).
Using the one_tailed procedure outlined in Section
3.4~l,
we have the
following:
1.
From Table 5.26, for (k-l) = 3 "experimental" treatments and
n = 6 subjects per treatment (including the "control"), a
62
= 5) corresponding to an experiment_wise
critical value) m*
error rate of 0.142) was selected .
.
2.
As noted) the placebo
the subscript) i
= 0)
(£.~.
"control") has been designated by
and the "experimental" drugs have been
designated arbitrarily as i = 1) 2) 3.
3.
The k-l
=3
combined "placebo" and "experimental" treatment
samples were formed as follows:
COMBINED SAMPLE
o
COMBINED SAMPLE
o and
and 1
(Placebo and 3lA)
Treat. No.
Data
o
2
(Placebo and 3lB)
Treat. No.
Data
(Placebo and 31C)
Treat. No.
Data
47
*
0
47
*
0
47
*
o
47
*
0
47
*
0
47
*
o
49
*
0
49
*
0
49
*
o
51
2
50
*
0
51
1
52
0
51
*
0
53
o
53
2
52
*
0
54
*
Median
e
and 3
o
*
"
COMBINED SAMPLE
-
Median
0
54
2
53
3
55
1
55
0
53
3
57
1
59
0
54
3
58
1
60
2
55
3
58
1
60
2
57
3
61
1
63
2
58
3
62
.
e
63
4.
The m_statistic vector was determined to be:
e~(xO)
where (since HI:
<
e~(xi)' i
= 1, 2, 3) mOl = 5, is
the count of the number of the "placebo"
(.:!:..~.
"control")
variate values (marked with an asterisk above) that are less
than the median of the combined !!placebo" and the first
"experimental" (3IA) treatment samples; m
02
=4
and m
03
=6
are the corresponding counts for the "placebo!! and the second
"experimental" treatment (3IB) and for the "placebo" and
third !!experimental" (3IC) combined samples, respectively.
5.
The
~-statistic
vector elements were then compared individ_
ually with the selected critical value (m*
= 5)
and the follow-
ing significance statements were made:
a.
Since mal
= m*
(5=5), the H :" Median (Placebo)
O
Median (3IA) was rejected; hence, the HI:
=
Median
(Placebo) < Median (3IA) was accepted.
b.
Since mal < m* (4 < 5), the
HD:
Median (Placebo)
Median (3IB) was not rejected.
c.
Since mal> m* (6 > 5), the
HD:
Median (Placebo) =
Median (3IC) was rejected; hence, the HI:
Median
(Placebo) < Median (3IC) was accepted.
These three significance statements are made jointly with an
error rate (one-tailed) of 0.142.
If the "control!! treatment had been a fixed dose of a standard
.
analgesic (say, codeine), a two-tailed test would have been more
.
e
64
appropriate and the procedure outlined in Section 3.4.2 would have been
followed.
The two-tailed version is similar to the above outlined one-
tailed example and) hence) does not warrant repetition.
3.5
Test Derivation
The rationale underlying the "treatments versus control" test
procedure outlined in Section 3.4 is essentially the same as that
discussed in Section 2.6.1 for the "all pairs" test procedure.
it is app r
0
p ria
t
Hence)
e to consider that it also is a "Maximum Two
Sample Statistic" type simultaneous extension of the two-sample
Westenberg~Mood
(21) 37J median test.
If one designates the i
=1
treatment in the "all pairs" case as a
"control" treatment) the "treatment versus control" m-statistic vector
is exactly the initial (k-l) element subset of the corresponding
[1 X
(~)J
"all pairs" m_statistic vector.
This relationship was
utilized in deriving the null distribution of the "treatments versus
control" m-statistic vector.
In fact both the exact and approximate
distribution of the corresponding "all pairs" and "treatments versus
control" m_statistic vectors were derived simultaneously) and hence)
the derivation of the latter statistic's distribution follows)
mutatis mutandis) the procedures outlined in Sections 2.5.2 and 2.5.3.
Tables of the null distribution of the "treatments versus control"
m-statistic vector
[~.~.
k
= 3) 4) 5j n = 2(1)15J as characterized by
its maximal and/or minimal element(s) are shown in Sections 5.5 and 5.6
for the one- and two-tailed procedures) respectively.
The validity of
the "nonparametric" Monte Carlo procedure is indicated in Table 3.2.
~
e
e
~
Table 302
e
Comparison of exact and lInonparametricll Monte Carlo null distribution probabilities
k= 3 treatments
One-tailed
~
n = 4 observations/treatment
Two-tailed
Critical Values
Exact
Monte Carlo
Difference
Critical Values
Exact
Monte Carlo
Difference
4
3
2
1
0.027
0.371
0.889
0.998
10000
0.026
0.371
0.889
0.998
1.000
+00002
+0. 001
0.000
0.000
0.000
0; 4
1; 3
2
0.053
0.689
1.000
0.051
0.688
LOOO
+0. 002
+0.001
0.000
a
k"=4 treatments
One-tailed
:
n = 3 observations/treatment""
Two_tailed
Critical Values
Exact
-
Monte Carlo
Difference
Critical Values
Exact
Monte Carlo
Difference
3
2
1
0.119
0.750
0.995
1.000
00119
0.747
0.996
1.000
0.000
+0.003
-0 .001
0.000
0; 3
1; 2
00234
LOOO
0.236
1.000
-0.002
0.000
a
k = 5 treatments
One-tailed
:
n = 2 observations/treatment
Two-tailed
Critical Values
Exact
Monte Carlo
Difference
Critical Values
Exact
Monte Carlo
Difference
2
1
0.387
0.978
1.000
0.387
0.977
1.000
0.000
+0. 001
0.000
0; 2
0.690
1.000
0.687
1.000
+0.003
0.000
a
~
1
0'\
\Jl
66
3,6
Power
The small sample power of this proposed fftreatments versus control ff
simultaneous .inference median test was investigated for the eleven
combinations of k, n and distributions shown in Table 3.3,
This table
also shows for each of these combinations the one- and two-tailed
ffExperiment_wise Error Rates ff for which the corresponding power was
estimated as well as the number of ffObservation Orderings Generated ff in
making these estimates.
Similarly to the generation of the m_statistic1s null distribu_
tions, the same computer runs were used to generate simultaneously the
Monte Carlo estimates of power for both the t'all pairs ff (see Section
2.6,1) and the fftreatmentsversus control ff procedures,
Tables 3,4 indi-
cates the degree of coincidence between the ffExperiment_wise Error
Rates ff generated by the t'parametric ff Monte Carlo ffpowerll program (viz.
5.9.3) for various dis tributions
(2::!:: by n setting ff the means of the k
treatment populations equal) and the ffexactll ones obtained by enumeration (viz. 5,9.1) as well as the ffapprox'imateff ones generated by the
"nonparametricff Monte Carlo ffnull distribution ff program (viz. 5.9,2).
The estimated second-order inverse polynomial regression coefficients for the ffpowerff response surface are shown in the tables in
Section 5.7,
Table 3,5 shows the maximum, mean and minimum absolute
deviations for the principal cases.
The tables in Section 5,8 show the ffstochastic differences ff that
correspond to given power levels [viz . . 50(,lO),9UJ and, thus, can be
used similarly to the corresponding "all pairs ff tables
5.4) in estimating the ffrequiredff sample sizes.
(.!:-'!:: Sec tion
67
Table 3.3
Combinations of k, n, distributions and experiment-wise
error rates for which the power was investigated and the
number of observation orderings generated for each
Experiment-wise
Error Rates
k
~e
n
Distributions
One-tailed
Two-tailed
Observation
Orderings
Generated
l.
3
6
2
Normal (0-.==1,1,1)
.002
.071
.004
.142
72,000
2.
3
10
2
Normal (0-.==1,1,1)
~
.001
.021
.002
.038
21,000
3.
3
14
2
Normal (0-.==1,1,1)
.006
.050
.012
.100
17,000
4.
4
6
2
Normal (0-.==1,1,1,1)
.004
.098
.007
.196
26,000
5.
4
10
2
Normal (0-. ==1, L, 1, 1)
.030
.198
.033
.192
39,000
6.
5
6
2
Normal (0-. ==L, 1, 1, 1, 1)
.004
.118
.007
.233
27,000
7.
5
10
2
Normal (O".==l,L,l,l,l)
.001
.021
.004
.083
52,000
8.
3
6
2
Normal (0-.==1,3,8)
.002
.071
.004
.142
18,000
9.
3
6
2
Normal (0-. ==2,2,2)
~
.002
.071
.004
.142
19,000
10.
3
6
Exponential
.002
.071
.004
.142
18,000
11.
3
6
Uniform
.002
.071
.004
.142
18,000
~
~
~
~
~
~
~
-i
e
-
Table 3.4
-
..
,
•
Comparison of exact and Monte Carlo (non-parametric and parametric) estimates of
experiment-wise error rates for k = 3 treatments and n = 4 observations per
treatment
One-Tailed
Method
-
Distribution
Observation
Orderings
Generated
Two-Tailed
(At least one m..
~~
- 4-
4 or 3
I
= --: )
4 or 0
-
Exact (Enumeration)
None
(34,650)
.027
.371
.053
Monte Carlo (Non-Parametric)
None
lO,OOO
.025
.369
.051
Monte Carlo (Parametric)
Normal
I, 000
.024
.368
.048
Monte Carlo (Parametric)
Uniform
1,000
.023
.376
.046
Monte Carlo (Parametric)
Exponential
1,000
.023
.376
.046
0'
00
"
e
69
Table 3.5
Absolute deviations between the "treatments versus control"
estimated inverse quadratic power surface and the Monte
Carlo generated powers for selected k, n, test type and
error rates
Error
n
3
6
6
6
6
Normal
Normal
Normal
Normal
One-Tailed
One_Tailed
Two_Tailed
Two_Tailed
10
10
10
10
Normal
Normal
Normal
Normal
3
3
3
14
14
14
14
4
4
4
4
Distributions
Estimated Absolute Deviation
powerEJ
Ma}t
.002
.071
.004
.142
.214
.368
.212
.816
.041
.053
.039
4049
.0lD .• 001
4015 .. 000
.011 .• 000
.014 .000
One-Tailed
. One-Tailed
Two-Tailed
Two-Tailed
.001
.021
.002
.038
.364
.495
.363
.497
.. 081
.064
.080
.066
.026
.023
.025
.021
Normal
Normal
Normal
Normal
One_Tailed
One-Tailed
Two-Tailed
Two_Tailed
.006
.050
.012
.100
.819
.938
.954
.948
.113
.. 054
.105
.044
4036
.025
.035
.021
.000
.002
,,000
.000
6
6
6
6
Normal
Normal
Normal
Normal
One-Tailec;l.
One-Tailed
Two_Tailed
Two_Tailed
.004
.098
.007
.196
.156 .
.697
.131
.894
.029
.031
.029
.049
.011
4010
.0lD
.014
.000
.000
.001
.002
4
4
4
4
10
10
Normal
Normal
Normal
Normal
One_Tailed
One-Tailed
Two-Tailed
Two-Tailed
.030
.198
.033
.192
1.000
.891
.833
.892
.050
.045
.052
.O43
.020
4010
.019
.009
.005
.000
.O02
.000
5
5
5
5
6
6
6
Normal
Normal
Normal
Normal
One-Tailed
One-Tailed
Two-Tailed
Two-Tailed
.004
.118
.007
.233
.222
.699
.212
.640
.049
.. 033
"O39
.045
.010
.010
.010
.013
.001
.001
.• 000
.000
Normal
Normal
Normal
Normal
One-Tailed
One-Tailed
Two-Tailed
Two-Tailed
.001
.021
.004
.083
.452
.894
.448
.835
.115
.075
.119
.067
.032
.023
.033
.025
.000
.002
.001
.001
3
3
3
3
3
3
3
3
~e
~ Test Types
Rates
k
5
5
5
5
10
10
6
10
10
10
10
aAll variances equal to one (1. e. cr~
b
--
~
= I,
Mean
all i).
Estimated power corresponding to maximum absolute deviation.
Min
.006
.ora
.002
.noo
70
3.7
Discussion
Any discussion of this proposed "treatmentsversus control"
simultaneous inference median test procedure
~ ~
and of its power
characteristics must closely parallel the corresponding discussion of
the "all pairs" version in Section 2.7 (q.v.).
This parallel is a
consequence) of course) of the close similarity between the two tests
with respect to their basic nature
(~.~.
the null and alternative
hypotheses) the assumptions) the procedures per se) etc.) and to the
derivation of the distribution of the test statistic.
The parallel
holds also in that the three most widely known nonparametric simultaneous inference alternatives to this proposed procedure are the
"treatmentsversus control ll versions of the alternative "all pairs" test
procedures discussed in Section 2.7 (viz. the rank-sum test of Steel
[30] and Nemenyi's [24] extensions of the Kruskal-Wallis and the
median test).
Parenthetically) it might be pointed out that it would have been
desirable to provide tables of the distribution of the test statistic)
mOil in which the sample size for the "control" sample was larger than
those of the "experimental" samples and then to have investigated the
effect of this sample size difference on the power characteristics.
71
4.
..
LIST OF REFERENCES
1,
Arbuthnott, J. 1710, An argument for divine providence, taken
from the constant regularity observed in the births of both
series .. Philosophical Transactions 27~ 186-190,
2,
Bradley, J. v. 1968. Distribution-free Statistical Tests,
Prentice-Hall, Inc" Englewood Cliffs, New Jersey,
3,
Brown, G, W. and A, M, Mood, 1951, On median tests for linear
hypotheses, pp, 159.-166, In Proceedings of the Second
Berkeley Symposium on Mathematical Statistics and Probability,
University of California Press, Berkeley,
4,
Dixon, W, J, and A, M, Mood. 1946, The statistical sign test,
J. American Statistical Association 41:557-566,
5,
Douglas, A, W. 1961. Some multiple comparison tests for two-way
classifications, Unpublished Master's thesis, Department of
Statistics, Cornell University, Ithaca, New York,
6,
Dunn, 0, J. 1964. Multiple comparisons using rank sums.
Technometrics 6:241-252,
Dwass, M. 1960. Some k-sample rank-order tests, pp. 198-202.
In I. 01kin, S, G, Ghurye, W. Hoeffding, W, G. Madow and
IL B, Mann (eds.), Contributions to Probability and Statis_
tics. Stanford University Press, Stanford, California,
8.
Finney, D. J" R. Latscha, B, M, Bennett and P. Hsu. 1963.
Tables for Testing Significance in a 2 X 2 Contingency
Table, Cambridge University Press, London and New York.
9,
Fisher, R, A. 1926, On the random sequence,
Royal Meteorological Society 52~250-258,
Quarterly J.
10.
Friedman, M, 1937, The use of ranks to avoid the assumption of
normality implicit in the analysis of variance, J, American
Statistical Association 32:675-701,
11.
Gabriel, K, R. and P, A. Lachenbruch. 1969, Nonparametric ANOVA
in small samples~ a Monte Carlo study of the adequacy of the
asymptotic approximation. Biometrics 25~593-596,
12.
Gibbons, J. D. 1964. On the power of two-sample rank tests of
the equality of two distribution functions. J. Royal
Statistical Society B26:293-304,
13,
International Business Machines, 1959. Random number generation
and testing. Form C20-80l1, IBM, Technical Publications
Department, White Plains, New York,
72
14.
Kendall, M, G. and A, Stuart. 1961. The Advanced Theory of
Statistics, Volume 2: Inference and Relationship, Charles
Griffin and Company, Limited, London.
15.
Kruskal, W, H, and W, A. Wallis, 1952. Use of ranks in onecriterion variance a rial y sis.
J, American Statistical
Association 47:583-621.
16.
Lehmer, D. H, 1964. The machine tools of combinatorics, pp, 531, In E, F, Beckenbach (ed,), Applied Combinational
Mathematics, John Wiley and Sons, Inc., New York.
17.
Lieberman, G, I, and D. B. Owen. 1961, Tables of the Hypergeometric Probability Distribution, Stanford University
Press, Stanford, California,
18,
Mann, H. B, and D, R, Whitney. 1947, On a test of whether one of
two random variables is stochastically larger than the other,
Annals of Mathematical Statistics 18:50-60.
19.
Mikulski) P, W. 1963, On the efficiency of optimal nonparametric
procedures in the two sample case, Annals of Mathematical
Statistics 34:22-32.
20,
Miller) R. G. 1966. Simultaneous Statistical Inference.
Hill Book Company) New York, New York.
21.
Mood, A, M. 1950. Introduction to the Theory of Statistics.
McGraw Hill Book Company, New York) New York.
22.
NeIder) J. A. 1966. Inverse polynomials) a useful group of
multi_factor response functions. Biometrics 22:128-141.
23.
Nemenyi} P, 1961, Some distribution-free multiple comparison
procedures in the asymptotic case (abstract). Annals of
Mathematical Statistics 32:921-922,
24,
Nemenyi) P, 1963. Distribution-free multiple comparisons.
Unpublished phD thesis) Department of Statistics, Princeton
University) Princeton, New Jersey.
25,
Pitman, E, J. G. 1937, Significance tests which may be applied
to samples from any population, J. Royal Statistical
Society) Series B 4~119-l30,
26.
Pitman, E, J. G, 1938. Significance tests which may be applied
to samples from any populations} III~ the analysis of
variance test, Biometrika 29:322-335,
27.
Rhyne, A, L, 1964, Some multiple comparison sign tests.
Unpublished phD thesis, Department of Experimental Statis_
tics) North Carolina State University at Raleigh.
McGraw
73
28.
Rhyne, A. L. and R. G. D. Steel. 1965. Tables for a treatment
versus control multiple comparisons sign test. Technometrics
7: 293-306.
29.
Savage, I. R. 1953.
related topics.
906.
30.
Steel, R. G. D. 1959. A multiple comparison rank sum test:
treatments versus control. Biometrics 15:560-572.
31.
Steel, R. G. D. 1959. A multiple comparison sign test: treat_
ments versus control. J. American Statistical Association
54: 767-775.
32.
Steel, R. G. D. 1960. A rank sum test for comparing all pairs of
treatments. Technometrics 2:197-207.
33.
Steel, R. G. D. and J. H. Torrie. 1960. Principles and Procedures of Statistics. McGraw Hill Book Company, New York,
New York.
34.
Walsh, J. E. 1962. Handbook of Nonparametric Statistics, I:
Investigation of Randomness, Moments, Percentiles, and
Distributions. D. Van Nostrand Company, Inc., Princeton,
New Jersey.
35.
Walsh, J. E. 1965. Handbook of Nonparametric Statistics, II:
.Resu1ts for Two and Several Sample Problems, Symmetry, and
Extremes. D. Van Nostrand Company, Inc., Princeton, New
Jersey.
36.
Walsh, J. E. 1968. Handbook of Nonparametric Statistics, III:
Analysis of Variance. ~ D. Van Nostrand Company, Inc., Princeton,
New Jersey.
37.
Westenberg, J. 1948. Significance tests for the median and
interquarti1e range in samples from continuous populations of
any form. Proceedings Wetenschappen51:252_261.
38.
Wilcoxon, F. 1945. Individual comparisons by ranking methods.
Biometrics 1:80-83.
39.
Wilcoxon, F. and R. A. Wilcox. 1964. Some Rapid Approximate
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Bibliography of nonparametric statistics and
J. American Statistical Association 48:844-
74
5.
5.1
APPENDICES
The Null Distribution of m..
1.1.
of k Treatments:
I
when Comparing All Pairs
One-tailed Critical Region
,.
e
..
•
e
"
Table Sol
The null distribution of m..
~~
n units per treatment:
rate
Critical
Values
(m*)
n
n_l
n-2
n-3
n-4
n-5
n-6
I
e
when comparing all pairs of k
= 3 treatments with
one_tailed critical region with an experiment-wise error
Number of Units Per Treatment (n) Equal to:
12
13
14
15
.000
.000
.000
.000
.000
.002
.000
.000
.000
.000
.000
.073
.030
.017
.005
.001
.000
.000
.377
.212
.109
.051
.022
.009
.005
.422
.255
.152
.074
.038
.453
.289
.183
2
3
4
5
6
7
8
9
10
.367
.126
.039
.011
.003
.000
.000
.000
.000
.498
.245
.102
.037
.014
.005
.323
.158
.11
.487
-...J
IJ1
.-
_
•
~
Table 5 02
_
•
The null distribution of m.. , when comparing all pairs of k
~~
n units per
rate
treatment~
=4
treatments with
one-tailed critical region with an experiment-wise error
Number of Units Per Treatment (n) Equal to:
Critical
Values
(m*)
2
3
4
5
6
7
8
9
10
11
12
13
14
15
n
0546
0214
.072
.022
0007
0001
0000
.000
.000
.000
0000
.000
0000
0000
0388
.179
.069
.027
.009
.003
.001
0000
.000
.000
0000
.489
0269
.129
.060
0027
0008
.004
0001
.000
.343
0194
.095
.044
.020
.007
.400
.238
.137
0068
.455
0289
n-1
n-2
n-3
n-4
n-5
'-I
Q'\
"
•
.
e
Table 5.3
The null distribution of m..
~~
n units per treatment:
rate
!
•
'"
e
when comparing all pairs of k
= 5 treatments with
one-tailed critical region with an experiment-wise error
Critfca1
Values
(m*)
2
3
4
5
6
7
8
9
10
11
12
13
14
15
n
.687
.305
.106
.035
.009
.003
.000
.000
.000
.000
.000
.000
.000
.000
.255
.113
.042
.015
.005
.001
.000
.000
.000
.000
.376
.193
.091
.034
.014
.005
.003
.001
.465
.262
.137
.067
.030
.014
.337
.196
.. 106
n-1
n-2
n-3
n-4
n-5
Number of Units Per Treatment (n) Equal to:
.398
~
~
78
5,2
The Null Distribution of m..
~~
Treatments:
I
when Comparing All Pairs of k
Two_tailed Critical Region
.,
e
..
..
e
Table 5.4
The null distribution of m..
~~
units per treatment:
rate
when comparing all pairs of k
=3
e
treatments with n
two-tailed critical region with an experiment-wise error
Number of Units Per Treatment (n) Equal to:
Critical
Values
(m*)
u
i
•
•
(m;:)
2
3
4
5
6
7
8
9
10
11
12
13
14
15
n
0
.600
.229
.074
.022
.006
.000
.000
.000
.000
.000
.000
.000
.000
.000
n-l
1
.422
.186
.069
.028
.009
.004
.001
.000
.000
.000 .. 000
n-2
2
.282
.134
.056
.023
.010
.002
.001
.000
n-3
3
.370
.198
.095
.043
.018
.010
n-4
4
.436
.271
.137
.073
n-5
5
.490
.320
.......
\0
"
e
.
Table 505
~
•
e
!O
e
The null distribution. of m.. , when comparing all pairs of k = 4 treatments with n
11
units per treatment:
rate
Critical
Values
two_tailed critical region with an experiment-wise error
Number of Units Per Treatment (n) Equal to:
u
(~
2
3
4
5
6
7
8
9
10
n
0
.772
.357
.131
.042
.013
.003
.000
.000
.000
n-1
1
.303
.125
.050
.. 018
n-2
2
.437
.222
n-3
3
n-4
4
n-5
5
(m:)
.11
12
13
14
15
.000
.000
.000
.000
.000
.005
.002
.000
.000
.000
.000
.108
.050
.016
.008
.002
.001
.328
.168
.080
.037
.014
.388
.235
.124
.456
co
o
..
e
JI
Table 5.6
~
JI
e
>
The null distribution of m.. , when comparing all pairs of k = 5 treatments with n
~~
units per treatment:
rate
Critical
Values
two-tailed critical region with an experiment-wise error
Number of Units Per Treatment (n) Equal to:
(~
2
3
4
5
6
7
8
9
10
11
12
13
14
15
n
0
.873
.476
.183
.064
.018
.006
.001
.001
.000
.000
.000
.000
.000
.000
n-l
1
.404
.194
.078
.029
.010
.002
.001
.000
.000
.000
n-2
2
.317
.161
.061
.028
.010
.005
.001
n-3
3
.416
.237
.117
.056
.027
n-4
4
.325
.182
(m~
e
•
00
.....
82
5.3
Table 5.7
Regression Coefficient Estimates for the
Inverse Quadratic Power Response Surface
when Comparing All Pairs of Treatments
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n = 6 units per
treatment when comparing all pairs of k = 3 normal treatment
populations
1) ]
TWO. TAIL
ONE. TAIL
.102
(10%)
.003
(1%)
.186
(20%)
.006
(1%)
REGRESSION
COEFFICIENTS
...
Po
+209.5239
+
6.0949
+194.1158
+
PI
+ 45.6860
+ 5.5491
+ 46.2702
+ 5.4898
P2
.482.5384
. 15.7803
.447.7937
8.9325
P11
- 18.0438
2.2241
- 18.1298
2.2854
~22
+266.3345
+ 8.6812
+246.7250
+ 4.6150
~12
. 18.5399
1.4850
- 19.0513
1. 3218
MODEL:
S2 S3
2
2
= Po + P1 S2 + ~2S3 + S11 S2 + P S + P S S
Power
22 3
12 2 3
where
S2 = Stochastic Difference between Populations 1 and 2
S3 = Stochastic Difference between Populations 1 and 3
3.1982
--
83
Table 5.8
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n = 10 units per
treatment when comparing all pairs of k = 3 normal treatment
populations
= 1);
3:N(~3J
2
rr3
ONE-TAIL
= l)J
TWO-TAIL
.002
(1%)
.030
(5%)
.004
(1%)
.056
(5%)
/3 0
+146.9996
+ 16. 7201
+145.5062
+ 14.5898
/3 1
+ 31. 7315
+
8.3011
+ 30.5737
+
/3 2
-341. 4481
- 41.8420
-337.1975
- 36. 7031
/3 11
8.8150
2.2624
9.0134
2.4101
/3 22
+192.5321
+ 24.6073
+189.5259
+ 21.4274
/3 12
- 19.2534
4.6442
- 17.6438
4.1630
REGRESSION
COEFFICIENTS
~-
MODEL:
S2 S3
Power
8.0910
2
2
/3 0 + /3 1 S2 + S2 S3 + /3 11 S2 + /3 22 S3 + /312 S2 S3
where
..
..
S2
= Stochastic
S3
= Stochastic Difference between Populations 1 and 3
Difference between Populations 1 and 2
84
Table 5.9
..
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n ;::: 14 units per
treatment when comparing all pairs of k ;::: 3 normal treatment
populations
2
1); 3: N(IJ. , 0"3 ;::: 1) ]
3
ONE-TAIL
TWO-TAIL
.009
(1%)
.074
(10%)
.018
(2%)
.137
(15%)
~O
+ 42.. 1148
+ 10.1249
+ 40.7816
+ 9.2257
~1
+ 13.2142
+
3.8929
+ 12.8644
+
~2
-105.7197
- 26.1534
-102.2907
- 23.9196
~11
8.2160
2.3611
8.0482
2.2415
~22
+ 60.7267
+ 15.2157
+ 58.6650
+ 13.9175
~12
0.7202
+ 0.3583
0.5990
+ 0.2629
REGRESSION
COEFFICIENTS
.-
MODEL:
S2 S3
Power
;:::
2
3.8043
2
~O + ~lS2 + ~2S3 + ~11 S2 + ~22S3 + ~12S2S3
where
S2 ;::: Stochastic Difference between Populations 1 and 2
S3 ;::: Stochastic Difference between Populations 1 and 3
.
-_
85
Table 5.10
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n = 6 units per
treatment when comparing all pairs of k = 4 normal treatment populations
2
2
2
[1: N(i-Lp 0"1 = 1) J. 2: N(i-L2 J 0"2 = 1) j 3:N(i-L3 J 0"3 = 1) j
2
1) ]
4:N(i-L4 J 0"4
ONE-TAIL
TWO-TAIL
.007
(1%)
.179
(20%)
.013
(1%)
~O
+ 41. 8643
+ 0.9745
+ 24.4907
-
0.4205
~1
+ 45.0339
+
2.5487
+>40.7018
+
1. 9936
~2
+ 13.8357
+ 0.4363
+ 16.4251
+ 0.1045
~3
.127.0262
3.9979
- 85.9276
+ 0.1227
~11
8.0284
0.4546
7.9269
0.6563
~22
4.8069
0.5246
4.5919
0.3778
13 33
+ 80.6193
+
2.4062
+ 56.4856
+ 0.1905
~12
+ 0.0602
+ 0.4935
0.1975
~13
- 34.4980
1. 5815
- 29.6869
1. 8667
~23
6.9081
+ 0.5629
9.8939
+ 0.0343
.303
(30%)
REGRESSION
COEFFICIENTS
J
~e
MODEL:
S2 83 84
8
Power = 130 + ~182 + 13 2 3 +
~3S4
+
1. 7083
2
2
2
+ ~11 82 + 13 22 83 + ~3384
+ ~1282S3 + ~138284 + ~238384
where
..
e
*'
8
2
= Stochastic Difference between Populations 1 and 2
S3 = 8tochastic Difference between Popula tions 1 and 3
S4 = Stochastic Difference between Populations 1 and 4
-e
86
Table 5.11
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n = 10 units per
treatment when comparing all pairs of k = 4 normal treatment populations
2
[l:N(lJ,p (J"1 = 1)
j
2:N(1J,2 J
4:N(1J,4 J
ONE-TAIL
.003
(1%)
TWO-TAIL
.060
(10%)
.005
(1%)
.108
(10%)
REGRESSION
COEFFICIENTS
*e
/30
+ 23.3836
+
2. 7179
+ 23.0669
+
2.5719
/3 1
+ 29.0214
+
4.6913
+ 28.7906
+
4.6124
/3 2
+ 0.3612
-
0.2033
+ 0.4402
-
0.1292
/3 3
- 66.4204
8.5740
- 65.6528
/3 11
1. 9018
- 0.1138
1. 9202
-
0.1191
/3 22
+ 0.2641
-
0.1471
+ 0.2726
-
0.1456
/3 33
+ 42.2807
+
5.6855
+ 41. 8277
+ 5.5165
/3 12
-
0.5007
+ 0.5378
0.4830
+ 0.5439
/3 13
- 26.1217
4.4052
- 25.8660
- 4.3177
/3 23
+ 0.1128
+ 0.6583
+ 0.0060
+ 0.5723
MODEL:
S2 S3S4
Power
=
-
8.2585
2
2
2
/30 + /3 1S2 + /3 2 S3 + /3 3 S4 + /3 11 S2 + /3 22 S3 + /3 33 S4
+ /312 S2 S3 + /313 S2 S4 + /323 S3 S4
where
Difference between Populations 1 and 2
S3
= Stochastic
= Stochastic
S4
= Stochastic
Difference between Populations 1 and 4
S2
".
e
..
Difference between Populations 1 and 3
-_
87
Table 5.12
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n = 6 units per
treatment when comparing all pairs of k = 5 normal treatment popu1a tions
2
2
[l:N(I-LF (J1 = 1)'J 2: N(I-L 2 J (J22 == 1).J 3: N(I-L 3 J (J3 = 1) ;
2
2
4:N(I-L4 J (J4 = 1) J. 5: N(I-L5 J (J5 = 1) ]
ONE-TAIL
.009
(1%)
.255
(25%)
TWO-TAIL
.018
(2%)
REGRESSION
COEFFICIENTS
"-
+ 0.0132
1.6985
+
fJ 1
+ 41. 7737
+
1. 7001
+ 37.8796
fJ 2
+ 4. 7959
-
0.4015
+ 5.6540
fJ 3
+
2.1579
-
0.3362
+
fJ 4
- 36.5599
- 0.8075
- 24.0416
13 11
8.7541
- 0.1994
6. 7820
fJ 22
2.0366
-
0.0791
2.0243
13 33
2.0579
-
0.0976
1. 9298
fJ 44
+ 27.9114
+ 0.7603
+ 20.8251
13 12
+
2.6599
+ 0.9316
+
2.5340
fJ 13
4.0099
+ 0.1636
-
3.6894
fJ 14
- 29.5764
1. 8033
- 28.2409
fJ 23
+ 0.3392
+ 0.3780
+ 0.1743
13 24
3.2812
+ 0.1376
3.9980
3.1074
+ 0.4667
fJ 34
_
2.9968
fJ O
+
+
2. 7393
2.1992
..
continued
88
Table 5.12 (continued)
MODEL:
=
2
~o + ~lS2 + ~2S3 + ~3S4 + ~4S5 + ~llS2
+
~33S~
+
~44S;
+
~12S2S3
+
~13S2S4
+ ~23S3S4 + ~24S3S5 + ~34S4S5
where
.,
=
Stochastic Difference between Populations 1 and 2
=
Stochastic Difference between Populations 1 and 3
=
Stochastic Difference between Popula tions 1 and 4
=
Stochastic Difference between Populations 1 and 5
89
""_
Table 5.13
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n = 10 units per
treatment when comparing all pairs of k = 5 normal treatment populations
[1: N(i-Ll'
2
1
(T
= 1)') 2: N(i-L2 )
4: N(i-L )
4
2
4
(T
2
(T2
= 1); 3: N(i-L 3 )
= 1) ). 5: N(i-L )
5
2
(T5
2
= 1) ;
(T3
= 1) ]
TWO-TAIL
ONE-TAIL
.091
(10%)
.010
(1%)
.161
(20%)
+ 0.8770
+ 5.9196
+ 0.7282
+ 20.2412
+
3.2539
+ 20.1995
+
3.0342
+ 4.4722
-
0.0298
+ 4.4086
-
0.0475
0.5231
-
0.4299
-
-
0.3810
3.8538
- 26.6016
1. 2974
.005
(1%)
REGRESSION
COEFFICIENTS
/3
~-
0
/3
1
/3
2
/3
3
/3
4
/3
11
/3
22
/3
33
/3
44
/3
12
/3
+
-
6.2168
- 27.2497
1.2928
-
0.2132
-
0.5116
-
0.0319
-
-
0.6827
-
0.1745
-
+ 20.5432
+
+
1.5540
+
13
-
/3
14
-
/3
0.4316
3.4081
0.2484
0.5136
-
0.6838
-
0.1717
3.0748
+ 20.1887
+
2.7633
1. 0834
+
1.5704
+
L 1005
0.3352
+ 0.1715
-
0.3520
+ 0.1861
19.5591
3.5220
- 19.5071
3.2646
23
+ 0.3312
+ 0.4334
+ 0.3360
+ 0.4289
/3
24
-
4.6298
-
0.4705
-
4.5726
-
/3
34
+
1. 8437
+ 0.6517
+
1. 7543
+ 0.5885
0.0377
0.4472
...
e
continued
90
Table 5413 (continued)
MODEL:
S2 S3 S4 S5
Power
where
S2
=
Stochastic Difference between Populations 1 and 3
S3
»e
Stochastic Difference between Populations 1 and 2
S4
=
Stochastic Difference between Popu1a tions 1 and 4
S5
=
Stochastic Difference between Populations 1 and 5
91
Table 5.14
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n = 6 units per
treatment when comparing all pairs of k = 3 normal treatment populations
2
= 2); 3:N(iJ,y 0"3 = 2)]
TWO-TAIL
ONE-TAIL
.003
(1%)
.102
(10%)
.186
.006
(1%)
(20io)
REGRESSION
COEFFICIENTS
.-
/30
+172.1916
+
7.1784
+172.5154
+ 4.5023
/3 1
+ 56.5597
+
4.5977
+ 55.5896
+
/3 2
-401. 5460
- 17.4368
-401. 6661
- 11. 1870
/3 11
- 12.1275
1. 4271
- 12.1626
1. 4094
/3 22
+225.6415
+ 9.6583
+225.3829
/3 12
- 39.5676
1.6767
- 38.4798
+
4.4890
6.0365
1. 5801
MODEL:
where
S2 = Stochastic Difference between Populations 1 and 2
S3 = Stochastic Difference between Populations 1 and 3
92
Table 5.15
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n ~ 6 units per
treatment when comparing all pairs of k ~ 3 normal treatment populations
(T
ONE-TAIL
.003
(1%)
2
3
= 8)]
TWO-TAIL
.102
(10%)
.006
(1%)
.186
(20%)
REGRESSION
COEFFICIENTS
130
+ 85.0193
+
6.8808
+ 84.2612
+ 4.8288
13 1
+ 28.9525
+ 0.2689
+ 29.4143
+ 0.5338
13 2
-192.3258
- 13.8541
-190.9891
9.3413
13 11
4.3669
0.6576
4.3847
0.6915
13 22
+106.5111
+
6.6616
+105.9385
+ 4.1494
13 12
- 22.9973
+
1. 6755
- 23.4633
+
~e
1.4581
MODEL:
where
,
e
...
I
S2
= Stochastic
Difference between Populations 1 and 2
S3
= Stochastic
Difference between Populations 1 and 3
93
Table 5.16
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n = 6 units per
treatment when comparing all pairs of k = 3 exponential
treatment populations
TWO_TAIL
ONE-TAIL
.102
(10%)
.003
(1%)
.006
(1%)
.186
(20/0)
REGRESSION
COEFFICIENTS
•
~e
130
+145.9320
+
7.4222
+140.9135
+ 4. 7427
13 1
+ 45.8296
+
3.5942
+ 46.6503
+
13 2
-337.2383
- 17.3235
-326.8149
- 11. 2077
13 11
- 14.2121
1.7271
- 14.1881
1. 5823
13 22
+186.3021
9.1898
+180.9063
+ 5.7808
13 12
- 25.7370
0.2719
- 26.6473
0.5110
+
3.6228
MODEL:
where
S2
= Stochastic
S3
~
Difference between Populations 1 and 2
Stochastic Difference between Populations 1 and 3
94
Table 5.17
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n = 6 units per
treatment when comparing all pairs of k = 3 uniform treatment populations
ONE-TAIL
.003
(1%)
TWO-TAIL
.102
(10%)
.006
(1%)
.186
(20%)
REGRESSION
COEFFICIENTS
'"
~e
~O
+222.9172
+
6.3291
+208.1513
+
2.3587
~1
+ 70.4148
+
6.9437
+ 72.4553
+
6.1501
~2
-511. 7522
- 16.3059
-480.8876
6.6923
~1l
- 11. 2313
1. 5094
- 11. 3832
1. 5085
~22
+285.8383
9.3673
+269.6453
~12
- 55.0759
3.9996
- 57.0288
+
+
3.6152
3.1370
MODEL:
where
S2
= Stochastic
Difference between Populations 1 and 2
Stochastic Difference between Populations 1 and 3
95
5.4
Table 5.18
Approximate Stochastic Differences among Treatment
Populations when Comparing All Pairs
of Treatments at Given Power Levels
Approximate stochastic differences among k = 3 normal
populations in the case of n = 6 units per treatment
when comparing all pairs of treatments at given power
levels
2
2
[1:N(1J. 1, 0"1 = 1) ; 2: N(1J. 2, 0"2
ONE-TAIL
.003
.102
(1%)
(10%)
POWER
S2
2
1) ; 3: N(1J. , 0"3
3
1) ]
TWO-TAIL
.006
.186
(1%)
(20%)
S3
S3
S3
S3
.905
.762
.796
.826
.854
.884
.906
.756
. 795
.828
.859
.887
.50
.60
. 70
.80
.90
.500
.500
.500
.500
.500
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.772
.812
.849
.888
.937
.768
.813
.853
.891
.928
.50
.60
.70
.80
.90
.700
.700
. 700
.700
.700
.772
.814
.855
.898
.956
.767
.815
.859
.900
.941
.50
.60
.70
.80
.90
.800
.800
.800
.800
.800
.806
.847
.888
.937
.806
.850
.891
.931
.50
.60
.70
.80
.90
.900
.900
.900
.900
.900
.906
.906
Are
96
Table 5.19
Approximate stochastic differences among k = 3 normal
populations in the case of n = 10 units per treatment
when comparing all pairs of treatments at given power
levels
2
[1: N(1J. 1 J cr1
= 1) ). 2: N(1J. 2 J cr22 = 1) ). 3: N(lJ.y cr32 = 1) ]
ONE-TAIL
.002
.030
(1%)
(5%)
~e
'"
e
.~
POWER
8
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.50
.60
. 70
.80
.90
.600
.600
.600
.600
.600
.50
.60
.70
.80
.90
.700
.700
. 700
.700
.700
.50
.60
.70
.80
.90
.800
.800
.800
.800
.800
.50
.60
.70
.80
.90
.900
.900
.900
.900
.900
2
TWO-TAIL
.004
.056
(1%)
(5%)
8
3
8
.875
.887
.900
.787
.810
.832
.855
.886
.875
.886
.899
.786
.810
.833
.856
.884
.905
.796
.823
.851
.885
.905
.795
.824
.852
.886
8
3
3
.909
.821
.849
.880
.931
.910
3
.796
.827
.857
.893
.797
.826
.855
.891
.909
8
.821
.850
.881
.926
.910
97
~e
Table 5.20
11
Approximate stochastic differences among k = 3 normal
populations in the case of n = 14 units per treatment
when comparing all pairs of treatments at given power
levels
2
[1: N(1J. ) 0"1
1
= 1) j
2
2: N(1J. 2 ) 0"2
= 1) .
ONE-TAIL
.009
.074
(1%)
(10%)
~e
.
e
/10'
J
2
3: N(1J. ) 0"3
3
= 1)]
TWO-TAIL
.018
.137
(2%)
(15%)
8
8
3
8
3
8
.500
.500
.500
.500
.500
.788
.800
.811
.820
.829
.710
.734
.756
· 777
· 798
. 788
.801
.812
.822
.831
· 707
· 732
· 755
.777
.800
.50
.60
. 70
.80
.90
.600
.600
.600
.600
.600
.807
.828
.851
.712
.742
.770
.799
.839
.807
.828
.852
.709
.740
· 769
.800
.842
.50
.60
.70
.80
.90
.700
.700
. 700
.700
. 700
.811
.836
· 706
.739
.770
.804
.863
.810
.837
· 703
· 736
.769
.804
.864
.50
.60
.70
.80
.90
.800
.800
.800
.800
.800
.50
.60
.70
.80
.90
.900
.900
.900
.900
.900
POWER
8
.50
.60
. 70
.80
.90
2
3
3
.821
.846
.821
.846
.832
.834
98
~e
Table 5.21
1t
Approximate stochastic differences among k = 4 normal
populations in the case of n = 6 units per treatment
when comparing all pairs of treatments at given power
levels
2
2
2
[1: N(/.Lp (J1 = 1) ). 2:N(/.L2) (J2 = 1) ; 3:N(/.L3) (J3
1) ).
=
2
4:N(1J.4' (J4 = 1) J
ONE:..:TAIL
... 179
.007
(20%)
(1%)
.-
_
...
iI
8
TWO-TAIL
.303
.013
(1%)
(30%)
8
4
8
.712
.755
. 794
.833
.877
.908
8
4
POWER
8
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.500
.500
.500
.500
.500
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.600
.600
.600
.600
.600
.700
.747
.793
.840
.899
.675
.741
.798
.848
.893
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.700
.700
.700
.700
.700
.733
. 782
.834
.900
.730
.791
.845
.893
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.800
.800
.800
.800
.800
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.900
.900
.900
.900
.900
2
8
3
4
.900
4
.685
. 746
.798
.843
.883
.933
.834
.885
.816
.881
.901
.908
.929
.954
continued
99
~-
Table 5.21 (continued)
ONE-TAIL
.179
.007
(1%)
(20%)
--
_
8<
8
8
8
TWO-TAIL
.013
.303
(1%)
(30%)
8
8
4
POWER
8
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.721
.774
.826
.882
.959
.691
.754
.809
.855
.896
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.700
. 700
.700
.700
.700
.756
.811
.869
.947
.748
.806
.858
.903
.50
.60
• 70
.80
.90
.600
.600
.600
.600
.600
.800
.800
.800
.800
.800
.847
.919
.854
.902
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.900
.900
.900
.900
.900
.50
.60
.70
.80
.90
.700
.700
. 700
.700
.700
.700
.700
. 700
.700
.700
.50
.60
.70
.80
.90
.700
.700
.700
.700
.700
.800
.800
.800
.800
.800
2
3
4
4
4
.943
.977
.709
.768
.825
.884
.958
.752
.808
.857
.899
.982
.801
.860
.929
.805
.857
.904
continued
ke
100
Table 5.21 (continued)
ONE-TAIL
. 179
.007
(20%)
(1%)
·e
POWER
8
2
8
3
8
4
.50
.60
.70
.80
.90
.700
.700
.700
.700
.700
.900
.900
.900
.900
.900
.978
.50
.60
.70
.80
.90
.800
.800
.800
.800
.800
.800
.800
.800
.800
.800
.983
.50
.60
.70
.80
.90
.800
.800
.800
.800
.800
.900
.900
.900
.900
.900
.944
.976
.50
.60
.70
.80
.90
.900
.900
.900
.900
.900
.900
.900
.900
.900
.900
.926
.942
.957
.971
.986
8
4
TWO-TAIL
.013 .
.303
(30%)
(1%)
8
4
8
4
.948
.973
1.000
.903
.805
.863
.926
.959
.982
1.000
.939
.957
.973
.987
1.000
.925
.939
.951
.960
.968
.803
.853
.897
.900
101
~e
Table 5.22
Approximate stochastic differences among k ~ 4 normal
populations in the case of n = 10 units per treatment
when comparing all pairs of treatments at given power
levels
2
2
2
[l:N(lJ.l' 0"1 = 1); 2: N(1J.2 J 0"2 = 1) ; 3: N(1J.3 J 0"3 = 1)·J
2
4: N(1J.4 J 0"4 = 1) ]
ONE-TAIL
.003
.060
(10%)
(1%)
.(
te
~
~
TWO-TAIL
.005
.108
(10%)
(1%)
4
8
4
.758
.788
.816
.844
.874
.871
.884
.896
.907
.920
.756
.786
.815
.843
.874
.866
.880
.893
.905
.920
.743
.777
.807
.838
.874
.866
.880
.893
.905
.920
.741
.775
.806
.837
.873
.700
.700
.700
. 700
.700
.861
.877
.891
.905
.922
.729
. 764
.797
.830
.869
.861
.877
.891
.905
.923
.728
.763
.797
.830
.869
.500
.500
.500
.500
.500
.800
.800
.800
.800
.800
.858
.875
.890
.906
.929
.821
.861
.858
.875
.890
.907
.929
.821
.862
.500
.500
.500
.500
.500
.900
.900
.900
.900
.900
POWER
8
8
3
8
4
8
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.500
.500
.500
.500
.500
.871
.884
.896
.907
.920
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.600
.600
.600
.600
.600
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.50
.60
.70
.80
.90
.50
.60
.70
.80
.90
2
.909
4
8
.910
continued
102
"e
Table 5.22 (continued)
ONE-TArL
.003
.060
(10%)
(1%)
·e
..
.If
TWO-TAIL
.005
.108
(10%)
(1%)
8
4
8
.768
.805
.840
.875
.918
.903
.923
.945
.766
.804
.839
.875
.918
.751
.790
.827
.864
.908
.895
.915
.936
.750
. 789
.826
.864
.908
POWER
8
2
8
3
8
4
8
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.903
.923
.945
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.700
.700
. 700
.700
. 700
.895
.915
.936
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.800
.800
.800
.800
.800
.888
.909
.929
.958
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.900
.900
.900
.900
.900
.50
.60
.70
.80
.90
.700
.700
.700
.700
. 700
.700
.700
.700
.700
.700
.912
.931
.950
.971
.50
.60
.70
.80
.90
.700
.700
.700
.700
.700
.800
.800
.800
.800
.800
.903
.923
.942
.962
.989
4
.814
.852
.897
.888
.909
.930
.958
4
.814
.853
.897
.904
.925
.953
.904
.925
.952
. 767
.808
.846
.883
.924
.832
.870
.912
.912
.931
.950
.971
.903
.923
.942
.962
.989
.766
.807
.845
.883
.924
.831
.870
.912
continued
.-
104
Table 5.23
"
*
Approximate stochastic differences among k == 5 normal
popula tions in the case of n == 6 units per treatment
when comparing all pairs of treatments at given power
levels
2
2
2
[l:N(!J. l , (Jl == 1) j 2: N(!J.2) (J2 == 1) j 3: N(!J.3' (J3 == l)j
2
2
4: N(!J.4' (J4 == 1) j 5:N(!J.S ' (J5 == 1) ]
ONE-TAIL
.255
.009
(25%)
(1%)
~e
"-
e
..
8
8
5
8
5
8
5
.892
.670
.720
.767
.814
.867
.898
.929
POWER
8
.50
.60
. 70
.80
.90
.500
.500
.500
.500
.500
.500
.500
.500
.500
.500
.500
.500
.500
.500
.500
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.500
.500
.500
.500
.500
.600
.600
.600
.600
.600
.640
.694
.746
. 799
.862
.50
.60
. 70
.80
.90
.500
.500
.500
.500
.500
.500
.500
.500
.500
.500
.700
.700
.700
.700
.700
.. 721
.778
.848
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.500
.500
.500
.500
.500
.800
.800
.800
.800
.800
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.500
.500
.500
.500
.500
.900
.900
.900
.900
.900
2
8
3
4
TWO-TAIL
.018
(2%)
.898
.886
.824
continued
105
~e
Table 5.23
(continued)
ONE-TAIL
.255
.009
(25%)
(1%)
.-
_
8
5
8
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.628
.686
. 743
.803
.878
.50
.60
. 70
.80
.90
.500
.500
.500
.500
.~OO
.600
.600
.600
.600
.600
.700
.700
.700
. 700
.700
.717
.781
.860
.50
.60
. 70
.80
.90
.500
.500
.500
.500
.500
.600
.600
.600
.600
.600
.800
.800
.800
.800
.800
.835
.50
.60
.70
.80
.90
,500
.500
.500
. .500
.500
.600
.600
.600
.600
.600
.900
.900
.900
.900
".900
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.700
.700
.700
. 700
. 700
.700
.700
.700
.700
. 700
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.700
.700
.700
.700
.700
.800
.800
.800
.800
.800
2
8
8
5
POWER
3
84
TWO-TAIL
.018
(2%)
8
5
.711
.779
.864
.944
.838
'"
..
continued
106
~e
Table 5.23 (continued)
ONE-TAIL
.255
.009
(25%)
(1%)
8
5
8
5
POWER
8
8
3
8
4
.50
.60
,70
.80
.90
.500
.500
,500
.500
.500
. 700
.700
.700
.700
.700
.900
.900
.900
.900
.900
.50
.60
.70
,80
.90
.500
.500
.500
.500
.500
.800
.800
.800
.800
.800
.800
.800
.800
.800
.800
.50
.60
,70
.80
,90
.500
.500
.500
.500
.500
.800
.800
.800
.800
.800
.900
.900
.900
.900
.900
.50
.60
,70
.80
,90
.500
.500
.500
.500
,500
.900
.900
.900
.900
.900
.900
.900
.900
.900
.900
.50
,60
.70
.80
.90
,600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.686
.747
.806
.865
.930
.50
.60
.70
,80
.90
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.700
.700
.700
. 700
. 700
.717
.778
.840
.909
2
'!WO-TAIL
.018
(2%)
8
5
.912
.905
.835
~
~e
...
e
"If
.919
.915
.927
continued
e
...
107
Table 5.23 (continued)
TWO-TAIL
ONE-TAIL
~e
.
e
-.l
.009
(1%)
.255
(25%)
.018
(2%)
8
5
8
5
8
5
POWER
8
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.800
.800
.800
.800
.800
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.900
.900
.900
.900
.900
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.700
.700
.700
.700
. 700
.700
.700
. 700
.700
. 700
.708
.773
.839
.914
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.700
.700
.700
.700
.700
.800
.800
.800
.800
.800
.811
.886
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
. 700
.700
.700
.700
.700
.900
.900
.900
.900
.900
.950
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.800
.800
.800
.800
.800
.800
.800
.800
.800
.800
.980
2
8
3
8
4
.814
.884
.955
.805
.884
continued
108
~e
c
Table 5.23 (continued)
ONE-TAIL
.009
.255
(25%)
(1%)
~e
....
e
'"
8
5
TWO-TAIL
.018
(2%)
8
5
POWER
8
2
8
3
8
4
8
5
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.800
.800
.800
.800
.800
.900
.900
.900
.900
.900
.959
.936
.980
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.900
.900
.900
.900
.900
.900
.900
.900
.900
.900
.926
.918
.952
.990
.50
.60
.70
.80
.90
.700
.700
.700
.700
.700
.700
.700
.700
.700
.700
.700
.700
.700
.700
.700
.50
.60
.70
.80
.90
.700
.700
.700
.700
.700
.700
.700
.700
.700
.700
.800
.800
.800
.800
.800
l.000
.50
.60
.70
.80
.90
.700
.700
.700
.700
.700
.700
.700
.700
.700
.700
.900
.900
.900
.900
.900
.950
.993
.942
.966
.988
1.000
.50
.60
.70
.80
.90
.700
.700
.700
.700
.700
.800
.800
.800
.800
.800
.800
.800
.800
.800
.800
.988
.966
.994
1.000
LOOO
.739
.802
.862
.924
.974
LOOO
.834
.897
.830
.897
continued
'_
--
"\
e
'1l,
111
Table 5.24 ( continued)
ONE-TAIL
.• 005
.091
(1%)
(10%)
TWO-TAIL
.010
.161
(1%)
(20%)
POWER
8
8
3
8
4
8
5
8
5
8
5
8
5
,50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.871
.890
.909
.941
.722
.757
.789
.822
.860
.871
.890
.909
.942
.717
.753
.787
.822
.863
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.600
.600
.600
.600
.600
.700
.700
.700
.700
.700
.863
.883
.903
. 704
.740
.775
.809
.851
.862
.883
.904
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.600
.600
.600
.600
.600
.800
.800
.800
.800
.800
.850
.870
.890
.915
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.600
.600
.600
.600
.600
.900
.900
.900
.900
.900
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.700
.700
.700
.700
.700
.700
.700
. 700
.700
.700
.868
.890
.912
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
. 700
.700
.700
.700
.700
.800
.800
.800
.800
.800
.856
.878
.899
.927
2
.737
.773
.810
.854
.850
.871
.891
.916
.838
.835
.921
.919
.701
. 740
.777
.814
.859
.868
.890
.912
.736
.775
.815
.862
.856
.878
.900
.927
.844
.847
continued
112
"e
Table 5.24 (continued)
ONE-TAIL
.091
.005
(1%)
(10%)
,
8
5
4
8
5
· 700
.700
· 700
.700
.700
.900
.900
.900
.900
.900
.904
.943
.905
.944
.500
.500
.500
.500
.500
.800
.800
.800
.800
.800
.800
.800
.800
.800
.800
.858
.880
.902
.926
.858
.880
.902
.927
.50
.60
· 70
· 80
· 90
.500
.500
.500
.500
.500
.800
.800
.800
.800
· 800
.900
.900
.900
.900
.900
.908
.936
.908
.937
.50
· 60
· 70
.80
· 90
.500
.500
.500
.500
.500
.900
.900
.900
.900
.900
.900
.900
.900
.900
.900
.906
.929
.907
.929
.50
• 60
· 70
· 80
· 90
· 600
· 600
· 600
· 600
· 600
· 600
· 600
.600
· 600
· 600
· 600
· 600
· 600
· 600
.600
.914
.936
. 960
· 764
· 803
· 840
.878
.926
.914
.936
. 960
· 759
.799
.838
· 878
.927
.50
· 60
.70
.80
.90
· 600
.. 600
.600
.600
.600
· 600
.600
.600
.600
.600
.700
· 700
.700
.700
.700
. 903
.926
.951
· 743
· 784
.823
.862
.911
. 904
.927
.951
· 738
· 780
.821
.862
.913
POWER
8
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.50
.60
.70
.80
.90
2
8
3
8
8
5
TWO-TAIL
.010
.161
(1%)
(20%)
.802
.850
8
5
.802
.853
I'f
"'e
~
e
?\
continued
114
~e
Table 5.24 (continued)
ONE-TAIL
.005
.091
(1%)
(10%)
~e
f'f
e
...
8
5
TWO-TAIL
.010
.161
(1%)
(20%)
8
5
POWER
8
2
8
3
8
4
8
5
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.800
.800
.800
.800
.800
.900
.900
.900
.900
.900
.920
.941
.965
.920
.941
.965
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.900
.900
.900
.900
.900
.900
.900
.900
.900
.900
.917
.9,36
.956
.917
.937
.957
.50
.60
.70
.80
.90
.700
.700
.700
.700
.700
.700
.700
.700
.700
· 700
.700
.700
.700
.700
. 700
.918
.937
.954
.970
.985
.50
.60
.70
.80
.90
.700
.700
.700
.700
.700
.700
· 700
.700
.700
.700
.800
.800
.800
.800
.800
.905
.926
.944
.960
.975
.50
.60
.70
.80
.90
· 700
.700
· 700
.700
· 700
· 700
.700
· 700
.700
.700
.900
.900
.900
.900
.900
.913
.931
.947
.962
.913
.931
.947
.962
.50
.60
.70
.80
.90
.700
.700
.700
.700
.700
.800
.800
.800
.800
.800
.800
.800
.800
.800
.800
.903
.924
.942
.958
.973
.903
.924
.942
.959
.973
8
5
.761
.806
.847
.888
.931
.918
.938
.955
.970
.985
.756
.802
.845
.887
.931
.827
.869
.912
.905
.926
.944
.960
.975
.825
.868
.912
.826
.869
.915
.824
.869
.916
continued
·116
5.5
rhe Null Distribution of m tqhen Comparing Treatments
Oi
Versus a Control: One-tailed Critical Region
e
~
Table 5.25
..
oil
e
~
2
3
4
5
6
7
8
9
10
11
12
13
14
15
n
.267
.088
.027
.007
.002
.001
.000
.000
.000
.000
.000
.000
.000
.000
.370
.175
.071
.026
.010
.003
.001
.000
.000
.000
.000
.000
.423
.234
.111
.049
.021
.008
.003
.001
.000
.000
.454
.274
.152
.071
.035
.015
.006
.002
.480
.310
.182
.108
.050
.026
.496
.335
.209
.128
n-2
n-3
n-4
n-5
n-6
e
The null distribution of m when comparing two treatments versus a control with
Oi
n units per treatment~ one_tailed critical region with an experiment-wise error
rate
Critical
Values
(m*)
n-l
.ol,
Number of Units Per Treatment (n) Equal
to~
.361
.....
.....
......
e
~
Table 5.26
Critical
Values
(m*)
n
n-l
n-2
n-3
n-4
n-5
~
"
e
~
AI
e
The null distribution of m when comparing three treatments versus a control with
Oi
n units per treatment: one_tailed critical region with an experiment_wise error
rate
Number of Units Per Treatment (n) Equal to:
2
.511
3
.175
4
5
6
7
8
9
10
11
12
13
14
15.
.053
.015
.004
.001
.000
.000
.000
.000
.000
.000
.000
.000
.344
.142
.050
.020
.006
.002
.001
.000
.000
.000
.000
.454
.222
.099
.038
.015
.006
.001
.001
.000
.275
.151
.070
.031
.012
.004
.359
.215
.100
.052
.410
.254
~
~
ex>
e
~
Table 5.27
n
n_l
n-2
n-3
n_4
n-5
n-6
e
'""
The null distribution of m when comparing tour treatments versus a control with
Oi
n units per treatment~ one_tailed critical region with an experiment-wise error
rate
Number of Units Per Treatment (n) Equal to:
Critical
Values
(Iri*)
t
~
e
~
2
3
4
5
6
7
8
9
10
11
12
13
14
15
.336
.119
.036
.011
.004
.001
.000
.000
.000
.000
.000
.000
.000
.000
.457
.226
.098
.036
.015
.004
.000
.000
.000
.000
.000
.000
.300
.151
.068
.030
.013
.004
.002
.001
.000
.345
.198
.108
.050
.023
.011
.004
.387
.237
.133
.076
.036
.417
.275
.161
.437
I-'
I-'
1.0
120
5.6
The Null Distribution of m when Comparing Treatments
Oi
Versus a Control: Two-tailed Critical Region
..
e
e
""
Table 5.28
,
.-
•
...
The null distribution of m when comparing two treatments versus a control with n
Oi
units per treatment: two-tailed critical region with an experiment-wise error rate
Number of Units Per Treatment (n) Equal to:
Critical
Values
(m*)
u
(m~)
2
3
4
5
6
n
0
.619
.234
.072
.022
.007
n-l
1
.437
.196
n-2
2
n-3
3
n-4
4
n-5
5
10
11
12
13
14
15
.000
.000
.000
.000
.000
.000
.000
.030
.008
.001
.001
.000
.000
.000
.000
.300
.136
.033
.026
.009
.004
.001
.001
.192
.215
.100
.046
.022
.009
.460
.234
.151
.072
8
9
.002
.000
.073
7
.315
....
I'-:l
t--'
,j
e
e
Table S~29
~rea~~:
Critical
Values
(v{>
n
0
11-1
1
11-2
2
Il-3
3
n-4
4
n-5
5
e
The DUll dis~ribu~ion of UUi when comparing three treabments versus a control with n
units per
(.~
..,
'l:
'"
two-tailed critical region with an experiment-wise error rate
lfumber of Units Per Treabment (n) Equal to:
.2
~381
3
4
5
6
7
8
9
10
11
12
13
14
15
.144
.047
.015
.004
.001
.000
.000
~OOO
.000
.000
.000
.000
.000
.271
.118
.050
.019
.006
.001
.000
.000
.000
.000
.000
.343
.183
.086
.021
.013
.006
.002
.001
.001
.400
.121
.123
.064
.029
.013
.005
.438
.278
.162
.090
.044
.500
.....
N
N
>
e
,
e
;>
Table 5.. 30
Number of Units Per Treatment (n) Equal to:
<n{.>
2
3
4
5
6
n
0
.690
.282
.093
.030
.007
n_l
1
.233
n-2
2
n-3
3
n-4
4
11.-5
5
u
e
The null distribution of m when comparing four treatments versus a control with n
Oi
units per treatment: two_tailed critical region with an experiment-wise error rate
Critical
Values
(m*)
.,
""
8
9
10
11
12
13
14
15
.002
.000
.000
.000
.000
.000
.000
.000
.000
.100
.037
.013
.004
.000
.000
.000
.000
.000
.358
.170
.083
.027
.011
4003
.002
.001
.466
.244
.128
.058
.025
.0lD
.529
.318
.179
.088
7
.385
I-'
N
W
124
5.7
Table 5.31
Regression Coefficient Estimates for the
Inverse Quadratic Power Response Surface
when· 'Comparing Treatments'
Versus a Control
Regression coefficient estimates for the inverse quadratic
power response surface in' theo.caseof n:= 6 units
per treatment when comparing two normal treatment populations versus a control
[O:N(i-LO'
2 2 2
1); I:N(i-L l , 0"1
1); 2:N(i-L2' CT2
0
0-
=
=
ONE_TAIL
= l)J
TWO-TAIL
.002
(1%)
.071
(10%)
.004
(1%)
.142
(15%)
130
+299.3523
+ 11. 9063
+293.6441
+ 7.2199
13 1
- 23.2622
1. 4643
- 20.3554
+ 1. 3772
13 2
-626.0419
- 22.8989
-615.4708
- 14.2957
13 11
- 15.8399
1. 9706
- 15.9439
2.1418
13 22
+318. 7619
+ 9.8235
+313.9633
+ 5.9252
13 12
+ 51. 7080
+ 5.6391
+ 48.6970
+ 2.7817
REGRESSION
COEFFICIENTS
..
MODEL:
where
::; Stochastic Difference between "Control" and "Treat_
ment 1"
..
= Stochastic Difference between "Control" and""Treat_
ment 2"
125
Table 5.32
...
Regression coefficient estimates for the inverse quadratic
power response surface in the . case of. n ==10 units·
per treatment when comparing two normal treatment populations versus a control
==
2
1); 2: N(1-L2' (J2 == 1)]
ONE-TAIL
REGRESSION
COEFFICIENTS
/30
'
TWO-TAIL
.001
.021
.002
.038
(1%)
(2%)
(1%)
(5%)
+209.5738
+ 25.9225
+209.0769
+ 22.6194
- 14.4273
3.7926
- 13.7832
1.1589
-442.0614
- 52.3276
-441. 5132
- 46.9109
- 10.7348
2.2771
- 10.7566
2.1488
+226.4492
+ 24.9220
+226.4354
+ 23.1015
+ 34.4487
+
+ 33. 7738
+ 5.6179
....
/3 11
8.7970
MODEL:
where
Sl == Stochastic Difference between "Control" and "Treatment 1"
S2 == Stochastic Difference between "Control" and "Treatment 2"
."
126
Table 5.33
Regression coefficient estimates for the inverse quadratic
power response surface in the. case of .. n·= 14 units·;
per treatment when comparing two normal treatment populations versus a control
= 1);
2
2:N(~2J ~2
ONE-TAIL
= 1)]
TWO-TAIL
.006
.050
.012
(1%)
(5%)
(1%)
.100
(10%)
+ 54.2452
+ 12.7795
+
-
REGRESSION
COEFFICIENTS
13 0
+ 54.4148
+ 14.1291
+ O. 7905
1. 4535
-122. 7781
- 30.7700
-122.6529
- 28.4146
6.2019
1. 7944
6~2785
1.8673
+ 65.8096
+ 15.7459
+ 65.8342
+ 14. 7606
+ 9.7680
+ 5.4049
+
+ 4.4105
1. 0794
9.5687
MODEL:
where
Sl = Stochastic Difference between "Control" and "Treatment 1 11
S2 = Stochastic Difference between "Control" and "Treatment 2 11
0.4664
127
Table 5.34
Regression coefficient estimates for the inverse quadratic
power response surface in the. case of n: =. 6 units .. '
per treatment when comparing three normal treatment populations versus a control
[O:N(!J.O}
2
2
O = 1) ; l:N(!J.p
eT
3: N(!J.3}
eT
eT
1
= 1) ; 2: N(!J.2}
2
2
= 1) ;
2
= 1) ]
3
TWO-TAIL
ONE-TAIL
.098
(10%)
.004
(1%)
eT
.007
(1%)
.196
(20%)
+ 71. 8772
+ 0.8014
+
6.4132
+ 0.0001
REGRESSION
COEFFICIENTS
....
.-_
130
+107.2291
+
fJ 1
+ 12. 7362
+ 0.3290
13 2
+ 10.3801
0.2315
+ 20.3132
+ 0.7413
13 3
-242. 7483
9.2002
-165.1047
1. 2231
13 11
6.4210
0.5122
6.2481
0.5251
13 22
6.2307
0.8392
6.0856
0.7563
13 33
+130.5087
+
3.9894
+ 87.8302
0.3994
13 12
+ 0.6954
+ 0.8259
+ 0.3464
+ 0.6937
13 13
3.3324
+ 0.5098
+
3.6072
+ 1. 0370
13 23
1. 3549
+ 1.5806
- 12.1363
+ 0.4788
4.4818
MODEL:
+ 1312S1S2 + 1313S1S3 + 1323S2S3
where
e·
8 = Stochastic Difference between "Control" and "Treatment 1"
1
S2 = Stochastic Difference between "Control" and "Treatment 2"
83
= Stochastic
Difference between "Control" and "Treatment 3"
re
128
Table 5.35
Regression coefficient estimates for the inverse quadratic
power response surface in the caseoL n = 10 units'
per treatment when'comparing three normal treatment populations versus a control
2
[O:N(j.LO' 0"0
= l)j
2
l:N(j.Ll' 0"1
2
3:N(j.L3' 0"3
= 1) j
2
2: N(j.L2' 0"2
= 1) ]
TWO-TAIL
ONE-TAIL
.198
(20%)
.030
(3%)
= 1) j
.033
(3%)
.192
(20%)
REGRESSION
COEFFICIENTS
fre
2.5258
+
7.6740
+ 1.6862
+ 0.8620
0.6816
+
1. 1634
~2
1. 2638
O. 7932
0.9756
-
~3
- 17.3934
4.7401
- 15.9619
0.0182
0.3189
0.1516
0.6178
~O
+ 8.5271
~1
~11
0.3597
~22
0.6394
~33
+ 8.4741
~12
+
+
-
0.3198
0.5290
3.3340
+
0.0442
0.1387
2.0707
+
7.9136
+
1. 2002
+ 0.9235
+
1.1301
+ 0.8482
~13
0.6268
+ O. 7676
0.9406
+ 0.3604
~23
+ 2.1764
1. 8736
+ 0.8235
+
+
1. 0858
+
MODEL:
where
SI = Stochastic Difference between "Control" and "Treatment I"
S2 = Stochastic Difference between "Contro 1" and "Treatment 2"
S3 = Stochastic Difference between "Contro I" and "Treatment 3"
1. 5391
129
Table 5.36
Regression coefficient estimates for the inverse quadratic
power response surface . in the caseo! -- n : = 6 unt ts -•..
per treatment when comparing four normal treatment populations versus a control
2 2 2
[O:N(lJ,O) 0"0 = 1); l:N(lJ,p 0"1 = 1); 2:N(j.L2) 0"2 = 1);
2
2
3:N(j.L3) 0"3 = 1); 4:N(j.L4) 0"4 = 1))
ONE-TAIL
.004
(1%)
TWO-TAIL
.118
(15%)
.007
(1%)
.233
(25%)
REGRESSION
COEFFICIENTS
~e
~O
+ 50.0183
+
1. 9349
+ 37.7603
~1
+ 35.4735
+ 0.6535
+ 34.2332
+
1. 5672
~2
-
0.3842
-
2.4081
-
0.1754
~3
- 0.9731
0.7769
+
3.2963
-
0.6552
~4
-127.0952
3.7501
-103.3254
+ 0.3931
~11
- 20.5078
-
0.6030
- 17.9516
~22
- 4.0416
- 0.2313
~33
-
5.3584
-
0.5390
-
~44
+ 63.2618
+
1.3249
+ 52.9807
-
/3 12
+
2.2495
+ 0.6678
+
2.1951
+ 0.6586
/3 13
-
5.6485
+ 0.566'3
-
5,>864'5
+ 0.5696
~14
- .5.3870
7.3150
1. 5733
~23
+
2.4317
+ 0.4371
+
2.4022
+ 0.2972
~24
+
8.1415
+ 0.4426
+ 5.6816
+ 0.2774
~34
+ 11. 3683
4.5609
-
+
0.2680
1. 3672
+
3.9190
5.3497
6.8492
0.2317
+
0.3991
0.1766
0.3658
0.4771
1.0560
,
e
'"
continued
re
130
Table 5.36 (continued)
MODEL:
where
·~e
'I
8
1
::;::
Stochastic Difference between "Control" and "Treatment 1"
S2
::;::
Stochastic Difference between "Control" and "Treatment 2"
S3
::;::
Stochastic Difference between "Control" and "Treatment 3"
S4
::;::
Stochastic Difference between "Control" and "Treatment 4"
132
Table 5.37
(continued)
MODEL:
Stochastic Difference between "Control" and "Treatment 1"
Stochastic Difference between "Control" and "Treatment 2"
Stochastic Difference between "Control" and "Treatment 3"
Stochastic Difference between "Control" and "Treatment 4"
133
Table 5.38
Regression coefficient estimates for the inverse quadratic
power response surface in the. case of n == 6 untts'
per treatment when comparing two normal treatment popu1a~
tions versus a control
=
2);
2
2:N(~2J ~2 =
ONE-TAIL
2)]
TWO-TAIL
.002
(1%)
,071
(10%)
.004
(1%)
.142
(15%)
~o
+228,6699
+ 11. 7124
+226.1976
+ 9.7993
~1
+
-
0.3343
+ 8.2694
0.0181
-483.6782
- 23.1633
-478,8503
- 19.1060
- 10,4882
1. 3765
- 10.5045
1.4266
~22
+250,7723
+ 10. 7486
+248.4300
+ 8.5404
~12
+ 9.4561
+
3.3949
+ 8.5404
+
REGRESSION
COEFFICIENTS
7,3910
MODEL:
where
8
1
= Stochastic
Difference between "Control" and "Treatment 1"
S2 = Stochastic Difference between "Control" and "Treatment 2"
3.1609
134
Table 5,39
Regression coefficient estimates for the inverse quadratic
power response surface in the case .of n.::: 6 units:
per treatment when comparing two normal treatment populations versus a control
ONE-TAIL
,071
.002 .
(1%)
(10%)
TWO-TAIL
,004
,142
(1%)
(15%)
REGRESSION
COEFFICIENTS
+ 96,2831
+
8.2770
+ 94.5705
+
6,0281
+ 19,4569
1. 6377
+ 20.3943
0.0398
-209,1321
- 15.3272
-206.0269
- 11. 5104
....
4.0226
-
0.5421
4.0804
0.6632
+111. 9494
+
6.7165
+110.5514
+ 5.1440
- 13.5401
+
3.5277
- 14.4491
+
MODEL:
where
Sl ::: Stochastic Difference between IIContro1 11 and IlTreatment 1"
S2 ::: Stochastic Difference between IIContro1 1l and IlTreatment 2 1l
1. 9953
135
Table 5.40
Regression coefficient estimates for the inverse quadratic
power response surface in the case of n= 6 units:
per treatment when comparing two exponential treatment
populations versus a control
[0 : EXP ( 01 ) ; 1:EXP(Ot ); 2:EXP(Ot )]
1
0
2
ONE-TAIL
TWO-TAIL
.142
(15%)
.002
(1%)
.071
(10%)
.004
(1%)
Po
+206.7965
+ 10.2484
+199.6620
+
PI
+ 0.8453
0.9899
+
3.0662
+ 0.5905
P2
-433.6376
- 19.4922
-419.5627
- 15.3979
Pn
- 11. 1051
1. 4414
- 11. 2810
1. 4833
P22
+222.0258
+
8.4815
+215.0257
+
6.8171
P12
+ 17.0378
+
4.1465
+ 14.9462
+
2.5278
REGRESSION
COEFFICIENTS
--
MODEL:
where
Sl = Stochastic Difference between "Control" and "Treatment 1"
S2 = Stochastic Difference between "Control" and "Treatment 2"
7.8471
136
Table 5.41
Regression coefficient estimates for the inverse quadcatic
power response surface in the case of n= 6 units"
per treatment when comparing two uniform treatme~t popula_
tions versus a control
ONE-TAIL
TWO-TAIL
.002
(1%)
.071
(10%)
.004
(1%)
.142
(15%)
+361. 2332
+ 11. 3657
+338.7452
+ 10.4742
- 12.5075
-
0.5782
2.1581
+ 1.5742
--747.9044
- 20.7071
-706.5016
- 20.6629
8. 7063
1. 2624
8.9858
1.4012
+382.7040
+ 8.6152
+363.9856
+ 9.4994
+ 28.4907
+
+ 17.8710
+ 1. 4214
REGRESSION
COEFFICIENTS
130
3.5006
MODEL:
where
Sl = Stochastic Difference between "Control" and "Treatment
S2
= Stochastic
IIi
Difference between "Control" and "Treatment 2"
137
5.8
Approximate Stochastic Differences among Treatment
Populations when Comparing Treatments Versus ,a
Control at Given PowerPLevels
"
Table 5.42
Approximate stochastic differences among k = 3 normal
populations in the case of n = 6 units per treatment when
comparing twa treatments versus a control at given power
levels
ONE.. TAIL
.071
.ooz.
(1%)
'
.
,.
(10%)
TWO_rAIL
.004
(1%)
.,142
(1;5%)
POWER
81
.50
.70
.80
.90
.500
.500
.500
.500
.500
.820
.851
.877
.902
.925
.809
.844
.873
.898
.922
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.807
.844
.879
.915
.958
.801
.843
.879
.914
.947
.50
.60
.70
.80
.90
.700
.700
,700
.700
.700
.784
.826
.868
.• 914
.783
.830
.872
.913
.957
.50
.60
.70
.80
.90
.800
.800
.800
.800
.800
.50
.900
.900
,.900
.900
.900
.60
.60
.70
.80
.90
8
Z
8
2
8
2
8
2
.840
.890
.805
.850
.895
.943
.907
.905
139
·e
Table 5,44
Approximate stochastic differences among k = 3 normal
popu1a tions in the case of n = 14 units per treatment
when comparing two treatments versus a control at given
power levels
2
2
[O:N(I-LO' (TO = 1); 1: N(1-L 1 , (T1
2
= 1) ; 2: N(1-L2, (T2 = 1) ]
ONE-TAIL
,006
,050
(1%)
(5%)
~e
..
e
..
'
8
POWER
81
8
.50
.60
.70
.80
.90
.500
,500
.500
.500
.500
,822
,836
,849
,861
.875
. 752
.778
.802
.825
.850
.821
.835
.848
,860
.875
· 745
.772
· 796
.820
.846
.50
.60
. 70
.80
.90
.600
.600
.600
.600
.600
,825
.846
.871
. 736
.766
.794
.825
.871
.825
,846
.871
.73L
.762
.792
.823
,872
.50
,60
.70
,80
.90
.700
.700
. 700
,700
.700
.816
,839
.871
.713
.744
.775
.808
,861
.816
.839
.872
.710
· 743
.775
.809
.868.
.50
,60
.70
.80
.90
.800
.800
,800
,800
,800
.50
,60
, 70
,80
.90
.900
,900
.900
.900
.900
2
8
TWO-TAIL
.012
.100
(1%)
(10%)
2
2
8
2
,816
.837
.872
.816
,837
.871
.814
.820
-_
140
Table 5.45
Approximate stochastic differences among k = 4 normal
populations in the case of n = 6 units per treatment
when comparing three treatments versus a control at given
power levels
2
[O:N(IJ,O' 0"0
=:
2
1); 1: N(lJ,l' 0"1
=:
1); 2:N(1J. , 0"2
2
2
2
3:N(1J,3' 0"3
=:
1)]
ONE-TAIL
.004
.098
( 10'70)
(1%)
,.
~-
_
•
TWO-TAIL
.196
.007
(1%)
(20%)
POWER,
81
8
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.500
.500
.500
.500
.500
.813
.848
.879
.908
.937
.798
.842
.875
.901
.922
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.600
.600
.600
.600
.600
.800
.841
.880
.919
.967
.789
.839
.878
.909
.935
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.700
.700
.700
.700
.700
.780
.826
.869
.916
.986
.771
.827
.872
.908
.937
.50
.60
.70
.80
·.90
.500
.500
.500
.500
.500
.800
.800
.800
.800
.800
.801
.846
.895
.966
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.900
.900
.900
.900
.900
2
8
3
8
1)'
'
=:
3
8
3
8
3
.946
.806
.856
.896
.929
.942
.960
.986
.916
.911
continued
Ae
141
Table 5.45 (continued)
,
ONE-TAIL
.004
.098
(10%)
(1%)
~e
-"
e
<I-
8
8
8
TWO-TAIL
.196
.007
(20%)
(1%)
8
8
POWER
8
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.788
.833
.877
.923
.987
.771
.831
.878
.916
.947
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.700
.700
.700
.700
.700
.766
.815
.864
.917
LOOO
.749
.815
.868
.911
.947
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.800
.800
.800
.800
.800
.839
.894
.981
.849
.897
.937
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.900
.900
.900
.900
.900
.925
.917
.50
.60
.70
.80
.90
.700
.700
.700
.700
.700
.700
.700
.700
.700
.700
.750
.801
.852
.908
1.000
.722
.796
.857
.908
.951
.50
.60
.70
.80
.90
.700
.700
.700
.700
.700
.800
.800
.800
.800
.800
.827
~ 884
.970
.835
.890
.938
1
2
3
3
3
3
.962
continued
Ae
142
Table 5,45 (continued)
ONE-TAIL
.098
.004
(10%)
(1%)
~e
':
.".
8
8
TWO-TAIL
.196
.007
(20%)
(1%)
8
8
POWER
81
8
.50
,60
,70
,80
,90
.700
.700
, 700
,700
.700
.900
.900
.900
.900
.900
,919
,915
.50
.60
.70
.80
.90
.800
,800
.800
.800
.800
.800
.800
.800
.800
.800
.811
.868
.945
.815
.877
.931
.50
.60
.70
.80
,90
,800
.800
,800
.800
.800
.900
.900
.900
,900
.900
,50
.60
.70
.80
,90
,900
,900
.900
.900
.900
.900
.900
.900
.900
,900
2
3
3
3
3
.969
.953
.906
.902
.939
.929
.960
144
·e
Table 5.46 (continued)
.
ONE-TAIL
.030
.198
(20%)
(3%)
,e
T
e
..
.
8
8
8
TWO-TAIL
.033
.192
(3%)
(20%)
8
POWER
8
.50
.60
.70
.80
.90
.600
.600
,600
.600
.600
.600
.600
.600
.600
.600
.810
.842
.871
.900
.934
.50
.60
.70
.80
.90
,600
.600
.600
.600
.600
· 700
· 700
· 700
,700
.700
, 793
.828
.860
.894
.938
,50
.60
.70
,80
.90
.600
.600
.600
.600
.600
.800
.800
.800
.800
.800
.810
.844
.880
.927
.50
.60
.70
.80
.90
,600
,600
.600
.600
.600
.900
.900
.900
.900
.900
.902
.903
.50
.60
.70
.80
.90
.700
.700
.700
. 700
.700
, 700
.700
.700
.700
.700
. 784
.820
.854
.888
.930
.782
.820
.854
.889
.931
.50
,60
.70
.80
,90
,700
.700
.700
.700
.700
,800
.800
.800
,800
.800
.803
.839
,876
.922
1
2
3
3
3
8
3
.633
.683
. 732
.781
.840
.807
.839
.869
.900
.935
.653
. 706
.762.828
. 708
.761
.824
.826
.859
.894
.938
.742.813
.803
.809
.844
.880
.926
.746
,813
.736
.812
.803
.840
.877
.923
continued
145
~e
Table 5.46 (continued)
ONE-TAIL
.198
.030
(20%)
(3%)
re
....
8
8
8
TWO-TAIL
.192
.033
(20%)
(3%)
8
POWER
8
,50
.60
,70
.80
,90
.700
, 700
,700
.700
.700
.900
.900
.900
,900
,900
.902
.904
.50
,60
,70
.80
,90
.800
.800
,800
.800
,800
,800
,800
,800
,800
,800
.832
,869
.913
,834
.872
.915
,50
,60
,70
.80
,90
,800
,800
,800
.800
.800
.900
,900
,900
,900
.900
.50
,60
,70
,80
,90
,900
.900
.900
.900
.900
,900
,900
.900
,900
.900
1
2
3
3
3
.900
8
3
147
·-~·e
Table 5.47 (continued)
ONE-TAIL
,004
, 118
(15%)
(1%)
'"
~e
1
e
~
8
8
4
8
8
4
8
4
POWER
8
,50
, 60
· 70
.80
.90
.500
,500
.500
.500
.500
· 600
· 600
· 600
.600
· 600
· 600
· 600
· 600
· 600
· 600
· 787
.836
.883
.928
· 977
.50
· 60
· 70
· 80
.90
.500
.500
.500
.500
.500
· 600
· 600
· 600
· 600
· 600
· 700
.700
· 700
· 700
· 700
· 755
· 811
.866
.925
LOOO
.786
.855
.909
.952
.50
.60
• 70
.80
.90
.500
.500
,500
.500
.500
.600
.600
.600
.600
.600
.800
.800
.800
.800
.800
.830
.896
1.000
.815
.886
.943
.50
,60
· 70
.80
.90
,500
.500
,500
.500
,500
,600
,600
.600
.600
,600
.900
,900
,900
,900
.900
.924
.914
.50
.60
.70
.80
.90
.500
.500
,500
,500
.500
.700
· 700
.700
,700
,700
.700
.700
.700
.700
.700
LOOO
. 768
.845
.907
.956
.50
.60
,70
.80
,90
,500
.500
,500
,500
.500
,700
.700
· 700
, 700
, 700
.800
,800
.800
,800
,800
.814
.885
1,000
,801
,880
,944
1
8
TWO-TAIL
.007
.233
(1%)
(25%)
2
3
4
.734
· 794
.853
.917
. 751
.821
.874
.916
.949
continued
..
_
148
Table 5.47 (continued)
ONE-TAIL
.118
.004
(15%)
(1%)
~-
_
7
-!"
8
8
4
8
8
4
POWER
8
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.700
.700
.700
.700
.700
.900
.900
.900
.900
.900
.919
.912
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.800
.800
.800
.800
.800
.800
.800
.800
.800
.800
.866
.974
.867
;938
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.800
.800
.800
.800
.800
.900
.900
.900
.900
.900
.50
.60
.. 70
.80
.90
.500
.500
.500
.500
.500
.900
.900
.900
.900
.900
.900
.900
.900
.900
.900
.50
.60
.70
.80
.90
.600
.600
.600
0600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.783
.837
.888
.938
.994
.767
.830
.879
.917
.948
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.700
.700
.700
. 700
.700
.751
.812
.871
.935
l.000
.726
.802
.862
.911
.950
1
8
2
8
TWO-TAIL
.233
.007
(25%)
(1%)
3
4
4
.913
.902
continued
149
~e
Table 5,47 (continued)
TWO-TAIL
ONE-TAIL
...
,004
(1%)
-Y'
--e
"7
e
)P
8
,118
(15%)
8
,007
(1%)
8
.233
(25%)
8
4·
POWER
8
,50
,60
, 70
,80
,90
,600
,600
,600
,600
,600
,600
,600
,600
,600
.600
,800
,800
,800
.800
,800
,837
,909
1,000
,834
.893
,942
,50
,60
.70
.80
,90
,600
,600
.600
.600
,600
,600
.600
,600
,600
.600
.900
,900
,900
,900
,900
.959
,92L
,50
.60
, 70
.80
,90
,600
,600
,600
.600
.600
· 700
,700
,700
~ 700
· 700
,700
,700
,700
,700
,700
.729
,793
.856
.925
1.000
,702
,786
.853
.907
.952.-
.50
,60
,70
,80
,90
.600
.600
.600
,600
,600
,700
· 700
,700
,700
,700
.800
,800
,800
,800
,800
.819
,894
1,000
.820
.886
.940
,50
,60
,70
,80
,90
,600
,600
,600
,600
,600
,700
.700
,700
, 700
· 700
,900
.900
,900
,900
,900
.943
.916
,50
,60
, 70
,80
,90
,600
,600
,600
,600
,600
,800
,800
,800
,800
,800
.800
,800
,800
.800
.800
,874
.982
.802
,874
,934
1
8
2
8
3
4
4
4
continued
150
~e
Table 5.47 (continued)
ONE-TAIL
.004
.118
(1%)
(15%)
..
'1"'
.e
l'
e
"
8
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.800
.800
.800
.800
.800
. 900
.900
.900
.900
.900
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.900
.900
.900
.900
.900
.900
.900
.900
.900
.900
.50
.60
.70
.80
.90
.700
.700
.700
.700
.700
.700
.700
.700
.700
.700
.50
.60
.70
.80
.90
.700
.700
.700
.700
.700
.50
.60
.70
.80
.90
.50
.60
.70
.80
.90
1
8
8
POWER
2
3
8
4
8
4
TWO-TAIL
..233
.007
(1%)
(25%)
84-
84- .
.918
.906
.700
. 700
.700
.700
.700
.717
.782
.846
.913
.996
.712
.789
.850
.900
.942
.700
.700
.700
.700
.700
.800
.800
.800
.800
.800
.810
.883
.978
.822
.881
.930
.700
.700
.700
,700
.700
.700
.700
.700
.700
.700
.900
.900
.900
.900
.900
.927
.909
. 700
.700
.700
.700
.700
.800
.800
.800
.800
.800
.800
.800
.800
.800
.800
.864
.957
.805
.870
.924
continued
Ie
152
Table 5.48
Approximate stochastic differences among k = 5 normal
popula tions in the case of n = 10 uni ts per treatment
when comparing four treatments versus a control at given
power levels
~
[0: N(IJ,O'
2
O
CT
= 1) ; 1: N(lJ,l'
2
3: N(1J,3' 0'3
2
1
CT
= 1); 2: N(1J,2'
~
\
r
e
2
2
= 1) ;
2
= 1) ; 4:N(1J,4' cr4 = 1) ]
ONE-TAIL
.021
.001
(2%)
(1%)
•
CT
TWO-TAIL
.083
.004
(1%)
(10%)
8
4
8
4
8
4
8
4
.500
.500
.500
.500
.500
.916
.928
.939
.835
.860
.883
.904
.926
.916
.928
.940
.961
.826
.851
.875
.898
.922
.500
.500
.500
.500
.500
.600
.600
.600
.600
.600
.914
.929
.819
.848
.875
.902
.935
.914
.930
.811
.841
.868
.897
.932
.500
.500
.500
.500
.500
.500
.500
.500
.500
.500
.700
.700
.700
.700
.700
.906
.923
.801
.832
.861
.892
.933
.906
.923
.794
.826
.856
.888
.931
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.500
.500
.500
.500
.500
.800
.800
.800
.800
.800
.893
.909
.929
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.500
.500
.500
.500
.500
.900
.900
.900
.900
.900
POWER
8
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.500
.500
.500
.500
.500
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.50
.60
.70
.80
.90
1
8
2
8
3
.905
.923
.811
.842
.873
.914
.893
.910
.930
.807
.838
.870
.912
.906
.925
continued
154
-e
Table 5.48 (continued)
ONE_TAIL
.021
.001
(2%)
(1%)
~
~e
f
e
~
8
TWO-TAIL
.004
.083
(10%)
(1%)
8
8
POWER
8
1
8
2
8
3
8
4
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.700
.700
.700
.700
.700
.900
.900
.900
.900
.900
.919
.920
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.800
.800
.800
.800
.800
.800
.800
.800
.800
.800
.893
.913
.936
.893
.913
.936
.50
.60
.70
.80
.90
.500
.500
.500
.500
.500
.800
.800
.800
.800
.800
.900
.900
.900
.900
.900
,50
.60
.70
.80
.90
,500
.500
.500
.500
.500
.900
.900
.900
.900
.900
,900
.900
.900
.900
.900
.50
.60
.70
.80
,90
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
,600
.600
.931
.961
.812
.847
.881
.918
.977
.931
.960
.808
.844
.879
.918
.991
.50
,60
,70
,80
,90
,600
,600
,600
.600
,600
,600
,600
,600
,600
,600
.700
.700
,700
,700
,700
.920
.948
.790
,828
.864
,904
.971
.920
.948
,788
.826
.863
.903
.974
4
.827
.864
.910
4
.917
.952
.918
.952
.912
.934
.913
.934
4
.825
.863
.910
continued
155
·e
Table 5.48 (continued)
ONE-TAIL
.001
.021
(2%)
(1%)
~e
I
e
.,.
POWER
8
.50
.60
• 70
.80
.90
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.800
.800
.800
.800
.800
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.600
.600
.600
.600
.600
.900
.900
.900
.900
.900
.50
.60
· 70
,80
.90
.600
.600
.600
.600
.600
.700
.700
.700
.700
· 700
. 700
. 700
.700
.700
.700
.917
.943
.50
.60
.70
.80
.90
.600
.600
.600
.600
.600
.700
.700
.700
.700
.700
.800
.800
.800
.800
.800
.903
.926
.50
.60
· 70
"80
090
.600
.600
.600
0600
.600
.700
· 700
.700
· 700
· 700
.900
.900
.900
.900
.900
.50
.60
.70
.80
.90
.600
0600
.600
.600
.600
.800
.800
.800
.800
.800
.800
.800
.800
.800
.800
1
8
2
.8
3
8
4
.905
.929
8
4
.806
.843
.883
.940
TWO-TAIL
.004
.083
(1%)
(10%)
8
4
.906
.930
4
.805
.843
.883
.940
.909
.934
.908
.933
.906
.905
.781
.821
.859
.899
.960
.918
.943
.778
.819
.858
.899
.962
.904
.927
.837
.878
.933
.837
.878
.933
.909
.933
.908
.932
.903
.902
.899
.921
.949
8
.831
.873
.926
.900
.922
.949
.831
.874
.927
continued
157
~e
Table 5,48 (continued)
ONE-TAIL
.001
.021
(1%)
(2%)
l'
,.e
t
e
."
POWER
8
.50
,60
.70
.80
.90
8
4
TWO-TAIL
.004
.083
(1%)
(10%)
8
4
8
8
3
8
4
.700
,700
.700
,700
.700
.800
.800
.800
.800
.800
,900
,900
.900
.900
.900
.907
.929
.960
.50
,60
.70
.80
.90
.700
.700
.700
.700
,700
.900
.900
.900
.900
.900
.900
.900
.900
.900
.900
.901
.922
.945
.902
.923
.946
,50
.60
,70
.80
,90
.800
,800
.800
.800
,800
.800
.800
.800
.800
.800
.800
.800
.800
.800
,800
.899
.921
.944
.975
.900
.922
.944
.972
,50
,60
,70
,80
,90
,800
.800
.800
.800
.800
.800
.800
,800
.800
.800
,900
.900
.900
.900
.900
.905
.926
.948
.906
.926
.948
,50
.60
.70
.80
,90
,800
.800
.800
,800
.800
,900
.900
.900
.900
.900
.900
.900
.900
.900
.900
.900
.920
.940
.967
.901
.921
.941
.965
.50
.60
.70
.80
.90
,900
.900
.900
.900
.900
.900
.900
.900
.900
.900
.900
.900
.900
.900
.900
.915
.934
.953
.916
.934
.953
1
2
8
4
.908
.929
.959
.901
.825
.870
.923
.829
.874
.925
IS8
S.9
•
S.9.1
Computer Programs
Enumeration of the Exact Distributions of miif and m
Oi
INTEGERA(S,5), B(2S,5), C(S,S), T(S), R(lO), Q(1O),TEST(1O,2)
DIMENSION N(5),NC(S), MID(10), NV(Sll,2)
COMMON A, B, C
C
C
C
INITIALIZATION
1 READ (I,SOI) NT, NeONT
S READ (I,S03) (N(J), J = I,NT)
NTOT = 0
DO 7 J = I,NT
7 NTPT = NTOT +N(J)
NREM = NTOT
DO 9 J = I,NT
NC(J) = NREM
NN ::: N(J)
NREM = NREM - NN
DO 9 I = I,NN
9 A(I,J) = 0
DO 11 I ::: 1, NTOI
11 B(I,I) = I
NTEST = 0
NTT = NT - 1
. DO 13 I = I,NTT
JJ = I + 1
DO 13 J = JJ,NT
NTEST = NTEST + 1
MID (NTEST) = (N(I) + N(J»/2
TEST(NTEST,I) = I
13 TEST(NTEST,2) = J
NREG = N(I) + 1
NVEC = 2
DO IS I
2,NREG
15 ··NVEC
NVEC*2
Q(1) = NVEC/2
NVEC = NVEC·_ 1
DO 17 I = 2,NREG
17 Q(I) = Q(I-1)/2
I::
I::
DO 19 J = 1,2
DO 19
I
19 NV(I,J)
NLINE
J
C
=1
I::
= 1,NVEC
=0
1
159
C
C
GENERATION OF POSSIBLE COMBINATIONS
99 T(J) = 1
101 CALL SELECT (N(J)} NC(J)} T(J)} J)
C
l'
IF (T(J)) 103}103}109
103 J = J _ 1
IF (J) 105}105}107
107 IF (T(J)) 103}103}101
109 JJ = J
J
=
J +
1
II = 1
c
L
= 1
I
=0
IF (J - NT) 113}121}121
=I +1
IF (I - A(L}JJ)) 117}115}117
115 L = L + 1
GO TO 113
117 B( II) J) = B(I) J J)
113 I
II=II+1
IF (I - NC(JJ)) 113}99}99
C
121 I = I + 1
IF (I - A(L,JJ)) 125}123}125
123 L = L + 1
GO TO 121
125 C(II,NT) = B(I}JJ)
II=II+1
IF (I - NC(JJ)) 121}127}127
C
C
C
DETERMINE TYPE OF STATISTIC VECTOR AND SET APPROPRIATE COUNTER
127 DO 129 K
129 R(K) = 0
K1 = 1
K2 = NTT
JJ
= 1,NREG
=1
130 DO 137 L = K1}K2
II = TEST(L,l)
12 = TEST(L,2)
NN = MID(L)
M = 1
MM = 1
DO 135 LL = 1, NN
IF (C(M,I1) - C(MM J I2)) 131 J 131 J 133
131 M = M + 1
GO TO 135
133 MM = MM + 1
135 CONTINUE
160
137 R(M) ::; 1
NCOUNT ::; 0
DO 139 L ::; 1,NREG
139 NCOUNT ::; NCOUNT + R(L)*Q(L)
NV(NCOUNT,JJ) = NV(NCOUNT,JJ) + 1
JJ =: JJ + 1
K1 ::; NT
K2 ::; NTEST
GO TO (130,130,141)} JJ
141 J =: NTT
NLINE ::; NLINE + 1
IF (NLINE - (NLINE/1000)*1000) 101,142}101
142 WRITE (3 J 505) NLINE
GO TO 101
C
C
PRINT RESULTS
C
105 DO 143 I ::; l}NREG
143 MID(I) = NREG - I
WRITE (3,547) (MID(I)} I ::; 1}NREG)
WRITE (3J 549)
DO 153 K ::; 1} 2
IF (K - 2) 147,145}147
145 WRITE (3,551)
147 DO 153,J ::; l,NVEC
NUM ::; J
MID(l) ::; NUM/O (1)
DO 151 I =: 2}NREG
NUM ::; NUM - MID(I) - l)*Q(I - 1)
151 MID(I) ::; NUM/Q(I)
153 WRITE (3 J 553) J} NV(J,K),(MID(L), L
IF (NCONT) 201,201}1
201 STOP
= l}NREG)
C
501
503
505
547
549
551
553
FORMAT
FORMAT
FORMAT
FORMAT
FORMAT
FORMAT
FORMAT
END
(12,11)
(1012)
(lX17,25H SAMPLE VECTORS PROCESSED)
(1H1) 21X, 6HNUMBER} 7X} 11HVECTOR TYFE/35X,1012)
(1HO,9X,21HTREATMENTS VS CONTROL/)
(1HO,9X,9HALL PAIRS/)
(10XI5, 6XI6, ax, 5012)
161
SUBROUTINE SELECT (K,N,T,J)
C
C
C
C
C
C IS A SAMPLE OF SIZE K FROM B, WHERE B IS OF SIZE N
CALL INITIALLY WITH T = 1
ON OUTPUT, IF T = K FURTHER SAMPLES REMAIN. IF T = 0, STOP.
INTEGER A(S,S), B(2S,S), C(S,5), T
COMMON A, B, C
IF (T - 1) I, I, 13
1 A(l,J) = 1
3 M = A(T,J)
C(T ,J) = B(M, J)
IF (T - K) 7} 5, 5
5 RETURN
= A(T,J) + 1
IF (A(T + l,J) - N) 9,9,11
9T=T+1
GO TO 3
llT=T-1
IF (T) 5, 5, 13
13 A(T,J) = A(T,J) + 1
IF (A(T,J) - N) 3,3,11
7 A(T + 1,J)
END
162
5.9.2
Generation of the Approximate Distributions of miit and m
Oi
This program generates random vectors for enumeration.
These
...
vectors are obtained from permutations of the patients 1, 2, 3, ... ,
N (N
total number of patients), which in turn are obtained from their
~
serial numbers by the lexicographic method of Lehmer [16].
The serial
numbers are random numbers that are generated by the power residue
method, using the multiplier 65,539 (i.e. 2
16
+ 3) [13].
DOUBLE PRECISION XN(44),Z,TOT
INTEGER TEST(55,2),R(2l),D(200),C(10,ZO)
DIMENSION NP(10),MID(55),NV(10,3,2),L(44),KZ(200),JX(44)
C
C
INITIALIZATION
C
1 READ (1,501) NT
IF (NT - 99) 5,3,5
3 STOP
5 READ (1,502) NP
N
K
~
~
0
0
DO 7 I ~ 1,NT
JJ ~ NP(I)
N = N + JJ
DO 7 J = 1,JJ
K ~ K + 1
7 KZ (K) ~ 1
NW=N-l
NTT = NT - 1
NTEST ~ 0
DO 9 I ~ 1,NTT
JJ = I + 1
DO 9 J = JJ,NT
NTEST ~ NTEST + 1
MID(NTEST) ~ (NP(I) + NP(J»/Z
TEST(NTEST,l) ~ I
9 TEST(NTEST, 2) ~ J
NREG = NP(l) + 1
NHALF ~ NREG/Z
DO 15 I = l,NHALF
DO 15 J ~ 1,3
DO 15 K = 1,2
15 NV (I, J) K)= 0
L(l) =: 12
163.
17
19
21
23
25
27
29
C
C
C
GENERATE FACTORIAL DIGITS SPECIFYING A PERMUTATION
51
53
55
57
59
C
C
C
L(2) == 19
DOll 1==3,5
L(I) = (L(I-1) + 6
DO 19 I == 6,12
L(I) = L(I-1) + 5
DO 21 I == 13,44
L(I) = L(I-1) + 4
DO 23 I = 1,45
IF (L(I) - N) 23,25,25
CONTINUE
NRAND = I
L(I) = N
LU = 0
DO 29 I = l,NRAND
JX(I) == 1
LL = LU + 1
LU == L(I)
DO 27 J = LL,LU
JX(I) == JX(I)~(J
Z == JX(I)
XN(I) == Z*4.656612875248D-10
READ (1,503) NGEN,IRAND
TOT == NGEN
DO 101 NLINE == 1,NGEN
D(l) == 0
LU == 1
DO 59 I == 1,NRAND
lRAND == lRANTh~65539
IF (IRAND) 53,55,55
lRAND == lRAND + 2147483647 + 1
Z == lRAND
IX == Z*XN(I)
IF (IX - JX(I» 57,51,51
LL == LU + 1
LU == L(I)
DO 59 J == LL, LU
JJ == IX/J
D(J) == IX - JJ'~J
IX == JJ
USE THE LEHMER TECHNIQUE TO FIND THE PERMUTATION
DO
JJ
LL
DO
IF
65 I == l,NW
== D(I)
== I + 1
63 J == LL,N
(JJ - D(J» 63,61,61
164
61 JJ = JJ + 1
63 CONTINUE
65 D(I) = JJ
C
C
C
SET UP STATISTIC VECTOR
DO 67 I = l)NT
67 R(I) = 1
DO 69 JJ = l)N
J = D(JJ) + 1
J = KZ(J)
I = R(J)
C(I)J) = JJ
69 R(J) = R(J) + 1
C
C
DETERMINE TYPE OF STATISTIC VECTOR AND SET APPROPRIATE COUNTER
C
127 DO 129 K = l)NREG
129 R(K) = 0
K1 = 1
K2 = NTT
JJ = 1
130 DO 137 LG = K1)K2
Il = TEST (LG, 1)
12 = TEST (LG, 2)
NN = MID(LG)
M = 1
MM
=
1
DO 135 LL = 1,NN
IF (C(M) Il) - C(MM, 12»
131 M = M + 1
GO TO 135
133 MM = MM + 1
135 CONTINUE
137 R(M) = 1
131,131,133
M = 0
DO 139 I = l)NHALF
+ R(I)
IF (M) 139,139,141
139 CONTINUE
Il = NREG
GO TO 145
141 DO 143 J = I)NHALF
143 NV(J)l,JJ) = NV(J,l)JJ) + 1
M = M
I1 = I
145 M = 0
DO 147 I = 1,NHALF
K = NREG - I + 1
M = M + R(K)
IF (M) 147,147,149
165
147 CONTINUE
12 = NREG
GO TO 153
149 DO 151 J = I,NHALF
151 NV(J,2,JJ) = NV(J,2,JJ) + 1
12
I
153 IF (12 - 11) 154,155,155
154 Il = 12
155 IF (11 - NREG) 156,159,159
156 DO 157 J = I1,NHALF
157 NV(J,3,JJ) = NV(J,3,JJ) + 1
159 JJ =: JJ + 1
K1 = NT
K2 = NTEST
GO TO (130,130,161), JJ
161 IF (NLINE - (NLINE/1000) *1000) 101,163,101
163 WRITE (3,505) NLINE
101 CONTINUE
.",
:::0:
C
C
PRINT RESULTS
C
165
167
169
171
WRITE (3,507) NT, (NP(I), I = l,NT)
WRITE (3,549)
JJ = 1
GO TO 167
WRITE (3,551)
WRITE (3,513)
DO 169 I = l,NHALF
D(I) = I - 1
Z =: NV ( I, 1, J J)
Z = Z/TOT
WRITE (3,515) NV(I, l,JJ), Z, (D(J), J
WRITE (3,517)
DO 171 I = l,NHALF
D(I) = NREG - I
Z = NV(I., 2, JJ)
Z = Z/TOT
WRITE (3,515) NV(I,2,JJ), Z, (D(J), J
WRITE (3,519)
DO 175 I = l,NHALF
D(I) = I - 1
=
1, I)
=
1, I)
=
1,12)
11=1+1
12 = 2*1
DO 173 J = Il,I2
173 D(J) = NREG - J + I
Z = NV ( I, 3, J J)
Z = Z/TOT
175 WRITE (3,515) NV(I,3,JJ), Z, (D(J), J
JJ = JJ + 1
GO TO (165,165,177), JJ
166
177 WRITE (3,555)
GO TO 1
C
..
'-r
501
502
503
505
507
FORMAT (12)
FORMAT (4012)
FORMAT (19)
FORMAT (lXI7,25H SAMPLE VECTORS PROCESSED)
FORMAT (lHl,9X,20HNUMBER OF TREATMENTS,I3//10X,28NUMBER OF
1PATIENTS/TREATMENT,lX,2013)
513 FORMAT (lH., 19X, 6HNUMBER, 9X, 11HPROBABILITY, 9X, 24HDIGITS
1PRESENT IN VECTOR,// lOX, 30HSINGLE-TAILED TEST, LOWER TAIL)
515 FORMAT (/18X, 18, 4X, F14.5, 11X, 3012)
517 FORMAT (j10X J 30HSINGLE-TAILED TEST, UPPER TAIL)
519 FORMAT (j10X J 15HTWO-TAILED TEST)
549 FORMAT (lHO,5X,21HTREATMENTS VS CONTROL/)
551 FORMAT (lHO J 5X,9HALL PAIRS/)
555 FORMAT (lH1)
END
167
5,9,3
Generation of the Approximate Powers
DOUBLE PRECISION C(20" 10)" AA(lO)" BB(lO), TOT, A, B, Z, ZZ, CMAX
INTEGER TEST(55,2), R(21), D(21)
DIMENSION NV(1O,3,2), KO(20,2), KDIST(10)
EQUIVALENCE (R(I),D(l»
COMMON lRAND
C
1 READ (1,501) NT,NP
IF (NT - 99) 5,3,5
3 STOP
5 WRITE (3,507) NT,NP
NTT =: NT - 1
NTEST =: 0
DO 9 I =: I,NTT
JJ =: I + 1
DO 9 J =: JJ,NT
NTEST =: NTEST + 1
TEST(NTEST,I) =: I
9 TEST(NTEST,2) =: J
NREG =: NP + 1
NHALF =: NREG/2
DO 15 I =: I,NHALF
DO 15 J =: 1,3
DO 15 K =: 1,2
15 NV ( I, J, K) =: 0
DO 51 I =: I,NT
READ (1,502) KDIST(I), AA(I), BB(I)
WRITE (3,509) I, KDIST(I), AA(I), BB(I)
KGO =: KDIST(I)
GO TO (17,19,26,21,23,25,27), KGO
17 BB(I) =: BB(I) - AA(I)
GO TO 51
19 BB(I) =: DSQRT(BB(I»
GO TO 51
21 BB(I) =: DLOG(BB(I)/(AA(I)*AA(I» + 1.0D+OO)
AA(I) =: DLOG(AA(I» - BB(I)/2,OD+OO
BB(I) =: DSQRT(BB(I»
GO TO 51
23 AA(I) =: AA(I) /BB(I)
BB(I) =: AA(I)*AA(I) "(BB(I) + 5.0D-01
GO TO 51
25 BB(I) =: BB(I)*2.0D+OO/(BB(I) - AA(I)*AA(I»
AA(I) =: AA(I)*BB(I) - AA(I)
BB(I) =: -LOD+OO/BB(I)
GO TO 51
26 AA(I) =: AA(I) - BB(I)
GO TO 51
27 AA(I) =: LOD+OO/AA(I)
51 CONTINUE
READ (1,503) NGEN,IRAND
168
TOT =: NGEN
WRITE (3 J 511) NGEN
C
.
52
53
54
55
56
57
),
.~
58
59
60
61
62
63
64
65
66
67
68
71
DO 161 NLINE =: 1J NGEN
DO 71 J =: 1 J NT
A =: AA(J)
B =: BB(J)
KB =: BB(J)
KGO :::; KOIST(J)
GO TO (52 J54 J57}59}62}65}67)J KGO
DO 53 K =: 1 J NP
CALL RANDOM (Z)
C(K,J) = A + B~'(Z
GO TO 71
DO 56 K = l}NP
ZZ =: O.OD+OO
DO 55 I =: 1 J 12
CALL RANDOM(Z)
ZZ =: ZZ + Z
C(K}J) =: A + B*(ZZ - 6.0D+OO)
GO TO 71
DO 58 K =: 1 J NP
CALL RANDOM(Z)
C(KJJ) = B - A*DLOG(Z)
GO TO 71
DO 61 K =: 1 J NP
ZZ =: O.OD+OO
DO 60 I = I} 12
CALL RANDOM(Z)
ZZ =: ZZ + Z
C(KJJ) =: DEXP(A + B*(ZZ - 6.0D+OO»
GO TO 71
DO 64 K =: l}NP
ZZ =: L OD+OO
DO 63 I = 1 J KB
CALL RANDOM(Z)
ZZ =: Zz*z
C(KJJ) = -DLOG(ZZ)!A
GO TO 71
DO 66 K =: 1 J NP
CALL RANDOM(Z)
C(KyJ) = A*Z**B - A
GO TO 71
DO 68 K =: 1J NP
CALL RANDOM(Z)
C(KyJ) =: B*(_DLOG(Z»**A
CONTINUE
C
DO 72 K =: 1yNREG
72 R(K) = 0
JJ =: 0
169
Kl =: 1
K2 =:: NTT
73 JJ =:: JJ + 1
DO 83 I =:: K1,K2
74
M =:: 1
DO 74 II =: l,NP
DO 74 J =:: 1,2
KO(II, J) =:: 0
DO 81 II =:: 1, NP
CMAX =:: -99999999.9
DO 79 J =:: 1,2
KJ =:: TEST (I, J)
DO 79 III =:: l,NP
IF (KO(III,J)) 75, 75} 79
75 IF (C(III}KJ) - CMAX) 79}79}77
77 CMAX =:: C(III}KJ)
KS =:: J
KW =:: III
79 CONTINUE
KO(KW,KS) =:: 1
81 M =:: M - KS + 2
83 R(M) =: 1
M
=::
0
DO 139 I
M
=::
M
(M)
=::
l,NHALF
+ R(I)
IF
139,139} 141
139 CONTINUE
11 =:: NREG
GO TO 145
141 DO 143 J =:: I}NHALF
143 NV(J,l,JJ) =:: NV(J}l}JJ) + 1
11
145
147
149
151
.
153
154
155
156
157
159
161
M
=::
=::
I
0
DO 147 I =:: l,NHALF
K =:: NREG - I + 1
M =:: M + R(K)
IF (M) 147, 147, 149
CONTINUE
12 =:: NREG
GO TO 153
DO 151 J =: I}NHALF
NV(J}2}JJ) =:: NV(J,2}JJ) + 1
12 =:: I
IF (12 - 11) 154}155}155
11 =:: 12
IF (n _ NREG) 156} 159} 159
DO 157 J == Il,NHALF
NV(J,3}JJ) == NV(J,3}JJ) + 1
Kl == NT
K2 == NTEST
GO TO (73}161)} JJ
CONTINUE
170
165
167
169
171
WRITE (3,549)
JJ = 1
GO TO 167
WRITE (3,551)
WRITE (3,513)
DO 169 I = 1,NHALF
D(I) = I _ 1
Z = NV(I, 1, JJ)
Z = Z/TOT
WRITE (3,515) NV(I, 1,JJ), Z, (D(J), J
WRITE (3,517)
DO 171 I = l,NHALF
D(I) = NREG - I
Z = NV(I,2,JJ)
Z = Z/TOT
WRITE (3,515) NV(I,2,JJ), Z, (D(J), J
WRITE (3,518)
DO 172 I = 1,NHALF
K = NV(I, 1, JJ) + NV(I, 2, JJ)
1, I)
= 1,1)
Z = K
Z = K*5.0D_01/TOT
172 WRITE (3,515) K, Z
WRITE (3,519)
DO 175 I = I,NHALF
D(I) = I _ 1
II=I+l
12 = 2*1
DO 173 J = II, 12
173 D(J) = NREG - J + I
Z = NV(I, 3, JJ)
Z = Z/TOT
175 WRITE (3,515) NV(I, 3,JJ), Z, (D(J), J = 1,12)
JJ = JJ + 1
GO TO (165,165,1), JJ
C
501
502
509
511
518
503
507
•
FORMAT (212)
FORMAT (I1,2F8.5)
FORMAT (/15X,3HTRT,I3,3X,4HDIST,I3,3X,10HPARAMETERS,2F14.4)
FORMAT (/10X,27HNUMBER OF VECTORS GENERATED,I9)
FORMAT (/10X,25HPOOLED SINGLE-TAILED TEST)
FORMAT (19)
FORMAT (lH1,9X,20HNUMBER OF TREATMENTS, 13//10X, 28HNUMBER OF
1PATIENTS/TREATMENT,lX,2013)
513 FORMAT(IH , 19X, 6HNUMBER, 9X, IIHPROBABILITY, 9X, 24HDIGITS
1PRESENT IN VECTOR, // lOX, 30HSINGLE-TAILED TEST, LOWER TAIL)
515 FORMAT (/ 18X, 18,4X, Fl4.5, 11X,30I2)
517 FORMAT (/ lOX, 30HSINGLE_TAILED TEST, UPPER TAIL)
519 FORMAT (/ lOX, 15HTWO-TAILED TEST)
549 FORMAT (lHO,5X,21HTREATMENTS VS CONTROL/)
551 FORMAT (lHO,5X,9HALL PAIRS/)
END
.e
171
SUBROUTINE RANDOM (2)
C
C
C
THIS SUBROUTINE GENERATES RANDOM NUMBERS BETWEEN 1 AND 0
DOUBLE PRECISION 2
COMMON IRAND
IRAND = IRAND * 65539
IF (IRAND) 1) 2} 2
1 IRAND = IRAND + 2147483647 + 1
2 Z = IRAND
Z = Z * 4. 6566128752458D-I0
RETURN
END
.e
172
5,9,4
Estimation of the Parameters for the Inverse Quadratic Power
·Surfaces
DOUBLE PRECISION TITLE(6)}SSY}Y}PROD}REGSS}RESSS}ZN}RMS}~S}F}R}
DOUBLE PRECISION SSADD} SDB} Z} YADJ}YSE} DIFF} DIFF2} SDIFF} SDIFF2
DOUBLE PRECISION SUMY} SSREG}REGMS} ERROR}RR}FF
DOUBLE PRECISION C(15}15)}SSP(15}15)}SUM(15)}B(15)}V(15)}X(4)
DIMENSION NIX(15)}L(15)
COMMON C}B
C
(~
1 READ(l) 501) NVAR} NOBS} (NIX(I)} I=l} 15)} (TITLE(J)} J=l} 6)
IF(NVAR-9)53}51}53
51 STOP
53 WRITE(3}502) (TITLE(J)}J=1}6)
REWIND 14
WRITE(3}503) NVAR
NCOL=(NVAR+l)*(NVAR+2)/2
NYES=O
DO 203 I=l}NCOL
IF (NIX(I» 20l} 20l} 203
201 NYES=NYES+l
L(NYES) =1
203 CONTINUE
WRITE(3}505) (I) I=l} NVAR)
XN=NOBS
DO 5 I=l}NCOL
SUM(I) =0. OD+OO
DO 5 J=l}NCOL
5 SSP (I) J) =0. OD+OO
SUMY=O,OD+OO
SSY=O,OD+OO
DO 11 I=l} NOBS
READ(1}507)Y} (X(J)}J=l}NVAR)
WRITE(14}507)Y~(X(J)}J=1}NVAR)
WRITE(3}509)I}Y} (X(J)}J=l}NVAR)
PROD=Y
DO 6 J=l}NVAR
6 PROD=PROD/X(J)
V(l)=PROD
K=l
DO 7 J=l}NVAR
K=K+l
7 V(K)=PRO!Y-(X(J)
DO 9 J=l}NVAR
DO 9 M=J} NVAR
K=K+l
9 V(K)=PROD*X(J)*X(M)
SUMY=SUMY+Y
SSY=SSY+Y*Y
DO 11 JJ=l} NCOL
173
·e
SUM(JJ)=SUM(JJ)+V(JJ)
DO 11 KK=I, JJ
11
SSP(JJ,KK)=SSP(JJ,KK)+V(JJ)~V(KK)
301 DO 303 I=I,NYES
I1=L(I)
B(I) =SUM(II)
DO 303 J=1,I
JJ=L(J)
303 C(I,J9=SSP(II,JJ)
DO 305 J=2, NYES
JJ=J-1
DO 305 I=1,JJ
305 C(I, J) =C(J, I)
C
CALL INVERT (NYES)
C
35
71
73
75
77
80
•
78
REGSS=O.OD+OO
DO 35 I=1,NYES
J=L(I)
REGSS=REGSS+B(I)*SUM(J)
SSY=SSY-(SUMY*SUMY/XN)
NDFT=NOBS-1
RESSS=SSY-REGSS
ZN=NYES
RMS=REGSS/ZN
NDFE=NOBS-NYES-1
DFE=NDFE
EMS=RESSS/(XN-ZN)
F=RMS/EMS
K=1
WRITE (3,502) (TITLE(J), J=1, 6)
DO 75 I=1,NCOL
IF(L(K) -1) 73, 71, 73
V(I) =B(K)
K=K+1
GO TO 75
V(I)=O.OD+oO
CONTINUE
WRITE(3,535) V(1),(V(I+1),I,I=I,NVAR)
JJ=O
DO 79 J=1,NVAR
JJ=JJ+NVAR-J+2
II=NVAR-J+l
DO 77 1=1, II
K=I+JJ
XCI) =V(K)
IF (J-2)80,78,90
WRITE(3,537) (X(I) ,J, I, I=J,NVAR)
GO TO 79
WRITE(3,537) (X(I-l) ,J, I, I=J, NVAR)
GO TO 79
174
•
90 IF(J-3)76,76,91
76 WRITE(3}537) (X(I-2),J,I,I=J,NVAR)
GO TO 79
91 WRITE(3,537) (X(I-3), J, I, I=J, NVAR)
79 CONTINUE
NJA=O
00 39 I=l,NYES
NJA=NJA+1
L(NJA) =L(I)
39 CONTINUE
REWIND 14
WRITE(3 J 502) (TITLE(J),J=l,6)
WRITE(3 J 545)
SDIFF=O.O
SDIFF2=O.0
DO 49 K=l, NOBS
READ (14,507) Y, (X(I),I=l,NVAR)
PROD=!. 0D+OO
Z=V(l)
00 41 I=l,NVAR
PROD=PROD*X (I)
41 Z=Z+V(I+1)*X(I)
JJ=NVAR+1
00 43 I=l,NVAR
DO 43 J=IJNVAR
JJ=JJ+1
43 Z=Z+V(JJ)3X(I)*X(J)
YADJ=PROD/Z
DIFF=Y-YADJ
DIFF2=DIFF*DIFF
SDIFF=SDIFF+DIFF
SDIFF2=SDIFF2+DIFF2
49 WRITE(3,547) K, Y, YADJ, DIFF J DIFF2
WRITE(3 J 549)SDIFF J SDIFF2
WRITE(3 J 502) (TITLE(J),J=l}6)
WRITE(3 J 525)
SSREG=SSY-SDIFF2
ERROR=SDIFF2/DFE
REGMS=SSREG/ZN
FF=REGMS/ERROR
WRITE(3,527) SSREG, NYES, REGMS J FF
WRITE(3,529) SDIFF2, NDFE, ERROR
WRITE(3,521) SSY, NDFT
RR = DSQRT(SSREG/SSY)
WRITE (3}533) RR
IF (NJA) I, 1,405
405 IF(NJA-NYES)407,l,l
407 NYES=NJA
GO TO 301
C
175
501
502
503
505
507
509
525
FORMAT (ll) I3) l5I1) 6A8)
FORMAT (lHl)9X)6A8)
FORMAT (lHO)9X) 29HINVERSE POLYNOMIAL REGRESSION) 15) lOH VARIABLES)
FORMAT (17HO
OBSERVATIONS/6HO NO.)5X)lHY)4X)2HX-)3XIl)9(7XI2)/)
FORMAT (F12,7)
FORMAT (2XI3)lX)11(FlO.4»
FORMAT (25HO
ANALYSIS OF VARIANCE/ lHO) lOX) 6HSOURCE) l4X)
2HSS) l2X) 2HDF) l2X) 2HMS) l4X) lHF)
1
527 FORMAT (lHO)9X)10HREGRESSION)4X)F14.5)5XI4)5XF14.5)2X)F12.5)
529 FORMAT ( lHO) 9X) 5HERROR) 9XF14.5) 5XI4) 6XF14.6)
531 FORMAT (lHO) 9X) 5HTOTAL) 9XF14.5) 5XI4)
MULTIPLE CORRELATION COEFFICIENT)F14.4)
533 FORMAT (//37HO
REGRESSION EQUATION/5X)12HPROD X / Y =)Fll.4)
535 FORMAT (24HO
14 {2H+) Fll. 4) 3H*X() 12) IH») 4X»
537 FORMAT (30X)4(Fl1.4)3H*X()I1)4H)i(X()I1)3H) +»
541 FORMAT (//lHO) 63X) l7HTEST SLOPE = ZERO/ l7X) 6HADD S$) l3X)
1
5HSLOPE) 7X) llHSD OF SLOPE) 8X) lHT) lOX) 2HDF)
543 FORMAT (lHO)11XF14.5)2XF14.4)2XF14.5)F13.4)6XI4)
ADJUSTED VALUES OF Y/7HO
OBS)9X)lHY)13X)5HY ADJ)
545 FORMAT (25HO
110X)9HY - Y ADJ) 6X) l4H(Y - Y ADJ)**2/)
547 FORMAT (lXI5)4(2X(F14.5»
549 FORMAT (lHO)37X)2(2XF14.5»
END