CALIFOR..~IA
STATE UNIVERSITY, NORTHRIDGE
THE EFFECT OF THE VIOLATION OF
INDEPENDENCE ON THE MM-."'1-1-\<JlUTNEY U TEST
A thesis submitted in partial satisfaction of the
requirements for the degree of Master of Science in
Health Science
by
Virginia L. Frohmberg
/
January, 1977
The thesis of Virginia L. Frohmberg is approved:
California State University, Northridge
December, 1976
-------·
-·~-------·-·
ii
ACKNOWLEDGEMENTS
I would like to express my deepest and most sincere appreciation
:to Dr. Edward F. Gocka, Chief of the Predictive and Evaluative Models
l
I
;Research Laboratory of the Veterans Administration Hospital, Sepulveda,
I
!california, for his continued support and guidance in the preparation
of this thesis.
I
To Dr. Bernard Hanes, I extend my gratitude for his
constructive criticism and helpful suggestions.
I also wish to thank
Dr. Roberta Solomon for her critical reading of the manuscript.
----..........
.)
iii
-----------------,
TABLE OF CONTENTS
!j
iii
!ACKNOWLEDGEMENTS
ABSTRACT
CH.
I
II
III
IV
v
v
INTRODUCTION
......................................... .
1
LITERATURE REVIEW
3
METHOD
5
The Mann-Whitney U Test
5
Simulation Procedure
6
Levels of Confidence
10
RESULTS
11
DISCUSSION AND CONCLUSION
17
:BIBLIOGRAPHY
19
21
APPENDICES
A
TABLES OF THE EXPERIMENTAL RESULTS
22
B
THE MONTE CARLO PROCEDURE
27
c
THE SIMULATION PROGRAM
35
D
GENERATION OF
E
THE
CUMULATI\~
CO~WUTATION
PROBABILITY DISTRIBUTION
OF A MANN-WHITNEY U
43
58
--·--------·-...-......-~--''
iv
1"''"'·~
..
---.~---·t-,
.......
~~-.r=--..----~---.-...--~~·---_..__.._
_____________---·-....,.,...._._..
<....,,_,_,.....
l
ABSTRACT
THE EFFECT OF THE VIOLATION OF
INDEPENDENCE ON THE MANN-WHITNEY U TEST
by
Virginia L. Frohmberg
January, 1977
The Mann-Whitney U test, a nonparametric alternative to the t
i test for independent groups, assumes-independently drawn random
i samples.
This study investigated the effects on the Mann-lfuitney U
test when non independently drawn random samples were used.
A Monte Carlo study was performed to simulate situations in which ·
'independence between groups was violated.
Two independent groups,
; composed of generated random normal variates were formed.
Increasing
percentages of correlated pairs were introduced to the groups which
' were originally independent.
For each percent of correlated pairs, a
. Mann-Whitney U value and a Student's t value were computed.
This
procedure was employed for sample sizes of 10 and 20.
As the pe1.·cent of correlated pairs increased, the number of U
values that fell into the critical regions (a.= .01, .05, and .10)
decreased.
This was found to be true for both sample sizes.
It was
demonstrated that the conclusions given by the Mann-Whitney U test are
v
,,.,.,,_.,,.......,.......,.,»..,_,._-.~-".._,..,.,,.,."""""'_""'"'""'--------~~~----------...---------~,
l
lnot effected with 10 percent correlated pairs, a rho of .7, and sample
sizes of 10 and 20.
In addition, the conclusions of the Mann-Whitney
U test are not effected by any percent of correlated pairs used in
lthia study, with an alpha of .01, a rho of .7, and sample sizes of
·10 and 20.
vi
CHAPTER I
I
l
INTRODUCTION
l
"If the Mann-Whitney test is applied to data which might properly
_be analyzed by the most powerful parametric test, the t test, its
i
!power-efficiency approaches 3/~
j
= 95.5 per cent as N increases, and is
·close to 95 percent even for moderate-sized samples.
It is therefore
•an excellent alternative to the t test, and of course it does not have
the restrictive assumptions and requirements associated with the t
.test." (20)
Of the many nonparametric techniques available for the analysis of
two independent samples, the Mann-Whitney U test is the best to use
!under the following two conditions: a) when the researcher wishes to
i
test whether the samples represent populations that differ in location
i or central tendency (as opposed to testing samples that represent
:populations differing in any respect at all, such as location or
dispersion or skewness) and b) when the level of measurement is at
: least ordinal.
An illustration of the procedure used in the computa-
tion of a Mann-Whitney U is given in Appendix E.
The application of this technique, though, is based on the
assumption of independent samples.
What effect does the violation of
independence have on the Mann-Whitney U test?
Correlated data may
arise in various ways, as for example when an investigator wishes "to
observe the effects (X.'s) of treatment A on L subjects, and at a later
l.
!
date observe the effects (Y. 's) of treatment Bon M subjects.
However,:
l.
____________
factors
such as a-·---·-shortage
the ·
··--·
------- ----.. of suitable
--------··patients may· -cause
- - · .... -- ..........
------------------------~
---·---------~--
_,
1
----~
!
l
..--------'"'
2
;experimenter to include a few subjects both times." (13)
Another example is when "Treatments A and B are compared using
I
j2M rats from the same litter.
Rather than assign the rats to treat-.
lments so that each set of M rats has probability
1/( 2~)
of being the
:'A rats', the researcher blocks (either implicitly or explicitly) on
!
isome factor.
t
This blocking may introduce dependencies; however, the
l
!blocking is not recorded and the blocks cannot be identified at data
analysis time." (13)
This study was concerned specifically with the Mann-Whitney U test
and how it responds, especially in the tails, to different percents of
.correlated pairs for small sample sizes.
There was also interest in
how the parametric counterpart to the Mann-Whitney U test, the t test,
;reacts under the same conditions.
The effect of correlated data on
jthe Mann-Whitney U test and on the t test were compared.
:
1
A criterion was needed to judge whether or not there was a statis-
!tically significant difference in the tails of the distributions
•obtained for each increasing percent of correlated pairs.
A test using
;the normal deviate z was utilized to find the significance of the size
of the difference between the proportions, at different points
(a= .01, .05, and .10) in the critical region.
,-----·-··--···-·~~..~..--~·~·~--··---·-··--··~~--~-.----·--·~--···---~·-·-·--~~,·~·~.
CHAPTER II
LITEP~TURE
REVIEW
· Boneau (1960) investigated the effects on Student's t test of the
!violation of the assumptions of normality of distribution and homageineity of variance.
lsizes
He
fo~nd
that the combination of unequal sample
and unequal variances produces inaccurate probabilities.
He also
discovered that if the two underlying distributions differ in skew,
the t values obtained tend to be skewed.
Although the t test is not
robust when used with both unequal variances and unequal sample sizes
or when used with two distributions that differ in skew, Boneau's
'final conclusion was that "the t test is a remarkably robust test."
In support of Boneau's conclusions, Edwards (1967) states that
jwhen a researcher has at least 25 subjects in each group and randomly
'assigns the subjects to the two treatments, he "need not spend a sleepless night worrying about such assumptions as homogeneity of variance
and normality of distribution."
Bradley (1968) endorses what he calls the "kernal of truth"
observed by Boneau and Edwards.
"The kernal of truth ••. is that
for most ••. parametric tests there are conditions under which a fairly
·'large' violation of an assumption produces impressively little distortion in the distribution of the test statistic."
He used the t test
'for independent groups as an example of a t2st that is remarkably
;insensitive to the violation of homogeneity of variance, provided that
'~no
other assumptions are violated and that the two samples are of equal
•size.
Bradley explains that "the 'degree' of a test's robustness
3
4
;against violation of a given assumption is strongly dependent upon
\
factors which are not involved in the statement of a test's assumptions'
I ••• and
. which are not mentioned in the usual allegation of robustness.";i
I
!As an example, two of the eleven factors he lists as being relevant in
'determining the robustness of a test are the location of the rejection
;region (whether left-tailed, right-tailed, or two-tailed) and the size
j
'
!of the significance level used (such as .05 or .01). "These factors
cause no distortion of Type I or Type II errors when all assumptions
are met, but greatly influence the distortion occurring under a given
.violation of assumptions- i.e., the factors interact with whatever
violation occurs."
each other.
In addition, these factors tend to interact with
Therefore, except in the case where only homogeneity of
variance or only normality of distribution is violated, nonparametric
;techniques are preferred over parametric techniques when underlying
!assumptions are violated.
Although the use of the Mann-Whitney U test does not
ass~me
homogeneity of variance, normality of distribution, or equaltiy of
sample size, it does assume that the two samples are independent •
. Hollander, Pledger, and Lin (1974) studied the behavior of the Mann'Whitney U test when pairing of X. andY. exists for some values of i.
1
1
A mathematical rather than an empirical investigation
was performed.
The results showed that the Mann-Whitney U test is "conservative when
ithe bivariate distribution governing the pairing is positively quadrant
dependent."
~~~·-,~
••.,.•.,.,...
..
~-----
,"_,__.....,,~
..
..,.~
......,..,..,. • ......,.,.,,.. ..........
- - - - -•
.,._..~<M.<.-~=-"'''-"'•'""...,.~-
.:.---~_,
_ _ __...._ _ _,._... _ _
.,...~--...--"'=-'"""'"---.,.....,.,...,_.,
...
_~,_,
.. .....--'-'--"•
~
;
CHAPTER III
METHOD
.The Mann-Whitney U
In 1947, Mann and Whitney published an article in which they
presented a nonparametric technique for testing the significance of a
difference between two independent samples (17).
This was a develop-
ment of the Wilcoxon two-sample test introduced by Frank Wilcoxon in
. 1945 (23).
These two tests have been shown to be linearly related and
thus yield equivalent results (1).
;
Kruskal showed that this technique
was developed at least four times prior to Wilcoxon's publication, the
'first being traced to Gustav Adolf Deuchler's publication in 1914 (16).
The calculation of the Mann-Whitney U requires combining the
;values of the two samples, ranking each value, and then summing the
·ranks for the values in Group 1.
- L:R
, where n
1
and n
2
1
are the sample sizes of Group 1 and Group 2, and L:R
1
is the sum of the ranks of the values in Group 1.
A different U value can be calculated using the sum of the ranks
of the values in Group 2.
The formula used in this case is:
- L:R
5
2
6
·-~-·-;,.,_,.-.,,.,... _ _ ,.,,.~,.,..."""""-~ ......::...t:.·<'.-·~-r • ..,..--~...... ~-.
.... . , , . _ . ..... ...__;:...., .....~--"-'----·-·-·- - - - - - -..-~~ .... _ . . . . . _ _ , . . ...,~·~-,-~~~·.-• .,
{
I
I
jwhere n
1
and n
2
are the sample sizes of Group 1 and Group 2, and ER
2
lis the sum of the ranks of the values in Group 2.
I
The smaller of the two U values is the one used to determine
I
~significance.
So that both values need not be calculated directly,
•the second value can be found indirectly using this formula:
.where n
1
and n
2
are the sample sizes of Group 1 and Group 2, u is the
1
U value of one of the groups, and u is the U value of the other group.
2
The smaller of the two values, u
1
or u , is the one whose sampling
2
distribution is the basis for the published tables.
:Simulation Procedure
To investigate the sampling distribution of the Mann-Whitney U
test when only the assumption of independence between the two groups
•is violated, a Monte Carlo study was performed to simulate the situations in which 10, 20, .and 30 percent of the pairs of values were
correlated.
The simulation program for two groups of 10 values each
. can be found in Appendix C.
Twenty random uniform numbers were generated and then converted to
random normal variates using the Box-Muller transformation (3).
The
i first 10 numbers were assigned to Group 1 and the next 10 were assigned
' to Group 2.
i
'
This represents a situation in which there are two inde-
pendent groups of 10 subjects each.
.
!.~ ...~ .... ~<.-<=<·-=,~~--,,., .... ~~.-.-,..--..~.-...-- ....,.,~ ...~-····"""'~--=---.,.._---,..,_.._....,..,__.,~... -~-~---------"'"""""------,.--- ...- - - - - - -.........,__~-·-~-~---·----·-'·"'~'~~J
7
The t value for independent groups was computed in a subroutine
!for the 2 groups of 10.
This t value was sent to another subroutine
!which placed it into one of twelve mutually exclusive intervals, as
jexplained below.
The Mann-Whitney U value was also computed for the 2 groups of
l~is
10.
U value was sent to a subroutine that placed it into one of
twelve mutually exclusive categories.
Then a random number, K, between 1 and 10 was generated.
This
number was used to randomly choose a value in Group 1 and its corresponding value in Group 2.
The Correlated Variables formula, used
to compute y is as follows:
:where
x1
is the randomly chosen value in Group 1,
x2
is the correspond-
ling value in Group 2, and pis the population correlation.
set at .7 for the entire study.
Rho was
A positive value of rho was chosen
because most medical research involves positive correlations.
for example, is positively correlated with most diseases.
. to replace
X~
...
Age,
Y was used
so that one of the pairs, or 10 percent of the values
were correlated with a rho of .7.
With 10 percent correlated data, the t value and the U value
:were again calculated.
These two values were each placed into one
of twelve mutually exclusive categories, similar to but separate from
the categories used when there were no correlated pairs.
Another pair of values was randomly selected and correlated,
8
!representing the situation in which 20 percent of the pairs are correj
llated.
Again the t and U values were calculated and put into frequency
Ijdistributions,
similar to but separate from the categories used
i
(preyiously.
Finally, another pair of values was randomly selected and correlated, resulting in 30 percent correlated pairs.
The t and U values
were again calculated and put into frequency distributions.
The
twelve categories of the frequency distribution for both the t and
the U values, expressed in terms of two-tailed probabilities are:
P<.Ol
.20<P<.30
.60<P<.70
.Ol<P<.05
.30<P<.40
.70<P<.80
.05<P<.l0
.40<P<.50
.80<P<.90
.10<P<.20
.50<P<.60
.90<P<l.OO
Due to the ease of categorization, the Kolgomorov-Smirnov test
was chosen to test for differences among the distributions.
Closer
inspection of the tails of the distributions was also possible because
of the use of this categorization.
The simulation program was iterated 2,000 times.
Therefore,
2,000 t values and 2,000 U values were calculated and placed into
separate frequency intervals for each percent of correlated pairs.
1
similar simulation program was used with two groups of 20.
Figure 1 illustrates the simulation procedure utilized.
A
9
FIGURE 1
ONE ITERATION OF THE SIMULATION PROGRAM
STEP 2:
:STEP 1:
Independent Data
10 Percent Correlated Pairs
X
y
X
y
xl
yl
x1
yl
x2
y2
x2
y2
x3
y3
x3
y
x4
y4
xs
y5
x6
x6
y6
x7
¥6
y7
x7
y7
xs
y8
xs
y8
x9
y9
x9
y9
xlO
y10
x10
y10
3
p=.7
x4 --------- y4
y5
xs
STEP 4:
/STEP 3:
!
20 Percent Correlated Pairs
X
y
X
1
yl
x2
y2
30 Percent Correlated Pairs
X
y
xs
y8
yl
p=.7
x2 --------- y 2
y3
x3
p=.7
x4 --------- y 4
y5
xs
y6
x6
p
x7 --------- y 7
y8
xs
x9
y9
x9
y9
x10
y10
x10
ylO
y3
p=.7
x4 -------- y4
y5
xs
x3
x6
x7
__E~.:.Z ___
y6
y7
x1
10
--------
!
I
;Levels of Confidence
I
To place the U and t values into frequency distributions reflec-
!ting the above-mentioned probabilities, levels of confidence were
!
needed.
For the Mann-Whitney U, reference to a table of the cumulative
'probability distribution for n=10 and m=10 (19) gave the levels of
I
jconfidence for 2 groups of 10.
A similar table was not available for
l
l
n=20 and m=20 so one was generated by the researcher (see Appendix D).
Reference to a table of the t distribution for 18 degrees of
. freedom (20) · gave the l·evels of confidence for 2 groups of 10.
values for 38 degrees of. freedom
could not be located.
Tabled
as needed for the 2 groups of 20
Therefore, linear interpolation
the tabled values for 30 and 40 degrees of freedom (8).
was done using
..,.,_.....,~·__ ,.,._~,..._~...,__-_.~,...._.,. _ _ ....._.,.c,.......,......,..~..._-...,.,-,,,..,_..,_~,_-·- - - - - - - - - - - - " - - ·_.....,_,.._~-- ,~.-...,.....,...,._,.
'
CHAPTER IV
i
j
RESULTS
· The purpose of this study was to investigate the effect of the
violation of between group independence, especially in the tails, on
the Mann-Whitney U test • . Thus, Table 1 shows the proportions of the
experimental sampling distributions falling at or below the critical
values of .01, .05, and .10.
All of the obtained Monte Carlo proper-
'tions were systematically smaller than that expected from the theoretical distribution (except for n=10, 0% correlated pairs) •. The normal
deviate z was used to test the significance of the difference in
proportions (20).
Each proportion containing a percent of correlated
ipairs was compared to the proportion containing no correlated pairs
ifor each sample size and for both the Mann-Whitney U and the Student's
'
t statistics.
Those proportions that reflected a statistically
significant difference at the .01 level are marked with a*· ·Another
'test, also using the normal deviate z (6), was applied to find the
·significance of the differences in proportions comparing each of the
four obtained proportions (0, 10, 20, and 30 percent correlated pairs)
'with the theoretical proportions.
Statistically significant differ-
•ences at the .01 level are marked with a II.
It can be seen from the
' table that there were three more statistically significant differences
; in proportions among the U values than among the t values.
The U
i
; statistic appears to be more sensitive to the violation of the assump-
1
i tion
of independence, thus making it less robust than the t statistic
:under these conditions.
For both the Mann-Whitney U and the Student's
11
I
12
TABLE 1
OBTAINED MONTE CARLO PROPORTIONS: THEORETICAL.CUTPOINTS
BY PERCENT CORRELATED PAIRS
U Statistic
;Sample
N=10
Theoretical Cutpoints
.05
.0505
.0415
• 0335 II
.0285*#
0%
10%
20%
30%
.01
.0095
.0095
.0065
• 0055 II
0%
10%
20%
30%
.0070
.0070
.0040 II
• 0025 II
0%
10%
20%
30%
t Statistic
Theoretical Cutpoints
.01
.05
.0090
.0505
.0090
.0440
.0060
.0405
.0045 If
.0335 II
0%
10%
20%
30%
.0080
.0070
• 0050 II
.0030 II
.10
.0980
•0825 If
.0775 II
.0640*11
N=20
•Sample
N=lO
.0435
.0360 II
.0295 II
.0240*/t
.0925
.0805
.0685
II
II
.0595*11
.10
.1090
.0930
.0840*11
.0695*1.!
N=20
.0475
.0405
.0315 II
.0915
.0840
.0750
.0245*11
.0605*11
·*Statistically significant at .01 level - comparing distribution
with distribution containing no correlated pairs
, #Statistically significant at .01 level - comparing distribution
with theoretical distribution
II
If
13
t statistics,
the number of proportions that were significantly
different, increased as the sample size increased.
Also for both
statistics, the number of significantly different proportions increased
l
as the percent correlated pairs increased and as the theoretical cutpoints within the critical region increased.
The experimental sampling distributions with 10, 20, and 30
'percent correlated pairs were modifications of the experimental
sampling distribution with no correlated pairs.
Thus, it would be more
meaningful to study the z values that compare each distribution with
correlated pairs, to the distribution with no correlated pairs.
Table
2 shows those z values and significance at the .01 and .05 levels is
indicated.
Reference to the table reveals the same pattern of signif-
icance for the U statistic when n=10 as when n=20.
similar for the t statistic.
This pattern was
There were no statistically significant
;differences for either statistic and for either sample size when there
:were 10 percent correlated pairs.
Similarly, for neither statistic
and for neither sample size, were there statistically significant
differences in the proportions falling at or below the .01 level.
20 and 30 percent correlated pairs,
For
there was a statistically signif-
icant difference at the .05 and .10 levels.
The sixteen experimental sampling distributions, results of the
· simulation program, are given in Appendix A.
A comparison of each
: distribution in which there existed a percent of correlated pairs with
the distribution in which none of the values were correlated was
1
performed for the U and the t values and for both sample sizes.
The
; Kolmogorov-Sm.irnov one-sample test was applied, treating the distribu-
14
TABLE 2
Z VALUES COMPARING THE TAILS OF EACH DISTRIBUTION WITH CORRELATED PAIRS
TO THE TAILS OF THE DISTRIBUTION WITH NO CORRELATED PAIRS
U Statistic
Sample
N=10
.01
10%
20%
30%
.9535
1.2713
10%
20%
30%
.9535
1.4302
0
Theoretical Cutpoints
.05
1.3059
2.4666*
3.1921**
.10
1.6338
2.1609*
3.5839**
N=20
1.0882
2 .0313*
2.8294**
0
1.2649
2.5298*
3.4785**
t Statistic
Sample
N=10
.01
10%
20%
30%
.9535
1.4302
10%
20%
30%
.3178
.9535
1.5891
0
Theoretical Cutpoints
.05
.9431
1.4510
2.4666*
.10
1.6866
2.6352**
4.16371~*
N=20
*
Statistically significant at .05 level
**Statistically significant at .01 level
1.0157
2.3215*
3.3372**
.7906
1. 7393
3.2677**
15
.tion in which there were no correlated pairs as the theoretical distril
lbution.
I
The results are shown in Table 3.
For both sample sizes and
!for.both the U and the t statistics, there was a statistically signifi
I
!icant difference in the experimental sampling distributions with 20
.and 30 percent correlated pairs.
!correlated pairs,
In the distributions with 10 percent
the Kolgomorov-Smirnov test showed no statistically
:significant difference for both sample sizes and for both the U and
the
t
statistics.
16
TABLE 3
KOLMOGOROV-SMIRNOV VALUES COMPARING THE DISTRIBUTIONS IN WHICH A
PERCENTAGE OF THE PAIRS ARE CORRELATED WITH THE DISTRIBUTION
IN WHICH NONE OF THE PAIRS ARE CORRELATED:
PERCENT CORRELATED PAIRS BY SAMPLE SIZE
U Statistic
Percent Correlated Pairs
Sample Size
20
10
30
10
0.0155
0.0365**
0.0510**
20
0.0215
0.0375**
0.0550**
t Statistic
Percent Correlated Pairs
Sample Size
10
20
30
10
0.0160
0.0400**
0.0550**
20
0.0245
0.0380**
0.0565**
· **Statistically significant at .01 level
CHAPTER V
DISCUSSION AND CONCLUSION
This investigation showed that the results given by the MannWhitney U test reflected the amount of spurious correlation present
and the level of alpha used for the test.
More specifically, there
was not a statistically significant difference from the ideal case
(independence of observations) in the results given by the Mann-Whitney
.U test when there were 10 percent correlated pairs for samples of size
10 and 20, a rho of .7, and an alpha of .01.
As the number of corre-
.lated pairs increased, the number of U values falling into the critical
regions of significance decreased.
Thus, there was a statistically
·significant difference in the results given by the Mann-Whitney U test,
when there were 20 percent correlated pairs (P<.OS) and when there
were 30 percent correlated pairs (P<.01), with both sample sizes, a
rho of .7, and an alpha of .05 or .10.
The results of the Student's t test corresponded to the results
of the Mann-Whitney U test.
As expected there was not a statistically
significant difference·in the results given by the t test when there
were 10 percent correlated pairs with samples of size 10 and 20, a
·rho of .7, and an alpha of .01.
This study also demonstrated that the results given by the Mann. Whitney U test were not statistically different,
~ven
when 20 or 30
:percent correlated pairs were present when an alpha of .01 was used
. with a rho of • 7 and a sample size of 10 or 20 •
As the level of alpha
increased from .01 to .10, the empirically derived proportions of U
17
18
!
;values falling into those regions decreased.
I!of
Therefore, with alphas
-
.05 and .10, the results given b y t h e Mann-Whitney U test showed a
!statistically significant difference (P<.05 for 20 percent correlated
1pairs;
P<.Ol for 30 percent correlated pairs) with both sample sizes
·and a rho of .7.
Again the results of the t test corresponded to the results of
;the Mann-Whitney U test.
The results given by the
t
test were not
statistically different with 20 or 30 percent correlated pairs, given
an alpha of .01.
To summarize, the results of the Mann-Whitney U test are not
significantly effected by the presence of 10 percent correlated pairs
for samples of size 10 or 20, and a rho of .7.
In addition, the
results of this test are not significantly effected by 20 or 30 per;cent correlated pairs if an alpha of .01 is used.
Therefore, in
applying the Mann-Whitney U test to data in which 20 or 30 percent of
ithe pairs are correlated, an alpha of .01 should be used in lieu of
an alpha of .05 or .10.
BIBLIOGRAPHY
Ii1•
I
:
Auble, Donavan. "Extended Tables for the Mann-Whitney Statistic,"
Institute of Educational Research Bulletin, Vol. 1, No. 2,
Indiana University, 1953.
2.
Boneau, C. Alan. "The Effects of Violations of Assumptions
Underlying the t Test," Psychological Bulletin, 57:49-64,
1960.
\3.
Box, G. E. P., and Mervin Muller. "A Note on the Generation of
Random Normal Deviates," The Annals of Mathematical
Statistics, 29:610-611, 1958.
4.
Bradley, James V. Distribution-Free Statistical Tests.
Cliffs: Prentice-Hall, Inc., 1968.
5.
Chay, S. C., R. D. Fardo, and H. Mazumdar. "On Using the BoxMuller Transformation with Multiplicative Congruential
Pseudo-Random Number Generators," Applied Statistics, 24:
132-135, 1975.
6.
Dixon, Wilfred J., and Frank J. Massey. Introduction to Statistical Analysis. 3rd edition.· New York: McGraw-Hill Book Co.,
1969.
7.
Edwards, Allen L. Statistical Methods. 2nd edition.
Holt, Rinehart, and Winston, Inc., 1967.
8.
Fisher, R. A., and Frank Yates. Statistical Tables for Biological,
Agricultural, and Medical Research. New York: Hafner
Publishing Co., 1963 .
• 9.
10.
Englewood
New York:
Gocka, Edward F. "Alternate Tests for Comparing Independent
Groups," Psychological Reports, 32:683-692, 1973.
., Chief, Predictive and Evaluative Models Research
Laboratory of the Veterans Administration Hospital, Sepulveda,
California. Personal Communication, 1976.
11. Gordan, Geoffrey. System Simulation.
Hall, Inc., 1969.
Englewood Cliffs: Prentice-
12. Gruenberger, Fred. Computing: An Introduction.
court, Brace & World, Inc., 1969.
New York: Har-
· 13. Hollander, Myles, Gordon Pledger, and Pi-Erh Lin. "Robustness of
the Wilcoxon Test to a Certain Dependency Between Samples,"
The Annals of Statistics, 2:177-181, 1974.
19
20
14.
International Timesharing Corporation.
Manuel. RM-17-0172-01, 1972.
15.
Kendall, Maurice G., and William R. Buckland. A Dictionary of
Statistical Terms. New York: Hafner Publishing C., 1960.
16.
Kruskal, William H. "Historical Notes on the Wilcoxon Unpaired
Two-Sample Test," Journal of the American Statistical
Association, 52:356-360, 1957.
17.
Mann, H. B., and D. R. lfuitney. "On a Test of Whether One of
Two Random Variables is Stochastically Larger Than the
Other," The Annals of Mathematical Statistics, 18:50-60,
1947.
I. T. S. EXFOR Reference
·18.
Neave, Henry. "Or.. Using the Box-Muller Transformation with
Muitiplicative Congruential Pseudo-Random Number Generators,"
~plied Statistics, 22:92-97, 1973.
'19.
Owen, Donald B. Handbook of Statistical Tables.
son-Wesley Publishing Co., Inc., 1962.
20.
Reading: Addi-
Siegel, Sidney. Nonparametric Statistics for the Behavioral
Sciences. New York: McGraw-Hill Book Co., 1956.
'21.
Snedecor, George W., and William G. Cochran. Statistical Methods.
6th edition. Ames: The Iowa State University Press, 1967.
• 22.
Teague, Robert. Fortran: A Discovery Approach.
Canfield Press, 1974.
:23.
Wilcoxon, Frank. "Individual Comparisons by Ranking Methods,"
Biometrics, 1:80-83, 1945.
24.
San Francisco:
Yee, Jean, Graduate Student in Biostatistics at California State
University, Northridge. Personal Communication, 1976.
CONTENTS OF THE APPENDICES
TABLES OF THE EXPERIMENTAL RESULTS
4a-4d
22
Experimental Sampling Distributions of the Mann-
.
Whitney
U and the Student's
t Statistics for
.
.
Sample Sizes of 10 and 20
B•
C.
D.
E.
23
THE MONTE CARLO PROCEDURE • • . • • • • • • • • • • • • • • • • • • • • • • • • • • • . • • •
27
Preliminary Investigations ••••••••.•••••••••••••••••••••
27
Random Number Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
Box-Muller Transformation...............................
31
Correlated Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
Derivation of Correlated Variables Formula •••••••.••••••
32
THE SIMULATION PROGRAM • • • . . • • • • • • • • • • • . • • • • • • • • • • • • • • • • • • • •
35
The Computer Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
GENERATION OF THE CUMULATIVE PROBABILITY DISTRIBUTION OF
THE MANN-WHITNEY U FOR 2 GROUPS OF 20 ••••••••••.••••••..••.
43
Generated Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
ILLUSTRATION OF THE
58
P~
COt~UTATION
21
OF A MANN-WHITNEY U •••••
APPENDIX A
TABLES OF THE EXPERIMENTAL RESULTS
22
23
TABLE 4a
EXPERIMENTAL SAMPLING DISTRIBUTIONS
MANN-WHITNEY U STATISTIC
SAMPLE SIZE OF 10
Percent Correlated Pairs
P Values
0
10
20
30
P<.01
19
19
13
1.1
.01<P<.05
82
64
54
46
.05<P<.10
95
82
88
71
.10<P<.20
192
199.
188
176
.20<P<.30
170
177
159
162
.30<P<.40
228
217
211
218
.40<P<.50
169
179
182
190
.50<P<.60
178
187
209
217
.60<P<.70
211
206
224
211
.70<P<.80
227
248
235
247
.80<P<.90
117
107
117
121
.90<P<l.OO
312
315
320
330
2,000
2,000
2,000
2,000
Total
24
TABLE 4b
EXPERIMENTAL
SPu~LING
DISTRIBUTIONS
MANN-WHITNEY U STATISTIC
SAMPLE SIZE OF 20
P Values
P<.Ol
0
Percent Correlated Pairs
10
20
30
14
14
8
5
.o1<P<.o5
73.
58
51
43
.05<P<.l0
98
89
78
71
.10<P<.20
197
201
203
175
.20<P<.30
204
195
176
182
.30<P<.40
208
195
203
208
.40<P<.50
153
171
176
174
.50<P<.60
212
210
236
238
.60<P<.70
237
230
226
242
.70<P<.80
205
195
226
224
.80<P<.90
207
243
210
208
.90<P<l.OO
192
199
207
230
2,000
2,000
2,000
2,000
Total
25
TABLE 4c
EXPERIMENTAL SAMPLING DISTRIBUTIONS
STUDENT'S
t
TEST
SAMPLE SIZE OF 10
P Values
0
Percent Correlated Pairs
10
20
30
P<.01
18
18
12
9
.01<P<.05
83
70
69
58
.05<P<.10
117
98
87
72
.10<P<.20
183
196
196
181
.20<P<.30
192
203
174
179
.30<P<.40
211
190
186
195
.40<P<.50
198
201
232
223
.50<P<.60
196
211
214
212
.60<P<.70
199
196
193
228
.70<P<.80
208
218
210
201
.80<P<.90
207
205
216
231
• 90<P<l.OO
188
194
211
211
2,000
2,000
2,000
2,000
Total
26
TABLE 4d
EXPERIMENTAL SAMPLING DISTRIBUTIONS
STUDENT'S
t
TEST
SAMPLE SIZE OF 20
P Values
0
Percent Correlated Pairs
10
20
30
P<.01
16
14
11
6
.01<P<.05
79
67
52
43
.05<P<.10
88
87
87
72
.10<P<.20
204
186
180
162
.20<P<.30
208
205
200
210
.30<P<.40
204
207
193
193
.40<P<.50
184
203
224
218
.50<P<.60
234
199
223
229
.60<P<.70
188
202
198
208
.70<P<.80
202
210
221
241
.80<P<.90
184
212
206
210
.90<P<1.00
209
208
205
208
2,000
2,000
2,000
2,000
Total
APPENDIX B
J
THE MONTE CARLO PROCEDURE
I
I
l
Preliminary Investigations
In an effort to generate random normal variates that best fit the
itheoretical normal distribution, many preliminary tests were run on a
variety of random number generators and variations thereof.
Finding
a perfect random normal generator was not the purpose of this thesis,
yet a good generator that was quick and easy to incorporate into the
simulation program was needed.
Due to the amount o£ work involved in
preparing a table look-up method (12) or constructing an original
congruential generator (12), these two methods were immediately ruled
out.
For each of the methods that were investigated, 40,000 random
normal variates were generated.
These variates were placed into a
'frequency distribution of 22 categories using a subroutine developed
by Jean Yee (24).
To evaluate the results given by each method, a chi-
square test was done comparing the frequency distribution of the generated variates to the theoretical normal frequency distribution.
: where 0 is the observed frequency - that ob·tained by the technique
'under investigation, and E is the expected frequency - that expected
j under the theoretical normal distribution.
27
The method that yielded
28
the smallest X2 value was chosen for implimentation in the Monte Carlo
study.
Geoffrey Gordon proposed the use of two subroutines, based on a
:multiplicative congruential method, to generate random normal variates
(11).
One subroutine generat·es random uniform numbers and the other
transforms them into random normal variates.
An initial value,
assigned by the programmer, was needed to start the computations.
Using two different initial values, two runs were made using this
The X2 values, for both initial values, came to over 10,000.
metl1od.
Observation alone verified that too many values were falling into the
tails of the distributions.
To determine whether Gordon's random uniform number generator
would work better in conjunction with another technique for trans.forming uniform numbers into normal variates, the Box-Muller
trans~
formation (3) was used.
x1
where
norrr~l
u1
and
u2
variates.
1
= (-2 ln
u )~ cos(2nU )
1
2
are random uniform numbers.
x1
and
x2
will be random
The Box-Muller transformation was chosen because it
. gives higher accuracy and is faster than other methods (3).
·two different initial values were utilized.
The resulting distribu-
• tions were both very obviously positively skewed, producing
·of over 8,000.
As before,
x2
values
Apparently a different random uniform number generator
was needed as well.
29
Replacing Gordon's subroutine for the generation of random
uniform numbers was the random number generator associated with the
batch mode of the ANSI FORTRAN compiler.
formation was used.
Again the Box-Muller trans-
The X2 value for this technique was 14.5.
In an effort to improve upon this method, a variation of the Box·Muller transformation employed by Neave (18) was tested.
Instead of
I
:using both equations given by Box and Muller, Neave used just the one
with the sine function.
The investigation of the distribution produced
by this variation gave a
x2
value of 26.4.
with the cosine function resulted in a
x2
Using instead the equation
value of 24.7.
In response to Neave's publication, Chay, Fardo, and Mazumdar (5)
indicated that reversing the order of
equation would give better results.
u1
and
u2
in the Box-Muller
This suggestion was applied to the
.method that had proved most successful, namely the original Box-Muller
transformation as proposed by Box and Muller.
The resulting X2 value
was 17.9.
The method that yielded the smallest
x2
value, is the method that
.produced random normal variates most closely fitting the theoretical
normal distribution.
rhus, the decision was made to use the random
number generator in the ANSI FORTRAN System Library and the Box-Muller
transformation as originally proposed, to generate random normal
, variates for the simulation program.
• Random Number Generator
The random number generator, made available under the 11ASTER ANSI
30
;FORTRAN IV library, is based on this algorithm:
A random uniform number, U, is returned such that O<U2_1.0.
Multiplicative congruential generators such as this, are called
pseudo-random number generators because it is knmm in advance what
numbers will be returned by the generator.
In an attempt to make this
generator more truly random, a way was devised to generate a non reproducible series of numbers.
The following statement will call the
random function:
U
=
RANDOM (i)
where RAlfDOM is the name of the function and i is the argument.
be a uniform random number.
U will
If i is negative, the generator is set
back to the beginning of the reproducible sequence each time the
function is called.
If i equals zero, the first number comes from the
beginning of the sequence and each time another number is called, it
is taken from where the last number left off.
If i is a positive
number, the generator is set back to the beginning of the reproducible
series as before.
With this argument, though, a certain number of
values are skipped before U is returned.
The number of values "burned
· off" is determined by the least significant portiC'n of the current
·value of the real-time clock in the computer.
Each subsequent number
called is reported only after a certain number of values are skipped .
. Thus, the use of this argument makes for a more truly random number
31
:generator, one that returns a non reproducible series of numbers.
I
1
iBox-Muller Transformation
The Box-Muller transformation takes random uniform numbers and
transforms them into random nonnal variates (3).
where
u1
and
val (0,1).
u2
are independent random uniform numbers from the inter-
x 1 and x 2 will be independent random normal variates with
a mean of zero and a variance of one.
Correlated Variables
To simulate two samples in which certain percentages of the pairs
are correlated, the Correlated Variables fonnula was used:
where x
1
is a value in Group 1, x
2
is the corresponding value in Group
2, and p is the population correlation.
·Correlated with X .
1
Y is a value that will be
32
Derivation of Correlated Variables Formula for Population
Let Y be a linear combination of x 1 and x •
2
y
=AX,.... + BX 2
.
Therefore:
( 1-1)
;Let
x1x2
ru
N(0,1)
and
x1 ,x 2
be independent.
Then
a2
x1
az
x2
az y
=1
1
A2 + B2
(1-2)
.
(1-3)
' By setting the following restriction,
(1-4)
• and substituting in Equation (1-3), we obtain
Then
33
Substituting this result in Equation (1-1) we get
(1-5)
The correlation of
x1
p
and Y is
=
(1-6)
Referring back to Equations (1-2) and (1-4) we see that
Therefore
.Substitution of these results in Equation (1-6) gives
2 k
=
p =
ALX 2
=
. But because
x1
and
x2
LX (AX + (1-A ) 2X
1
1
2
N
1
1
+ (1-A 2 )~ LX
N
are independent,
x
1 2
34
Then
p
·AL.x 2
1
= __....;:..._
(1-7)
N
Since
1
substitution in Equation (1-7) results in
p
A •
By substituting this in Equation (1-5), we obtain
APPE11D IX C
THE SIMULATION PROGRAM
The following is the computer simulation program for two groups
of 10.
A similar program was used for two groups of 20.
35
36
N 0 01
N 0 02
N 0 03
N
N
N
N
N
N
0 04
0 C5
0 06
D C7
0 08
c cq
N 0 10
N 0 11
N Q 12
N 0 13
N " 14
N
0
N 0
~l
C THIS IS A COMPUTER SI~UL~TION ~RJG~AH TO OSTAIN THE FREOU~NCY DISTRIBUTIONS OF
C THE T ANO THE MANN-WHITNEY U FJR J 1 1C 1 2Ji AhD 30 PERCENT CORRELATED PAIRS
CPENSION R.At,QI'~O I, '<.ANO'i.M12JI ,lNT 1.~01, SUHI3C I
OIM<::NSIJN IrHTI12loi>H1JTI121 ,INTUI121,INT10UI121
0 I '1 ': NS I 0 N IN TT1 ( 12 I , PI 1 0T 1 ( 12 I t IN TU1 ( 12 I , IN H U1( 1 21
OI~Et.:SiON HHT2U21 ,IN1GT21121 ,INTU2(12l,INHU2!121
:Hi'1ENSION I~lTT3!121 ,INluT3112l ,INTU3112l tlll1CU31121
P HO = • 7
REMl IS0,101 SAHPSZ
1C FC9~AT IF.L ~')
00 20 I=t,3C
20 ISU'HII =0
ao
30 I=1,12
H<T10TIII"'O
tt.l 0 Tt (I I = 0
HH':T? IT l =0
15
16
30 It-.11TIIIl=O
OC4:li=1.t2
ll 17
N C 18
N C 1'3
N IJ 20
N
a
21
N 0 22
N C 23
N 0 24
11 0 2?
N ~ 26
tl 0 27
r: o 28
1J 0 29
N C .3C
XHtSUlii=O
!N!CUJ.(Il =G
It-.1cU2ili=G
«.~
u.uu~o 1 =o
oo 1~:: '1=1, zor.o
C TO GEh~PATE 20 ~ANOOH UNIFORM NUMBERS BETWEEN 0 AND 1
5C
C TO
DO 5C l(:1,2Q
~'N~IKI=PAN00~11l
TQtN3F2~1'1 OANDCH UNIFORM
L=l.
RAti091"(L+il
&C L"'L+2
N C 35
N G 36
II C 3'1
N C 38
N C J'J
il 0 <.Q
7C
n 5 1::
N 0 .,1
.,z
N 0 47
~
9~
C TO
c
5r. 55
~~
1CO
H~
N 0 08
TO
~ ~ ;~
120
c.-•
E.2
63
N 0 i'b
n.o on 1 1 1 • 1 o •o+ 1. o
1 PAIR OF
~ALUES
(10
PE~CENTI
!Kt.NORH,S~~PSZ,Tl
FOI.ST1~T
1~C J=1 .. 12
IT.I'lTTll
DC
It;tOT11Jl=I'l10Tt!Jl+HiTT1!Jl
Cl-iLL
'-";ltlf'·;:~r.T(-\,~'iOr'i,St..'iP.SZ,N,U)
CALL FJISTlQU (U,INTU!l
00 11: J=l, 12
If,10U11Jl~I'i1CU:!Jl+IN1"U1!Jl
~A~10~LY 3~L[CT A~J CC~~EL~T~
K2=~ANJJ~(l)~tC.O+l.G
lK2.~0.K1l GO TO 120
1 MORE PAIR OF VALUES 120 PERCENT!
IF
K2V=«2•1r
13~
a
N 0 7~
N 0 7-5
K 1 =;::
RANOR~IK2Yl=RHO"~ANORMIK21+1SQRTI1-~H0 4 •2l•~ANC~M!K2YI)
65
6&
C S8
f. g
N G ;· G
N C 71
N 0 7 C.
N r n
I~T10U(Jl=INT1DUIJI+I~TU!Jl
PA~~OHLY SEL~CT ~NO CORRELATE.
CALL TEST (~;..r-;QO;M,SCd1PSZ,Tl
T=A'JS(TI
CALL rJISTlQT !T,I~TT21
b~
N 0 67
t<
I~A~ORH,3u,2J,-3,500,0.25,INTl
(JI =ISU:HJI +II;T !Jl
CALL
N C %
N 0 57
.N
lo~'!
T=~.'lS (T)
g~~
N 0
;-j 0
N C
N 0
N 0
N C
NU~BERS
=:SORT I 1·2l•ALOGIRANOIII I I •SINI2"' 3,1.,15g265 .. ~AND (1+111
CALL F~~O~IST
DC 7~ J=1,30
CALL TToST
N 0 50
~
NO~~AL
K1V~~1•1G
RANOC~(K1Yl=FHQ•RA~Q~M(~1l>tSQRTI1-RH0••21•RANC~H!K1Yil
1.5
N !) 4<J
~1
~
~ANDOH
N='·l
CALL TToST IRANJRH,SA~PSZ,Tl
T=.!i3S Ill
CALL FJIST1°T !T,INTTI
oo ilO J=1.12
SC !NT10T!Jl=I~110TIJI+l~TTIJI
CALL ~·.<NtniHT (~,\·JOFM, St.H?SZ, ~l.UI
CALL FJIST10U (U,INTUI
co go J=t.12
N 0
N 0 43
N 0 4b
TO
CA~~F~Ill=SD~l((-21•ALOGIR~NJ!Illl•COS(2•3.14153265•RANDII+1ll
N 0 31
N 0 32
N C ,.4
N G .,5
NU~9~RS
00 &0 !=1.20.2
rJG 13 ... J=l, 1.2
H:ET21Jl =l'l1CT2!Jl+l~><TT21Jl
CALL
CAL~
c
1'->r,
D:J
~~NN~~TI~A~O~~,;~~~SZ,N,Ul
F~IST1CU
14·=·
J=-1.12
IU,INTu2l
H:l.O!J2 Jl=l,H·JtJ2(JltHITU2!Jl
TO P~~~QY Y SiLfCT A~J CO~~EL~T~ 1
1~0 ;(1-='\A;J OH11l"H.u+1,G
! F ( '< ~ Ul, K 1l GO TO 1:; ~
IF tKJ lQ,K.?l GO TO 150
MO~E
PAIR OF VALUES !30 PERCENT)
37
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
lN
l N
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
0077
OC78
CCBO
0081
0082
K3Y=K 3+ 10
RANO~" I K3Y I =RHO• RAtiORM I K3 I+ !SQRT U•RH0 .... 2 I •RANORH (I(JYt I
CALL TTEST IKANORM,5AHP5Z,TI
T=A3SITI
CALL. FDIST10T IT, ItHT31
0~79
one 3
160
o~e"
0085
OC86
OJ87
OD88
170
MANNWHT!~ANO~H,SA~?SZ,N,Ul
F~IST10U
IU,INTu31
~~16~j~~~~f~IOU31JI+INTU31JI
!80 CONTINUE
WF.ITE 161,1901
190 FORMAT 17~H1 TH~ FOLLOWING IS THE FREQUENCY DISTRIBUTION FOR THE G
CENfgATEO NO~MAL NU~B~~S//1
DO 21G I::t 3G
w~:T£ 161,ZOCI ISU~III
200 FC~~AT !2X,!91
210 GOtH VlUC:
WRITE lb1,22CI
220 FCP~AT 111CH1 THE FDLLOHINi IRE THE FREQUENCY DISTRIBUTIONS FOR TH
CE T VALUES WITH G,1Qf2J, AND 30 PERCENT CORRELATED PAIRS!
WR!T': 1&1,2.%1 lNTlO
WRIT~ 1&1,23CI !N10T1
Wf'!TE !61 ,23L I !N10T2
WRITE 161,21CI IN1DT3
230 FO~MAT I//31H G~EATER THAN 0~ EQUAL TO 2,878,29X,I9/51H GREATER TH
CAN OR EDU~L TO 2,191 AND L~S5 THAN 2.878,9X,Ii151H GR::ATER THAN 0~
C [QUAL TO 1,73~ AND LESS TH~N 2.101,9X,I9/51H G~fATEF THAN OR fQUA
CL TO 1,330 AND LESS THA~ lo73*t9X,Iq/51H GREATER THIN D~ EQUAL TO
C1.~67 A"D LESS THA~ 1o33Jo3X,!Yf50H GqEATER THAN OR EQUAL T~ ,662
CANO Lrss THAN 1,0&7,1&X,I9141H G~EATE~ TH~N 0~ lQUAL TO .&88 MNC L
CESS THI~ .862 11X.l'3/4'3H G<EATE'< THAN OR EQUAL TO , 53<, AIID LESS TH
CAN o6!3,11X,I~/~9H GR~ATE~ TH~N OR EQUAL TO o392 AND LE55 THAt< ,53
('<,11X.I9/4'3H GR<::ATo~ THAll )il. t.QUAL TO .257 AtiO LESS T>iAN o3'32 1 11X,
CI9/49H Gg~ATF~ THAN DR :;:QU~L TO ,127 ANQ LESS T~AN .257,11X,I9/4EH
C G;EATE~ THAh OR EQUAL TO 0 hNO LESS THAN ,127 1 1~X,I91
1
1
2~0 ~~~~~T ~Ii~~I THE FOLLJWINi ARE THE FREQUENCY DISTil.IBUTIDNS FOR T~
CE U VALUFS WITH Q,to,za, A~D 30 PERCENT CG~RELATED PAIRS!
WgiTf lo1,2501 INTlOJ
W~ITE 1&1,25CI IN10U1
WR!TE (f>l ,23il I IN1QU2
WFITE 151 1 25CI IN1QU3
250 F09~AT !//27H L~SS THAN 0~ eQUAL TO 1&,0,33X,I9/49H LESS THAN ORE
CQUAL TO 23,1 ANJ G~EATER T~A~ 16,0 11X,I9/4'lH LESS TH~N D~ ~QUAL T
CO 27,C A~Q GR~ATER THA~ 23,0,11X,I~/~'lH LLSS T~AN DR €QUAL fD 32,3
C ANG GQEATE~ TH~N 27,(,11X,I9/~9H LESS THAt< OR EQUAL TO 35.~ AND G
CR~ATER THfN 32.1,11X,I'l/~1~ L~SS TH~t< OR EQUAL TO 3R.O ANJ GREATE~
C THAN 35,1,11X,IY/~9H LESS T~~N OR FQUAL TO Ll,Q A~D GR~ATER THAN
C38,0o11X,I'l/49H LESS THAN )R EQUAL fo ~2.J ANJ GREATER THAN 4G,~,t
C1Xt I91'+1H LESS THAN OF EQUh TO l,<t,O AND G~C:A~U T.JAN ~2.C, 11X,I9
C/4gH LESS THAN OR EQUAL TO ~b,O AND G9EATf R THAN 4-o0t11X,l9/~9H L
CESS THAW OR EOUAL TO 47,0 AW~ G~EATER THtN 46.0,11X,l3/50H LESS TH
CAN 0~ ~QUAL TU l=~.J A~G G~cATER THAN ~7.G,1CX,I91
.
STOP
END
OJ89
OC90
O~'l1
0092
0~93
CQ94
C Q95
O:J%
oog?
Cc'lS
0099
0 1C C
0101
0102
01G3
010"
C105
01Cb
01J7
0108
ltl 010 9
Ltl c 11 c
I.N 0111
LN
Ul
Ul
LN
Ltl
LN
LN
LN
LN
Ul
LN
LN
Ltl
LN
LN
LN
LN
LN
LH
LN
LN
LN
~2t5~j!~l;lh~GT31Jl+INTT3!JI
CALL
CALL
0112
0113
011!+
0115
Q 116
0117
011!1
0119
0120
0121
ll122
C123
0124
0125
0126
0127
0128
C129
C!30
0131
C132
0133
US~SI
FORTRA~
· NO EP?OiiS
JIAGNOSTIC RESULTS FOR FTN.HAIN
38
I
I
I
I
!
'
LN o·o01
LN
LN
LN
Ul
LN
LN
LN
LIJ
LN
Ul
LN
Ul
LN
LN
LN
Ul
LN
LN
LN
l..N
Lfi
O~C2
0003
OOCit
OU05
COCo
COC7
OG~8
0~1}9
0010
0 011
0112
CC13
on ..
OH5
COlo
0 ~ 17
ufJ18
CD19
ono
C~l1
0122
U~ASI
c THI~U~~~~~~~¥N~R~3¥1SJH~R'~~~5='~~~~~fN~5=x~~5 INTO A FREQUENCY DISTRIBUTION
c N=NUH1ER OF INTE~VALS INCLUOI~G FIRST AND LAST AS OPEN INTE~VALS
H=~UHdfR OF ~LEHiNTS IN ~~~AY R
B=LOWER BOUND ON RANGE
OF EACH INTERVAL
c SIZI=WIOTH
H=20
3=-3.~00
SIZI=0,25
c N=30
c
DIMENSION
RIZOI,INTE~\11301
DO 10 J=1,N
10 INTERIIIJI=O
DO 30 I=1 1 H
J" 1
.. - . -·---
--··--··
GO TO 20
IF IRIII,GT,(B+IN-2l•SlZIIIlNT£RVINI:INTER\IINI+1
GC TO 30
20 J=J+1
L=J-1
IF IRUI.GT.IB+L•SIZIII GO TO 20
IF
IF
(~III.GT,B.A~O.~III.LE.IB+IN-21"SIZIII
I R II I • Lt. , 61 ! N TE ~II I 11 =ItlT E '<. V I 11 +1
INTcF.v!JI=INTERVIJI+t
:SO CONT!NU£
REYUI<.II
END
FORTF.AN
NO ERF.OP.S
~!AGNOSTIC
RESULTS FOR
FR~QO!ST
39
LN OCG1
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
L'l
LN
LN
LN
LN
0~02
cn3
c
01l C Lo
0005
GOGb
0007
cCGd
0009
CHJ
~911
0012
0~13
CJ110
QH5
CQ1b
SU9'\CUTINE TTEST !DATA, SA'I?SZ,TI
THIS SUB~OUTIN~ C0'1PUTSS A T-TEST FOR THO INDEPENDENT G~OUPS OF EQUAL SiiE
DIM:NSION DATA 12CI
SU'1=0,0
10 ~s~;~U~~~A}~III
AHEANl=SU'I/SAHPSZ
SUM=O.J
00 20 1=11,20
20 SU'1=SU'1+DATAIII
AMEANZ=SU~/SAMPSZ
D£'/1=0.0
DO 3~ 1=1,10
3C O€V1=~cV1+1DATAtii•AH~AN1l••z
DEV2=C.C
DO <>9 1=11,20
~0
C~17
CHe
0'1'3
CG2C
0021
0022
D~V2=~=V?t(nATA(li·AH~AN2l••z
VA"=
lc 1 ~V1H;". V?l/12"
ISA'1>'SZ•11 I
VA~niF=SQqJ((2•~ARI/SIHPSZI
T=IA~~AN1•AMEAN21/JAKu!F
P".TUPN
E~lO
USASI FDF-TPAN DIAGNOSTIC RESULTS FOR
NO
ER~ORS
TT~ST
40
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
0001
f:OC2
OOCJ
oryo4
CQ05
00~6
OOC7
O~C8
O·JGq
CHG
~111
CC12
C~1J
0014
~015
0~16
LN or.t7
LN Q:j18
LN OC1'l
LN o,;zo
LN 0021
ltl G0?2
LN CG23
Lt¥ G124
\.N C025
LN OJ2c
Ul 0~27
L"l C028
LN
D~Z'l
lN
0~32
LN Q03G
LN 0231
LN 0~33
LN OG3'+
LN C~35
l.N oa3c
LN 0~37
Ltl CJJe
LN 0~3'3
LN oc.:.ry
LN 0341
LN 0042
Ui 0043
LN 01!.4
Ul OC 45
LN Q~i.o
LN GC47
LN CJ't8
L~i
C~4q
LN CG50
IN 0~51
LN
Ul
LN
CC52
CC'~J
C'15 t.
LN C055
\.N OCS&
Ul OU57
C
SUBROUTIN:: "'ANN\oiHT liiANOi<H,SAMPSZ,L,AMf.NNl
THIS suq~OUTINE CO~PUTES THE MANN-WHITNEY U FOR TWO GROUPS OF EQUAL SIZE
OIM~NSION
RANUR"'!20I,J!20I,RANK!20I,~RNCRM!2GI
!'0 10 I=1,20
10 RkNO~M!Il=R~NCRHIII
DO 20 I=1,10
2C J!Il=C
00 30 I=11,20
30 Jl!l=t
00 90 M=l,l'l
I=M
GO TO 70
41l I=I-1
GO TO 70
5C HRlT~ !61,601 L,R~NORMII+ll
60 FCR~AT ISH ON THE ol3,12H ITERATION, oF13,11,34H WAS FOUNO TO OCCU
CP THREe TIMES
l
:;roP
n IF !R"~0RM!Il.GT,R~NOR.Htl+1ll GO TO 80
IF II,GT.!I GO TO ~a
GO TC gc
~0
R=R'(~l0;(~t!+1l
D~NOC~(itl)=hRNORM!Il
R?NJR~1! I I
=R
K=JII+ll
JII+li=JIII
J!II=I(
IF II.GT.ll GO TO 40
9C CO>iTI:JUE
SUM=O,O
CC l.JS I=1,2C
RA'lKIII=C,O
IF !I .:::0,11 GO TO 12C
IF (RCN~~~!Il,E~.PRNO~~II+ll) GJ TO 110
IF (PC~O~~!Il.~S.RRNORM!I-111 qA~K!!l=~ANKII-ll+1o0
IF!~FNO~H!l-ll.EO,RRNCR~I!ll RANKIIl=RANKII-1)
IF !I.L0:.2l GO TO 1~C
IF !h'~OR4!l-2l,fO,RRNORM!I-1llRANK!Il=Rf.NK~Il+,5
100 IF IJ!!l.~Q.Ol
SUM=SU~+RANK!ll
GO TO 13G
110 IF ~~~NCR~!I+11.EO,RRNORH(I+2ll GO TO 5C
RANK!Il=RINKII-11+1,~
RANK!Il=!~ANK!I-1l+RANKII1+2,JI/2,0
IF !I.L0",21 GO TO 100
IF ~~~~oq~II-21.EO.RkNORH!I-1ll RANK!II=RANKII)+,5
GO TO iJC
12C IF tR~NO~M!II.E~.RRNCR~!I+ll l RANK!Il=1o5
IF !RF:IJG'<"'II+1l,EC,R~N0RM!I+2ll GO TO !$0
IF CR.a>lKIIl.t.Q.1,5l GO TO lOG
ll.ANW:IIl=l,G
GG T~ 1:0
130 CC'HIME
A~ANN=ISAHPSZ 4 SAMPSZl+!SA~?SZ•!SAMPSZ+l,Clli2,0-SUH
IMA~P<=~AMP3Z•SAMPSZ-AHAN~
IF !AHA'IP":,LT.A'11H•Nl AHAN'l=A'Ii\NPR
!iETU~N
END
USASI FOR.H.AN OIAGNOST IC RESULTS FOR !'lANNWHT
NO ERFOFS
41
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
LN
Ul
LN
LN
LN
LN
LN
COOl
0002
0%3
OJ04
OJ05
OCC7
0~08
000'3
0010
0011
0 HZ
C~ 13
O~C6
001'+
OH5
0016
C017
0018
0019
C
SUB;;OUT
THIS SU~RO
OIHEN3I
DO 10 I
10 I NT I I I=
IF <T, G
IF (T,G
IF !T.G
IF !T,G
IF !T.G
IF !T.G
IF !T.G
IF IT.G
IF (i,G
IF (T.G
IF !T.G
IF !T,G
RFTURN
NS FDIST1CT «T.INTI
TINE PLACES T VALUES INTO A FREQUENCY DISTRIBUTION
N I~T!121
1.12
, 2. ll7 81 HJT I 1l = 1
.2.101,ANQ.T.LT,2.8781 INT!21=1
.t.~34.ANO,T.LT.2.1D11
INTI31=1
.t.33G,AN~.T.LT,t,7341
INTI41=1
.1.0t7oANO.T.LT.1.33GI INT!51=1
,.B62.A!IIO.TnT.t.a'>71-!NT!61=1
•• &~&.AND,T.LT,,8621 INTI71=1
•• 534,A'IO,T.LT,,6881 INT!81=1
,,3g2.tNQ,T,LT •• 53<1 !NTI'31=1
•• 257.ANO.T,LT •• 3'321 INTI101=1
•• t27.A!IID,T.LT,,25TI I!IITI111=1
,J,O,ANJ,T,4T.,1271 INTI121=1
E~O
USASI FORTRAN DIAG'IOSTIC RESULTS FOP. FOIST1DT
NO E"R.RDRS
42
LN
LN
LN
LN
LN
LN
LN
LN
LN
d<
LN
LN
LN
U!
LN
LN
LN
Cr!O!
0002
OOC3
cJ t '+
oc~
O~C6
CC07
0006
CH9
0010
OH1
0012
OH3
0014
G015
OH6
OG17
LN OQ18
LN C019
c
SUHOUTINE FDIST1CU lU,INTJI
THIS SUBPOUTINE PLl\C~S u VALU~S
DI"!~NSlUN INTUt121
DO 10 1=1,12
1C INTU!Il=O
IF tU.LC:.16,0l :NTUt11=1
IF tU.Ll.?.3,C,ANO,U,GT.1&,Ql
IF IU.LC:,27,0,A~O.U,GT,23.01
IF (U,LE,32,G,AND,U,GT.27,01
INTO A FREQUENCY DISTRIBUTION
INTUt21
INTUt31
INTUI41
IF tU.LE,35,G,A~o.u.;r,32,QI INTUt51
IF (U.LE.38.C.Ar~o.U.GT.35.CJ I'ITUt&l
IF w.u;. ~o .c ,ANo.u.p. Ja.
INTUI71
IF tu.L~.~2.c.A~o.u.~r.~o. , INTUt81
IF tU,LE.~~.C..ANO,U,GT,*2.3l INTUt91
IF tu.u:.46 ,Q ,A<W,U,GT ,44, il I I'ITUI10
IF IU.L~o47,D,ANQ,U,GT.<t&.CI INTU 111
IF !U,LE,15~.U.ANQ,U,GT.47,GI INTU! 1
f<ETUR:I
8'
END
USASI fOKTKAN OIAGN05TIC RESULTS FOR FOIST1CU
OflJ,LGO
NO ERRORS
1
1
1
1
i
1
1
1
=1
=1
1=1
APPENDIX D
GENERATION OF THE CUMULATIVE PROBABILITY DISTRIBUTION
OF THE MANN-WHITNEY U FOR 2 GROUPS OF 20
Whereas a table of the cumulative probability distribution of the
Mann-Whitney U was not available for n=20 and m=20, one was generated
by computer simulation.
To test the procedure, a cumulative probabil-
ity distribution was first generated for n=lO and m=lO.
A Kolgomorov-
Smirnov test, applied to the resulting values of the computer output,
verified that there was not a statistically significant difference from
the mathematically derived values found in published tables.
A cumulative probability distribution was then generated for n=20
and m=20 (see following table).
The critical values of this computer-
generated table were compared to published critical values.
matched perfectly.
The two
The other levels of confidence were then read off
the computer output and used in the study.
43
44
CUMUJ..,ATIVE
u
VAL'QES.
\;
•
~
2
3
4
5
b
7
..
FREQUENCY
I'ROBA.BILJTY: :
J
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c
0
0
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4&
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I'ROBABILITY
PROBABILITY
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413
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VALUES
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FREQlJENCY
')"
CUMULATIVE
PROBABILITY
PROBABILITY
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5
a.J01oo67
O.J240JJ
O.J256o7
0.021667
6
o.. oozoJud
5
7
8
G.JJ16E67
C:.CC!26667
O~U34333
122
123
124
12S
126
127
1.2
u.O:J4D0ou
G.ll3':l?i33
4
6
3
D.OJ13333
10
G.OJ33333
q
12?
6
12'3
13!_!
g
J.003000U
C.J020JCO
O.OJ30llD:J
0.00366b7
u.o3g667
u.041o67
0.')42667
o.o+6oJJ
D.0490J;J
11>\
11 'J
12C
121
131.
132
133
134
135
C.027667
0.029333
O.<J31G67
D.Ou23333
O,.DJ20JUO
G.OJ1GOGO
11
13
16
D.Ji1600JD
14
G.lJJLt6667
O.G51DOJ
0.05400:']
0 • 0 57 66 7
o.os3E67
o.o6goo:J
0.073667
0.,0053333
12
G.. OC400JC
14
(1.0046667
..
---
.
·~·-·-· ~-
-
0.077667
O.O(l2333
·--~----~-~-·-~---
·····-·-··-··
47
u
VALUES
136
137
1 3 i~
139
14li
141
142
143
141-+
PROBABILITY:
FREQUEl\lCY
22
22
147
148
26
14~
21
O.O>HOO<J
."1
\. .OC6o667
G.OG33333
20
10
14
22
15
15
16
22
17
1 t}~
140
CUMULATIVE
PROBABILITY
o.oqz:L53
0.097000
0.104333
O.OC46667
o.OC7333.3
D.:J050DDO
0.109333
0.114333
(;. 0 0 50 0 0 0
0.0053333
0.11g667
G.JiJ73333
0.127JJJ
0.132667
0.14CJJJ
1).141333
0.159:333
0 • 1 i) ~ 'J uQ
l.l.Gd5b667
JeOJ?3333
O.JJ73333
0.012GGOO
O.J0-31J667
O.J07GOOG
36
0.175G~J
0.1l3J667
15D
17
151
152
2.'1
u.JtlSfiE67
i! .. J Cl 3 3 3 3
?2
J.ll073333
1':>3
.S6
OeJ12JuLiU
154
155
156
157
15d
159
26
u.JrJ8ot.67
0.21f)[:JJ
vL
';('>
t).1J106667
0.228667
34
0.011333.)
25
37
O.O'J83333
0.0123333
D • 2 1~ U J J J
0.248333
0.260667
26
34
42
,).0Ci36667
0.011.3333
iSG
1 (, 1
lee
163
164
165
166
167
if>"
16::1
17Q
17 .1.
172
17J
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0.11J·JiJO
G.1'J7333
u. C:J'-:15.} ..3
0.26933~5
0.2iG667
(!.294667
O.Qli.jOQOG
zg
:].f'J196667
O.J1400CO
C.0106667
42
32
64
50
J •3
a.367eoo
J.01E:6G67
J. Ji3l1JGG
Si
4333
0.3290JG
ll.3SJ)S3
0.0213333
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ij
0.31!-S33,~
0 • 3 ·~ ,J GJ C
o.Jq?'JJO
0.413333
0. 4 27 J J 0
0.4430.Jl)
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J.J17Gauo
o.ult:,:n33
41
43
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62
0.01EldOG
O.J176h67
D.J206667
1r4
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J~CJ1g3333
0.4·'!133.3
0 • '? 'l 0 bb 7
17S
176
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0.0146667
O.Q18JDOO
0.51S333
0.533333
0.0136667
0.460667
177
173
54
51
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1 7 Cj
56
0.01U6667
0.5~5E67
1BU
49
0.0163j33
0.6D2'JOJ
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o.o17o;;co
0~550333
O.Ji66667
0.567.J'JJ
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48
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VALUES
181
182
183
184
135
186
187
183
18g
1Y!J
1g1
19?
193
1 g l+
195
1911
1<37
1 g :s
1')g
FREQUENC¥
64
54
sg
0.021333"3
o.62:n~n
J.Oif'dJGOC
0.()196667
0.6413)3
0.6G100J
0.677!JJQ
0.636667
0.713335
0.733667
0 .755667
0 • 774033
o.t-J?tor
0. iH~66 7
4~
G.J16G~1JO
59
50
61
i.l.0196667
6t)
56
64
I)J
CUMULATIVE
PROBABILITY
PROBABILITY
G.G1bobb7
J.i)2J3333
O• .JZ2~JS0
o~Dlo6o67
tJ • J Z L~ 3 .3 ,)
o.ozooooo
G.0223:533
o.~38CJJO
Ge•J19GGOO
0.135700)
G.Ot9,)33:::>
o.c213333
0.376333
O.t397667
72
IJ.l:24GJQ:J
ll.9?.16G7
77
0. 0 2 56 ~b 7
0.0223333
0 .9'-.t7333
O.YbCJ667
67
57
53
64
67
62
O.. GZ\.16667
o~ggo333
o.oog&667
O.O:JOOGOO
202
0
0
2C3
0
u.OGOOOOC
204
205
0
;}$\.1(1
00000
1.JOJOOG
1.002003
1.0Q:JOQQ
1.00GOJJ
i.OJJOJJ
Q.O:JOOOOG
1.G(;!.l0'1G
2Gb
0
3
1 .. lJ'JjQJO
2C7
()
O.OODOGJO
J.OODDGOC
20l:l
r·
O.JJ~OOOC
1.0JOOGD
1.0CJCJJ
20G
29
2]1
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21G
211
u
o.ocooooo
1 • ..J:J.:ODO
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0
0
G.O'JiJOQOO
1.:JJO·~JJ
1.GHJJJG
1 • JOGO<Ju
l.G'JJJD'}
o.oooooou
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213
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G ~ J 0 :JGJGO
216
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D.OGOOGOG
t.OGGG.JJ
1.JOJOJG
217
0
O.OCOOOOG
1.0J;JGOJ
213
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O.OJJOGGC
l.O'JOCOO
0
0
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o.ooso ':J 0
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O.Oi.'GJ,JOO
1.DOOCJJ
O.OJ'JC00G
1.0JCCOO
o.os~'u
1.J:)<J(ll)J
1.0'JJOJJ
1 • JJ;.JGO'J
21CJ
22:J
221
222
223
22~
225
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t.OGGGOQ
00 0
G.OOClOOOQ
O.OGGO ~ 0 IJ
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~-~·-··--·-
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49
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VALUES
FREQUE~CY
227
0
0
228
22g
!.;
226
2 30
231.
232
233
234
Q
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0
0
0
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l?ROBABIL:rTY
0 • OG 008(}0
o. 00 00 G:JO
0. oc OOJOO
o. :J!] OD 'JOG
o. 10 GOOOli
o. 00 on ooo
JofJO 0 0 0 0 0
G.Oj JOG GJ
CUMULATIVE
PROBABIL:rTY
1. OJOOJJ
1 • OQUOJO
1 • 000DOJ
1. uJJJJo
1. OOJOO:J
1. J:JiJOOO
1. JJOO,JO
1 • Q ] >J 'j i) 0
OJOOJJ
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0 0 C:.J OGJOC
J. 00 noooo
Q. QQ QQ QOQ
1. OOJQQO
1. OlJOOO
237
2 3d
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G.OG DOJCO
1.JQ'JGQG
0
J.JOJO 00 0
2 3g
2 4D
0
c. JG OQOOC
1 • JJOiJiJJO
o.fJG CCLJGG
Q. J:J
242
0
0
0
1.0 002QG
1 • Q.J!JGIJ J
4:~
Q
2 ,)5
?36
241
2
244
245
2 [~6
247
24R
244
25J
0
0
n
0
u
0
0
0
0
0
25.1
252
()
253
254
25S
0
J
2S>b
257
J
a
2 5!1
0
2sq
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26'::
0
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0. <l D OOOOD
u. 0 J 00 uo:
o.oo 0 fJ 'J 0 0
u. OJ 0 0 0 0 0
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o. OUOOQOO
o.. DJOOOOO
0. c \J co C0 G
Oe llJOOOGO
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a.. !)GOGO GO
u.oo GO GOO
o. J·~DCOCO
J. )J 0 0 GG0
c, .. Jf') OQOOG
u. JO OOOOG
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a. ,J J OOQiJG
u.QOQQJOO
o.oooocoo
261
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262
D
263
264
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0. ,) J
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267
26'3
26g
0
0
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266
27
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1. OJGJUO
1. O'JJOOO
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1.JJ!JGJJ
1. OJOOJJ
1 • JOOOJ 0
1. 3JOOJO
1. OOJOJD
1. JD.;GJC
1. 0 J J J ,) 'J
1 • OOJJJO
1 .. JJllGOQ
1. JQJGJJ
1. Cl:JJQ)J
1
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1· Q c':.~LlJJ
1 • J GQ ;~ ·J 0
1 • J 1JJJO
i.O·JGOJO
1. 0 JJ:1GJ
1 .O;iOGQQ
1. OJOJOO
1 • J uDGCJJ
1. ;JJOOJO
1 • 0 :JQGJO
J.oo COJCC
1. O;JJQ.Jj
D. 0000000
J.OO 0 0 GGJ
1. GOiJJ:)i.i
1. 0000JJ
1. OGJOG J
u. 00 UOOOD
50
u
CUMULATIVE
VALUES
271
272
27)
2 7lt
275
276
277
2 7"
2 7 C1
21H
231
2:12
283
234
FREQUENCY
o.auoooou
a
c.ooooooo
O.OOGOuOu
PROBABILITY
1.0JGOOO
1.00'JllJO
0
O..:JOOODDO
1.000000
0
0
w.JOOOOGO
O.OOOOJOG
2'35
236
0
0
287
0
8
28.1
zgg
zgo
zg1
2-::12
293
zg4
295
zgr:
0
0
PROBABJLI.TY-
l..u'J:JQDO
J
O.. OOOOODC
1.00:)003
1.0000JO
1.003000
Q
O.JOOODiJO
1.000000
0
0
u .. OuOJ'2:JQ
1.JOJOOJ
fJ.,JLJGOJCC
1.JOQOJO
[)
o.JJOJuoa
:l
O.OJOOOuO
0
0
0
G
0
0
0
l.QO'JOOJ
i.OOJOQJ
!].0000000
1..0JflOOJ
G.O:JCJOJG
1.uJu.JCJ
UsJti000i)Q
J.dQO·J!:GO
1.0BOJG
o.ooooaoo
1.00JOJD
i.GOJJQO
J.\lfiOOGOO
fl.OJGOOOG
O.. CdOOOOO
Q.O!JOOOOO
O.uJOOQOO
O.QOCCCGO
D.GJCOOuG
J.GGDOGQO
l.J:JJ,Ji)J
1.JJlGJO
1sODOOOJ
1.GOOOJJ
1.000UOO
1.000000
1.0'000~0
1.00:JQJJ
O.JJO'JOJO
t.JOJOJO
29l
u
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1.0ClQ0f)0
290
:)
o.::<JDOJGO
t.iJJJQ;)iJ
2 39
3 .J l
0
Q"~JOO:JOOQ
l.'J!JJC:JD
Q
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i.G·::;~n.~
301
G
O,JJUOGOC
312
l!.llOOGOQJ
3 ']3
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1.Jd.JCJ00
1. n ooc J 'j
30 4
0
o.ooooooo
305
306
•J.'JJOO OOJ
3J7
0
0
0
308
a
309
J
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310
311
312
313
314
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O.OOCiOJOO
1.000i]JQ
i.OOJOJJ
o.ooooJJa
LJ:JJOJO
O.QGQGJOO
u.OGOOJOO
J.GuOO'JJG
1.GJJOJO
1. !J 0 ,: G G0
l.JJJOJO
u.OGUGOGO
i.:JG~ODO
O.ODGOJOO
0
0
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O.OJOOJOJ
D.OOOOJJiJ
J.OCOO.JOO
u.uGuCuJO
1.JOJOtl1J
1.0iJ.JOOO
1· J J '~
J Q0
l.>J.J:JJG
51
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l'
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I vALUES
316'
317
FREQUENCY .
PROBABILITY
n
o ;o a·c·o·'JG u
0
G.OOGOOJO
CUMULATIVE
PROBABILITY
1 • ·o ·o :1 b n·o ·
o.ooooooo
32C
0
o.ooooono
1.\J·JJOiJO
i.,Cj..;l)JO
1.00:1000
321
322
0
0
O.DOOOJDO
1.0u.JODO
J.OOOOOJO
1.iG~JCC
u.GOOOOOD
1.JJJ>).i0
O.OOOJQuO
31~
319
323
324
325
32b
327
a
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O.OiJOtlGvO
J
D.UOOOJGO
J
3 2!3
u
32'3
jl
3 3J
"
0
331
332
333
334
35
J
O.JJOOOOO
1 .. JJJJOJ
OoGOGCODG
1.0JJODO
1.JO.OJ0
;J.OCOOOJO
O.OOOOJUO
O.OGOOJCO
t.;~Jc:i\JGG
O.COGGJOU
0.00\JOOQO
l.O.JJGGC
O.GOJOtlOU
336
3 37
G.OOOOG:JG
CJ.CuGGOOO
0
1.00JOOD
1.GJJUJG
o.ooooa:Jo
u
1 .. ou.:Joo
i.J!J,\)!)0
o.oonoaoo
::~
338
33-=3
1 .. 0DJO!JO
o.oooooou
O.DOOOOOD
!.OQJOOD
1.0GJJOG
1.0JJJOC
1 .. JJ.:OGG
lcOJJO'JO
i.OOJGGO
34C
G.OGOOOJO
1.JJJJl)G
341
342
343
G.DOOOG:JO
1.JJJOOO
G
352
353
354
355
356
357
358
.359
36C
1.dJ~JeJ
1.uJJOJO
O.;JOOODDG
1.:.JJj;Ji)Q
J
o.ouooooo
;j
G.COOOOJLJ
t.:JJJJCu
l .. ~cuCISG
344
345
346
347
348
.3 49
3 5;J
351
O.CGOOOGO
D.OiJQOOOO
Q
o.coaaaoo
LJ
O.GJ;JO,JJG
1.J:3~JCQ
u
o.coaouoo
1.0JJJDG
G
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O.OGOOOJG
O.OOOOOJO
1.DJ1GJG
u
O.GOQOOOO
J
O.OGJOOOD
1. G .J ~' J J i.J
l.OJJJOO
i.OJJG,JO
i.DJjiJ!Jt
u.ooooooa
1 .. \JJJOGG
D
ll.QGOOJJQ
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O.,JQGOOOC
LJ
O.OOJOJOO
D
O.QGJOJOO
1.JJ;J;JG
l.OJJDGC
t.OJJJJG
1 .. 00JrlGO
0
O.OOOOJGO
O.GOOOOOu
1.CJ:OJi.i
1.GJ..i0-JO
52
u
PROBABILITY
---
VALUES
FREQUENCY
361
Q
o.ooooouo
362
0
0.0000000
363
364
365
366
367
368
369
370
3li
372
j
3 73
374
175
3 76
3 77
373
3 7 '3
3 8-J
3d1
.382
383
384
385
u.uuuliuuu
J
0
G
O.DDOOOOO
{).OOOOJOO
:1
D.DuOO;JQO
O.OOJOOJlJ
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J
ll
O.GOOtJOOO
1.00CQOG
1.J.J)000
i • \j u.J \;';} ~
1.00JOOU
1.000000
1.GtJ.-OGG
i.lHJOQO
1.JOJOOO
O.GOQCOJO
1.JJJOJU
o~co_Jocoo
1.JJ1iJ,JD
u.iJOiJOOOO
1.JGJJ0C
1.tjJJJGG
OeGOQOQCJO
0
J
CUMULATIVE
PROBABILITY
o.ooooooo
O.OOJJGDO
J.DuJOQOO
O.JOOOJOO
1.GJJOGG
1.QJJODG
i.JJJDJD
l.JJJJ)C
1. 0 Q .j !) J Q
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o.ooooooG
0
J
J.GQOOJCG
t.Jo;nJc
O.OJOOOQO
i.U·JJQOG
i .. OJJOiJL<
1. D J .} J J C
O.OOOOQJO
.
-'1
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O.OOOOJOv
O .. GOOD\JJO
O.OOOOOOJ
i.OuJ:JuU
1.0uj0JG
1.0J3UOG
1.J'J100G
0
o.ooooooo
3d6
Q
O.OGODOJO
387
313H
u
O.DDIJOJDD
1.J3~JGC
0
O.JOOOJOO
D.OOO\lJOO
1.0JJOQD
1.;JJJGCC
u
3d9
390
0
3g1
3Y2
393
;)'34
395
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397
398
399
'• CD
401
4)2
4G3
4\)4
405
Q
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u~uOOOJOO
o.uooocoo
1.JOJUJG
u.OJGOOOO
G.OOGDJJO
1.0JJGiJD
1 • .JLO'JG
1.JJJGOO
1.GJJGDU
o.oocoooo
u .. OOOOJOG
O.OOODOOCJ
J.OOOOJOti
1.JCLJGDO
· O.GOUGJOO
1.GGJO\JC
0
IJ.OOOOJOO
\}
O.GOOJuOG
1.iJJJGCC:
1.GO:O'JG
1.0GJUOG
1.GJ.i000
0
J
0
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0
0
0
1.uJJJJO
o.ooooooo
o.ooooooo
O.GOOGQOC
0.0000000
O.OGJOJG0
O.DOOOOOD
1.0GJOOG
leOJJOUC
1.0Ll00D
53
u
CUMULATIVE
VALUES
J;'RRQUENCY
4J8
409
()
40&
4 07
410
0
0
411
(,
412
O.OOOOJDO
O.OOOOJOO
1.0J.JODG
l.J:)}GQC
0.0000000
1.~;;:;.;()0(;
O .. OOOOGOJ
1.JOJOJG
O.GOGOCJOO
1.00JOGG
1.00aOGU
1.Q:LOJC
o.ooooooo
o.oooooco
u.ooooooo
41:3
414
415
l?ROBA.Bll.ITY
PROBABILITY
1.,jJ~Q0(;
O.DOOu:JOO
l.JQ~OJG
G.OOQOJOJ
G.:JCOOJGO
0.0000;}00
1.J].JG•JG
416
0
D
417
J
413
G
419
G
o.coooooo
1.'}J)GOG
420
21
422
Ll
O.,OOOOOGD
G.:JOOQQJC
1. 0 .J.J DOD
0
0
0
0
t.l.,QQGOODD
1.fJJ}!!Ol
le\JJJJGC
t+
423
42L•
425
426
u
427
42g
0
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429
0
0
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4 31
432
433
434
0
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435
0
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436
437
0
O.GO::JGJOO
u .. 0uOOJ;JO
O.OOOOJOG
o.ooooooo
O.OOQOOJO
O~GCOOJOO
O.GOOOOOG
Q.,OOuGuQQ
0.0000\.iJG
O.CO:lOiJJO
i.GJ.JOOG
1.JJ,JOOO
l..JJJGJD
1.UJiOOG
1.0JJOJO
1.GJJ:)JQ
i.OJJJOG
1.0.)J0llC
1.CdJDJG
1.00:!tJGG
1.CGJ\l0(J
O.OOOGJCI;J
t.:)J~DOO
O.OOGOJQO
l •.}jJ;JJO
l.iJJJJGD
O.uOGOJJC
!.Oi.)JQ{JO
t.O'JJJDD
o.ooooaoo
o.. ooooouo
G.OOOOJOlJ
l.(i.J.~OOC
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VALUES:
451
452
45,~
454
4-55
456
4-57
it 58
459
460
4 61
462
463
461+
FREQUENCY
j
o.oaooo3J
0
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0
0
0
0
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Ll
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a
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468
469
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0
0
471
472
0
0
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J
O.OOOOJQD
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CUMULATIVE
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1.J)JGGO
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t .. DOJOOO
t.OJJOOU
u.DOOOGGG
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55
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495
L• 99
5 00
J?ROBAB.ILITY
CUMULATIVE
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J
O.COODJOC
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0
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FREQ_UENCY ,
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535
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56
CUMULATIVE
u
VALUES
FREQUENCY .
'51+1
l'ROBABILITY
PROBABILITY
o.·o·ounD·d o
51t2
543
51t4
(j
J.OOOOiJuO
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545
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CUMULATIVE
VALUES
',li'REQUENCY
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'5136
5 87
588
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592
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PROBABILITY
1.i)J][JJ(j
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APPENDIX E
AN ILLUSTRATION OF THE CO}WUTATION OF A MANN-WHITNEY U
Experimental
Control
4
7
8
5
2
9
To find the U value for the above data, the scores must be ranked
as follows:
2
C
4
E
5
C
7 8
E E
9
C
In considering the control group, for example, the U value is the
number of experimental scores that precedes each control score.
The
first control score, 2, is not preceded by any experimental scores.
The next control score, 5, is preceded by one experimental score.
The final control score, 9, is preceded by three experimental scores.
•Therefore,
u
0 + 1 + 3
4
A different U value is found by counting the number of control scores
that prece.des each experimental score.
The smaller of the two values
is the one whose sampling distribution is the basis of the tables of
significance.
Because this method is very tedious fer large sample sizes, an
. alternate method, giving equivalent results, has been devised.
The
·scores are ranked as before, but in addition a number associated with
·the rank is assigned to each score:
58
59
Rank
Score
Group
1 2
2 4
C E
3
5
4
5 7
C E
6
8 9
E C
The ·formula for the calculation of the U value is as follows:
u
where n
1
and n
2
(1-1)
are the sample sizes and ER
of the group whose sample size is n 1 •
ranks of the control
gr~up
and n
1
1
is the sum of the ranks
Using ER 1 as the sum of the
as the sample size of the control
group,
u
(3)(3)
+ 3 (3 + 1)
2
- 10
u =5 •
ER
1
and n
1
could have been used to refer to the statistics for the
experimental group rather than the control group.
Formula (1-1) will
give different U values depending on which group is used.
The smaller
of the tvm values is the one whose sampling distribution is the basis
of the tables of significance.
Therefore, the following formula is
used to verify that the U value obtained either from Formula (1-1) or
from the first method is the smaller:
where n
1
and n
either method.
2
are the sample sizes and
u~
is the U value computed by
Substituting the results of Formula (1-1) for U~,
60
u
= (3)(3) - 5
u
=
4 .
Thus we see that the value computed in Formula (1-1) was not the
smaller value.
By Formula (1-2) we arrived at the smaller U value,
,the same value produced by the first method.