Download The Probability Plot Correlation Coefficient Test for the Normal

RESOURCES RESEARCH, YOLo 22, NO.4, PAGES 587-590, APRIL 1986
WATER
..
The Probability Plot Correlation Coefficient Test for the Normal,
Lognormal, and Gumbel Distributional Hypotheses
RICHARD
M.
VOGEL
!
Department of Civil Engineering,Tufts University, Medford, Massachusetts
Filliben (1975) and Looney and Gulledge (1985) developed powerful tests of nonnality which are
conceptually simple and computationally convenient and may be readily extended to testing nonnonnal
distributional hypotheses. The probability plot correlation coefficient test consists of two widely used
tools in water resource engineering: the probability plot and the product moment correlation coefficient.
Many water resource applications require powerful hypothesis tests for nonnormal distributions. This
technical note develops a new probability plot correlation coefficient test for the Gumbel distribution.
Critical points of the test statistic are provided for samplesof length 10 to 10,000.Filliben's and Looney
and Gulledge's tests were originally developedfor testing the normal hypothesis for sample sizeslessthan
100. Since many water resource research applications require a test of nonnality for samplesof length
greater than 100, Filliben's test is extendedhere for samplesof length 100 to 10,000.
INTRODUCnoN
water
for
both
and
resource
normal
and
Looney
and
flow
research
investigations
nonnormal
require
hypotheses.
Gulledge
[1985J
Filliben
developed
y
P
lot
correlation
o
distributions.
tests
a~sociate~
I
.
t
PPCC
)
following
t
t
e
I.
es
t
attractive
t
s
mb
t
a
.
t
IS
.
.
IC
IS
I
Ines
probability
and
test
3.
The
a
test
ndam
y
t
t
easy
II
en
0
.
a
correlation
y
d
un
d
erstan
simple
b
t
e
th
concep
is
readily
compares
it
basis
[1975J
is
I
ure
emp
only
for
as
is
requires
6.
The
this
d
and
to
fi
t
allows
e
a
pro
b
a
l
these
attractive
.
d
.
Istn
this
PPCC
10,000
the
to
and
provide
a
the
utIon.
t'
new
,
best
in
both
on
that
water
for
the
and
Data,
to
and
various
of
o~
ex~~-
If
too
It
estimation
a
procedures.
recent
the
quantile
of
general
this
problem
1986
by
the
American
Geophysical
an.nu~1
dlstn-
,
FI~lIben
s
PPCC
procedure
[1984J
useful
for.
testing
autoregressive
most
moving
sequences,
fitting
in
Union.
of
Snede-
[1982J
assessing
the
number
of
which
recom-
goodness
of
investigators
are
fit
have
based
upon
such
as
[1977J
been
infor-
the
tests
and
observed
plots
of
a
were
pro-
Looney
data
Similarly,
the
and
3J
ice
the
moments
dam
Re-
a
basis
use
probability
evalu-
Agency
of
probability
plots
distribution
arises
for
safety
of
from
the
annual
combined
flooding.
Water
Resources
on
of
as
Management
storm-induced
U.S.
Committee
Prob-
National
EJ
in
the
which
and
the
and
Emergency
elevations
jam
plot),
curves
the
efficient
procedures,
decisions
by
Federal
of
more
fitting
(probability
D
recommends
resource
probabil-
theory,
recommended
the
flood
method
display
frequency
water
fitting
engineering
Appendixes
determination
of
in
curve
make
graphical
flood
Appendix
are,
not
[1985,
of
in
for
graphical
recently
Council
widely
approaches
than
use
used
analytic
would
the
Advisory
0043-1397j86j005W-4291$05.00
literature.
152-156J,
Mage
plots
have
to
Although
5W429l,
statistics
pp.
LaBrecque
procedures
[1982,
flood-
for
probability
hydrologists
ations.
and
the
and
tests
distributions
ability
Wood
59-63J,
A
While
search
study,
in
[1982,
pp.
plots
without
mention
widely
[1975J,
extrapolation
and
of
of
of
normality
Wichern
plots
Filliben
statistical
distributions
type
39
[1985J.
many
estimates
Wallis
example
reviews
the
probability
[1984J
LOT
used
in
by
ity
Water
Com-
to
of
al.
P
goodness-of-fit
Probability
peak
the
numerous
precision
.use
streamflow
distribution,
investigations.
annual
by
perhaps
of
et
of
ROBABIUTY
probability
hypothesized
maximum
number
F
Kottegoda
of
annual
and
contained
effects
Copyright
"
Gumbel
P
[1980,
use
a
posed
litera-
Advisory
combinations
[1985J
Paper
t..
.
generalized
fitting
to
are
Cochran
mation
distri-
observed
summarized
studies,
the
provide
Tnomas
fi
Rossi
floodflows.
test
models
plots
mend
probability
represents
peak
Johnson
proposed
100
resource
[Interagency
1982J,
Other
parameter
[1985J
of
study,
17
compared
from
f
contrIbutIon
the
when
A)
example,
of
Filliben's
length
theoretical
sequences
study.
de~ived
the
flood
outlier-detection
hypothesis
Probability
re-
Gumbel
water
which
[1974J
Water
have
for
future
dlstnb~tlon
a
PPCC
(ARM
For
nonnor-
extend
of
existing
Bulletin
comprehensive
here,
the
describes
Council's
mittee
0
.
annual
Filliben's
normal
a
coef-
and
to
test
determine
Beard's
Resource
ness
by
preliminary
HE
normal
samples
PPCC
of
to
streamflows.
.I
t
portion
sought
distribution
]'
"
significant
has
d
study
using
a
.
fact
of
to
of
found
average
results
undertaken
normality
as
Gulledge
A
I
pro-
bution.
ture
test
In
valuable
reco~m~nded
normality
sequences
the
(correlation
tests
was
for
the
Italy
T
b
numerical
require
study
test
and
goo
a
the
In
.
of
a
features
often
hypotheses,
original
..
e
provIded
~pproxlma~e
[:985J
of
cor
applications
mal
hypoth~sls
employed
..
also
performed
estimation
Iity
and
of
[1985J.
comparison
plot)
to
records
test
tests
studies
parameter
b
t
other
Gulledge
the
h
to
test
Given
.
Into
Kottegoda
tech-
form.
source
have
sought
some
in
seven
power
Looney
(probability
ficient)
h
examlne.t
Incorporated
flo~dflow
testing
shown
with
empirical
invariant
oye
graphical
of
and
test
d
ce
new
type
probability
button,
extendible
favorably
the
The
rarely
of
assume~
'.
coefficient.
'
Filliben
to
could
.
been
whIch
since
hypotheses,
test
on
5.
test
.
les
e,
for
The
normality
by
this
had
s:
note.
4.
.
stu
pIe,
e-
coefficient.
correlation
statistic
I
simp
simple
distributional
nical
d
reque?cy
II
ua
computationally
of
nonnormal
t
u
the
is
computation
f
w
plot
The
an
be
y
features:
concep
0
t
co
2.
may
studies
probability
'
cause
of
a
which
f
the
Th
.
c?oice
~rovldes
distrIbutIon
these
the
test
norma
have
1
the
:hls.paper
Gumbel
general,
determine
coefficient
(
which
to
with
dlstnbutIo~.
r
In
hypothesis
e~ro~s
probabil-
fi
frequency
include
[1975J
powerful
s
it
tests
Water
to
Council
Data,
fit
the
1982J
Log-Pearson
[Interagency
advocates
the
type
use
IIIdistri-
587
!
Many
(
;" :::"~~~~~~
~~':"Y*~
-
I
588
-
'
--
VOGEL: TECHNICAL NOTE
bution to observed floodflow data, their recommendations
also include the use of probability plots. Clearly, probability
plots play an important role in statistical hydrology.
Since the introduction of probability plots in hydrology by
Hazen [1914], the choice of which plotting position to employ
in a given application has been a subject of debate for decades; Cunnane [1978] provides a review of the problem. Although the debate regarding which plotting position to
employ still continues, most studies have failed to acknowledge how imprecise all such estimates must be. Loucks et al.
[1981, p. 109] document the sampling properties of plotting
positions and raise the question whether differencesin the bias
among many competing plotting positians are very important
consi~ering their large variances [see Loucks et al., 1981, pp.
Here the M i correspond to the median (or mean) valuesof the
ith largest observation in a sample of n standardized random
variables from the hypothesizeddistribution.
Filliben's PPCC test was developed for a two-parameter
normal (or log normal) distribution. Generalized PPCC tests
may be developedfor any one- or two-parameter distribution
which exhibits a fixed shape.However, distributions which do
not exhibit a fixed shape such as the gamma family or distributions with more than two independent parameters are not
suited to the construction of a general and exact PPCC test.
For example, the PPCC test for normality presented here
could be employed to test the two-parameter lognormal hypothesis, however, the test would not be suited to testing the
three-parameter lognormal hypothesis. Use of the critical
179-180].
points
This
technical
note
need
not
aqdress
that
issue,
since a probability plot is used here as a basis for the construction of hypothesis tests,rather than for selecting a quantiIe of the cumulative distribution function as the design event.
A probability plot is defined as a graphical representation of
the ith order statistic Y(i) v,ersusa plotting position which is
simply a measureof the location of the ith order statistic from
the standardized distribution. One is often tempted to choose
the expected value of the ith order statistic, E(Y(IJ,as a measure of the location parameter. However, Filliben argued that
f h
t t.
1'
.
. t d ' th 1 .
compu a lona InconvenIencesassoclae WI se ectlon 0 t e
of the
test
statistic
the ith orderstatistic.Approximationsto the expectedvalueof
the ith order statistic, E(y(i))' are now available for a wide
variety of probability distributions (see,for example, Cunnane
[1978]).Looneyand Gulledge[1985] and Ryan et al. [1982]
define a probability plot for the normal distribution as a plot
of the ith order statistic versusan approximation to the mean
value of the ith order statistic. There appearsto be no particu-
larly convincingreasonwhy one should usethe order statistic's mean or median as a measureof the location parameter
when constructing a probability plot for the purpose of hypo thesis testing.
THE PROBABILITY
PLOTCORRELATION
COEFFICIENT
TEST
If the sample to be tested is actually distributed as hypothesized,one would expect the plot of the ordered observations
here
or
in
the
work
by
Filliben [1975] for testing the three-parameter lognormal hypothesis will lead to fewer rejections of the null hypothesis
than one would anticipate. This is becauseonly two parameters are estimatedin the construction of the PPCC testsdeveloped here, yet three parameters are required to fit a threeparameter log normal distribution.
Filliben's Testfor Normality Extended
.
,.
.
Filliben employedan estimate of the order statistic median
order statistic mean can, in general, be avoided by choosing to
measure the location of the ith order statistic by its median,
Mv instead of its mean, E(y(i)).Filliben chose to define a probability plot for the normal distribution as a plot of the ith
order statistic versusan approximation to the median value of
f provided
Mi
= cI>-I(FY(Y(IJ)
(2)
.
.
..
..
.
In (1); here cI>(x)IS t~e ~um~latlve distrIbution. function of t?e
stan~ard normal, dlst.n?utlon, and. F Y(y(i) IS equal to ItS
median value,which Fllllben approximated as
= 1. - (0.5)1/0
F Y(y(i))= (I - 0.3175)/(n+ 0.365)
i. = 1
f Y(y(I).) = (0.5)I/n
i=n
f y{y-)
~
(I)
1= 2, "',
n - 1 (3)
Filliben's approximation to the median of the ith order statistic in (3) is employed in this study. The Minitab computer
program[Ryan et al., 1982]andLooneyand Gulledge[1985]
implement the PPCC test by employing Blom's [1953] approximation to the order statistic means for a normal population. Hence the tables of critical points which Ryan et al.
[1982] and Looneyand Gulledge[1985] providediffer slightly
from Filliben's results.
Filliben tabulated critical values of f for samplesof length
100 or less.In Monte-Carlo experiments one is often confron-
versusthe order statisticmeansor mediansto be approxi-
ted with the ne~~for te~tsof no~m~litywith samplesof.g:eat~r
mately linear. Thus the product moment correlation coefficient which measures the degree of linear association between two random variables is an appropriate test statistic.
Filliben's PPCC test is simply a formalization of a technique
used by statistical hydrologists for many decades; that is, it
determines the linearity of a probability plot. Prior to the
introduction of Filliben's PPCC test of normality into the
length. Thus crItical pOints(or signIficancelevels)for FJillbens
test statistic were computed for samples of length n = 100,
200, 300, 500, 1000, 2000, 3000, 5000, and 10,000.This was
accomplished by generating 10,000 sequencesof standard
normal random vari~bles each of length ~ and ~pplying (1~
(2), and (3) to obtain 10,000 corresponding estImates of r,
denoted f;, i = 1, .'., 10,000.Critical points of the distribution!
water resources literature by Loucks et al. [1981, p. 181], determination
ofsubjective
the linearity
of a probability plot was largely a
graphicaland
procedure.
of f were obtained by using the empirical sampling procedure
.
(4)
r.p -- r(IO.OOOp)
Y(i)
Filliben's PPCC test statistic is defined as the product where r~ denotesthe pth quantile of the distribution of f and
moment correlation coefficient between the ordered observa- . f(lo.ooOp)denotes the 10,000p largest observation in the setions Y(i)and the order statistic medians M; for a standardized quence of 10,000generatedvalues of r. As the sample size,n,
normal distribution. His test statistic becomes
becomesvery large, the percentagepoints of the distribution
of r approachunity and, in fact, becomeindistinguishable
0
L
r=
'r~
(Y(i)
- Y)(Mi - M)
i= I
(1)
~ - ii)2 f
A.j i?-1 0'1;)
(M
- Y)- J~I \JVlj - JV1)_~2
from that value. Therefore it is more convenient to tabulate
the percentagepoints of the distribution of (1 - f). The results
of these experiments are summarized in Table 1, which also
provides a comparison with Filliben's results for the casewhen
.
~
~A~,:.:c
i
Ii
4"""
I
0
I
\
.
~
I
.
VOGEL: TECHNICAL NOTE
TABLE 1. Critical Points of 1000(1
- f) Where, is the Nonnal
ProbabilityPlot CorrelationCoefficient
S. .fi
L I
Igm cance eve
n
0.005
0.01
0.05
0.10
0.25
0.50
13.
13.0
6.96
4.75
11.
10.7
5.83
3.98
8.
7.84
4.26
2.98
6.
5.54
3.07
2.18
589
where the cumulative prvbabilities F,,(yJare generatedfrom a
uniform distribution over the interval (0, 1).
In this case the test statistic is defined as the product
moment correlation coefficient between the ordered observations Yi and Mi using
.
;,
100*
100
200
300
21.
21.3
11.2
7.61
19.
18.8
9.85
6.46
1,000
2.66
2.45
1.76
1.46
1.09
0.746
2,000
1.23
1.09
0.816
0.698
0.533
0.400
500
;
,
3,000
5,000
10000
,
4.62
4.18
0.846
0.493
0252
.
3.04
0.752
0.450
0.226
2.52
0.546
0.343
0.174
1.89
0.468
0.293
0.150
MI
1
-
(FY(Y(iJ)
(8)
where f ,,(Y(IJis Gringorten's [1963] plotting position for the
Gumbel distribution:
1.38
0.363
0.228
0.117
= Fy-
0.276
0.171
0.0890
i - 0.44
f y(y(iJ
= n + 0 12
(9)
.
Gringorten's plotting position was derived with the objective
0 f sett Ing
'
This table is based upon 10,000replicate experiments.The first row,
'"
75]
IA
Id
h
marked *; gives Fill/ben's [19
resu ts. n examp e ocuments t e
use of this table. The 10th percentile of rs distribution when n = 500
. d
.
is etermrned from
F (y ) = F
y(o)
y (E(y(oj))
(10)
where E
(y distribution.
) is the expected
value of the largest observation of
a Gumbel (0).
Thus Gririgorten's plotting position is
"0 = i - 2.52 X 10-3 = 0.99748
oniy unbiased for the largest observation. Cunnane [1978] rec-
.. I . t
b
I. h d b
t.
ommends the use of Gringorten's plotting position over sevI nterpoIatlon
. 0f t he cntlca
porns may e accompIS e y no rng
.
.
.
. .
that In (n)and In (1000(1- f» arelinearlyrelatedfor eachsignificance eral competing alternatives for use with the Gumbel dlstnlevel.
bution.
For testing the Gumbel hypothesis the test statistic is given
by (1) with MI obtained from (6), (7), (8), and (9). Since critical
n
= 100
in the nrst two rows of the table. The agreement is
points of this test statistic are unavailable
in the literature
generally very good; discrepanciesseemto be due to Filliben's
even for small samples,critical points (or significance levels)
rounding off of the values he reported.
were computed for sample sizes in the range n
.
A Probability Plot Correlation Coefficient
Test for the GumbelDistribution
This was accomplished by generating 10,000 sequencesof
Gumbel random variables (using (7)) each of length nand
applying equations (I), (6), (7), (8), and (9) to obtain 10,000
As discussedearlier, an important and distinguishing property of Filliben's PPCC test statistic in (1) is that it is extendible io some nonnormal distributional hypotheses.In this section a probability plot correlation test for the extreme value
type I distribution is presented.The extreme value type I distribution is often called the Gumbel distribution, since Gumbel
[1941] first applied it to flood frequency analysis. Its CDF
may be written as
corresponding estimates of , denoted 'I' i = 1, "', 10,000.
Critical points of, were obtained by using the empirical sampiing procedure given in (4). The results of these experiments
Fy(y) = exp(-exp (-(a + by))
a. = Y -;()"i72}i7t
(6a)
"
.6
where
., Eulers
Y IS
= ~7t
s"
constant
(y = 0:57:21)
an
d
yan
d
s" a:e
Significance
Level
n
0.005
]
1982
,
.
In
h
t
.
IS
h
case,
t
h
e
met
d
0
137.
91.6
74.0
49.6
32.0
194.
80.9
61.0
47.4
48.3
37.8
33.3
25.4
21.7
16.9
85.9
71.4
40.6
31.1
21.4
13.8
(6b)
50
60
70
73.7
66.6
57.2
61.1
53.3
49.4
35.4
31.5
28.0
27.1
24.0
21.3
i8.2
16.1
14.4
12.1
10.6
9.43
h
80
90
59.7
53.0
47.5
44.6
25.3
23.6
19.6
18.1
13.1
11.9
8.61
7.97
100
200
300
48.3
30.1
22.5
40.4
23.7
18.1
22.1
13.4
9.79
16.9
10.2
7.49
11.2
6.73
4.91
7.39
4.45
3.23
500
153
t
e
moments
Yi
-I.
Fy
(y.)
= -a
-In
(-In
b
"
(F y(yJ))
122
..
mum likelihood estimators, which require a numerical algorithm to solve the resulting system of nonlinear equations.
Method of moments estimators are employed here, since they
are computationally convenient an~ they have nC} impa:t
upon the hypothesis tests(see(11) which follows). ThIS CDF IS
unique becausesequencesof Gumbel random variables may
~e ~onveniently generatedby noting that (5) can be written in
itS Inverse form as
=
0.50
40
f
0
0.25
114.
99.2
'
Ier t han t he correspondIng
'
,1,0008.236.66.
estimators are much SImp
maXI2,000
4.77
Burges,
0.10
156.
..
and
0.05
10
I
.
0.01
20
30
sample mean and standard devIatIon. Although maXImum
likelihood estimators of a distribution's parameters are usually
preferred over method of moments estimators, [see Lettenmaler
10 to 10,000.
TABLE 2. Critical Points of 1000(1- f) Where, is the Gumbel
ProbabilityPlot CorrelationCoefficient
(5)
Method of moments estimators of the parameters a and b are
given by
=
(7)
3.82
6.67
5.01
3.33
2.20
3 78
2.09
21.61"
.92
193
.
1.09
128
.
0.736
3,000
3.23
2.61
1.50
1.16
0.779 0.528
5,000
1.95
1.59
0.975 0,756 0.507 0.344
10,000
1.12
0.858 0.525 0.414 0.277 0.190
This tableis basedupon 10,000replicateexperiments.
An example
documentsthe useof this table.The 10thpercentage
point of, when
n = 1000is detenninedfrom
"0 = 1 - 2.92X 10-3 = 0.99708
Interpolationof the critical points may be accomplished
.by.noting
-
that In (n) and In (1000(1 f» arelinearlyrelatedfor eachsignIficance
level.
i
i
1.~'J!~'j'"",,";"
':'
590
:
VOGEL: TECHNICAL NOTE
are summarized in Table 2, where again, as in Table 1, the
percentagepoints of (1 f) are more convenient to tabulate.
Interestingly, the PPCC test is invariant to the fitting pro-
-
rZ(~:
"-'C '-"
cedure employed to estimate a and b in (7). This result is
evident for the Gumbel PPCC test when one rewrites the test
statistic as
cov (y M.)
r =
I'
I
1/2
Blom,
~
EFERENCES
G., Statistical
Estimates
and
Transformed
Beta
Variables,
"
,..! i
pp.
68-75,143-146,
JohnWiley,NewYork, 1953.
Cunanne,C., Unbiasedplotting positions-A review,J. Hydrol.,37,
cov In [ -In [V J], In
FederalEmergencyManagementAgency,Flood insurancestudyGuidelinesand specifications
for study contractors,Washington,
D. C.,September
1982.
(
=
R
Beard,L. R., Flood flow frequencytechniques,Tech.Rep.CRWR//9, Cent.for Res.in Water Resour.,The Univ. of Tex.at Austin,
Austin,1974.
[Var (yJ Var (MJ]
'.
-
Acknowledgments. The author is indebted to Jery R. Stedingerfor
his helpful suggestionsduring the course of this research.
.
.
'
205-222, 1978.
[Var (In [-In
[VI]]) Var n - n
(11)
i
i :c~
Filliben, J. J., Techniques for tail length analysis, Proc. Coni Des. Exp.
here the V are uniform random variables generated to equal
F (,, ) A I tt
t.
roperty of this test is that the t t t t" . Y'.-rl. n a rac Ive p
.
. .
. es s a IS
tiC m (11) does not depend on either of the distrIbution parameters. This result is general in that it applies to any PPCC test
for a one- or two-parameter distribution which exhibits a fixed
shape .
SUMMARY
,
:
:i'::
~-~~":c:
"'.'1'
The probability plot correlation coefficient test is an attractive and useful tool for testing the normal, lognormal, and
Gumbel hypotheses. The advantages of the PPCC hypothesis
tests developed in this technical note include:
Th PPCC
.
f
.d I
d
I .
1.
e
.test ~onslsts 0 two. .WI e y use too s m
water resourceengIneerIng:the probabll1ty plot and the product moment correlation coefficient.Since hydrologists are well
acquainted with both these tools the PPCC test provides a
.'
d
f I 1
.
. I
conceptua~ly simp e, a~tractlve, an power u a ternatlve to
other possiblehypothesIstests.
2. The PPCC test is flexible becauseit is not limited to
any sample size. In addition, the test is readily extended to
nonnormal hypotheses, as was accomplished here for the
.
. ..
...
Gumbel hypothesIs. Critical pOInts for the test stattstlc r m (1)
could readily be developed for other one- or two-parameter
probability distributions which exhibit a fixed shape.
3. Filliben [1975] and Looney and Gulledge [1985] found
h h PPCC
"
r
t:
bl .
t at t e
. test .or nor~a Ity compare~ ~vora y, m terms
of power, with sevenother normal test statIstics.
4. The PPCC test statistic in (1) does not depend upon the
procedure employed to estimate the parameters of the probability distribution.
5
Wh
.l
h .
h
.
1
h
d
I
d
h
PPCC
.
tec
mca
as
eve ope
t e
test
statistic
forlet the ISpurposes
ofnote
constructing
composite
hypothesis
i
nicipal water supply, Trans. Am. Soc. Civ. Eng., 77, 1547-1550,
1914.
Interagency Advisory Committee on Water Data, Guidelines for determining flood flow frequency,Washington, D. C., 1982.
Johnson,.R. A., ~nd D. W. Wichern, Afplied Muilivariate Statistical
AnalysIs,Prenllce-Hall~En.glewoodC~lffs,.N. J., 1982. .
Kottegoda,N. T., Investigationof outliersm annualmaximumflow
series,J. Hydrol.,72,105-137,1984.
Kottegoda,N. T., Assessment
of non-stationarityin annual series
throughevolutionaryspectra,J. Hydrol.,76,381-402,1985.
LaBrecque,J., Goodness-of-fit
testsbasedon nonlinearityin probability plots,Technometrics,
/9(3),293-306,1977.
Lettenmaier,D. P., and S.J. Burges,Gumbel'sextremevalueI distribution: A newlook, J. Hydraul.Div. Am.Soc.Civ.Eng.,/08(HY4),
502-514,1982.
Loo~ey,S'.W., and T. R. G~~ledge,
Jr., Use of the correlatior.coefficlentwith normalprob~bliltyplots,Am.Stat.:39(1),75-79,1985.
Loucks, D. P., J. R. Stedmger, and D. A. Haith, Water Resources
SystemsPlanning and Analysis, Prentice-Hall, Englewood Cliffs, N.
J., 1981.
Mage, D. T., An objective graphical method for testing normal distributional assumptionsusing probability plots, Am. Stat., 36(2), 116120, 1982.
National ResearchCouncil,Safetyof Dams-Flood and Earthquake
Criteria,NationalAcademyPress,Washington,D. C., 1985.
Rossi,F., M. Fiorentino,and P. Versace,Two-component
extreme
valuedistributionfor flood frequencyanalysis,WaterResour.Res.,
20(7),847-856,
1984. .
..
Ryan,
T.
Manual
A.
Jr.,
B.
Minitab
vemberi982.'
L. Jomer,
and
B.
Project
Penn.
State
F. Ryan,
Mln/tab
Univ.,
University
Reference
Park
No-
,
tests, the PPCC test statistic in (1) can readily be employed to
compare the goodnessof fit of a family of admissible distributions. That is, a sample could be fit to a number of reasond
d..
f h
bl d . .b . ".
Snedecor,
(J. W., and Cochran,W. G., StatisticalMethods,7th ed.,
Iowa StateUniversity~ress,Ames,.
1980.
.
Thomas,W.O., Jr.,A uniformtechmqu.e
for flood fre~uency
analys1s,
J. Water Resour.Plann.Manage.Dlv. Am. Soc.CIV.Eng.,///(3),
PPCC could be used to compare the goodness of fit of each
distribution. Filliben [1972] has found the PPCC to be a
promising criterion for selection of a reasonable distribution
Wallis,J. R., and E. F. Wood, Relativeaccuracyof log PearsonIII
procedures,
J. Hydraul.Eng.Am.Soc.Civ. Eng.,///(7), 1043-1056,
1985.
function
among several
competing
alternatives.
6. Filliben's
PPCC test
of normality
has recently been in-
R. M. Vogel
Departmentof Civil Engineering
Medford.
MA 02155.
" Tufts University'
a e Istn utIon.unctIons,an corresponmg estImates
0 t e
corporated into the Minitab computer program [Ryan et al.,
1982]. Although Ryan et al. [1982] recommend the use of
Blom's [1953] plotting position. rather than Filliben's approximation
given
in
(3),
the
Mimtab
computer
program
could
readily be employed to implement the tests reported here.
...
-
ArmyRes.Dev..Test.,/8th, 1972.
Filliben, J. J., The probability plot correlation coefficient test for normality, Technometrics,/7(1), 1975.
Gringorten, I. I., A plotting rule for extreme probability paper, J.
Geophys.Res.,68(3),813-814, 1963.
Gumbel, E. J., The return period of flood flows, Ann. Math Stat.,
/2(2),A.,
163-190,
1941.
Hazen,
Storage
to be provided in impounding reservoirs for mu-
~..
.
','
~---
"
",,'
,~;
321-337,
1985.
(R ecelve
. d Sep t em b er, 4 1985 ;
revised
September
27,1985;
acceptedNovember 4, 1985.)
,
~
I
r
,--
.
WATER RESOURCES RESEARCH, VOL. 23, NO. 10, PAGE 2013, OCTOBER 1987
!
Correction to "The Probability Plot Correlation Coefficient Test
for the Normal, Lognormal, and Gumbel Distributional Hypotheses" by
Richard M. Vogel
In the paper "The Probability Plot Correlation Coefficient
Table 2 should be revised as follows: The critical point of
Test for the Normal, Lognormal, and Gumbel Distributional
1000(1- f) for n = 20 and a significance level of 0.01 should
Hypotheses" by R. M. Vogel (Water Resources Research, read 94.0 instead of 294.
22(4), 587-590, 1986), the following corrections should be
Equation (11) should read
made.
Equation(1)shouldread
r=-~"
cov { In[-ln(U;)],ln
n
L
r=
[
n
~
(y(i)
;=1
L.. (y(i)
ML-
[Var (y;) Var (M;)] 1/2
-
Y)(M1- M)
~
n
={
(1)
-
[ -In (;+a:12
i-O.44
)J}
[ (
-
Var[ln(-ln(U;)]Var
In -In
j= 1
;+a:12
(Received July 16, 1987.)
I
1
!
j
i
,
Copyright 1987by the American Geophysical Union.
Paper number 7WS031.
0043-1397/87/007W-S03IS02.00
I
--
2~
-2
- YJ
L.. (M j - M) J
i= 1
I
(i-0.44 ))J}
Co
;
2013
!~c
{
1/2
(11)