The Probability Plot Correlation Coefficient Test for the Normal, Log ...

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

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resource

normal

and

Looney

and

flow

research

investigations

nonnormal

require

hypotheses.

Gulledge

[1985J

Filliben

developed

y

P

lot

correlation

o

distributions.

tests

a~sociate~

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PPCC

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correlation

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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

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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

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examlne.t

Incorporated

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favorably

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The

rarely

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coefficient.

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Filliben

to

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hypotheses,

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had

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note.

4.

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coefficient.

correlation

statistic

I

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II

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general,

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hypothesis

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--

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

--

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i= 1

I

(i-0.44 ))J}

Co

;

2013

!~c {

1/2

(11)