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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)