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Statistics 27 3 TESTS OF NORMALITY AND OTHER GOODNESS-OF-FIT TESTS Ralph B. D' Agostino, Albert J. Belanger, and Ralph B. D' Agostino Jr. Boston University Mathematics Department, Statistics and Consulting Unit Probability plots and goodness-of-fit tests are useful tools in detennining the underlying disnibution of a population (D'Agostino and Stephens, 1986. chapter 2). Probability plotting is an informal procedure for describing data and for identifying deviations from the hypothesized disnibution. Goodness-of-fit tests are formal procedures which can be used to test for specific hypothesized disnibutions. We will present macros using SAS for creating probability plots for six disnibutions: the uniform. normal, lognonnai, logistic, Weibull. and exponential. In addition these macros will compute the skewness <.pi;> . kurtosis (b2), and D' Agostino-Pearson Kz statistics for testing if the underlying disttibution is normal (or ·lognormal) and the Anderson-Darling (EDF) A2 statistic for testing for the normal, log-normal, and exponential disnibutions. The latter can be modified for general distributions. standardized deviates, :z;. on the horizontal axis. In table I, we list the formulas for the disnibutions we plot in our macro. If the underlying distribution is F(x), the resulting plot will be approximately a straight line. TABLE 1 Plotting Fonnulas for the six distributions plotted in our macro. (PF(i-0.5)/n) Distribution Unifonn cdf F(x) Vertical Axis x-11 for 1'<%<1' +o G PROBABILITY PLOTS ~ P...~ ~;j ··I( i-3/8) 11+1(4 In(~~ Say we desire to investigate if the underlying cumulative disnibution of a population is F(x) where this disnibution depends upon a location parameter )l and scale plll31lleter a. not necessarily the mean and standard deviation. Further. let Weibull 1-exp( -(.!.t) Logistic [I <ap{-11'(%-jl)/oi3>J·1 ln(lw) 6 Horizontal Axis 7, II ·-1( 1-3/8) 11+1{4 In( -In( 1-p,)) %-1' F(z)=G(-)=G(z) Exponential 0 where z=(x-p)/a. Further, say we have a random sample of observations of size n with ordered observations ~n~...~Dl" A probability plot is a plot of X(o 011 ~=G- 1 (F.f~u))=G- 1 (p~ o·•o where is the inverse transformation of the standardized disttibution of the population (hypothesized distribution) under consideration and F.O is the empirical cumulative defined here as: i~.S F~'=p,=• (I)' n (1) (see D'Agostino and Stephens. 1986, p 34). In our plots we place the data (~il) on the vertical axis and the l-ap( -%/6)) -ln(l-p,) The macro PROBPLOT takes as input the data set and produces probability plots for the six distributions mentioned above. We use the rank procedure to order the data to produce the ordered .standardized rants. When observations are equal we use the means of the rants (ties=mean option)as in D' Agostino and Stephens, chapter 2. Using these rants, i, we compute p,=(i-0.5)/n and then the inverse transformation distributions for the six dislributions. Probability plots are then produced. Since the normal probability plot is the most widely used we describe it in detail now. This plot consists of the ordered observations on the vertical axis and the standard normal deviates on the horizontal axis. We use Blom's approximation when defining the nonnal cumulative in order to enhance the linearity of the NESUG '91 Proceedings 27 4 Statistics plot. The plot is thus XIII on Z=e-1( i-3/8) n+l/4 where ~ 0 is the ith ordered observation from the ordered sample may reflect the presence of outliers, mixlmes in data, or· truncation (censoring) in the data. The reader is referred to D' Agostino and Stephens (1986) chapter 2 and D' Agostino, Belanger and D' Agostino (1990) for further details. Probability plots are only informal techniques for evaluating the underlying distribution of data. Next we provide several statistical tests which provide a more formal approach. GOODNESS-OF-FIT TESTS and Z is such that j-3/8 z 1 --=! n +1/4 -.;2ft --'-<! z' 2dJ& for i:l,...,n The figure in Appendix A contains a nonnal probability plot of sample data with the expected sll'aight line going through the +'son the graph. In programming the macro to create this plot we took advantage of two options in the Proc Rank procedure. The fust. was the "ties=mean" option which chooses the mean rank when there· are observations with the same value (see D' Agostino and Stephens. chap 2 for further discussion) and second the "normal=blom" option which will find the standardized cumulative nonnal Blom rank automatically. The pagesize and linesize options allowed the axis to be wider than the traditional Proc Univariate nonnal probability plot. For the lognormal distribution we provide two plots, the first after taking logs of the raw data and the second after taking the logs of (observed data- estimated lambda ). Lambda corresponds to the third parameter of a three parameter lognormal distribution whose density is Unless lambda is close to zero. the probability plot will not be a straight line for a lognormal distribution when one takes the logs of the data. The macro will automatically produce both plots and gives as output the estimated value of lambda (D' Agostino and Stephens, 1986, p. 53) so the user can decide which plot is more appropriate. If the raw data contains values less than or equal to zero. the macro will automatically add the absolute value of the minimum plus .01 to each value 01 the data set for calculating the logs of the data. Probability plots will fonn approximately a straight line if the underlying distribution is the hypothesized distribution. Deviations from linearity help to determine properties of the underlying distribution such as if it is skewed and/or thick tailed. Other deviations NESUG '91 Proceedings A population, or its random variable X. is said to -- be normally distributed if its density function is given by Here p and o are the mean imd standard deviation, respectively, of it. Of interest here are the third and fourth standardized moments given by and B:= E(X-~&) =E{X-~&f [E(X-11ff a4 4 . where E is the expected value operator. These moments measure skewness and kurtosis, respectively, and for the normal distribution they equal 0 and 3, respectively. A positive third moment correspond to a skewness to the right (ie a longer right tail) and a negative skewness corresponds to skewness to the left. Kurtosis, (the word means curvature) is a measure of tail thickness. A kurtosis larger than 3 on a unimodal distribution indicates thicker or heavier 1llils than the normal distribution, while kurtosis less than three on a unimodal distribution indicates lighter tails than the normal. The sample estimates of these moments have been shown to be useful statistics to test whether data is normally distributed (D' Agostino et al 1990). For a sample of size n, ~.... .X,. the sample estimates of and .pr; and B: are respectively. Statistics These are related to following: where If:' v~~· and x is the sample me311 t·2) ll(n-1) (n-2)(n-3) and bnosis I . n(n .. l)L (X-Xyt _ 3(n-lf and bz via the ll1 and ~"' (n+l)(n-1) 8%+ i=EX,fn. Values of 0 (for the third moment) and 3 (for the fourth moment) would indicate that the underlying population of a data set was normally distributed. Their expected values under normality are 0 and 3(n-l)/(n+l) respectively. These statistics can be used to test. formally if the underlying distribution is normal (D' Agostino and Stephens. 1986, chapter 9). If they lead to rejecting the normal distribution they automatically indicate the type of nonnormality present in the data. For instance, if the third moment is negative this indicates that the data is negatively skewed or if the fourth moment is greater than 3(n-l)/(n+l) this indicates heavy tails in the population distribution. Thus, the signs and magnitude of these statistics are both useful here. We present tests for normalitv using these statistics in our macro as well as an omnibus te~t using the K1 statistic. (Omnibus here means that the test will detect deviations from normality due to either skewness or kurtosis). Much of the programming for these tests involved finding the third and fourth moments using the output from SAS's Proc Univariate procedure. The skewness and kurtosis statistics calculated in the procedure · are the Flsher g statistics defined as: .fDt 27 5 3(n-l) (n+l) • Thus, once we transformed the statistics we can perform the normality tests. The second type of formal tests we programmed into the macro are EDF (Empirical Distribution Function) tests. For a random sample of size n, with data X 1•••• and the order statistics defmed as ~ 1 ,SX(2,s··::;~.,; let the distribution of X be F(x). EDF statistics measure the difference between F.(x) and F(x) where Fu(x) is defined here by: .x•. more precisely Note, F.(x) here is defined differently than for the probability plots (formula ( 1)). . In our macro we used the Anderson-Darling (1954) A1 statistic which uses a quadratic measure of discrepancy between F.(x) and F(x) when it is calculated. This test falls in the class given by the Cramer-von Mises family z (n·l)(n-2)(n-3)r (n-2)(n-3) -'Nhere where is the sample variance. -.oo is [{F(x) }{ 1-F(x))]'1• See D' Agostino and Stephens chaprer 4 for an in depth di.w·nssion of EDF statistics. In order to compute the A2 statistic, we used the computing formulas suggested in D' Agostino and Stephens NESUG '91 Proceedings 276 Statistics (chap. 4). Using the Probability Integral Transformation (PIT), z,;F(x), we know that if F(x) is the true distribution of X 1hen Z will be uniformly distributed on [0,1). We calculate values of Z,=F(X;) i=l •.•.,n from our sample X1, ... .X.,. Fe'(z), the EDF of the values of Z, is then found. Using these values we compute A2 as follows: We compute this statistic for testing the normal. For the lognormal. and exponential distributions. statistic the exponential distribution we calculated for estimates with assuming the origin was zero and then 1986, Stephens. the origin. We follow D'Agostino and procedure on page 141. where we have the exponential distribution For the first case we assume n is 0 and use the sample mean for ~. and in the second we estimate n and ~ using the formulas uniform, logistic, exponential and Weibull all are far from . linear indicating that these data probably do not follow any of these distributions. When we look at what the goodness-of-fit tests produced. we see that the skewness test for normality is rejected (p=.Ol), while the kurtosis test is not (p=.59). The K 2 test, which combines the two, rejects normality at 2 p=.03. The Anderson-Darling A tests confirm these results, with the statistic to test for normality having a p value <.005. The test for the lognormal distribution is not rejected with .15<p<.25. Finally, the tests· for the exponential distribution are rejected at p<.Ol. for the cases when the origin is assumed known and when it was estimated. From this output, the investigator might decide that the data were lognormally distributed and proceed with an analysis after raking logarithms of the data. The output from the macro can be found in the figures and in table 3 in Appendix A. The macro itself is listed in Appendix B. Thus, we have provided a useful macro written for use with SAS which will provide both a graphical display of the data as well as several formal tests for determining the underlying distribution of a population. Table 2 Systolic Blood Pressure Data from the Framingham Heart SIUdy For the lognormal distribution we calculate the EDF statistic only without the l we computed in the probability plotting. We also present the A2 statistics both in their unmodified and modified forms where the modifications are made using suggestions by Stephens in D' Agostino and Stephens (chap. 4). The user could test for other distributions as well if they modify the macro so they can input the Z; values for the distribution of interest An example of the output from this macro is presented below. Table 2 contains a sample of 67 systolic blood pressures from a sample of 67 subjects from the Framingham Heart Srudy. The data are presented in a stem-and-leaf plot with descriptive statistics, also. From examining the seven probability plots computed one can see that the data appears most likely to follow the lognormal distribution. After examining the normal probability plot one can see that the data seems to form two straight lines, one for z-values below zero, and one for z-values above zero. This could be an indication of a mixture of two normals. The lognormal probability plot forms nearly a straight line, however when we examine the lognormal plot with estimate of lambda the data form a sttaight line except at the lower tail. This should prompt the investigator to check the lowest point as a possible outlier. Finally. the other four plots,the NESUG '91 Proceedings Slem-11111-leaf plot 21 20 19 18 17 16 15 14 13 12 11 10 9 Nwnber 0 08 0006 003 0008 0046 00124688 000002244689 000024467888 0000446889 046888 0 2 4 3 4 4 8 12 12 10 6 I The clescriplive 118tiJtics an:118111ple size, n=67, mem:l37.15; slanciard devialion=25.63; stewnes....768; ll:wlosia=3.08. This work was funded by a grant from the National Heart, Lung and Blood Instirute to R. B. D' Agostino (RO 1 HL 40423..()3) REFERENCES D'Agoslino, R.B., Belanger, AJ., and D' Agostino Jr., R.B., (1990) "A Suggestion for Using Powerful and Iofonnative Tests of Normality." The Ameticao St.atislician,44.316-321. D'Agostino. R.B., and Stephens, M.A. (1986), GoodMss-of·fit T«hniquu, New York: Man:e1 Dekker. 277 Statistics APPENDIX A: OUTP\IT FROM THE MACRO 0 I .... .. ·-.Ii ....i ., I I .....I I , I I 5 t. 0 G I 0 ! 0 t.& • a .. ..... . ....II ! ll I ~ ~ E $ .. £ g tl s •.a • E ll 'I E g .....I ...I .. ,. ··' I. I .... ....I 'I I I .... I ... ......... .... t. -· I Q' E ....I ....I .... ...... ..... I I I ...... ...... .... .... .... ••• 1.~ 1.~ J.J J,3 I •&o.l •lol •• J loJ lol "'' ..... lol loJ J,J .... I "' .....I I " ....I : I "' ....I .. I c c . ....I ....I : ... .I .... t. 0 -- 4ol'"' G '1 5 I K G 5 ! 1.1 • t. ll A ll £ I lol • A '7 t. C' ... ... ... ... •.. ... ... ... ... ... ... I •••• •••• •"-1 ..,.. .... ... ••• .... ~.. ••• G~~%1'!~1111 ••• J.t UIG•IfOIIIQl. Z•'llU.ll1: (tAIIIIDAooUI .. I I I -i 0 I s E II 'I E D ] 0 I s E • ....I 1' IC a ~a u•:.I 1' .r. t. Q' IC .... :I -·I .. i. -I.... --II· =.:: I I .. : ...... . I. •1.1 1-VlU.lll: ... ... ... ... ... ... ... ~.J.L •I.J •1,1 'L•I •\•I .... 1 1.1 1.1 1.1 :,..,c-:r.:e :-,u:..""E S...J 1.1 1.1 1.1 1 oDa v•r• z-v;um: cu~ o~ ranqe 278 Statistics APPENDIX B: THE MACRO ~MA~O PROBPLOT(VAR.DATAI: DATA: SET &DATA: T=1; KEEP T &:VAR; PROC SORT: BY &:VAR: PROC RANK TIES-MEAN OurooAA: VAR &:VAB:. RANKS~ PROCUNIVAR.!ATE DATAaAA NORMAL PLOT: VAR &:VAR; OU'I'PIJT OuralOC.STAT MIN•MIN NaN MEAN-XBAR S'ID-S P:5_.., P9~5 MEDIAN-MEDIAN SKEWNESS=01 ICURTOSIS-G2; ·•·• •&.a •lei ·~·• •Lol ...l t.a •·• ;..J ~.J 1.1 a.J a• .a W!%Str...:. :-U'.u,t,-;; l obs vere ou~ ~~ :anqe. TABLE3 HORMAIJTY TI!ST I'OR VAIUAIILJ! SPP H-67 uh0.711392 SQRTBI .0.76821 Z.l.S6017 1"".0105 Z. O..S~OIJ p.o.599:S 02a41Mtj 82.-3.0&216 !t•o:!oOIISQ (: DP) • 6.!30"..!1 PoO.Ql:9 ·-··----·----····---··-··--------·· DATA lOC.STAT: SET JOCSTAT; T•1; UMBDAooi(P95•P5)-{MEDIAN•MEDIAN))((P9$+1'5-(2"MEDIAN)); ALPHA•Mnii-1/N; BETA.-N"'CBAR-MIN')J(N-1); DROP P9S ~ MEDIAN: DATA M: MERGE AA JOCSTAT: BY T: LOGVARE=LOO&:VAR-LAMBDAI: IF MlN>O niE."' LOOV AR=LOO(&:VARJ; ELSE DO: LOGY AR=LOO(&:V AR+ABS(MIN)+.Ol): FILE PRINT: PUr 'WARNING SOME OFn!E DATA HAS VALUES <zOI All.' I 'DATA HAS BEEN TR.A.'ISFOR.'w!ED IN THE LOG-NORMAL A.'fO'I 'WEIBlJIL PLOTS BY MAKING AJ.J.. VALUES POSmVE.'; END: CAJ.J.. SYMPurC'LAl>'IB' .LEFT(Pt..'TCLAMBDA.BESTI.)Il: PROC UNIVARIATE DATADAA NOPRJNT; VARLOOVARE LOGVAR; OU'I'PIJT Ot.rr•BB MEAN•LXBARE L.XBAR S1'DaLSE LS: EDP 'IUrS USIHu AHlll!llSOH·DAIWNO A.SQUAJU!Il STAns!IC CRmCAL VALliES I'OR NORMAl. DIStlUII1lriOH W!'IH MEAN AHll VAIUI.l<CI! ~ VA1UA.1112 •SPP SIGNli'IC.U<a! LJ!VU AI..PIIA .D5 .D25 .01 JlOS .23 .15 .10 UPPER TAIL .301 .0111 ,j61 All .7:52 .ITl l.a!S US. LOWB TAIL .119 .)It .249 .z:e .111 .Ill) .139 .so BDP n'A'Itl'r!CI'OR1JIII NORMAL DIS11UBUUQHOIODD'II!Dl 1.1714 EDP n'A'Itl'r!C POl!. 1JIII !IOilMAL DIS11UBU'DQH (l1l6IOD1IIIID) 1~511 l!Dl' STA.mt!C !'OR 'Ill!! toG-NORMAL DISTRI3IITION (MODII'IED) D.n24 l!Dl' S'tA.'t!SnC 1'0111JIII t:.OG-HORMAL DIS'!liiBITDON (IJNMOD!FIEII) IUOii5 atmC\1. V ALliES POR 'Ill!! 1!XP0N!1mAL DISnllllltiOH. OIIIUIN mowN AND SCAU1 ~ .:Z:S .:111 SICINII'ICA.HCI UV2L AIJ'HA ~ .10 .D5 .IIZS .D1 JlOS .D11Z5 UPPER TAIL .1M .116 .916 1.DG 1.321 I ~I I.J59 l.lU ~Jol !.OWD TAIL .302 .311 .:111.2141 .lDI J1l ~:10 l!Dl' STA.'Itl'r!C P0R 1JIIII!XPONI!Il1IAI. DIS11UBITDOII lii.!IS3 1!llP STA.'Itl'r!C POR 'Ill!! I!XPOH1!HltAL DIS'!liiBITDOH ()IODII'II!III :11.1411 aaiCAI. V ALliES POR 'Ill!! !XP0M!1mA1. DISniiiU'IION. OIIJGIH AHDSCAUitJICHOWior M4f .23 H 5 10 IS ::D 25 :10 100 .$OS ..ftS .1125 .410 .110 - . .735 SICIIIIPICANCZ UV2L 1oLP11A ~5 .10 Jl5 .D1S .111 Ul'l'!ll TAIL .sss .w .7%5 .... 319 .7.rl .920 1.11051 1.3S2 .720 .116 I.ID 1.191 1_.95 .7J7 .161 L.OQ 1.:47 l..sa:J .D& .890 1.1197 I.JI7 1.635 .131 .965 u97 ~- 1.rn .175 I.Dal I.:SO 1..510 I..ISS .916 I.DQ I.J%1 1..591 I.?S. I!S'tiMA1BS PORioLPIIA • l.J216 AND B!TA ••7.1636 EDP STA1lS'!lC !'OR '!HI! :e:xPOHI!NTL\!. DIS'!llllllrnOH 6..0107 EDP STA1lS'!lC I'OR 'Ill!! :e:xPOHI!NTL\!. DIS'!liiBliTIOH IMODIF!Eill ~I NESUG '91 Proceedings DATA BB: SET BB: T•1; DATA M; MERGE M BB: BY T; Ploo(l4S}/N: UNIPORMZooPi; WEIB"UI.t.Z-LOO<·LOO<l·Pil>: LOGIS'I7.oo(SQRT(3)13.141S926Sl"LOG(Pi/(1·PI)l: EXPONENZ--1..00(1-Pil: NORMALZI-PROBNORM((&:VAR·XBAR).IS); LOONCIIIZI-PROBNORMfiLOGVAR-UBARJILS); EXPOHI!ZI>o1-EXP(-(&:VARIXBARll: EXI'OHMZI-1-EXP(-((&VAR-AIJ'HA)IBEI'Al); .NORMAU.S-(lJN)"((l"_."'_·I)"LOO<NORMALZI)+ <Z.N+1-2•_N_.)"LOO( 1-NORMAU!)); LOGNORAS-1:1/Nl"((l•_.'f_-ll"LOGILOGNORZI>+ Cl"N+1·2•_.'f..)"LOG(I·LOGNORZn); EXPONEAS-< 1/N)"((l"_.'f_-1l 0 LOOIEXPONEZn+ (2-N+ 1-2•_."'..)"LOG(l-EXPONEZill: EXI'ONMASoo(1/Nl"((2" _."'_·ll"LClG(EICPONMZI>+ . (2"N+1-2"_!'IJ•LOG(1-EXPONMZill: PROC MEANS NOPI!.OO; VAR NORMALAS LOGNORAS EXPONEAS EXPONMAS: OU'I'PUT OUT•ANDERSON SUM•NASQUARE LASQUARE EASQUARE EMA.SQUAR N-N: DATA ANDERSON; MERGE ANDERSON XXSTAT; BY N: NASQUARE-N-NASQUARE: MASQUARE-NASQUARE•(i+(.75~•N))); LASQUARE-N-LASQUARE: MLASQUAR-LASQUARE"f I +(.751N)+(l.:!.51(N•N))); EASQUARE-N-EASQUARE: MEASQUAR-EASQUARE"fl+(.6/Nl); EMASQIJAR-N-EMASQUAR; MMEASQUA-£MASQUAR•(!+(.6/Nl); DROPT J'REQ.._nPE..; DATA; SEI'XXSTAT: DO _z_a-1.0.1; _X..•XBAR+_z_"S:OUTPtJI': END: KEEP ..X....):_; DATA: MEROE AA J,..o.S'I'..; Statistics PROC RANK T!ES=MEAN NORMALooBLOM OUT=>AA: VAR .tVAR LOGVAR LOGVARE; RANKS BLOMRANK LOOBLOM LOGEBLOM; OPllONS LS-10 PS>D60: PROC PLOT NOL£GEND: LABEL BLOMRANK-'NORMALIZED RANK' LOGBLOM •'LOG-NORMAL RANK' LOGEBLOM•"LOG-NORMAL RANK (LAMBDA..t.LAMII)" .tVAR a'OBSERVED VALUE' LOGY AR •'LOG OF OBSERVED VALUE' LOGYARE •'LOG OF OBSERVED VALUE (USING LAMBDA)' UNIFORM:z.a'UNIFORM RANK' WEIBUJ..I.Z,.'WEIBULL RANK' LOGISTZ a'LOGISnC RANK' EXPONENZ='EXPONENTIAL RANK': PLOT <I:VAR*BLOMRANK-'•• .J<..•.J._•'+'/ OVERLAY HAXIS-3 TO 3 BY .5: PLOT LOGVAR*LOGBLOM•'*' I HAXIS-3 TO 3 BY .5; PLOT LOGVARE*LOGEBLOMa'*' /HAXIS • -3 TO 3 BY .5; PLOT AVAR•l.JNIFORMZ-'*' I HAXIS=O TO I BY .1: PLOT .tVAR*LOGISTZ='*' I HAXIS-3 TO 3 BY .5; PLOT AVAR•EXPONENZ-'*' I HAXIS..O TO 4 BY .S; PLOT LOGVAR*WEIBUll.Z='*' I HAXIS~3 TO 3 BY .5; DATA ANDERSON; SET ANDERSON; SQRTBI=(N-2)1SQRT(N*(N-l))*GI; Y.SQRTB I*SQRT((N+ ll*(N+3 )!(6*(N-2))); BETA2-3*(N*N+27*N-70)*(N+I)*(N+3)/ ((N·2)"(N+Sl'"!N+7)*(N+9)); W.SQRT(·l+SQRT(2*CBETA2·1))); OELTAai/SQRT(LOG(W)); ALPHA.SQRTC2/(W*W-1)); Z..B1•DELTA •LOGCY/ALPHA+SQRT!CY/ALPHA)**Z+l)); 82-3*(N-l)!(N+l}+(N·2)*(N-3)K(N+l)*(N-1))*G2: MEANB2a3*(N·llKN+I); VARB2a24*N*(N-2)*(N-3)!({N+I)*(N+I)*(N+3)'"(N+S)); X-<B2·MI!ANB2)1SQRT(V ARBl); MOMENTa6*(N*N-5*N+2)1((N+7)*(N+9))* SQRT(6*(N+3)"(N+5)1(N*(N-2)*(N-3))): Aa6+8/MOMENT*C21MOMENT+SQRT(l+41(MOMENT**Z))): ZJI2oo( 1·2K!I*A).(( 1·2/A)I ( 1+X*SQRT(2/(A-4))))**(113))/SQRT(2/(9*A)); PRZBI-.2*(1-PROBNORM(ABS(Z..Bl))); PRZB2a2*(1·PROBNORM(ABS(Z..B2))); an'l'Uf•Z..BI*Z..B1-+Z..B2"Z..B2: PRCHI-1-PROBCHI(OU'l'EST,l); l'll.E PRINT: PUr @2 "NORMAUTY TEST FOR VARlABLE I<VAR " N•/ @20 G1..S..S @33 SQRTBI..S..S @.50 OZ.'Z..B1 8.5 • P.' PRZBI 6.41 @20 m..&..S @33 BW..S @.50 'Za' Z...B2 8.5 ' P.' PRZB2 6.41 @33 'K**2aCHISQ (2 OF) •' OU'l'EST 8.5 ' P.' PRCHl 6.41 @10'...· - · - - · - - - · - · - - · - · - -....- -......... ,, @ 10 'EDF TESTS USING ANDERSON-DARLING A..SQUARED STATISTIC'/ @5 'CRITICAL VALUES FOR NORMAL DISTRIBUTION wrm MEAN AND VARIANCE'/ @20 "UNKNOWN VARIABLE • I<VAR "/ @W' ~ @10 'I SIGNIFICANCE I..EVEl.. ALPHA I'/ @10 '1.50 .:!5 .15 .10 .05 .02!1 .01 .005 1'/ @10 'I UPP£R. TAIL 1'/ @IO '1.341 .470 .561 .631 .752 .rn 1.035 1.159 r1 @10 'I LOWER TAIL 1'/ @10 'I .341 .249 .226 .188 .160 .139 .119 1'/ @10' 'II @10 'EDF STATISTIC FOR 1liE NORMAL DISTRIBU'TlON (MODIFIED) ' 279 MASQUARE 6.4 I @110 'EDF STATISTIC FOR 1liE NORMAL DISTRIBUTION <UNMODIFIED) ' NASQUARE 6.4 II @110 'EDF STATISTlC FOR 1liE LOG-NORMAL DISTRIBUTION (MODIFIED) ' MLASQUAR 6.4/ @110 'EDF STATISTIC FOR 1liE LOG-NORMAL DISTRIBUTION <UNMODIFIED) ' LASQUARE 6.4 II @10 ................- .........................- -........, @110 'ClUTICAL VALUES FOR 1liE EXPONENTIAL DISTRIBUTION, ORIGIN '/ @120 'KNOWN AND SCALE UNKNOWN'/ @110 ·--~~=-----=---------'! @110 'I SIGNIFICANCE LEVEL ALPHA I'/ @10 'I .:!5 .20 .15 .10 .0$ .02!1 .01 .005 .002.51'/ @110 'I UPPER TAIL I'/ @10 'I .736 .816 .916 1.062 1.3211.591 1.959 2.244 2..5341'/ @110 'I LOWER TAIL 1'/ @110 '1.342 .312 .2&0 .249 .20& .178 .150 I'/ @IW' W @110 'EDF STATISTIC FOR 1liE EXPONENTIAL DISTRIBUTION' EASQUARE 6.4/ @110 'EOF STATISTIC FOR 1liE EXPONENTIAL DISTRIBU'TlON (MODIFIED) • MEASQUAR 6.4 I @l(Y ... _ ................................... ...__........ , @110 'CRII'ICAL VALUES FOR 1liE EXPONENTIAL DISTRIBUTION, ORIGIN'/ @120 'AND SCALE UNKNOWN'/ @W ~ SIGNIFICA.'IICE LEVEL A1J'HA 1'/ N .:!5 .15 .10 .OS .02!1 .01 1'/ UPPER TAIL 1'/ ' .460 .j55 .621 .725 .848 .9&9 1'/ 10 .j4S .660 .747 .920 1.068 1.352 I'/ IS .575 .720 .816 1.009 1.198 1.495 1'/ 20 .60& .757 .861 1.062 1.267 1.580 1'/ 25 .625 .784 .890 1.097 1.317 1.635 1'/ so .6&0 .838 .965 1.197 1.440 1.775 1'/ 100 .710 .815 l.OOS 1.250 1.510 1.85.S 1'/ '@W iDfiaity .736 .916 1..062 1.321 1..591 1.959 I'/ @10 '// @110 'ESTIMA11lS FOR ALPHA • ' ALPHA 7.4 'AND BETA a ' BETA 7.41 @10 'EDF STATISTIC FOR 1liE EXPONBNnAL DIS11UBUTION' EMASQUAR 6.4 .I @10 'EDF STATISTIC FOR 1liE EXPONI!NTIAL DIS11UBUTION @110 @110 @10 @10 @10 @10 @10 @10 @10 @10 'N=' (MODIFIED) • MMEASQUA 6.4; RUN; ...MEND PROBPLOT; ~ ~ ,. Euaaple of. - t o Cllec:ute lbo ......... ollcwe: ., ~ ~ 'M'ROBPLOT(SPF.OATAl) */ ~ NESUG '91 Proceedings