Aalyss of Varace wth Webull Data Lahaa Watthaacheewaul Abstract I statstcal data aalyss by aalyss of varace, the usual basc assumptos are that the model s addtve ad the errors are radomly, depedetly, ad ormally dstrbuted about zero mea ad costat varace. For aalyzg data whch do ot match the assumptos of the covetoal method of aalyss, we have two choces. We may trasform the data to ft the assumptos, or we may develop ew methods of aalyss wth assumptos whch ft the orgal data. If we ca fd a satsfactory trasformato, t wll almost always be easer to use t rather tha to develop a ew method of aalyss. I aalyss of varace wth Webull data, the data should frst be trasformed to ft all the assumptos requred. The wellow Box-Cox trasformato ca use to get the ormalty but caot trasform the observatos that equal zero. I the sets of Webull data, the observatos may be zero. To cope ths problem, a alteratve trasformato s proposed. Whe the trasformed data have met the requred assumptos of ormalty ad homogeety of varaces, we the ca apply the aalyss of varace to test the equalty of the populato meas or the treatmet effects of the orgal Webull populatos. Moreover, umercal studes of the powers of the tests obtaed from ANOVA of the trasformed data are also gve. Keywords Webull Data, The Box-Cox trasformato, The alteratve trasformato I. INTRODUCTION I the aalyss of varace (ANOVA) the usual basc assumptos are that the model s addtve ad the errors are radomly, depedetly, ad ormally dstrbuted about zero mea ad equal varaces. Wth some specfc sets of data, the basc assumptos are ot satsfed so aalyss of varace caot be appled approprately. Tuey [] suggested that aalyzg data whch do ot match the assumptos of the covetoal method of aalyss, we have two alteratve ways to go about. We may trasform the data to ft the assumptos, or we may develop some ew methods of aalyss wth assumptos fttg the orgal data. If we ca fd a satsfactory trasformato, t wll almost always be easer to use the covetoal method of aalyss rather tha to develop a ew oe. Motgomery [] suggested that trasformatos are used for three purposes, stablzg respose varace, mag the dstrbuto of the respose varable closer to the ormal dstrbuto, ad mprovg the ft of model to the data. Choosg a approprate Ths wor was supported part by the Faculty of Scece, Maejo Uversty, Chag Ma, Thalad. L. Watthaacheewaul s wth the Faculty of Scece, Maejo Uversty, Chag Ma, Thalad (phoe: 66-5-87-55; fax: 66-5-878-5; e-mal: lahaa@mju.ac.th). trasformato depeds o the probablty dstrbuto of the sample data. For example, the square root trasformato s used for Posso data ad the logarthmc trasformato s used for logormal data. Moreover, we ca use the relatoshp betwee the stadard devato ad the mea for stablzg varace. Furthermore, we ca trasform the data by usg a famly of trasformatos studed for a log tme. May authors have studed the trasformatos of the data to meet the requremets of the aalyss of varace []- [6]. The Box-Cox trasformato for ANOVA s the form X j, 0 Yj for x j > 0 () l X j, 0 where X j s a radom varable the j th tral from the th dstrbuto, Y j the trasformed varable of X j, ad a trasformato parameter. It s ofte used to trasform the data to fulfll the requremets but t mght ot be satsfactory some cases. Dosum ad Wag [7] dcated that the Box-Cox trasformato should be used wth cauto some cases such as falure tme ad survval data. Joh ad Draper [6] showed that the Box-Cox trasformato was ot satsfactory eve whe the best value of trasformato parameter have bee chose. Moreover, the codto of observato s that the value of t s greater tha zero. I the sets of Webull data, the some observatos may be zero. I order to cope wth ths problem, the alteratve trasformato s proposed. I ths paper, the two parameter Webull dstrbuto s vestgated. II. THE WEIBULL DISTRIBUTION The Webull dstrbuto s a cotuous probablty dstrbuto. It s amed after Walodd Webull who descrbed t detal 95. The probablty desty fucto of a two parameter Webull radom varable X s α α x α x β, 0;, > 0 ( ) e x αβ f x β β () 0, x<0 where α s the shape parameter ad β s the scale parameter.
The mea s β Γ +. It s useful may felds such as α survval aalyss, extreme value theory, weather forecastg, relablty egeerg ad falure aalyss. Moreover, t s used to descrbe wd speed dstrbuto, the partcle sze dstrbuto, ad so o. Furthermore, t s related to the other probablty dstrbuto such as the expoetal dstrbuto whe α [8]. A alteratve test procedure for testg the equalty of scale parameters of Webull populatos wth a commo shape based o sample quatles was preseted ad the power of ths procedure was show to be qute good umercally several stuatos [9]. III. AN ALTERNATIVE TRANSFORMATION A trasformato for ay sets of Webull data to ormalty wth equal varaces proposed here s the form Substtute the values of ˆ μ ad ˆ σ to the lelhood equato (4). Thus for fxed, except for a costat, the maxmzed log lelhood s f( ) l L( xj ) + 0.0 + 0.0 l Xj c Xj c j j + ( ) l Xj + 0.0c Hece, dl L( ) d j Xj + 0.0c l Xj + 0.0c j Xj + 0.0c Xj + 0.0c j j + (7) where th Xj + 0.0c, 0 Yj l Xj + 0.0 c, 0 () X j s a radom varable the j th tral from the Webull dstrbuto, Y the trasformed varable of j X j, c the rage of the value of X j from the th Webull dstrbuto, ad a trasformato parameter. The lelhood fucto relato to the observatos s gve by L x J x ( πσ ) 0.0 (, ) exp X +. ( ; ), j c μ σ j μ σ j where J( y; x) y j x j j (4) X + 0.0c X + 0.0c l X + 0.0c j j j j j Xj + 0.0c Xj + 0.0c j j l Xj 0.0c j + + + The maxmum lelhood estmate of s obtaed by solvg the lelhood equato Xj + 0.0c l Xj + 0.0c j Xj + 0.0c Xj + 0.0c j j X + 0.0c X + 0.0c l X + 0.0c j j j j j Xj + 0.0c Xj + 0.0c j j l Xj 0.0c 0 j + + + + (8) (9) j j + 0.0 Xj c x Xj 0.0c. j + For a fxed, the MLE s for μ ad σ are ad ˆ X μ j j + 0.0c 0.0 0.0 ˆ X + j c Xj + c. j j σ (5) (6) Sce appears o the expoet of the observatos, t s cosdered to be too complcated for solvg t. The maxmzed log lelhood fucto s a umodal fucto so the value of the trasformato parameter s obtaed whe the slope of the curvature of the maxmzed log lelhood fucto s early zero []. Hece we ca also use the umercal method such as bsecto for fdg the sutable value of. IV. AN EXAMPLE For the purpose of llustratg the examples, oly three Webull populatos, each of sze 4,000, are geerated wth shape parameters ad scale parameters as follows. The shape parameter of the frst populato s 0.5 ad the scale parameter s,000. Wth the secod populato, the shape parameter s
0.8 ad the scale parameter s,500. The shape parameter of the thrd populato s 0. ad the scale parameter s,800. Supposg that three radom samples of sze 0 are tae from each Webull populato, the sample data are show Table I. TABLE I THREE RANDOM SAMPLES OF SIZE 0 TAKEN FROM EACH OF THREE WEIBULL POPULATIONS Sample Sample Sample 579.6908 84.7686 640.84 4.74 40.976 48.47 0.0458 59.66 4874.669.009 78.005.674 8.7890 989.694 5990.77 8575 4.699 00.874 085.7657 0786.7084 4.9.05 489.4 7.868 54.7 655.945 558.407 4.876 4.0990 759.756 8.645 45.787 678.46 755.5777 9445.876 657 776.64 57.488 50.484 6.0706 47.96 48.7 609.549 477.80 400.6656 45.846 74.967 4960.0986 84.5990 075.55 59.48 00.64 769.5 976.0678 4849.4784 55.8477 6995.885 54.7 890.889 886.0844 The Normal P-P plot for each sample s preseted Fg. -. The results show that each sample of data s o-ormal..00.5.5 Fg. Normal P-P plot of data from Sample The Webull P-P plot for each sample s preseted Fg. 4-6. The results show that each sample of data s Webull..00.5.5 Fg. 4 Webull P-P plot of data from Sample.00.00.00.00.5.5.5.00 Fg. 5 Webull P-P plot of data from Sample.5.00 Fg. Normal P-P plot of data from Sample.00.00.5.5.5.00 Fg. 6 Webull P-P plot of data from Sample.5.00 Fg. Normal P-P plot of data from Sample The value of trasformato parameter s -0.0698 by the bsecto method.
Hece, the trasformato for ths Webull data set s -0.0698 Xj + 0.0c Y j. (0) -0.0698 The trasformed data are show Table II. TABLE II THE TRANSFORMED DATA Sample Sample Sample 6.05 4.4056 6.6085 5.096 6.899 5.408 4.805 5.6805 6.477 4 6.08 5.794 4.566 5.96 6.575 5.98 4.7986 5.747 5.998 6.856 6.880 7 5.6 5.78 4.6584 6.0969 6.540 4.550 5.0994 5.978 4.680 5.776 6.5988 5.4758 6.784 5.57 6.444 5.984 5.90 5.79 6.0495 6.0 6.5546 5.7697 7.59 4.8497 4.66 7.604 6.57 5.594 5.94 5.079 6.645 6.06 6.4406 5.77 7.097 4.6584 5.9078 6.7707 I geeral, the usual basc assumptos, ormalty each group of data ad homogeety of varaces, should be valdated before ANOVA s appled ad so these assumptos should be tested. The ormal P-P plot for each sample of trasformed data s preseted Fg 7-9. The results show that each sample of trasformed data s ormal..00.5.5 Fg. 7 Normal P-P plot of trasformed data from Sample.00.00.5.5 Fg. 8 Normal P-P plot of trasformed data from Sample.00.5.5 Fg. 9 Normal P-P plot of trasformed data from Sample The Levee statstc F * L of trasformed data s.57. For sgfcace level α 0.05, F0.05,,57.5. Sce * FL.57 <.5, They have a costat varace. The ANOVA assumptos of the trasformed data are checed ad are vald. Subsequetly the trasformed data are used to test the equalty of the populato meas usg ANOVA. The results are show Table III. TABLE III ANOVA TABLE FOR H0 : μ μ μ Source of Varato df Sum of Squares Mea Square F-rato Betwee treatmet 5.80.95 5.48 Wth treatmet 57 0.684 0.58 Total 59 6.54 The F test statstc, F 5.48, ad F0.05,,57.5. Sce F 5.48 >.5, there s a sgfcat dfferece at least oe par amog the three populato meas..00.00 V. A NUMERICAL STUDY I order to atta the most effectve use of the proposed trasformato, we set the values of parameters ad the sgfcat value as follows: ) umber of the populatos, ) sample sze from the th Webull populato 0, 0, 0, 50, ) β scale parameter of the th Webull populato
s betwee 000 ad 4000, 4) α shape parameter of the th Webull populato s betwee ad.5, 5) Sgfcat level 0.05. The Webull populatos of sze N 4,000 (,,) are geerated for dfferet values of parameters α, β show Table IV. TABLE IV THE VALUES OF PARAMETERS α AND β Values of Parameters α ad β α.5, α.5, α.5, β 000, β 000, β 000 α.5, α.5, α.5, β 000, β 500, β 000 α.5, α.5, α.5, β 000, β 000, β 000 4 α.5, α.5, α.5, β 000, β 000, β 4000 5 α.0, α., α.5, β 000, β 000, β 000 6 α.0, α., α.5, β 000, β 500, β 000 7 α.0, α., α.5, β 000, β 000, β 000 8 α.0, α., α.5, β 000, β 000, β 4000 From a Webull( α, β ),,000 radom samples, each of sze, are draw. The we trasform each set of the sample data to ormalty by the proposed trasformato. The dffereces amog the populato meas are measured by the coeffcet of varato (C.V.) show Table V. TABLE V THE COEFFICIENT OF VARIATION AMONG THE POPULATION MEANS μ μ μ C.V.(%)) 90.745 90.745 90.745 90.745 54.79 805.4906. 90.745 805.4906 708.59 5 4 90.745 805.4906 60.98 64.47 5 0000 940.6559 90.740 5.7 6 0000 40.984 805.4906 8.66 7 0000 88.7 708.59 45.85 8 0000 88.7 60.98 6.8 A. Chec Valdty of Assumpto The results of the goodess- of-ft tests ad the tests of homogeety of varaces wth,000 replcated samples of varous szes are show Table VI to Table IX. TABLE VI BY THE ALTERNATIVE TRANSFORMATION WITH 0 of Trasformed Data TABLE VII BY THE ALTERNATIVE TRANSFORMATION WITH 0 TABLE VIII BY THE ALTERNATIVE TRANSFORMATION WITH 50 Levee Test 0.8595 0.86797 0.8448 0.58 0.85 0.80454 0.8098 0059 0.80749 0.870 0.875 05986 4 0.88064 0.80966 0.878 0.5758 5 0.87 0.840045 0.809 0.46488 6 0.89 0.840 0.84 0.48794 7 0.84745 0.8669 0.8069 0.57945 8 0.8698 0.887 0.85086 0.5844 of Trasformed Data Levee Test 0.7087 0.7087 0.76705 0.496945 0.684 0.6486 0.6559 0.7486 0.657 0.677 0.57767 0.86 4 0.554 0.500 0.590 0.454 5 0.76650 085 0.68876 0.8976 6 0.744558 0.7747 0.6979 0.40884 7 0.74598 0.684544 0.566 0.56666 8 0.74099 0.70077 0.59849 0.5887 of Trasformed Data Levee Test 0.596 0.5469 0.6408 0.565 0.5705 0.458770 0497 0.8900 0.487750 0.468 0.4698 0.74 4 0.4499 0.097 0.665 0.96 5 0.658 0.66944 0.499 0.0807 6 0.67854 0.686797 0.406 0.77 7 0.64887 054 0.46788 0.5450 8 0.608 0.569 0.5506 07
TABLE IX BY THE ALTERNATIVE TRANSFORMATION WITH 0, 0, 0 We have see that, all sets of the Webull data trasformed by the alteratve trasformato ca be checed by the K-S test ad for homogeety of varaces by the Levee test. Furthermore, they always meet all the requred assumptos for ANOVA. B. Powers of the ANOVA Test We trasform each set of the sample data to ormalty ad homogeety of varaces by proposed alteratve trasformato. The the trasformed data sets are used to test the equalty of the populato meas by ANOVA. The power of the F-test as obtaed from ANOVA gve by Pata [0] s β( μ,..., μ ) ( ) pf df Fα ( μ μ) t ( ) σ μ μ e σ. t 0 tb! ( ) + t, ( ) ( ) + t ( ) t ( ) + t ( ) ( ) F + F ( ) ( ) F α df () where μ yj, μ yj, ad σ ( yj μ ). j of Trasformed Data j j Levee Test 0.85954 0.796768 0.7858 0.49555 0.89 0.779097 0.70564 0.4498 0.89868 0.767 0.6489 0.85555 4 0.80668 0.695800 0.56457 0.686 5 0.8588 0.80746 0.670 0.5649 6 0.80560 0.8099 0.647 0.4894 7 0.80795 0.7809 0.6750 0.5474 8 0.8850 0.7786 0.6646 0.5056 The results of the power of the ANOVA tests wth,000 replcated samples of varous szes are show Table X. TABLE X POWERS OF THE ANOVA TESTS OF EQUALITY OF MEANS USING TRANSFORMED DATA Power of the ANOVA Test 0, 0 0 50 0, 0 0.047946 0.0500 0.0506 0.04878 0.8658 0.6698 0.8644 0.9087 0.58486 0.9747 0.979087 0.670549 4 0.76907 0.980 0.999780 0.86675 5 0.0655 0.06798 0.077095 0.07057 6 0.47068 0.89 0.98057 0.659 7 0.66 0.89947 0.96976 0.70698 8 0.86969 0.98594 0.998056 0.87996 We see that the power of the ANOVA test creases as creases. Furthermore, whe the dffereces amog the populato meas are larger, hgher powers of the tests are obtaed. VI. CONCLUSION The alteratve trasformato as proposed ths paper s appled to trasform Webull data to Normal data wth costat varace. The umercal results dcated that the Webull data sets trasformed by the alteratve trasformato always meet the assumptos requred applcato of ANOVA. The power of the test depeds o the sample szes, ad also o the shape ad scale parameters of the populatos. REFERENCES [] W. Tuey, O the comparatve aatomy of trasformatos, Aals of Mathematcal Statstcs, vol., 957, pp. 55-540. [] D. C. Motgomery, Desg ad Aalyss of Expermets, 5th ed. New Yor: Wley, 00, pp. 590. [] G. E. P. Box ad D. R. Cox, A aalyss of trasformatos (wth dscusso), Joural of the Royal Statstcal Socety, Ser.B. vol. 6, 964, pp.-5. [4] J. Schlesselma, Power Famles: A Note o the Box ad Cox Trasformato, Joural of the Royal Statstcal Socety, Ser. B. vol., 97, pp.07-. [5] B. F. J. Maly, Expoetal Data Trasformatos, Statstca. vol. 5, 976, pp.7-4. [6] J. A. Joh ad N. R. Draper, A alteratve famly of trasformatos, Appled Statstcs, vol. 9(), 980, pp.90-97. [7] K. A. Dosum, ad C. Wog, Statstcal tests based o trasformed data, Joural of the Amerca Statstcal Assocato, vol. 78, 98, pp. 4-47. [8] N. L. Johso, ad S. Kotz, Cotuous Uvarate Dstrbutos. New Yor: Wley, 970. [9] A. Chaudhur, ad N. K. Chadra, A Test for Webull Populatos, Statstcs & Probablty Letters, vol. 7, 989, pp. 77-80. [0] P. B. Pata, The o-cetral χ ad F-dstrbutos ad ther Applcatos, Bometra, vol. 6(), 949, pp. 0-.