Proceedgs of the World Cogress o Egeerg 4 Vol I, WCE 4, July - 4, 4, Lodo, U.K. A New Famly of Trasformatos for Lfetme Data Lakhaa Watthaacheewakul Abstract A famly of trasformatos s the oe of several methods to aalyze the data that do ot correspod wth the assumpto. A well-kow famly of trasformatos ofte used may studes was proposed by Box ad Cox. However, Box-Cox trasformato s ot always applcable. It should be used wth cauto some cases such as falure tme ad survval data. The smple case, some observatos the set of falure tme data may be zero but the value of observato the codto of Box-Cox trasformato s greater tha zero. I ths case, Maly trasformato may be approprated tha Box-Cox trasformato because t was proposed as a famly of expoetal trasformatos that egatve x values are also allowed. I ths paper, a ew famly of trasformato s proposed to maage wth the problem as metoed ad Maly trasformato were compared the lfetme data those have expoetal gamma ad webull dstrbuto. They were vestgated for some sets of the lfetme data. It s foud that the proposed trasformato ad Maly trasformato have ot dfferet effcecy sese of ormalty. The proposed trasformato performs better tha Maly trasformato sese of homogeety of varaces for some data set of webull dstrbutos ad expoetal dstrbutos whe the sample szes are large. Idex Terms Maly trasformato, proposed trasformato, homogeety of varaces, lfetme data, ormalty I I. INTRODUCTION N statstcal data aalyss, may statstcal procedures requre data to be approxmately ormal. If the data are ot ormally dstrbuted, a trasformato that trasforms the data set to acheve ormalty s used. Tukey [] suggested that whe aalyzg data that do ot match the assumptos of a covetoal method of aalyss, there are two choces; trasform the data to ft the assumptos or develop some ew robust methods of aalyss. Motgomery Mauscrpt receved February 4, 4; revsed March 7, 4. Ths work was supported part by the Faculty of Scece, Maejo Uversty, Chag Ma, Thalad. L. Watthaacheewakul s wth the Faculty of Scece, Maejo Uversty, Chag Ma, Thalad (phoe: 66--87-; fax: 66--87-; e-mal: lakhaaw@yahoo.com; lakhaa@mju.ac.th). [] suggested that trasformatos are used for three purposes; stablzg respose varace, makg the dstrbuto of the respose varable closer to a ormal dstrbuto ad mprovg the ft of the model to the data. There are several alteratves for trasformg such as trasformatos based o the relatoshp betwee the stadard devato ad the mea. Furthermore, t s possble to trasform the data usg a famly of trasformatos already extesvely studed over a log perod of tme, e.g. Box ad Cox [], Maly [4], ad Joh ad Draper []. A well-kow famly of trasformatos ofte used prevous studes was proposed by Box ad Cox. Doksum ad Wog [6] dcated that the Box-Cox trasformato should be used wth cauto some cases such as falure tme ad survval data. Joh ad Draper [] showed that the Box-Cox trasformato was ot satsfactory eve whe the best value of trasformato parameter had bee chose. II. A FAMILY OF TRANSFORMATIONS A famly of trasformatos appled over a log perod ca be used for data from ay populato so that the trasformed data are ormally dstrbuted. Let X be a radom varable dstrbuted as o-ormal,y the trasformed varable of X, x the value of X, c the rage of data set ad a trasformato parameter. Box ad Cox [] gave a smple modfed form of the power trasformato to avod dscotuty at =. They cosdered X, Y = l X, = for. () Ths has become well kow as Box-Cox trasformato. Maly [4] suggested a oe parameter famly of expoetal trasformatos exp( X ), Y = X, =. x> Ths s a useful alteratve to Box-Cox trasformatos because egatve x values are also allowed. It has bee () ISBN: 978-988-9-7- ISSN: 78-98 (Prt); ISSN: 78-966 (Ole) WCE 4
Proceedgs of the World Cogress o Egeerg 4 Vol I, WCE 4, July - 4, 4, Lodo, U.K. foud partcular that ths trasformato s qute effectve at turg skew umodal dstrbutos to early symmetrc ormal dstrbutos. Yeo ad Johso [8] proposed a famly of modfed Box ad Cox trasformato [ X + ], x, l[ X + ], x, = Y = () [ X + ], x <, l[ X + ], x<, = I ths paper, the alteratve famly of trasformatos for lfetme data s proposed ths form X +, x, Y = l + X, x, =. III. LIFETIME DATA Lfetme data are mportat relablty aalyss ad survval aalyss. It s ofte of terest to estmate the relablty of the system/compoet from the observed lfetme data. Webull Expoetal ad Gamma dstrbutos are volved lfetme data. The Webull dstrbuto s a atural startg pot the modelg of falure tmes relablty, materal stregth data ad may other applcatos. The probablty desty fucto of a two parameter Webull radom varable X s (4) x x β, ;, > ( ) = e x β f x β β (), x< where s the shape parameter ad β s the scale parameter. It s related to the other probablty dstrbuto such as the Expoetal dstrbuto whe =. The probablty desty fucto of oe parameter Expoetal radom varable X s x β e, x ; β > f( x) = β (6), x< where β s the scale parameter. Gamma dstrbuto s the commo choces of fralty dstrbuto lfetme data models. x β x e, x ; β > f( x) = β Γ( ) (7), x< where s the shape parameter ad β s the scale parameter. IV. ESTIMATION OF THE TRANSFORMATION PARAMETER For several groups of data, the value of () ad () eed to be foud so that the trasformed varables wll be depedetly ormal dstrbuto wth homogeety of varaces. The probablty desty fucto of each Y j s the form f( yj μ, σ ) = exp ( ) y j μ, (8) ( πσ ) σ where μ s the mea of the th trasformed populato data, σ the pooled varace of all trasformed populato data ad y j the observed value of Y j. For (), the lkelhood fucto relato to the observatos x j s gve by L ( μ, σ, xj) = k exp( ) exp. ( ; ) xj μ J yx ( πσ ) σ = j= where J( y; x) = for μ ad σ are k y j x = j= j exp( ) ˆ = xj μ j =. For a fxed, the MLE s ad k exp( ) exp( ) x j xj ˆ σ = = j= j= Substtute ˆ μ ad ˆ σ to the lkelhood equato (9). Thus for fxed, the maxmzed log lkelhood s l L( x ) = j k exp( xj ) exp( xj ) l π l = j= j= k - + xj, = j= () except for a costat, the maxmum lkelhood estmate of s obtaed by solvg the lkelhood equato d l L( ) = d k k x j xj xj e xj e e xj = j= = j= j= k k x j xj e e = j= = j= k + + xj =. = j= () Smlar procedures yeld the same results for (4), the maxmum lkelhood estmate of s obtaed by solvg the lkelhood equato (9) ISBN: 978-988-9-7- ISSN: 78-98 (Prt); ISSN: 78-966 (Ole) WCE 4
Proceedgs of the World Cogress o Egeerg 4 Vol I, WCE 4, July - 4, 4, Lodo, U.K. d l L( ) = d k x = j= k = j = + + k = j= k = j= () Sce appears o the expoet of the observatos, t s cosdered to be too complcated for solvg t. The maxmzed log lkelhood fucto s a umodal fucto so the value of the trasformato parameterr s obtaed whe the slope of the curvature of the maxmzed log lkelhood fucto s early zero []. Hece we ca also use the umercal method such as bsecto for fdg the sutable value of. I order to atta the most effectvee use of the two trasformatos, we set the values of parameters ad the sgfcat value as follows: k = umber of the populatos =, = sample sze from the th populato s betwee ad 8, β = scale parameterr of the th Webull Expoetal ad Gamma populatos s betwee ad, = shape parameter of the th Webull ad Gamma populato s betwee ad 4, the sgfcat level =.. The graph of Webull Expoetal ad Gamma dstrbutos are show Fgure 7. ( j + ) ( xj + ) ( xj + ) ( l x + l x + j ( j ) ) j= V. SIMULATION STUDY k ( = j= =. x + l x + j ( ) ( ) j xj + ) Fg.. Graph of Webull dstrbutos whe shape parameters ad scale parameters are dfferet....8 =,, β=.6 =, β= =.4 =4,β=. 4 =,, β= =, β= = 4, β=... Fg.. Graph of Webull dstrbutos whe shape parameters are dfferet ad scale parameters are the same. 4 β= β= β= = 4.8.6.4. =, β= =,β= =,β= Fg. 4. Graph of Expoetal dstrbutos whe scale parameters are dfferet..4 4 Fg.. Graph of Webull dstrbutos whe shape parameters are the same ad scale parameters are dfferet.. =, β= =,β= =,β= Fg.. Graph of Gamma dstrbutos whe shape parameters are the same ad scale parameters are dfferet. ISBN: 978-988-9-7- ISSN: 78-98 (Prt); ISSN: 78-966 (Ole) WCE 4
Proceedgs of the World Cogress o Egeerg 4 Vol I, WCE 4, July - 4, 4, Lodo, U.K..4. Fg. 6. Graph of Gamma dstrbutos whe shape parameters ad scale parameters are dfferet..4. Fg. 7. Graph of Gamma dstrbutos whe shape parameters are dfferet ad scale parameters are the same. As a umercal study, Webull, Expoetal ad Gamma populatos of sze N =, ( =,,) are geerated for dfferet values of parameters β,. The, radom samples, each of sze, are draw. Each set of the sample data was trasformed to ormalty by the proposed trasformato ad Maly trasformato. The results of the goodess-of-ft tests sese of ormalty wth, replcated samples of varous szes are show Table I III for Webull data. Smlarly, the results are show Table IV for Expoetal data ad the results are show Table V-VII for Gamma data. TABLE I USING DATA TRANSFORMED BY THE TWO TRANSFORMATIONS WITH WEIBULL DATA WHEN =, β =, β =, β = Trasformatos Maly Maly Maly Maly =, β= =, β= =, β= =, β= = 4, β= =4,β= Averages of the p-values for of Trasformed Data 8 8,,,,.886.84.7487.7.69.94.88.87.884.8.69.9.48.44.7699.699.888.87.869.86.94.6.687.644 TABLE II AVERAGES A OF THE P-VALUES FOR K-S TEST OF NORMALITY USING DATA TRANSFORMED BY THE TWO TRANSFORMATIONS WITH WEIBULLL DATA WHEN =, =, = 4, β =, β =, β = Trasformatos Averages of the p-values for of Trasformed Data Maly.84.84.879.89.896.897 Maly.77.7..779.696.8 Maly 8.7.48.77 8.76.88.866 Maly,,.86.797.67,,.847.7897.49 TABLE III N USING U DATA TRANSFORMED BY THE TWO TRANSFORMATIONS WITH WEIBULL DATA Trasformatos Maly Maly Maly Maly TABLE IV N USING U DATA TRANSFORMED BY THE TWO TRANSFORMATIONS WITH EXPONENTIAL DATA WHEN β =, β =, β = Trasformatos Maly Maly Maly Maly A WHEN =, =, = 4, β = 8 8,,,, 8 8,,,, Averages of the p-values for of Trasformed Data.8.84.78.8.844.77.79.7.7.774.68.466.474..49.6.84.898.787.779.64.847.747.94 Averages of the p-values for of Trasformed Data.7.74.8.894.799.88.7.478.688.749.6644.74.9.88.48.649.48.479.79.688.68.879.794.79 ISBN: 978-988-9-7- ISSN: 78-98 (Prt); ISSN: 78-966 (Ole) WCE 4
Proceedgs of the World Cogress o Egeerg 4 Vol I, WCE 4, July - 4, 4, Lodo, U.K. TABLE V USING DATA TRANSFORMED BY THE TWO TRANSFORMATIONS WITH GAMMA DATA WHEN =, β =, β =, β = Trasformatos From Table I to VII, we see that the results from both of two trasformatos the averages of the p-value of K-S test are small dfferet each stuato. Moreover, the averages of the p-value of K-S test decrease as the sample szes crease. TABLE VI USING DATA TRANSFORMED BY THE TWO TRANSFORMATIONS WITH GAMMA =, =, = 4, β =, β =, β = DATA WHEN TABLE VII USING DATA TRANSFORMED BY THE TWO TRANSFORMATIONS WITH =, =, = 4, β = GAMMA DATA WHEN Averages of the p-values for of Trasformed Data Maly.76.7777.799.7688.78.794 Maly.8.6.69.94.648.667 Maly 8.9.8.498 8..74.446 Maly,,.784.78.66,,.784.799.668 Trasformatos Averages of the p-values for of Trasformed Data Maly.77.777.7767.774.776.78 Maly.88.66.6.89.9.68 Maly 8.468.98.46 8.8..44 Maly,,.774.687.8,,.78.68.88 Trasformatos Averages of the p-values for of Trasformed Data Maly.77.7749.7844.7776.778.784 Maly.976.6.698.648.679.647 Maly 8.7.6.69 8.88.68.746 Maly,,.79.7.647,,.797.7.6 For the check of valdty sese of homogeety of varace, the results of the Levee test wth, replcated samples of varous szes ad data are show Table VIII. TABLE VIII AVERAGES OF THE P-VALUES FOR LEVENE TEST USING DATA TRANSFORMED BY THE TWO TRANSFORMATIONS Data Maly Webull (Case I).47.487 =.44.4 β =, β =, β = 8.8.7,,.944.999 Webull (Case II).74.6 =, =, = 4.44. β =, β =, β = 8.84.67,,.99.87 Webull (Case III).9.7 =, =, = 4.9.68 β = 8..,,.6.47 Expoetal (Case IV).4.49 β =, β =, β =.89.498 8..4,,.4.87 Gamma (Case V).66.697 =.96.664 β =, β =, β = 8.7.976,,.94.669 Gamma (Case VI).67.676 =, =, = 4.48.9 β =, β =, β = 8..94,,.849.688 Gamma (Case VII).6.7 =, =, = 4.6696.678 β = 8.674.67,,.66.667 From Table VIII, for Case I to VII, we see that averages of the p-value of Levee test of proposed trasformato are hgher tha them of Maly trasformato each of sample szes. I case I ad IV whe the sample szes are large, proposed trasformato performs better tha Maly trasformato at sgfcat level.. For Case III, we see that both proposed trasformato ad Maly trasformato work well wth oly the small sample sze. Moreover, the averages of the p-value of Levee test decrease as the sample szes crease. VI. CONCLUSION The effcecy of the proposed trasformato s compared wth Maly trasformato sese of ormalty ad homogeety of varace. Both of them ca trasform the lfetme data to correspod wth the basc assumptos some stuato. I sese of ormalty, t s foud that the proposed trasformato ad Maly trasformato have ot dfferet effcecy. The proposed trasformato performs better tha Maly trasformato sese of homogeety of varaces for some data set of webull dstrbutos ad expoetal dstrbutos whe the sample szes are large. ISBN: 978-988-9-7- ISSN: 78-98 (Prt); ISSN: 78-966 (Ole) WCE 4
Proceedgs of the World Cogress o Egeerg 4 Vol I, WCE 4, July - 4, 4, Lodo, U.K. REFERENCES [] W. Tukey, O the comparatve aatomy of trasformatos, Aals of Mathematcal Statstcs, vol. 8, o., pp. -4, Sep. 97. [] D. C. Motgomery, Desg ad Aalyss of Expermets, th ed. New York: Wley,, pp. 9. [] G. E. P. Box ad D. R. Cox, A aalyss of trasformatos (wth dscusso), Joural of the Royal Statstcal Socety, Ser.B. vol. 6, o., pp.-, Apr. 964. [4] B. F. J. Maly, Expoetal Data Trasformatos, Statstca. vol., o., pp.7-4, Mar. 976. [] J. A. Joh ad N. R. Draper, A alteratve famly of trasformatos, Appled Statstcs, vol. 9, o., pp.9-97, 98. [6] K. A. Doksum, ad C. Wog, Statstcal tests based o trasformed data, Joural of the Amerca Statstcal Assocato, vol. 78, o. 8, pp. 4-47, Ju. 98. [7] N. L. Johso, S. Kotz, ad N.Balakrsha. Cotuous Uvarate Dstrbutos, d ed. vol.. New York: Wley, 994. [8] I. Yeo ad N. R. Johso, A ew famly of power trasformatos to mprove ormalty or symmetry, Bometrka, vol. 87, o., pp.94-99,. ISBN: 978-988-9-7- ISSN: 78-98 (Prt); ISSN: 78-966 (Ole) WCE 4