MINIMUM VARIANCE STRATIFICATION FOR COMPROMISE ALLOCATION

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1 MINIMUM VARIANCE STRATIFICATION FOR COMPROMISE AOCATION Med Ram VERMA Jadis Prasad Sig JOORE Raeev Kumar AGNIHOTRI ABSTRACT: I tis paper we as cosidered te problem of optimum stratificatio for two sesitive quatitative variables we te data from differet strata ave bee collected by scrambled respose tecique ad a auxiliary variable ave bee used as a stratificatio variable. I tis paper we ave obtaied te limitig expressio for te trace of geeralized variace covariace matrix, expressio for umber of strata ad expressio for approximate sample size [ ]. Te paper cocluded wit a umerical illustratio. KEYWORDS: Sesitive variable; scrambled respose; optimum stratificatio; AOSB (Approximately Optimum Strata Boudaries). Itroductio I stratified samplig te mai aim is to get estimators of te populatio parameters for te caracter uder study wit maximum precisio at miimum cost. Te precisio of a estimator of populatio mea/total depeds ot oly o te sample size ad samplig fractio but also o te variability or eterogeeity amog te uits of te populatio. Apart from icreasig te sample size oe possible way to estimate te populatio mea wit maximum precisio is to divide te populatio ito certai umber of groups, wic are more omogeeous witi, ad te selectig samples from eac of te group idepedetly. Tese groups are called strata ad te wole procedure is called stratified samplig. Stratificatio i sample survey is well kow device to icrease te precisio of te estimators. I stratified samplig efficiecy of te estimator of populatio parameters depeds o several factors suc as coice of stratificatio variable, umber of strata, determiatio of strata boudaries ad allocatio of sample sizes to te differet strata. Oce it is decided about te total umber of strata ad te procedure of allocatig sample sizes to differet strata, te problem of optimum stratificatio may be cosidered to cosist of determiatio of optimum strata boudaries. Idia Veteriary Researc Istitute, Divisio of ivestock Ecoomics, Statistics ad Iformatio Tecology, Izatagar (Bareilly) U.P., Idia. address: mrverma9@yaoo.co.i / medramverma@rediffmail.com Uiversity of Jammu, Departmet of Statistics, Jammu (J&K), Pi 80006, Idia. St. Jo s College Agra (U.P.), Departmet of Statistics, Pi 800, Idia. 78 Rev. Bras. Biom., São Paulo, v.0,., p.78-9, 0

2 Te pioeerig work i tis field was doe by Daleius (950), Daleius ad Gurey (95), Daleius ad Hodges (957), Sig ad Sukatme (969) ad several oter researc workers also cosidered te problem of optimum stratificatio wit respect to a auxiliary variable closely related to te study variable. Gos (96) cosidered te problem of optimum stratificatio wit two caracters uder proportioal metod of allocatio assumig stratificatio variable as idetical to te estimatio variable uder cosideratio. It is urealistic to assume te distributio of study variable is kow i advace. Rizvi et al. (000) cosidered te optimum stratificatio for two caracters usig proportioal metod of allocatio by takig a auxiliary variable as stratificatio variable. Rizvi et al. (00) cosidered te case of optimum stratificatio for two study variables i case of compromise metod of allocatio. Te radomized respose tecique is used to procure trustworty data for estimatig te proportio of people wit a sesitive caracteristic. Several researc workers exteded te tecique sice its itroductio by Warer (965). Greeberg et al. (97) exteded te metod to te case we te resposes to te sesitive questios were quatitative rater ta qualitative. Eicor ad Hayre (98) itroduced a scrambled radomized respose tecique, wic does ot cotai te difficulties of Greeberg et al. (97) urelated questio metod. Te scrambled radomized respose tecique ivolves te respodet multiplyig is sesitive aswer Y by a radom umber S from te kow distributio ad givig te scrambled respose Z= Y.S to te iterviewer wo does ot kow te particular values of te radom umber S. Maaa et al. (994) developed te teory for determiatio of optimum strata boudaries for a sesitive variable usig scrambled respose tecique. Verma et al. (007) cosidered te problem of optimum stratificatio for two sesitive variables usig equal allocatio metod. I te preset paper we ave cosidered te problem of optimum stratificatio we two sesitive variables are preset i te survey ad data is collected by scrambled radomized respose tecique for te compromise metod of allocatio ad auxiliary variable is take as stratificatio variable. Tis strategy is importat we we are dealig wit stigmatized quatitative variables. For istace let Y be te Icome uderstated i icome tax retur ad Y be te Expediture. Tese two variables ca be stratified by usig a auxiliary variable X (Eye estimated value of property) as te stratificatio variable. For teoretical developmet, let us assume tat tere be a populatio of size N wic is divided ito strata of N, N,, N uits respectively so tat N = N ( =,, =, ). For drawig a stratified SRSWR (Simple Radom Samplig Wit Replacemet) sample of size, te sample of sizes,,, are to be draw from respective stratum so tat = =. et Y ( =, )be two sesitive quatitative variables. et Y deote te value of sesitive variable Y for te -t stratum. Suppose S be scramblig radom variable idepedet of Y ad wit fiite mea ad variace. Te respodet geerates S usig some specified metods ad multiplies te sesitive variable value Y by S. Te Rev. Bras. Biom., São Paulo, v.0,., p.78-9, 0 79

3 iterviewer tus receives te scrambled aswer Z = Y. S. Te particular values of S are ukow to te iterviewer but its distributio is kow. I tis way te privacy of te respodets is ot violated. et E( S ) = θ ad V ( S ) = γ ; E( Y ) = µ adv ( Y ) = ; y wereθ ad y γ are kow to te iterviewer but Y ad S are idepedet, we ave: If Z ( ). y µ y ad y are ukow. Sice E Z = µ θ. () ( ) ( ) V Z = θ + γ + µ. γ. () y y deote te value of scrambled respose for -t sesitive variable i te -t stratum ad samplig witi eac stratum is SRSWR, te ubiased estimator of µ is y ˆ µ y Z θ =, were: i i= Z = Z ; () Z i = Scrambled respose for te -t sesitive variable for i-t elemet i -t stratum. as: werew were Hece ubiased estimator of populatio mea N N st = y = y W ˆ µ ; Y uder scrambled respose is give = = Proportio of te elemets preset i te -t stratum. Variace of estimator C -t stratum. y st is give by: ( st ) = y ( + ) + ( µ y ) (4) = V y W C C γ = is te coefficiet of variatio of te -t scramblig variable S i θ 80 Rev. Bras. Biom., São Paulo, v.0,., p.78-9, 0

4 Allocatio i stratified samplig Optimum allocatio: Te guidig priciple i determiatio of is to coose tem i suc away so as to miimize te cost for a desired precisio or maximum precisio for a give cost, tus makig te most effective use of te resources available. Te allocatio of te sample to te differet strata accordace wit tis priciple is called te priciple of optimum allocatio. A special case of tis allocatio is te Neyma optimum allocatio were it is assumed tat te ivestigatig costs per elemet are te same for all te strata. Te umber of uits allocated to -t stratum for tis allocatio is give by: =. W S. W S = Tis metod provides te best allocatio a dissuitable i cases were te stratum variace differ muc. Compromise allocatio: I multivariate stratified samplig were more ta oe caracteristics are to be estimated, a allocatio wic is optimum for oe caracteristic may or may ot be optimum for oter caracteristics. I suc situatio a compromise is eeded to work out a usable allocatio, wic is optimum i some sese for all caracteristics. Suc a allocatio may be called as Compromise Allocatio because it is based o some compromise criterio. Te problem of allocatio to strata wit several caracteristics was first cosidered by Neyma (94). Sukatme et al. (984) reviewed te problem of allocatio wit several caracteristics as give by several researc workers. Tey ave sow umerically tat all te compromise allocatios, as compared by tem, are more efficiet ta proportioal allocatio. However te compromise allocatio based o te trace of te variace-covariace matrix is most efficiet. Hece we ave cosidered te case of compromise allocatio based o miimizatio of trace of variace-covariace matrix. I te -t stratum, te sample size are determied i suc a way so tat for give total sample size (wic amouts to fixed total cost were te cost per uit i eac stratum is same) V ( y st ) is miimized were V ( y st ) is te variace for -t sesitive = variable. If fiite populatio correctio factor ca be eglected te te variace expressio for -t sesitive variable is give by (5). We ave to miimize: ( st ) = y + + µ y V y W [ ( C ) C ]. (5) = ( ) V y st = V ( yst ) + V ( yst ). (6) = Rev. Bras. Biom., São Paulo, v.0,., p.78-9, 0 8

5 Now miimizig (6) subect to te coditio = is give by: were: ( ) = y y y W K + K y y W Ky + K y = K = + C + µ C ( =, ). = te optimum value of Usig tis value of we ave obtaied te variace expressio for compromise allocatio. Uder compromise metod of allocatio, te optimal variace of te estimated populatio mea of te sesitive variablesy is give: W K V y W K K = K y + K = y y ( st ) = y + y ( =, ). (8) ; (7) Variace uder super populatio model et us ow assume tat te populatio uder cosideratio is a radom sample from a ifiite super populatio wit same caracteristics. Furter we assume tat te study variables are liearly related wit te auxiliary variable X so tat te regressio of Y o X is give by te liear model: ( ) ; Y = c X + e (9) were: c ( X ) is a real valued fuctio of X, e E e X = E( e e X, X ) = 0 for x ' ( ) 0 ; is a error compoet suc tat: x ad V ( e X ) = φ > 0 for all x ( a, b) were ( b a) <. It may be oted tat E( e ( X ) c ( X )) = 0 but E( c ( X ) c ( X )) 0 ad E( e ( X ) e ( X )) 0. If te oit desity fuctio of ( X, Y, Y ) i te super populatio is fs ( x, y, y ) ad te margial desity fuctio of Xis f(x), te uder model (9) it ca be easily see tat: x x ( ) W = f x dx ; x y = c = x µ µ W c ( x) f ( x) dx ; 8 Rev. Bras. Biom., São Paulo, v.0,., p.78-9, 0

6 x c = W c x f x dx c x ( ) ( ) µ ; x cc = W c x c x f x dx c c x ( ) ( ) ( ) µ µ ; = + µ y c ϕ. were x, x ) are te boudaries of te -t stratum, ( te fuctio (x) variable. φ ad (x) Te variace expressio of model (9) are give by: µ φ is te expected value of φ is te coditioal variace fuctio of te -t sesitive y st W K for compromise allocatio uder super populatio = V y = W K + K = K c + K = c c ( st ) c ; (0) c were: W K = V y = W K + K = K c + K = c c ( st ) c ; c () K = ( + µ )( + C ) + µ C ( =, ). c c ϕ c We assumed tat stratificatio variable is cotiuous wit pdf f(x), a x b ad te poits of demarcatio formig strata are x, x,,x. et us deote te optimum poits of stratificatio as x } te correspodig to tese strata boudaries te { geeralized variace G, te determiat of variace covariace matrix, wic is a fuctio of poit of stratificatio is miimum. Now geeralized variace G is give by: = = G () It is cumbersome to obtai eve approximate solutio obtaied troug miimizatio of G uder compromise metod of allocatio; terefore, we ave cosidered te miimizatio of trace of variace covariace matrix for te purpose of obtaiig miimal equatios ad teir solutio. et us deote te trace of variace covariace matrix by tr (G) wic is give by: Rev. Bras. Biom., São Paulo, v.0,., p.78-9, 0 8

7 tr( G) = +. () Usig (0) ad () i () tr (G) ca be expressed as: c c () = tr( G) = W K + K Verma et al. (00) obtaied te solutio of te miimal equatios obtaied by miimizig te trace of variace covariace matrix (). Tey proposed followig cumulative cube root rule for obtaiig approximate optimum strata boudaries. Cumulative M ( x ) Rule: If te fuctio M ( x) = P( x) f ( x) is bouded ad its first two derivatives exists for all x i (a,b) wit ( b a) <, te for a give value of takig equal itervals o te cumulative cube root of M ( x ) will give approximately optimum strata boudaries (AOSB), were: * * ( φ x + φ x )( c x + c x ) + ( φ x + φ x ) * * ( φ ( x) + φ ( x) ) 4 ( ) ( ) ( ) ( ) ( ) ( ) P( x) = ; (4) ad: φ = φ + φ C + c C, * φ = φ ' + φ ' C + c c ' C, φ = φ + φ C + c C φ = φ ' + φ ' C + c c ' C ; *, wereφ is te first order derivative of * φ ad φ is te first order derivative of 4 imitig form of te trace of te variace covariace matrix * φ. For obtaiig te limitig expressio for te trace of variace covariace matrix tr( G ) as defied i (), we give te followig lemma for bivariate case, wic ca be proved by usig te series expasio of te various terms ivolved i it, exactly as for te uivariate case discussed i Sig ad Sukatme (969). For tis purpose we impose certai regularity coditios o te fuctios f ( x ), φ *( x) ad c ( x) as give below. A fuctio ω( x) belogs to te class of fuctios Ω if it satisfies te followig regularity coditios: (i) 0 < ω( x) < (ii) ω ( x), ω ( x) ad ω ( x) exist ad are cotiuous for all x ( a, b) ( b a) <., were 84 Rev. Bras. Biom., São Paulo, v.0,., p.78-9, 0

8 emma : Uder regularity coditios, for -t stratum we ave x x * * k c + c φ + φ = x 96 + x W K K ( x) ( x) f ( x) dx P( x) f ( x) dx[ O( k )]; were P( x) is defied i (4). Teorem : If te AOSB are obtaied by usig cumulative cube root rule M ( x) te limitig expressio for te trace of variace covariace matrix tr( G ) is give by: β tr( G) = α +. Were b * * [ ( x) ( x)] f ( x) dx ad a α = φ + φ b β = P( x) f ( x) dx ] 96 a Proof: Now makig use of te emma 4. i te expressio (), we ave: as: x b * * k tr( G) = [ φ ( x) + φ ( x)] f ( x) dx + P( x) f ( x) dx[ + O( k )]. a = 96 x (5) Now usig te result (.8) of Sig ad Sukatme (969), te equatio ca be put b x * * tr( G) = φ ( x) φ ( x) f ( x) dx P( x) f ( x) dx}. + + a 96 = x (6) Now if te strata boudaries are determied by makig use of cumulative cube root rule M ( x) te for =,,, we ave: x b (7) x Terefore, equatio (6) reduces to: P( x) f ( x) dx = P( x) f ( x) dx. a were: β tr( G) = α + ; (8) Rev. Bras. Biom., São Paulo, v.0,., p.78-9, 0 85

9 b b * * α = φ ( x) + φ ( x) f ( x) dx ad β = P( x) f ( x) dx] a 96. a Now takig limit as o bot sides of (8) we get: α im tr( G) =. (9) From te above relatio it may be cocluded tat wit a icrease i te umber of strata, te trace of geeralized variace covariace matrix decreases ad as te umber of strata becomes large eoug, tr( G ) teds to α. However if umber of strata goes to ifiity te te sample size goes more faster to ifiity, because we ave to select miimum oe uit from eac stratum ece tr( G) 0. But i geeral practice umber of strata are always fiite ece te equatio (8) is useful for determiig te trace of variace covariace matrix. 5 Optimum umber of strata Te trace of te variace- covariace matrix of te estimator y st as give i (8) as a approximately miimal value for te give umber of strata ad fixed total cost. Now to obtai approximately optimum stratificatio it remais to fid a optimum value for, te umber of strata to be costructed. Te variace (8) is oly te fuctio of as α ad ß are costats for a give populatio ad for te give auxiliary variable x. Now equatig to zero te partial derivative of te trace of te variace covariace matrix tr( G ) as give i (8) wit respect to we get. α + β = 0. (0) 6 Approximate expressio for sample size [ ] After te strata boudaries ave bee obtaied by cumulative M ( x) rule of Verma et al. (00) for te umber of strata satisfyig (0), te sample size [ ] allocated to te -t stratum is give by (7). Sice te fuctios f(x), C(x), ad φ(x) are kow a priori, te parameter W, µ, ad µ ca be evaluated ad te value c c ca be determied. Te total sample size is φ = =. It may sometime tedious to determie [ ] from (7) because of itegratios ivolved i it. We ow obtaied te approximate expressio for te sample size [ ]. 86 Rev. Bras. Biom., São Paulo, v.0,., p.78-9, 0

10 Teorem : If te AOSB are obtaied by usig cumulative cube root rule M ( x) te te approximate expressio for te sample size [ ] allocated to -t stratum is give by: * * k = ( φ ( x ) + φ ( x )) + P( x ) W. β 96 α + Proof: For obtaiig te approximate expressio for te sample size [ ] allocated to -t stratum we used emma 4.. Terefore, if te terms of uder stratum is give by: 4 O( m ) are eglected, te sample size i te -t x x * * k = φ ( x) + φ ( x) f ( x) dx + P( x) f ( x) dx ; β 96 α x x + Were: = Te () is approximately give by β W Kc + K. c α = + () * * k = ( φ ( x ) + φ ( x )) + P( x ) W. β 96 α + Were: x + x+ x =. * * ( φ x + φ x )( c x + c x ) + ( φ x + φ x ) * * ( φ ( x ) + φ ( x )) 4 ( ) ( ) ( ) ( ) ( ) ( ) P( x ) =. If optimum poits of stratificatio { x } are obtaied by usig te cumulative cube root rule M ( x) te te equatio () ca be used for determiatio of optimum sample size. Rev. Bras. Biom., São Paulo, v.0,., p.78-9, 0 87

11 7 Imperial stud To determie approximately optimum strata boudaries (AOSB) by te use of cumulative cube root rule we cosider tat stratificatio variable x follows followig probability desity fuctio: Uiform distributio f ( x) = x Rigt triagular distributio f ( x) = ( x) x Expoetial distributio f x e x x+ ( ) = <. Te rages of bot uiform ad rigt triagular distributios are fiite wereas rage of expoetial distributio is ifiite. We ave cosidered tat sesitive study variables Y are related wit te stratificatio variable x as Y = a + x + e, Y = a + x + e. Te coditioal variaces of te error terms i.e. V ( e / x ) adv ( e / x ) are to be assumed to g g be of te forms A x ad A x respectively were A, A > 0, g ad g beig costats. Here we ave take values of g = ad g =. Te values of A ad A were determied for te values g, g ad ρ, ρ by usig te followig formulae. Were ρ ad ρ are te correlatio coefficiets betwee te sesitive study variables Y ad Y wit stratificatio variable x. β ( ρ ) = ad A x g ρ E( x ) A β ( ρ ) ; x = g ρ E( x ) x is te variace of te stratificatio variable x. For te purpose of umerical illustratio we ave assumed ρ = 0.9, ρ = 0.7, C = 0. ad C = 0.. For fidig out te approximately optimum strata boudaries (AOSB), te rages of uiform, rigt triagular ad expoetial distributios were divided ito 0 classes of equal widt. Te Approximately optimum strata boudaries (AOSB) obtaied by te use of cumulative cube root rule ( ) M x as give i equatio (4) alog wit te relative efficiecy of stratificatio wit o stratificatio. Tese strata boudaries ca furter be used for determiatio of trace of variace- covariace matrix ad optimum sample sizes. From Table it is clearly evidet tat for te uiform distributio as te umber of strata icreases relative efficiecy icreases from 7.07% (=) to.% (=6). From Table we observed tat for te rigt triagular distributio relative efficiecy icreases from 9.45% (=) to 60.4% (=6). From Table it is idicated tat i case of expoetial distributio relative efficiecy icreases from 79.9% (=) to 45.7% (=6). Table to Table ca furter be used for fidig te geeralized variace ad te approximate sample sizes. 88 Rev. Bras. Biom., São Paulo, v.0,., p.78-9, 0

12 Table - AOSB for uiform distributio No. of Strata Approximately optimum strata boudaries (AOSB) tr( G ) Percet Relative Efficiecy Table - AOSB for Rigt triagular distributio No. of Strata Approximately optimum strata boudaries (AOSB) tr( G ) Percet Relative Efficiecy Table - AOSB for Expoetial distributio No. of Strata Approximately optimum strata boudaries (AOSB) tr( G ) Percet Relative Efficiecy Coclusio I te preset paper we ave cosidered te problem of optimum stratificatio we two sesitive variables are preset i te survey ad data are collected by scrambled radomized respose tecique for te compromise metod of allocatio ad a auxiliary variable is take as stratificatio variable. Tis strategy is importat we we are dealig wit stigmatized quatitative variables. For istace lety be te Icome uderstated i icome tax retur ad Y be te Expediture. Tese two variables ca be stratified by Rev. Bras. Biom., São Paulo, v.0,., p.78-9, 0 89

13 usig a auxiliary variable X (Eye estimated value of property) as te stratificatio variable. Te optimum poits of stratificatio ca be foud out. For example i our umerical example we ave cosidered tat te stratificatio variable followed uiform, rigt triagular ad expoetial distributios. Furter optimum strata boudaries ca be used to fid te geeralized variace ad approximate sample sizes. I geeral if te distributio of te auxiliary variable is kow ad iformatio o te two sesitive variables is collected by scrambled respose te i tat case preset metodology ca be used for fidig te geeralized variace ad approximate sample sizes. Ackowledgemet Te autors would like to tak two leared referees ad te Editor for teir valuable suggestio to brig te paper i te preset form. VERMA, M. R.; JOORE, J. P. S; AGNIHOTRI, R. K. Estratificação de variâcia míima para alocação de tarefas. Rev. Bras. Biom., São Paulo, v.0,., p.78-9, 0. RESUMO: Neste artigo estudou-se o problema da estratificação ótima para duas variáveis quatitativas sesitivas com dados obtidos a partir de diferetes camadas pela técica de resposta embaralada e uma variável auxiliar usada como variável de estratificação.foi obtida a expressão limitate para o traço da matriz variâcia covariâcia geeralizada, as expressões para o úmero de estratos e para o tamao aproximado da amostra.o artigo apreseta uma ilustração umérica. PAAVRAS-CHAVE: Variável sesitiva; respostas embaraladas; estratificação ótima; AOSB (imites Estratos Aproximadamete Ótimos). Refereces DAENIUS, T. Te problem of optimum stratificatio. Skad. Akt., Oslo, v., p.0-, 950. DAENIUS, T.; GURNEY, M. Te problem of optimum stratificatio II. Skad. Akt., Oslo, v.4, p.-48, 95. DAENIUS, T.; HODGES JR., J.. Miimum variace stratificatio. J. Am. Stat. Assoc., Baltimore, v.54,., p.88 0, 957. EICHHORN, B.H.; HAYRE,. S. Scrambled radomized respose metod for obtaiig sesitive quatitative data. J. Stat. Pla. Iferece, Amsterdam,v.7,.4, p.07-6, 98. GHOSH, S. P. Optimum stratificatio wit two caracters. A. Mat. Statist., Betesda, v.4,., p , 96. GREENBERGE, B. G.; KUBER, R. R.; ABERNATHY, J.R.; HORVITZ, D. G. Applicatios of radomized respose tecique i obtaiig quatitative data. J. Am. Stat. Assoc., New York, v.66,.4, p.4-50, 97. NEYMAN, J. O te two differet aspects of represetative metods: Te metod of stratified samplig ad metod of purposive selectio. J. R. Stat. Soc., Cicester, v.97,.4, p , Rev. Bras. Biom., São Paulo, v.0,., p.78-9, 0

14 MAHAJAN, P.K.; GUPTA, J.P.; SINGH, R. Determiatio of optimum strata boudaries for scrambled respose. Statistica, Rome, v.54,., p.75-8, 994. RIZVI, S. E. H.; GUPTA, J.P.; SINGH, R. Approximately optimum stratificatio for two study variables usig auxiliary iformatio. J. Idia Soc. Agric. Stat., New Deli, v.5,., p.87-98, 000. RIZVI, S.E. H.; GUPTA, J.P.; BHARGAVA, M. Optimum stratificatio based o auxiliary variable for compromise allocatio. Metro, Rome, v.50,.-4, p.0-5, 00. SINGH, R.; SUKHATME, B. V. Optimum stratificatio. A. Ist. Stat. Mat., Heidelberg, v., p.55-58, 969. SUKHATME, P. V.; SUKHATME, B.V.; SUKHATME, S.; ASOK, C. Samplig teory wit applicatios..ed. New Deli: Iowa, Idia Society of Agricultural Statistics, p. VERMA, M. R.; SINGH JOORE, J. P.; AGNIHOTRI, R. K. Approximately optimum stratificatio for two sesitive variables i case of compromise metod of allocatio. Sri aka J. Appl. Stat., Abigdo, v.4, p.-, 00. VERMA, M. R.; SINGH JOORE, J. P.; AGNIHOTRI, R. K. Optimum stratificatio for two sesitive quatitative variables usig equal allocatio metod. Braz. J. Mat. Stat., São Paulo, v.5,., p.7-85, 007. WARNER, S.. Radomized respose: a survey tecique for elimiatig evasive aswer bias. J. Am. Stat. Assoc., New York, v.60, p.6-69, 965. Received i Approved after revised i Rev. Bras. Biom., São Paulo, v.0,., p.78-9, 0 9