Using Principal Components in Multivariate Stratification Scheme

Transcription

1 Iteratoal Joural of Aled See ad Teolog Vol. o. ; Mar Usg Pral Comoets Multvarate Stratfato Seme Aataku Fadeke Sola Deartmet of Stats Federal Uverst of Agrulture PMB Abeokuta gera Abrat I samle surves more ta oe oulato araters are emated ad tese araters ma be of ofltg ature. Stratfed samlg as bee desged to esure tat all mortat vews are rereseted samles. I multvarate ratfed samle desg orrelato s osdered amog tere varables. A varato of oe varable wt lower orrelato s more mortat ta oters. Otmal alloato mult-tem s develoed as a multvarate otmzato roblem b fdg te ral omoets. A sear was made for a set of mutuall uorrelated varables Y Y... Y ea oe beg a lear ombato of te orgal set of varables X X... X. A emral ud from a ouseold surve oduted Abeokuta Sout ad Iebu ort loal govermet areas were used. Te data about te ouseolds are avalable for four araters or varables tat are related to te surve. Tese araters lude ouato ome umber of deedats ad te eduatoal level. Ea of te two loal govermet areas wt a samle sze of ouseolds ea were radoml seleted usg smle radom samlg teque makg a total of ouseolds. Te eads of te ouseolds were tervewed. Te ud adoted a aroa based o te fat tat ts metodolog s more real uder te ambt of multvarate aalss. Usg Slus software te varae- ovaraes matres were omuted. Te ral omoet aalss esured tat te varae-ovarae matrx was deomosed ad te egevalues ad egevetors alulated from te multvarate data reresetg formato from te ouseolds were omuted. Kewords: Pral omoets multvarate ratfato otmal desgs samlg alloato.. Itroduto Samlg metods are desged to rovde vald setf ad eoomal tools for resear roblems. Aordg to Ks (965 ad ut ad Trell ( samlg las a vtal role resear desg volvg uma oulato ad ommads reasg atteto from soal sets ems egeers aoutats bologs ad medal rattoers. Samlg roblems are equall materal to rattoers egaged marketg ommere dur ubl ealt boats eduato ubl admrato eooms soolog atroolog solog oltal See ad eve soal workers. Samlg metods are develoed as meas to a ed orgatg subatve resear roblems eseall te soal sees ad ter alatos (Ks 965; ut ad Trell. A workg kowledge of ratal samlg metods wt a uderadg of ter teoretal bakgroud eed to be a requremet for quattatvel oreted udets te soal sees as well as alled felds. It s elful altoug dffult to searate samlg desg from te related atvtes volved surve resear. Te samle desg overs te tasks of seleto ad emato for makg feree from samle value to te oulato value. Beod ts are te roblems of makg ferees from oe surve oulato to aoter ad geerall broader oulato wt measuremets free from error. Dfferet samlg desgs would result dfferet adard errors ad oosg te desg wt te smalle error s te ral am of samlg desg. A effetve samlg teque wt a oulato reresets a arorate extrato of useful data w rovdes meagful kowledge of te mortat asets of te oulato (ara ad Cortez 6. Probablt samles are usuall desged to be measurable tat s so desged tat atal feree to oulato values a be based o measures of varablt usuall adard errors omuted from te samle data. I geeral tere s eed to devse a samlg seme w s eoomal ad eas to oerate tat elds ubased emates ad mmzes te effets of samlg varato.

2 Ceter for Promotg Ideas USA Usuall samle surves more ta oe oulato araters are emated ad tese araters ma be of ofltg ature. Stratfed samlg as bee desged to esure tat all mortat vews are rereseted samles. Stratfato s a meas of samle desg b w te oulato of tere s dvded to grous alled rata aordg to some kow arater(s. Stratfed samle desgs are emloed for several reasos. Tese lude: to rease te reso of emates for te wole oulato for oe or more ke data tems beg olleted te surve; to obta more rese emates for tereg domas; to allow te use of dfferet samlg o-resose adumet edtg or emato metods for domas wt dfferg araters affetg te oe of metod ad to faltate admrato of te surve. Stratfed samlg s alwas more rertve ta smle radom samlg.. Te Multvarate Stratfato Seme Moreover te otext of ratfed samlg some multvarate aroaes ave bee roosed wereb te samle sze ad ts alloato wt rata take dverse araters to osderato (Sukatme et al. 98 ad Artaar ad Dodge 98. Te multvarate ratfed samle desg s dfferet wt two es from te uvarate ratfed samle desg. Te fr e s to dede rata usg ratfed varables w are multvarate. Te seod e s to dede a samle sze ad te otmal alloato tat s to dede a samle sze of ea ratum usg tere varables w are multvarate. Tere are ma metods for alloato to rata te multvarate ratfed samle desg. Te fr metod s te roortoal alloato ad te seod s te multvarate alloato usg oe tere varable w s seleted of multvarate tere varables. Te trd s a omromse alloato w s a wegted average of samle sze of rata usg dvdual alloato (Cora 977; Catteree 97. Te fourt s te otmal alloato for a loss futo of arater values w ombe varaes of all varables (Ks 976; Sukatme et al 98; Betel 989; Ka ad Asa ; ad Daz-ara ad Cortez 8. I multvarate ratfed samle desg orrelato s osdered amog tere varables. Te fr metod s a omromse alloato wegted b orrelato oeffets or ovaraes ad te seod metod s te otmal alloato for a loss futo of arater values of varae-ovarae matrx. A varato of oe varable wt lower orrelato s more mortat ta oters. Te trd metod s to use wegted obet futo b te mortae of tere varables a matematal rogrammg ad of mortae s te error of emato te orrelato oeffet or te ovaraes. Te multvarate ratfed samle desg s used for mult-obetve surves w tere s dfferee amog te mortae of tere varables. Te roblem of alloato wt more ta oe arater ratfed samlg s ofltg ature as te be alloato for oe arater wll ot geeral be be for oters. Some omromse mu be reaed to obta a alloato tat s effet for all araters. Ts roblem was fr osdered b ema (9 Daleus (957 os (958 Koka ad Ka (967 Ka ad Asa ( Ka Jaa ad Asa (997. Attemts were made for a aetable alloato b eter suggeg ew rtera or exlorg exg rtera furter. Oe of te roblems of ratfato s tat loss reso te emate of a arater reases f te arater a ratum s ot terall omogeous. To refra from te rease loss of reso s to assg a maxmum wegt to te t arater. Otmal alloato mult-tem s develoed as a multvarate otmzato roblem b fdg te ral omoets. Ts ould be doe b determg te overall lear ombatos tat oetrate te varablt to few varables. Te obetves of te ud are to:. Fd alloato mult-tem ratfed samlg usg ral omoets. Determe w of te omoets aouts for mo of te varato te model roosed ad. Comare te alloato te two zoes were te emral data were olleted.. Metodolog Te roblem of alloatg samle to varous rata ma be vewed as mmzg te varaes of varous araters subet to te odtos of te gve budget ad tolerae lmts o erta varaes. Te roblem turs out to be olear rogrammg roblem wt several lear obetve futos ad sgle ovex orat. Pzada ad Maqbool ( solved te resultg lear rogrammg roblem troug Cebsev aroxmato. Te rtera bed te Cebsev aroxmato are to fd a soluto tat mmzes te sgle wor. 5

3 Iteratoal Joural of Aled See ad Teolog Vol. o. ; Mar. Otmum Alloato va Mult-obetve Otmzato Te emator of te oulato mea multvarate ratfed samlg for te -t arater s defed as Were W (. s te samle mea ratum of te -t arater ad s te value obtaed for te -t ut ratum of te -t arater. Te Var ( s defed usg te oulato varaes S... w are usuall ukow ad terefore tese are subtuted b te samle varaes s... defed as s ( (. Ad tus Vˆ ar( s subtuted b te emated varae Vˆ ar( w s gve b W s W s Vˆ ar( (.. Otmal Desgs Multvarate Stratfed Samlg Te roblem of otmal alloato of a ratfed samle we several varables are measured o ea ut a be summarzed as follows. Suose tat ( Y Y... Y.... k s a sefed arametr futo of te ukow values te t ratum a obetve s to emate learl deedet futos from te ratfed samle k L ( Y a.... To roeed wt a alloato Koka ad Ka (967 suggeed te mmzato of te total samlg o su tat orats o ea varae C C k C var( L Y..... Te odtos Koka ad Ka (967 moses o ea varae are su tat all Lˆ satsfes te roortoal loseess requremet amel { L ˆ L L } for... ad Draer ad uttma (968 remarked tat t ould be of tere to osder sef lear ombato elemets of. Ts leads to te followg: Let te vetor Y be ormall drbuted wt mea µ ad Covarae.... k were L of s a vetor ad s a o sgular matrx. Draer ad uttma (968 suggeed tat te fr ral omoet s to be ose so as to make ts varae as large as ossble. 6

4 Ceter for Promotg Ideas USA Dae-based Metod Ts metod s a vetor of deal goals w s determed wt te ull formato. O ma oasos te vegator omes u aga te roblem tat o ateedets are avalable wt w to address t. Wt ts metod t s ossble to obta te otmum values mmzg te dae betwee te otmum ad te vetor of targets smultaeousl. B lettg v to be te deal ot or goal for te obetve Vˆ ar(... te vetor of targets V s gve as V v : : v A great advatage of ts metod s tat ts vetor of targets V a be alulated wtout addtoal formato. Ts s doe b mmzg searatel ea obetve Vˆ ar(... su tat te vetor V s defed as te vetor of ts dvdual mma w s aeved o resolvg te followg olear mmzato rogrammes for tegers (Rao 985 5: m Var ˆ ( subet to C... For =. We te vetor V as bee eablsed oe roeed to exame te roblem to be otmzed wt te ew obetve futo amel m d subet to (Vˆ ar(.... V C Were d orresods to a wegted orm. More geerall we te wegted orm L s osdered te roblem to be otmzed takes te followg form m Var( subet to... v C q q wt q ad w s te wegt or rort gve to ea obetve. B takg wt q we ave te followg roblem (. (.5 (.6 q 7

5 Iteratoal Joural of Aled See ad Teolog Vol. o. ; Mar 8... (.7 subet to ( Var m C v As v are oat for all = te roblem s redued to... (.8 subet to Var( m C Wt q oe eed ol to take to aout te maxmum devato ad so te roblem to be otmzed s... (.9 subet to Var( max m... C v Ad for q= te roblem s... (. subet to Var( m C v Alteratvel aoter dae as bee roosed b Kur ad Corell (987:... (. subet to (Var( m C v v

6 Ceter for Promotg Ideas USA I all tese otmzato metods te o rerto C as bee utlzed. owever o some oasos te rertos do ot al to te os but to te avalablt of ma-ours for arrg out a surve or sml to te total tme avalable for erformg te surve... Lear Comouds A lear omoud Y Y... Y or Y of a radom vetor X s a lear ombato of ts omouds su tat Y ax were a ( a a... a s a vetor of real oats. Suose tat X s a radom vetor a s a vetor of oats ad Y ax te te mea of te lear omoud s E( Y E( ax E ( a X Tus E( Y a Te varae s defed as a E ( a a X Var( Y E[( ax a ]. were a ( X s a salar ad so s also equal to ( X a. Tus var( Y E[ a( X ] = E[ a( X ( X a] = a [ E( X ( X ] a = a a For a radom vetor X ad a Te E[ A X ] A ad Var( A X AA. Pral Comoet Aalss matrx of real oats A We sear for a set of mutuall uorrelated varables Y Y... Y ea oe beg a lear ombato of te orgal set of varables X X... X. Oe of te motvatos for determg su a olleto s of f we derve a set tat oetrates te overall varablt to te fr few varables t s eras easer to see wat aouts for te varato te data. Ideed f a few of te { Y } seem to aout for mo of te varato te data te t ould be argued tat te effetve dmesoalt s less ta P ad ts ould result a smlfed aalss based o a smaller set of varables. 9

7 Iteratoal Joural of Aled See ad Teolog Vol. o. ; Mar.. Fdg Pral Comoets Suose tat X X X... X s a radom vetor wt mea ad ovarae matrx. Te te ( ral omoets of X defed b ( Y... Y... Y Y are mutuall uorrelated. ( Y Var( Y... Var( Y. ( Var ( Y a X a X... a Y Y satsfes te followg odtos: X a X. Were a a a... a s a vetor of oats satsfg ( a = a k k a a = for all.... I addto te t ral omoet Y a X s te lear omoud of X tat maxmzes Var Y subet to beg uorrelated wt te reedg omoets Y Y.... Se te Y Y ax s a lear omoud Var( Y Var( a X ( = a a... To derve te fr ral omoet ofy we ave Var( Y aa Te dea s to selet a su a wa tat Var ( Y s as large as ossble subet to te orat a a Ts s a adard roblem oraed otmzato ad ma be solved usg te metod of Larage multlers. To use ts metod te Laraga s formed as L ( a aa ( aa (. Te requred a s te value of a tat s a atoar ot of (.. ow defe a (... a a... It ma be sow tat a( aa a a( aa a A atoar ot of (. mu satsf: a ( L ( a a

8 Ceter for Promotg Ideas USA Se a ( L ( a a ( a a a ( a a It follows tat a satsfes Tat s a a a a ( a (. A o-trval soluto ( a to te above exs f ad ol f Were s te determat oerator. Tus δ mu be a Ege value of wt a beg ts orresodg Ege vetor: Se s a smmetr matrx te tere a be u to dt Ege values. Se s ostve (sem defte te all of ts Ege values are o-egatve. Assume for te momet tat te Ege values of... are all dt Tat s... Var ( Y Var ( a X = a a = a ( a (. Usg (. w s equal to a a wll take ts large value at se ts s te value of te large Ege value wt a beg te Ege vetor orresodg to.. Te Emral Stud A emral data from a ouseold surve oduted Abeokuta Sout ad Iebu ort loal govermet areas were used. Te data about te ouseolds are avalable for four araters or varables tat are related to te surve. Tese araters lude ouato ome umber of deedats ad te eduatoal level. Ea of te two loal govermet areas wt a samle sze of ouseolds ea were radoml seleted usg smle radom samlg teque makg a total of ouseolds. Te eads of te ouseolds were tervewed. Results. Parameter Emato Te rmar oer all samle surves s te dervato of ot emates for te arameters of ma tere. Te ot emate for te roortoatel alloated samle data are obtaed b addg u te observato over all te rata ad dvdg b te samle sze. owever equall mortat s te dervato of te varaes of te above emates. Te samlg varae s deed oe of te ke dators of qualt samle surves ad emato. Varae emato s rual ssue te assessmet of te surve results. Statal software Slus was used te aalss of data. Te summar emates of te samle ats for Abeokuta Sout ad Iebu ort samles are as sow Tables. ad..

9 Iteratoal Joural of Aled See ad Teolog Vol. o. ; Mar Table.: Summar Emates of Abeokuta Sout Samle Stats Ouato Iome Deedat Sze Eduatoal Level Mea ( V srs ( Var(o ( V mod Table.: Summar Emates of Iebu ort Samle Stats Ouato Iome Deedat Sze Eduatoal Level Mea (.8... V srs ( Var(o (..8.. V mod. Multvarate Stratfed Samlg Te roblem of otmum alloato multvarate ratfed samlg as bee examed atal lterature but te solutos roosed ave bee artular ases of a mult-obetve otmsato teque. Usg our data set for Abeokuta ad Iebu te geeral mult-obetve otmsato rogramme as (.79 s Subet to m Var ˆ ( Var ˆ ( m ˆ Var ( Furtermore we osder te followg two rogrammes for te o lear mmzg of tegers: Subet to ad = Subet to m ˆ V ar ( m ˆ V ar (

10 Ceter for Promotg Ideas USA Te ud adoted a aroa based o te fat tat ts metodolog s more real uder te ambt of multvarate aalss. Te fr e s to omute te matrx of varae-ovaraes of te vetor (.... Usg Slus software te varae- ovaraes matrx s as Tables. ad.. Te Egevalues of te ovarae matrx of Abeokuta ad Iebu data set s as sow Table.5 wle te Ege vetors are as sow Tables.6 ad.7. Table.: Varae-Covarae Matrx of Abeokuta Data Set Ouato Iome Deedat Sze Eduatoal Level Ouato Iome Deedat Sze Eduatoal Level Table.: Varae-Covarae Matrx of Iebu Data Set Ouato Iome Deedat Sze Eduatoal Level Ouato Iome Deedat Sze Eduatoal Level Table.5: Egevalues of te Covarae Matrx of Abeokuta ad Iebu Data Set Egevalues Abeokuta Iebu ( Table.6: Ege Vetors of Abeokuta Data Set Table.7: Ege Vetors of Iebu Data Set Results Based O Pral Comoet Aalss Te ral omoet aalss esured tat te varae-ovarae matrx was deomosed ad te egevalues ad egevetors alulated from te multvarate data reresetg formato from te ouseolds. Te ral omoets were omuted from te ud o te bass of te samle ovarae matrx ad te result for Abeokuta samles are Y.5X.8X.X. 86X Y.7X.7X.95X. 8X Y.756X.67X.6X. 7X Y.6X.6X.86X. 7X

11 Iteratoal Joural of Aled See ad Teolog Vol. o. ; Mar wt orresodg samle varaes ad.59 resetvel. Tus te total varae s.599 ad te ral omoets Y Y Y Y suessvel aouts for 9.% 5.8%.9% ad.% of te total varae. Smlarl te ral omoets based o te samle orrelato matrx for Abeokuta are gve b Y.8X.566X.7 X. 68X Y.X.9X.887X. X Y.75X.565X.97X. 6X Y.6X.59X.7X. 7X X Were X for S Te samle varaes of te ew ral omoets Y Y Y Y are ad.86 resetvel. I ts ase te total varae s ad te ral omoets aout suessvel for.%.7%.% ad.7% of te total varae. B usg te Ege futo we foud tat te Ege values of te samle ovarae matrx were ad.59. Te square root of tese values s te adard devatos of te ral omoets ad s ad.9 resetvel. Te ral omoets o te bass of te samle ovarae matrx for Iebu samles are Y.6X.X.X. 78X Y.55X.X.99X. 5X Y.57X.795X.6X. 7X Y.76X.X.7X. 55X wt orresodg samle varaes ad.66 resetvel. Tus te total varae s.5 ad te ral omoets suessvel aouts for 5.%.%.% ad 8.7% of te total varae. Smlarl te ral omoets based o te samle orrelato matrx for Abeokuta are gve b Y.8X.59X.X. 6X Y.59X.58X.85X. 89X Y.557X.689X.5X. X Y.6X.8X.X. 755X X Were X for S Te samle varaes of te ew ral omoets Y Y Y Y are ad.658 resetvel. I ts ase te total varae s ad te ral omoets aout suessvel for 8.9%.7% 9.% ad 9.% of te total varae. B usg te Ege futo we foud tat te Ege values of te samle ovarae matrx were ad.66. Te square root of tese values s te adard devatos of te ral omoets ad s ad.558 resetvel.

12 Ceter for Promotg Ideas USA Referees Artaar T.S ad Dodge Y. 98. Matematal rogrammg o ats. A Wle-Itersee Publato Jo Wle & Sos I. Betel F Baes ad Mmax redto fte oulato. Joural of Statal Plag Catteree S. 97. A ud of otmal alloato multvarate ratfed surves. Skad Akt Cora W Samlg Teques ( rd Edto ew York Wle. Daleus T Samlg Swede: Cotrbutos to te metods ad teores of samle surve rate. Almav ad Wksell Stokolm. Daz-ara J.A ad Cortez L.U. 8. Mult-obetve otmsato for otmum alloato multvarate ratfed samlg. Surve Metodolog Vol. o 5-. Daz-ara J. A. ad Cortez L. U. 6. Otmum alloato multvarate ratfed samlg: mult-obetve rogrammg. Commuao Tea: Commuaoes Del CIMAT. 6(7 8-. Draer.R ad uttma I Some Baesa Stratfed Two Pase Samlg Results. Bometrka os S.P A ote o Stratfed Radom Samlg wt Multle Caraters. Col.Stat. Bull ut. ad Trell S.. Stratfed samlg. Covetr Uverst Press. tt:// (aessed Februar 8. Ka M..M ad Asa M.J.. A ote o Otmum Alloato Multvarate Stratfed Samlg. Sout Paf Joural of atural See Ka M..M Jaa. ad Asa M.J Determg te otmum luer sze. Joural of te Ida Soet of Agrultural Stats. Vol. 5 ( -9. Kur A.I. ad Corell J Resose Surfaes; Desg ad Aalss. Marel Dekker ew York Y. Ks L Surve samlg. ew York Wle. Koka A.R ad Ka S.U Otmum alloato multvarate surves. A aaltal soluto. Joural of Roal Statal Soet. Seres B emaa J. 9. O te two dfferet asets of te reresetatve metods. Te metod ratfed samlg ad te metod of urosve seleto. Joural of Roal Statal Soet Przada S. ad Maqbool S.. Otmal Alloato Multvarate samlg Troug Cebsev s Aroxmato. Bullet of te Malasa Matematal See Soet (6. Sukatme P.V Sukatme B.V Sukatme S. ad Asok C. 98. Samlg Teor of Surve wt Alatos. rd Edto. Ames Iowa: Iowa State Uverst Press. Rao J.. K. 5. Iferee from ratfed samles Joural of Amera Statal Assoato Rao J.. K.. Small area emato. ew York Wle. Rao J.. K Codtoal feree surve samlg. Surve Metodolog. II 5. 5