VARIANCE ESTIMATION FROM COMPLEX SURVEYS USING BALANCED REPEATED REPLICATION

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1 VAIANCE ESTIMATION FOM COMPLEX SUVEYS USING BALANCED EPEATED EPLICATION aeder Parsad ad V.K.Gupta I.A.S..I., Lbrary Aveue, New Del For ematg te varace of olear atcs lke regresso ad correlato coeffcets ratfed samplg desgs, te Balaced epeated eplcato (B) metod as receved specal atteto, altoug oter procedures lke learzato (Taylor s seres expaso metod), Jackkfe repeated replcatos ad te Bootrap metod are also avalable te lterature. B metod volves formg replcato by coosg oe of te uts selected from eac ratum to form a replcato. Eac of te replcatos provdes a emate of te olear atc. Te procedure s repeated may tmes to get more able emator. For two prmary selectos per ratum, samplg wt equal or uequal probabltes ad wt replacemet, McCarty (966, 969) proposed te balaced repeated replcatos metod tat volves formg alf-samples by radomly selectg oe prmary samplg ut from te two uts eac ratum ad sowed tat usg te colums of Plackett ad Burma (946) plas two symbols ( ad ) for re-samplg from eac of te ratum, tere s o loss effcecy for lear atcs. Ts s llurated wt te elp of a example, te sequel: Cosder tat te sample desg cos of a smple radom sample wt replacemet of sze = selected from a ratum wt populato sze N, for =,..., L (=4). Furter let N = 5; N = 6; N3 = 8 ad N 4 = 6 : N= N + N + N3 + N4 = 5. Te values of te caracterc uder udy for te uts selected from te 4 rata are respectvely: Stratum Selected Observatos Stratum mea = y W / = N W y 45, , , , y = 4 W y (a ubased emator of populato mea) 5.6 = It s well kow tat a ubased emator of te varace of y s vˆ ( ) y = P ( y y ) were W /( ) = P ad y = y., /

2 Te computatos volved te emato of a ubased emator of te varace of y are gve te sequel. Stratum Selecte d Obs. Strata mea= y ( W y y ) P / () () (3) (4) (5) (6) (4)*(6) 45, , , , vˆ ( y ) = P ( y y ) Now cosder a Plackett ad Burma Pla for 4 8 rus: Usg te above, get 8 repeated samples, oe correspodg to eac ru or row of te array. If te symbol te colum s, te select ut from ratum ad ut for ratum. Tus te 8 repeated samples are gve as Eac of te sub-sample meas are obtaed as for sample wll be deoted by were = y = P y. Sgle observato ratum y, were s oe of, L,. Here. Te varous computatos volved are sow below: P = W as epeated Samples P y P y P y 3 II-34

3 P 4 y y Now 8 y y = =5.6 ad Var( y )= Terefore, te results of repeated replcatos gve ubased emator for populato mea wt same varace,.e., wtout ay loss effcecy. Te set of replcatos tat aceve te full precso s called a Balaced Set ad te metod, terefore, s termed as Balaced epeated eplcatos. Sce te metod coss of selectg oe of te two uts, te umber of uts selected for eac replcato s exactly oe alf te total sample sze. Hece te omeclature Balaced Half sample metod s commoly used to descrbe te McCarty's case of B. Te plackett ad Burma plas are otg but Hadamard matrces or ortogoal arrays of regt two two symbols. A bref descrpto of ortogoal arrays s gve te Appedx. Gurey ad Jewett (975) used ortogoal arrays to exted ts to a prme umber p of prmary selectos from eac ratum, by formg a set of balaced sub samples. Gupta ad Ngam (987, Bometrka, 74(4), ) exteded te metod of balaced repeated replcatos to arbtrary umber of prmary selectos per ratum desgs. It s sow tat mxed ortogoal arrays of regt two are balaced subsamples eeded for varace emato. Te algebra of te use of mxed ortogoal arrays varace emato s gve te sequel. To smplfy expoo atteto wll be cofed to te emato of a populato mea from a ratfed radom sample. Exteso to more complcated uatos ca be adled by usg lear approxmatos. We suppose te sample desg to cos of a smple radom sample wt replacemet of sze selected from a ratum wt populato sze N, for =,..., L. Te measuremet o te t member of ratum wll be deoted by y, for =, L,, so tat a ubased emator of te populato mea s y L = W y = /, were W = N N, N = ( N + L+ NL ) /. A ubased emator of te varace of y vˆ ( ) y = P ( y y ), s gve by were W /( ) P ad = =. y y / Te same emator of var ( ) y may be calculated by te use of balaced subsamples wt observato per ratum coructed by te use of mxed ortogoal arrays of regt two. I effect, tese mxed ortogoal arrays of regt two defe balaced subsamples, so tat te II-343

4 exece of suc a array mples te exece of balaced subsamples wt te reque propertes. To see ts defe a set of subsamples wt oe observato per ratum. Te sgle observato ratum for sample wll be deoted by y were s oe of, L,. We ca equally wrte y = δ (, ) y, were δ ( k, l) = f k = l ad δ ( k, l) = oter wse. For te t subsample let y = P y. Te te average of tese terms may be wrtte as y = L = P = y (, ) δ, were δ (, ) subsamples. If ts s coat for all wt, δ (, ) = µ, te, as (, ) = we ave tat represets te umber of tmes tat ut from ratum occurs over all = µ or = µ, wece y = P y. Cosder ow ( y y) = y = S + T = = = y y L L wt S = P δ (, ) y, T = P P y y (, ) δ (, ) = = δ. δ, As oly oe ut s selected from eac ratum eac subsample, t s readly deducted tat = P y δ, δ, represets te umber of tmes tat ut from ratum appear same subsample as ut from ratum. If ts s coat, say µ for all pars (, ) from rata (, ), te, as δ (, ) δ (, ) = = µ, = = S. I summato T, te term ( ) ( ) we ave = /( ) = P µ. Ts allows us to wrte T P y y, L wece S + T = P ( y y ) + y Tus = = ( y y) = P ( y y ) = vˆ ( y ) as requred. Tus, a clear relatosp exs betwee te mxed ortogoal arrays of regt two ad te set of balaced repeated replcatos. Te metod of Balaced epeated eplcato s applcable II-344

5 weever te mxed ortogoal array of regt two exs. However, te mxed ortogoal arrays of regt two do ot always ex for all te combatos of symbols. Ts puts a severe rercto o te applcato of B metod real sample survey uatos. Ks ad Frakel (97, 974) troduced a metod called groupg metod to overcome te obacle of oavalablty of a balaced set te case of arbtrary umber of prmary selectos. Te metod s to dvde te sample szes wt eac ratum at radom to two equal subsamples, ad te to form balaced repeated replcatos treatg eac subsample as f t s oe member of a sample of sze. Ts metod s commoly called as grouped balaced alf sample metod or grouped balaced repeated replcato. I fact, te repeated replcatos based o ts metod are partally balaced. Wu (99) ave sow tat te groupg metod s very effcet to emate te varace of a o-lear atc. Gupta ad Ngam (987) advocated te use of ortogoal ma effect plas wt uequal frequeces geeral ad wt proportoal frequeces partcular. Wu (99) dscouraged te use of proportoal frequecy plas te B as tese result effcecy loss ematg te varace of a o-lear atc. Wu dvocated te use of ear ortogoal arrays wc mo of te colums are ortogoal. Dadapa (996) vegated te valdty of Wu's atemet regardg te use of proportoal frequecy plas. He as sow tat certa proportoal frequecy plas ca be used wt o loss effcecy for ematg te varace of a lear atc. Furter, tey ave also sow tat te varace emate usg proportoal frequecy plas s asymptotcally coset for a o-lear atc tat s expressble as a geeral smoot fucto of populato meas. efereces Dadapa, A. (996). Varace emato from complex survey data usg proportoal frequecy plas. Upublsed P.D. Tess, I.A..I., New Del. Gupta, V.K. ad Ngam, A.K. (987). Mxed ortogoal arrays for varace emato wt uequal umbers of prmary selectos per ratum. Bometrka, 74, Gurey, M. ad Jewett,.S. (975). Coructg ortogoal replcatos for varace emato. J. Am. Stat. Assoc., 7, Ks, L. ad Frakel,.M. (97). Balaced repeated replcatos for adard errors. J. Am. Stat. Assoc., 65, Ks, L. ad Frakel,.M. (974). Iferece from complex samples (wt dscusso).. J. oyal. Stat. Soc., B36, McCarty, P.J. (966). eplcato: A Approac to te Aalyss of Data from Complex Surveys, Vtal ad Healt Statcs, Seres, No. 4, Wasgto, D.C., US Departmet of Healt Educato ad Welfare, Natoal Cetre for Healt Statcs. McCarty, P.J. (969). Psuedo replcato: alf samples. ev. It. Stat. I., 37, Plackett,.L. ad Burma, J.P. (946). Te desg of optmum mult-factor expermets. Bometrka, 33, Wu, C.F.. (99). Balaced repeated replcatos based o mxed ortogoal arrays. Bometrka, 78, II-345

6 Appedx Ortogoal arrays Let S be te set of s symbols (or levels). Tese s levels are deoted by,,..., s-. Defto : A N k Array A wt etres from S s sad to be a OA wt s levels, regt t ad dex λ ( t k) f every N t subarray of A cotas eac t-tuple based o S exactly λ tmes as row. Te ortogoal arrays are deoted by OA (N, k, s, t) or OA (N, s k, t) wt N, k, s ad t as te parameters, were N s te sze of te array, or te umber of rus, or te umber of treatmet combatos, k s te umber of corats or te umber of factors, s s te umber of symbols or levels ad t s te regt of te ortogoal array. Example : Let us cosder a array I ts array f we see ay tree colums, we fd tat all possble combatos,,,,,,, & of symbols & appears same umber of tmes,.e. oe. Ts property of te above array makes t to be called ortogoal arrays (OA) of regt 3 ad dex uty. We all te factors a ortogoal array are ot at same level, ortogoal arrays are deoted by OA (N, s s s k, t), were fr factor s at te level s, secod factor s at te level s ad so o. II-346

7 k Defto : A mxed ortogoal array OA( N,s k s k...s v,t ) s a array of sze N k, were k = k +k k v s te total umber of factors, wc te fr k colums ave symbols from {,,..., s -}, te ext k colums ave symbols from {,,..., s -}, ad so o, wt te property tat ay N t subarray every possble t-tuple occurs a equal umber of tmes as arrow. Example : A array OA (8, 4. 4, ). I ts array te fr factor s at four levels ad te factors to 5 are at two levels eac. 3 3 II-347