ON SOME ASPECTS OF CLUSTER SAMPLING

Transcription

1 O O APT OF LUTR APLIG PRAJIT PAL.c. (Agrcultural tatstcs, Roll o. 446 I.A..R.I., Lbrary Aveue, ew Delh- harperso: Dr. U.. ud Abstract: luster saplg s a procedure fro e populato of partcular cluster (a group of saplg uts sze we draw e saple by RWOR of at sze. It s used whe e lst of uts s ot avalable. A saplg procedure uequal cluster saplg for fxed saple sze, e uber of uts e tal saple of selected clusters exceeds e plaed sze of uts s dscussed. A schee for dscardg e excess uber of clusters fro e tal saple of clusters s also preseted. Ths procedure, takg to accout splcty ad practcal feasblty, ca be used practce u-stage uequal cluster saplg desgs for arrvg at e fxed saple sze of eleets. The reducto e varace effcecy of estator o accout of e decrease e saple sze copared to at of e tally selected usual u-stage uequal cluster saple of a relatvely larger sze s copesated by ts creased cost effcecy. Key words: luster aplg, luster, ffcecy, aple ze.. Itroducto I rado saplg, t s presued at e populato has bee dvded to a fte uber of dstct ad detfable uts defed as saplg uts. The sallest ut to whch e populato ca be dvded s called a eleet of e populato. A group of such eleets s kow as a cluster. Whe he saplg ut s a cluster, e procedure s called cluster saplg. If e etre area cotag e populato uder study s dvded to saller segets ad each eleet e populato belogs to oe ad oly oe seget, e procedure s soetes called area saplg. Geerally, detfcato ad locato of a eleet requres cosderable te. However, oce a eleet has bee located, e te take for surveyg a few eghborg eleets s sall. Thus, e a fucto cluster saplg s to specfy clusters or to dvde e populato to approprate clusters. lusters are geerally ade up of eghborg eleets ad, erefore, e eleets w a cluster ted to have slar characterstcs. As a sple rule, e uber of eleets a cluster should be sall ad e uber of clusters should be large. After dvdg e populato to specfed clusters, e requred uber of clusters ca be selected eer by equal or uequal probabltes of selecto. All e eleets selected clusters are e euerated. For a gve uber of saplg uts, cluster saplg s ore coveet ad less costly. The advatages of cluster saplg are at, ollecto of data for eghborg eleets s easer, cheaper, faster ad operatoally ore coveet a observg uts spread over a rego. It s less costly a sple rado saplg due to e savg of te joureys, detfcato, cotacts, etc.

2 O oe Aspects of luster aplg Whe e saplg frae of eleets ay ot be readly avalable whch s e geerally e case large scale surveys. The dsadvatage of t s at e effcecy of cluster saplg relatve to sple rado saplg s less. For ay types of populatos a lst of uts (eleets s ot avalable, for e.g. to estate e dstrct crop yeld e lst of dvdual farers s ot avalable we use cluster saplg. The eod of cluster or area saplg s applcable such cases. However, w clusters of uequal szes, e uber of uts to be ultately observed s ot uder cotrol. Ths ay result e uber of uts to be observed beg uch larger a plaed, leadg to operatoal dffcultes, partcularly whe ere are budgetary, traed apower ad oer costrats. I e usual cluster saplg, e uber of clusters s fxed ad e uber of uts s a varable whe clusters are uequal. But soe stuatos deeper observato of uts volvg use of sophstcated struets ad costly cosuable ateral s volved, t ay be desrable to fx e uber of eleets advace. I agrcultural surveys households are uts, soetes reuerato s pad to e selected households as a otvato to provde better qualty of data. I such stuatos t would be desrable to keep e ultate saple sze of uts to be observed, a far as possble, close to e plaed sze. Oe possble approach to s proble would be to dscard a sub-saple of clusters fro e tally selected clusters w a sutable procedure whch would, as far as possble, esure e requred saple sze of uts wout coplcatg e desg ad retag e ubased character of e estator. The procedures for dscardg e excess uber of clusters fro aogst e tally selected clusters ad for e cosequet estato of populato paraeters are dscussed. xaple.: To estate e wheat producto a dstrct cotag te blocks a group of ree vllages ca be fored as a cluster. ow saple of ree blocks are take at rado ad used to estate e wheat producto. otatos We shall assue at ( e populato cossts of clusters of eleets each ad ( clusters are selected fro clusters by sple rado saplg wout replaceet. Let, j be e value of e characterstc uder study for e j eleet e cluster, j,,, ;,,,. j, e ea per eleet of e cluster j., e ea of cluster eas j, e populato ea j (j., e ea square betwee eleets e cluster j

3 O oe Aspects of luster aplg w b (., e ea square w clusters, e ea square betwee cluster eas As a estator of, cosder e ea of cluster eas e saple, aely, y y. learly, y s a ubased estator of, w ts varace gve by V(y b ce, (s b ( y. y b A ubased estator of ( y V V(y ( s b ffcecy of luster aplg V(y R V(y b b s. The Proble I practcal stuato we ay get e clusters of uequal szes. Let e populato be coposed of clusters, e cluster cosstg of eleets (, ad at a saple of clusters s draw fro t by e eod of sple rado saplg wout replaceet. Furer, let us deote by e average cluster sze e populato, by e uber of eleets e selected clusters, by e saple average cluster sze ad e plaed uber of eleets to be fally observed e saple. As per e pla, has already bee selected ad t s assued at s greater a. Hece, e proble s ow to work out e approxate uber of clusters to be

4 O oe Aspects of luster aplg rejected so at e uber of eleets of e tally selected clusters ad e plaed sze of uts, by e average cluster sze e populato. Thus, e approxate uber of clusters to be rejected fro e tally elected clusters would be gve by (say If s a fracto, t s approxated to e ext lower teger. Havg worked out e approxate uber of clusters for rejecto aely, we select clusters fro e tally selected clusters by e eod of sple rado saplg wout replaceet ad reject e fro e. Let e uber of clusters fally selected, by rejectg clusters fro e tally selected clusters be deoted by. learly ad are rado varables. To fd e approxate uber of clusters to be rejected, alteratvely e average cluster sze based o e saple, at s, ay also be used. The approxate uber of clusters to be rejected s case would be gve by (say Here too, ad are rado varables as well as. A rd alteratve would be to estate e average cluster sze fro a larger saple of e populato wout apprecably addg to e cost of equry ad utlze t for deterg e uber of clusters for rejecto. tll aoer approach would be to select a cluster fro e tally selected clusters ad rejectg e tll as close to e plaed uber of uts. I e followg sectos, e procedures for estato of populato paraeters based o e frst two schees suggested above for dscardg e excess uber of clusters has bee descrbed.. stato of Populato Total whe e Average luster ze e Populato s Kow The proposed estator of, e populato total for e study character y, s gve by Ŷ (., y j j y j deotg e j eleet of e cluster. Ŷ s evdetly a ubased estator for. The varace of e estator Ŷ s gve by V(Ŷ ( 4

5 O oe Aspects of luster aplg (. ad j y j The expectato of requred e above equato ay be obtaed to a secod order of approxato utlzg e usual techque as follows: (( ( ( ( ad ( ( Hece a expresso for up to e order of s gve by Therefore, (Ŷ V (. The secod ter e varace expresso at (. above s postve ad us t shows e crease varace due to rejecto of clusters over at of u-stage uequal cluster saplg w o rejecto of clusters. 5

6 O oe Aspects of luster aplg A ubased estator of e varace (. s gve by s (Ŷ Vˆ (.4 s 4. stato of e Populato Total whe Average luster ze e Populato s ot Kow A ubased estator of, e populato total for e study character y, s gve by Ŷ (4. s as defed earler ad ts varace s gve by s ( (Ŷ Vˆ (4. ( ( (4. ow (, ad proceedg e sae way as above, e expected value of s obtaed. ubsttutg e above equato we get ( [ / σ ] (4.4 σ ad also ( To obta e expected value of, we proceed e sae way as above ad substtutg e above equato we get, ( σ ( (4.5 Fro equatos (4., (4.4 ad (4.5, we obta 6

7 O oe Aspects of luster aplg σ σ Thus, e varace of e estator s gve by Ŷ (Ŷ V (4.6 s as defed earler. The varace expresso at (4.6 above dcates at f e uber of clusters to be rejected s based o average cluster sze as obtaed fro e saple, ere s a reducto e varace as copared to e stuato whe e average cluster sze s based o e etre populato. A ubased estator of e varace preseted at (4. s gve b s (Ŷ Vˆ (4.7 s as defed uder (.4. s 5. Relatve ffcecy of e uggested Procedure Let deote e varace correspodg to procedure descrbed ad usual cluster saplg procedure respectvely. We defe relatve varace effcecy (RV of e descrbed procedure over e usual procedure as RV V ad V V V. osder e cost fucto as s e cost of e survey apart fro e overhead cost of plag ad aalyss. s e cost per cluster for prelary operatos, such as, jourey, detfcato, cotact, etc.volved coductg e survey a cluster ad s e cost of surveyg oe ut. I geeral s expected to be cosderably less a. However, certa cases ay be large, partcularly whe observatos o e uts su able ateral ad us s lkely to be suffcetly ore a. The relatve cost effcecy of e suggested procedure over e usual cluster saplg procedure s defed as R e suggested procedure uder xpected cos t e usual cluster saplg procedure uder xpected cos t 7

8 O oe Aspects of luster aplg Ad, e preset stuato, e R of e suggested procedure whe e average cluster sze s based o e etre populato over e usual cluster saplg procedure s. However, whe e average cluster sze s estated fro e saple e R of e suggested procedure over e usual cluster saplg procedure reduces to It s see at R (% of e suggested procedure bo e cases wll be ore a. 6. prcal Illustrato To deostrate e usefuless of e suggested saplg procedure, uercal llustrato s preseted. For s purpose, a populato of clusters a tehsl s take. The eleets of e clusters were holdgs of varyg szes ad e character uder study beg area uder wheat crop uts of oe-te of a hectare durg e rab seaso. It s desred to estate e area uder wheat crop e tehsl fro a saple of 5 clusters, subject to e codto at e saple selected clusters should ot cota ore a 5 holdgs. A saple of 5 clusters usg sple rado saplg wout replaceet s draw ad e total uber of holdgs costtutg e 5 clusters was 58. Hece order to be close to e plaed sze e populato was 4. ad at estated fro e saple of 5 clusters was.9. Followg e procedure dscussed above, e uber of clusters to be dscarded worked out to uder each of e two cases. The total uber of holdgs e fally selected clusters, after rejectg e clusters fro e tally selected 5 cluster was 48, whch was close to e plaed sze. The estate of total ad ts varace as well as e relatve varace effcecy ad relatve cost effcecy of e proposed estators over e usual procedure, are preseted e followg table. Usual procedure uggested procedure Ŷ V (Ŷ Relatve Varace ffcecy (% Relatve ost ffcecy (% Average cluster sze kow Average cluster sze estated Thus, as expected, because of e reducto of e saple sze uder e suggested techque copared to e usual procedure, e proposed estator s less effcet a e usual procedure. O e oer had e suggested procedure s ore effcet a e 8

9 O oe Aspects of luster aplg usual procedure fro cost effcecy pot of vew. Thus, e suggested techque esures achevg e plaed sze of e saple ad e reducto ts varace effcecy s copesated by ts creased cost effcecy. Refereces ochra, W.G. (977. aplg techques rd d. (ew ork, Joh Wley. ukhate, P.V. ad ukhate, B.V. (97. aplg Theory of urveys w Applcatos, d d, Ida ocety of Agrcultural tatstcs, ew Delh-. P..ehrotra (987. O uequal cluster saplg for fxed saple sze The tatstca, 6,