# CLUSTER SAMPLING DR. SHALABH DEPARTMENT OF MATHEMATICS AND STATISTICS INDIAN INSTITUTE OF TECHNOLOGY KANPUR 1

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1 amplng Theory MODULE IX LECTURE - 30 CLUTER AMPLIG DR HALABH DEPARTMET OF MATHEMATIC AD TATITIC IDIA ITITUTE OF TECHOLOGY KAPUR

2 It s one of the asc assumptons n any samplng procedure that the populaton can e dvded nto a fnte numer of dstnct t and dentfale unts, called samplng unts Thesmallest unts nto hch h thepopulaton can e dvded are called elements of the populaton The groups of such elements are called usters In many practcal stuatons and many types of populatons, a lst of elements s not avalale and so the use of an element as a samplng unt s not feasle The method of uster samplng or area samplng can e used n such stuatons In uster samplng dvde the hole populaton nto usters accordng to some ell defned rule Treat the usters as samplng unts Choose a sample of usters accordng to some procedure Carry out a complete enumeraton of the selected usters, e, collect nformaton on all the samplng unts avalale n selected usters Area amplng In case, the entre area contanng the populatons s sudvded nto smaller area segments and each element n the populaton s assocated th one and only one such area segment, the procedure s called as area samplng

3 Examples In a cty, the lst of all the ndvdual persons stayng n the houses may e dffcult to otan or even may e not avalale ut a lst of all the houses n the cty may e avalale o every ndvdual person ll e treated as samplng unt and every house ll e a uster The lst of all the agrcultural farms n a vllage or a dstrct may not e easly avalale ut the lst of vllage or dstrcts are generally avalale In ths case, every farm s samplng unt and every vllage or dstrct s the uster Moreover, t s easer, faster, cheaper and convenent to collect nformaton on usters rather than on samplng unts In oth the examples, dra a sample of usters from houses/vllages and then collect the oservatons on all the samplng unts avalale n the selected usters 3

4 Condtons under hch the uster samplng s used Cluster samplng s preferred hen o relale lstng of elements s avalale and t s expensve to prepare t Even f the lst of elements s avalale, the locaton or dentfcaton of the unts may e dffcult A necessary condton for the valdty of ths procedure s that every unt of the populaton under study must correspond to one and only one unt of the uster so that the total numer of samplng unts n the frame may cover all the unts of the populaton under study thout any omsson or duplcaton When ths condton s not satsfed, as s ntroduced Open segment and osed segment It s not necessary that all the elements assocated th an area segment need e located physcally thn ts oundares For example, n the study of farms, the dfferent felds of the same farm need not le thn the same area segment uch a segment s called an open segment In a osed segment, the sum of the characterstc under study, e, area, lvestock etc for all the elements assocated th the segment ll account for all the area, lvestock etc thn the segment 4

5 Constructon of usters The usters are constructed such that the samplng unts are heterogeneous thn the usters and homogeneous among the usters The reason for ths ll ecome ear later Ths s opposte to the constructon of the strata n the stratfed samplng There are to optons to construct t the usters equal sze and unequal sze We dscuss the estmaton of populaton means and ts varance n oth the cases Case of equal usters uppose the populaton s dvded nto usters and each uster s of sze n elect a sample of n usters from usters y the method of R, generally WOR o total populaton sze M total sample sze nm Let y : Value of the characterstc under study for the value of j th element (j,,m n the th uster (,, y M y M j mean per element of th uster 5

6 6

7 7here hch s the mean sum of square eteen the uster means n the populaton Estmaton of populaton mean Frst select n usters from usters y RWOR Based on n usters fnd the mean of each uster separately ased on all the unts n every uster o e have the uster means as mean of all such uster means as an estmator of populaton mean as y Consder the, y,, yn Bas Thus y n n y n E( y E( y n n Y n Y y Y s an unased estmator of ( snce R s used Varance The varance of y can e derved on the same lnes as dervng the varance of sample mean n RWOR The only dfference s that n RWOR, the samplng unts are samplng unts are y, y,, y n n n Var( y and Var( y s n n Var( y E( y Y n n ( y Y ote that n case of RWOR, y, y,, yn hereas n case of y, the

8 Estmate of varance Usng agan the phlosophy p of estmate of varance n case of RWOR, e can fnd n n ( s Var y n here s ( y y s the mean sum of squares eteen uster means n the sample n n Comparson th R If an equvalent sample of nm unts ere to e selected from the populaton of M unts y RWOR, the varance of the mean per element ould e here M nm Var( ynm M nm f n M -n f and ( y Y M n Also Var( y n M j f n 8

9 Consder M j ( M ( y Y M ( y y + ( y Y j M M ( y y ( y Y j j + M ( + M ( here M ( M j y y s the mean sum of squares thn usters n the populaton s the mean sum of squares for the th uster The effcency of uster samplng over RWOR s Var( ynm E Var( y M M ( + ( ( M M Thus the relatve effcency ncreases hen s large and s small o uster samplng ll e effcent f usters are so formed that the varaton eteen the uster means s as small as possle hle varaton thn the usters s as large as possle 9

10 Effcency n terms of ntra ass correlaton The ntra ass correlaton eteen the elements thn a uster s gven y E ( y Y ( yk Y ρ ; ρ E( y Y M M ( M M M ( M M M j k ( j M j M M j k( j ( y Y( y Y ( y Y M M k ( y Y( y Y k M M j k( j ( y Y ( y Y ( M ( M k Consder ( y Y ( y Y M Y M j M M M ( ( ( y Y + y Y yk Y M j M j k( j M M M ( y Y( yk Y M ( y Y ( y Y j k( j j 0

11 or ρ( M ( M M ( ( M ( M or + ρ( M M ( The varance of no ecomes y Var( y n n M [ + ( M ρ] n M M n For large,, and so M Var( y [ + ( M ρ] nm The varance of sample mean under RWOR for large s Var( ynm nm The relatve effcency for large s no gven y Var( ynm E Var( y nm [ + ( M ρ] nm ; ρ + ( M ρ ( M

12 If M then E, e, R and uster samplng are equally effcent Each uster ll consst of one unt, e, R If M >, then uster samplng s more effcent hen or or E > ( M ρ < 0 ρ < 0 If ρ 0, then E, e, there s no error hch means that the unts n each uster are arranged randomly o the sample s heterogeneous ρ ρ ρ In practce, s usually postve and decreases as M ncreases ut the rate of decrease n s much loer n comparson to the rate of ncrease n M The stuaton that ρ > 0 s possle hen the neary unts are grouped together to form uster and hch are completely enumerated There are stuatons hen ρ < 0 Estmaton of relatve effcency The relatve effcency of uster samplng relatve to an equvalent RWOR s otaned as E M An estmator of E can e otaned y susttutng the estmates of and nce y y s the mean of n means y from a populaton of means y,,,, hch are dran y n n RWOR, so from the theory of RWOR,

13 n Es ( E ( y yc n ( y Y s Thus s an unased estmator of s nce s the mean of n mean sum of squares dran from the populaton of mean sums of squares n n,,,,, n E( s E n so t follos from the theory of RWOR that s Thus s an unased estmator of 3

14 Consder y Y M ( M j or M ( M ( y y + ( y Y j M ( y ( y + y Y j ( M + M( M + M ( ( An unased estmator of can e otaned as ˆ ( M s + M( s M so n n ( y s Var y Var ( y nm ˆ n n M here s ( y y n ( n 4

15 An estmate of effcency E s M ˆ ( M s + M( s E M( M s If s large so that M ( M and M M, then M E + M M M and ts estmate s ˆ M s E + M M Ms 5

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