AN OVERVIEW OF VARIOUS SAMPLING SCHEMES AND DETERMINATION OF SAMPLE SIZES

Transcription

1 : A Overvew of Varous Samplg Schemes.. A OVERVIEW OF VARIOUS SAMPLIG SCHEMES AD DETERMIATIO OF SAMPLE SIZES.. Tag Ida Agrcultural Statstcs Research Isttute, ew Delh-00. ITRODUCTIO The prme objectve of a sample surve s to obta fereces about the characterstc of a populato. Populato s defed as a group of uts defed accordg to the objectves of the surve. The populato ma cosst of all the households a vllage / localt, all the felds uder a partcular crop etc. We ma also cosder a populato of persos, famles, felds, amals a rego, or a populato of trees, brds a forest depedg upo the ature of data requred. The formato that we seek about the populato s ormall, the total umber of uts, aggregate values of varous characterstcs, averages of these characterstcs per ut, proportos of uts possessg specfed attrbutes etc. The data ca be collected two dfferet was. The frst oe s complete eumerato whch meas collecto of data o the surve characterstcs from each ut of the populato. Ths tpe of method s used cesuses of populato, agrculture, retal stores, dustral establshmets etc. The other approach s based o the use of samplg methods ad cossts of collecto of data o surve characterstcs from selected uts of the populato. The frst approach ca be cosdered as ts specal case. A samplg method s a scetfc ad objectve procedure of selectg uts from the populato ad provdes a sample that s expected to be represetatve of the populato. A samplg method makes t possble to estmate the populato parameters whle reducg at the same tme the sze of surve operatos. Some of the advatages of sample surves as compared to complete eumerato are reducto cost, greater speed, wder scope ad hgher accurac. A fucto of the ut values of the sample s called a estmator. Varous measures, lke bas, mea square errors, varace etc. are used to assess the performace of the estmator. The ma problem sample surves s the choce of a proper samplg strateg, whch essetall comprse of a samplg method ad the estmato procedure. I the choce of a samplg method there are some methods of selecto whle some others are cotrol measures whch help groupg the populato before the selecto process. I the methods of selecto, schemes such as smple radom samplg, sstematc samplg ad varg probablt samplg are geerall used. Amog the cotrol measures are procedures such as stratfed samplg, cluster samplg ad mult-stage samplg etc. A combato of cotrol measures alog wth the method of selecto s called the samplg scheme. We shall descrbe bref the dfferet samplg methods the followg sectos:. SIMPLE RADOM SAMPLIG Smple radom samplg (SRS) s a method of selectg '' uts out of '' uts such that each oe of the possble o-dstct samples has a equal chace of ts beg chose. I 5

2 : A Overvew of Varous Samplg Schemes.. practce, a smple wa of obtag a smple radom sample s to draw the uts oe b oe wth a kow probablt of selecto assged to each ut of the populato at the frst ad each subsequet draw. The successve draws ma be made wth or wthout replacg the uts selected the precedg draws. The former s called the procedure of samplg wth replacemet whle the latter samplg wthout replacemet. The uts the populato are umbered from to. A seres of radom umbers betwee ad are the draw ether b meas of a Table of radom umbers or b meas of a computer program that produces such a Table. It ca be see that smple radom samplg wthout replacemet (SRSWOR), the probablt of selectg the uts the sample s equal for all the uts. Let Y be the characterstc of terest. The uts that comprse the populato are deoted b,,...,. Let the populato mea, Y, deote the parameter of terest. Let us draw a smple radom sample, wthout replacemet, of sze. We deote b, the sample mea, a ubased estmator of the populato mea Y. I other words, the average value of over all possble samples ( C ths case) s equal toy. Also, the samplg varace of s gve b V ( ) S ( ) (.) where S ( Y ) s the populato mea square. A ubased estmator of ths samplg varace s gve b Vˆ ( ) ( ) s (.) where s ( ) s the sample mea square. 6

3 : A Overvew of Varous Samplg Schemes.. We ow take a example to llustrate the cocepts of ubasedess of sample mea ( ) ad sample mea square (s ). Suppose we have a populato of = 5 uts, sa U, U, U 3, U 4, U 5. The characterstc uder stud, Y s the eld kg per plot. We wat to estmatey, the populato mea of Y o the bass of a sample of sze = draw b SRSWOR. The data ad the aalss are as gve the Table below: Uts ( U ) U U U 3 U 4 U 5 Total Values of (kg / plot) Populato Mea ( Y ) 30 kg PopulatoMea Square( S ) Y Y = Uts the sample Sample Observatos Sample Mea Sample Mea Square (s ) U, U 3, U, U 3 3, U, U 4 3, U, U 5 3, U, U 3 8, U, U 4 8, U, U 5 8, U 3, U 4 5, U 3, U 5 5, U 4, U 5 30, Average 300/0 = /0 = 4.5 A smlar approach apples whe samplg s wth replacemet (SRSWR). I ths case, there are possble samples. The estmator of populato mea, samplg varace of the estmator, ad estmator of the samplg varace are gve as (.3) ( ) (.4) V 7

4 : A Overvew of Varous Samplg Schemes.. ad ˆ s ( ) (.5) V where ( Y ) s the populato varace ad s s the sample mea square. Cosder all possble samples of sze whch ca be draw from a gve populato. I SRSWOR scheme, there wll be all C possble samples. For each sample, oe ca compute a statstc, such as the mea, stadard devato etc., whch wll var from sample to sample. I ths maer, oe ca obta a dstrbuto of the statstc whch s called ts samplg dstrbuto. For estmatg the populato total Y, we have a estmator Y ˆ /. (.6).e., the populato sze multpled b the sample mea. Ths estmator ca be expressed as Y w / ˆ, where w /. The costat / s the samplg weght ad s the verse of the samplg fracto /. The estmator of samplg varace of Yˆ s gve b Y ˆ Vˆ.. Vˆ ˆ V (.7) ad the estmator of stadard error of Yˆ s gve b SEˆ Y ˆ SEˆ.. SEˆ (.8) 8

5 : A Overvew of Varous Samplg Schemes.. From the above, t s evdet that uder Smple Radom Samplg Wth Replacemet (SRSWR), ) the sample mea s ubased for the populato mea Y, ) sample mea square (s ) s ubased for the populato varace, ad ) V =. Lke-wse, uder Smple Radom Samplg Wthout Replacemet (SRSWOR), ) the sample mea s ubased for the populato mea Y, ) sample mea square (s ) s ubased for the populato mea square ( S ), ad ) V = S 3. ESTIMATIO OF POPULATIO PROPORTIO Sometmes, the uts the populato are classfed to two groups () havg a partcular characterstc ad () ot havg that characterstc. For example, a crop feld ma be rrgated or ot rrgated. If t s rrgated, we sa that t possesses the characterstc, rrgato. If t s ot rrgated, we sa that t does ot possess the partcular characterstc of rrgato. If we are terested estmatg the proporto of rrgated felds, the populato of felds ca be defed wth varate as havg value f the feld s rrgated, a value 0, otherwse. If the total umber of rrgated felds be out of, the (.) Thus, Y P proporto of rrgated felds the populato (.) 9

6 : A Overvew of Varous Samplg Schemes.. ad. P (.3) Thus the problem of estmatg a populato proporto becomes that of estmatg a populato mea b defg the varate as above. If uts out of a radom sample of sze possess that characterstc, the proporto of rrgated felds the sample s gve b p. Thus. p (.4) Hece a ubased estmator of P s gve b P ˆ ( ) p sample mea (.5) I samplg wthout replacemet, the varace of p s gve b ( ) P. Q V ( p). (.6) ( ) where Q = P. I samplg wth replacemet, the varace of p s gve b P. Q V ( p) (.7) I samplg wthout replacemet, a ubased estmator of V(p) s gve b 0

7 : A Overvew of Varous Samplg Schemes.. ( ) p. q V ˆ( p). (.8) ( ) where q = p. I samplg wth replacemet, a ubased estmator of V(p) s gve b p. q V ˆ( p) (.9) ( ) 4. EMPIRICAL EXERCISE The data gve below pertas to the average eld of wheat crop ( qutals) pertag to 08 vllages a Block of a Dstrct: Vllage Sl. os. Yeld ( qutals) Select a radom sample of sze 0 b smple radom samplg wthout replacemet (SRSWOR) ad estmate the average eld alog wth ts stadard error o the bass of selected sample uts.. Set up 95% cofdece terval for the populato mea. SOLUTIO: As the populato sze = 08 s a three dgt umber, so for selectg a smple radom sample of sze = 0, we shall select three-dgt radom umbers from the Radom umber Table (from 000 to 97, whch s the hghest multple of 08 up to 999) as follows:

8 : A Overvew of Varous Samplg Schemes.. Radom umber Samplg Ut Sl. o. (Remader of Radom umber/08) Yeld (q) Estmate of Populato Average eld = ˆ 9 Y 9. 0 q Estmate of Populato Total Yˆ q The estmate of stadard error of Yˆ s gve b SEˆ Y ˆ SEˆ.. SEˆ ˆ where SE. s where s q 0 So, s q ˆ Hece, SE ) The 95% cofdece terval for populato mea s gve b t ˆ 0.05/(0 9) df SE

9 : A Overvew of Varous Samplg Schemes.. So, the 95% cofdece terval for populato mea s ( to ).e. (0.3, 38.09). It ca be clearl see that the populato mea 330 Y q s cotaed ths cofdece terval. It ma be metoed here that 08 out of total umber of possble samples.e. 08 C 0, the populato mea wll be cotaed such lke cofdece tervals correspodg to 95% of the total umber of samples. 5. STRATIFIED RADOM SAMPLIG I radom samplg techques, the selecto s based o radom mechasm. Most of the tmes, radom umbers are used for selecto purposes. The smplest method of selecto s smple radom samplg (SRS) whch ever sample gets a equal chace of selecto. I SRS, uts are selected wth equal probablt at ever draw. We have see that the precso of a sample estmate of the populato mea depeds ot ol upo the sze of the sample ad the samplg fracto but also o the populato varablt. Selecto of a smple radom sample from the etre populato ma be desrable whe we do ot have a kowledge about the ature of populato, such as, populato varablt etc. However, f t s kow that the populato has got dfferetal behavour regardg varablt, dfferet pockets, ths formato ca be made use of provdg a cotrol the selecto. The approach through whch such a cotrolled selecto ca be exercsed s called stratfed samplg. I case of SRSWOR, the samplg varace of the sample mea s V ( ) ( ) S Clearl, the varace decreases as the sample sze creases whle the populato varablt S decreases. ow oe of the objectves of a good samplg techque s to reduce the samplg varace. So we have to ether crease or decrease S. Apart from the sample sze, therefore, the ol wa of creasg the precso of a estmate s to devse samplg procedure whch wll effectvel reduce S.e. the heterogeet the populato. I fact, S s a populato parameter ad s heret wth the populato, therefore, t caot be decreased. Istead, the populato ma be dvded to umber of groups (called strata), thereb, cotrollg varablt wth each group. Ths procedure s kow as stratfcato. I stratfed samplg, the populato cosstg of uts s frst dvded to sub-populatos of szes,,, uts respectvel. These sub-populatos are o-overlappg ad together the comprse the whole of the populato.e. =. These sub-populatos are called strata. To obta full beeft from stratfcato, the values of s must be kow. Whe the strata have bee determed, a sample s draw from each stratum, the drawgs beg made depedetl dfferet strata. If a smple radom sample s take each stratum the the procedure s termed as stratfed radom samplg. As the samplg varace of the estmate of populato mea or total depeds o wth strata varato, the 3

10 : A Overvew of Varous Samplg Schemes.. stratfcato of populato s doe such a wa that strata are homogeeous wth themselves wth respect to the varable uder stud. However, ma practcal stuatos, t s usuall dffcult to stratf wth respect to the varable uder cosderato especall because of phscal ad cost cosderato. Geerall the stratfcato s doe accordg to admstratve groupgs, geographcal regos ad o the bass of auxlar characters correlated wth the character uder stud. Let,,, deote the szes of strata, such that the populato). (total umber of uts Let j deotes the j th observato the th stratum. Deote Y j j as populato mea. Aga, S ( j Y ) = populato mea square, j S ( j Y ) = populato mea square the th stratum, j where Y j j = populato mea for the th stratum. We select a smple radom sample of sze from the frst stratum, of sze from the secod stratum,..., of sze from the th stratum ad so o such that = (sample sze). The populato mea Y ca be wrtte as Y Y P Y (4.) 4

11 : A Overvew of Varous Samplg Schemes.. where P. Sce wth each stratum, the samples have bee draw b SRSWOR, ( ), j sample mea for the th stratum s a ubased estmator of Y ad obvousl j st P (4.) The weghted mea of the strata sample meas wth strata szes as the weghts, wll be a approprate estmator of the populato mea. Clearl st s a ubased estmator of Y, sce st E( ) E( P ) P E( ) P Y Y (4.3) Sce the sample the th stratum has bee draw b SRSWOR, so V - S The samplg varace of st s gve b V ( ) P V ( ) P P Cov(, ) (4.4) st j j Sce the samples have bee draw depedetl each stratum, so Cov(, ) 0 ad so j V ( st) P V ( ) P ( ) S (4.5) 5

12 : A Overvew of Varous Samplg Schemes.. Sce the sample mea square for the th stratum, the populato mea square for the th stratum, V ( st ) s gve b s j ( j ) ubasedl estmates S, t follows that a ubased estmator of Vˆ( st ) V ( st ) P ( ) s (4.6) From the above, we see that samplg varace of stratfed sample mea depeds o S 's, varablt wth the strata whch suggests that the smaller the S 's,.e. the more homogeeous the strata, greater wll be the precso of the stratfed sample. 6. EMPIRICAL ILLUSTRATIO The data gve below pertas to the total geographcal area 0 vllages of a Block. Treatg ths as populato of 0 uts, stratf ths populato three strata takg stratum szes to be vllages wth geographcal area, 50 ha or less, vllages wth geographcal area betwee 50 ad 00 ha ad vllages havg geographcal area more tha 00 ha. A sample of 6 vllages s to be selected b takg vllages each stratum. Compare the effcec of stratfed samplg wth correspodg ustratfed smple radom samplg. Vllage Sl.o.: Geographcal : Area ( ha.) Vllage Sl.o.: Geographcal : Area ( ha) SOLUTIO: Clearl = 0, = 6. 6

13 : A Overvew of Varous Samplg Schemes Populato Mea Y 9ha 0 0 Populato Mea Square S ( Y ) ( Y 9680 ) 506ha 9 Samplg Varace of Smple Radom Sample Mea V ( ) ( ) S 590ha ow stratf the populato accordg to gve strata szes to followg three strata: STRATA Sl. o. UITS I (less tha 50 ha) II (betwee 50 ad 00 ha) III (more tha 00 ha) Clearl, = 8, Y = 03.3 ha, S = 070 ha = 7, Y = ha, S = 06 ha 3 = 5, Y 3 = 76.0 ha, S 3 = 330 ha From each stratum, a sample of vllages has bee take so = = 3 =. ow, V ( st) P ( ) S 67ha Obvousl the stratfcato has reduced the samplg varace of the sample mea from 590 ha ( case of SRSWOR) to 67 ha ( case of stratfed samplg).e. a reducto of about 89 per cet. I stratfed samplg, havg decded the strata ad the sample sze, the ext questo whch a 7

14 : A Overvew of Varous Samplg Schemes.. surve samplg expert has to face s regardg the method of selecto wth each stratum ad the allocato of sample to dfferet strata. The expresso for the varace of stratfed sample mea shows that the precso of a stratfed sample for gve strata depeds upo the 's whch ca be fxed at wll. The gudg prcple the determato of the 's s to choose them such a maer so as to provde a estmate of the populato mea wth the desred degree of precso for a mmum cost or to provde a estmate wth maxmum precso for a gve cost, thus makg the most effectve use of the avalable resources. The allocato of the sample to dfferet strata made accordg to ths prcple s called the prcple of optmum allocato. The cost of a surve s a fucto of strata sample szes just as the varace. The maer whch cost wll var wth total sample sze ad wth ts allocato amog the dfferet strata wll deped upo the tpe of surve. I eld estmato surves, the major tem the surve cost cossts of labour charges for harvestg of produce ad as such surve cost s foud to be approxmatel proportoal to the umber of crop cuttg expermets (CCEs). Cost per CCE ma, however, var dfferet strata depedg upo labour avalablt. Uder such stuatos, the total cost ma be represeted bc c where c s the cost per CCE the th stratum. Whe c s same from stratum to stratum, sa c, the total cost of a surve s gve b C c. The cost fucto wll chage form, f travel cost, feld staff salar, statstcal aalss etc. are to be pad for. To fd optmum values of (cost fucto beg C c ), cosder the fucto V ( st ) C (4.7) where s some costat. ow, V ( st) C P ( ) S c = P S +. c. + terms depedet of = ( P S / c ) + terms depedet of 8

15 : A Overvew of Varous Samplg Schemes.. Clearl, V ( st ) s mmum for fxed cost C, or cost of a surve s mmum for a fxed value of V ( st ), whe each of the square terms o rght-had sde of the above equato s zero.e. P S = c (=,,...,) or = P S. c (=,,...,) (4.8) From the above, oe ca easl fer that: the larger the stratum sze, the larger should be the sze of the sample to be selected from that stratum, the larger the stratum varablt, the larger should be the sze of the sample from that stratum, ad the cheaper the cost per samplg ut a stratum, the larger should be the sample from that stratum. The exact value of for maxmzg precso for a fxed cost C 0 ca be obtaed b evaluatg /, the costat of proportoalt as P S c C0 (4.9) P S c ad the total sample sze as C 0 ( P S / P S c c ) (4.0) The allocato of the sample sze '' to dfferet strata accordg to the above equato s kow as optmum allocato. Whe c s the same from stratum to stratum.e. c = c (sa), the cost fucto takes the form C = c., or other words, the cost of surve s proportoal to the sze of the sample, the optmum values of 's are gve b 9

16 : A Overvew of Varous Samplg Schemes.. P S P S (=,,...,) (4.) The allocato of the sample accordg to the above equato s kow as ema Allocato. O substtutg for V ( st ) expresso (4.5), we obta V ( st) ( P S ) P S (4.) where the subscrpt stads for stratfcato wth ema Allocato. Aother logcal approach of allocato appears to be to allocate larger sample szes for larger strata.e. or =. (=,,...,). The allocato of sample sze accordg to ths equato s kow as proportoal allocato ad V( st ) ths case becomes VP ( st) ( ) P S (4.3) where the subscrpt P dcates the stratfcato wth proportoal allocato. 7. CLUSTER SAMPLIG A cluster ma be defed as a group of uts. Whe the samplg uts are clusters, the method of samplg s kow as cluster samplg. Cluster samplg s used whe the frame of uts s ot avalable or t s expesve to costruct such a frame. Thus, a lst of all the farms the dstrcts ma ot be avalable but formato o the lst of vllages s easl avalable. For carrg out a dstrct level surve amed at estmatg the eld of a crop, t s practcall feasble to select vllages frst ad the eumeratg the elemets ( ths case farms) the selected vllage. The method s operatoall coveet, less tme cosumg ad more mportatl such a method s cost-wse effcet. The ma dsadvatage of cluster samplg s that t s less effcet tha a method of samplg whch the uts are selected dvduall. The effcec of cluster samplg procedure creases as the heterogeet betwee uts belogg to same cluster creases. Cluster samplg becomes more effcet tha elemet samplg f the uts pertag to same cluster are egatvel correlated. Let there be clusters the populato. Further, let M be the sze of each cluster. We deote b Y the character uder stud. We defe 30

17 : A Overvew of Varous Samplg Schemes.. j = value of the characterstc uder stud for j th elemet, (j=,,,m) the th cluster, (=,,,) M th. j the mea per elemet of the cluster, M j Y.. the meaof the cluster meas the populato, M Y.. j. the mea per elemet the populato. M j It s clear that Y. Y... Ths s so sce the clusters are of equal sze. If, however, the clusters var sze, the two meas wll geerall ot be equal. We deote b cl, the estmator of Y.. as cl cl M. j the meaof cluster meas a sampleof clusters (5.) M j s a ubased estmate of Y.. ad ts varace s gve b Clearl V ( cl ) ( ) Sb ( 5.) where S b (. Y.) s the mea square betwee cluster meas the populato. The varace of the mea of cluster meas terms of tra-class correlato c (tracluster correlato betwee elemets belogg to the same cluster), f s suffcetl large, s gve b V ( cl S ) M ( M ) c (5.3) 3

18 : A Overvew of Varous Samplg Schemes.. where S M M j ( j Y ).. s the mea square betwee elemets the populato. If a equvalet sample of M elemets were selected from the populato of M elemets b SRSWOR, the varace of the mea per elemet M would be V ( M ) S M M ( ) (5.4) Accordgl, the relatve effcec of a cluster as the ut of samplg compared wth that of a elemet s gve b V ( M ) S R. E. (5.5) V ( ) M S cl b Thus, the relatve effcec of cluster samplg creases wth decrease both S b ad M. Sce relatve effcec of cluster samplg depeds upo M, t s atural to determe the optmum sze of the cluster. As the case of stratfed samplg, the optmum sze of the cluster ca be determed b mmzg varace for a fxed cost or vce-versa. A smple cost fucto ca be C = c M +c d (5.6) where C = total cost of surve operato, c = cost of eumeratg a elemet cludg the tme spet ad cost of trasportato wth clusters, c = per ut cost of travellg a ut dstace betwee clusters, ad d = the dstace betwee the clusters. 3

19 : A Overvew of Varous Samplg Schemes.. B mmzg the varace subject to the above cost fucto, we obta c M (5.7) c C Thus, M wll be smaller f c ad C are large ad c s small. 8. MULTI-STAGE SAMPLIG Geerall, elemets belogg to the same cluster are more homogeeous as compared to those elemets whch belog to dfferet clusters. Therefore, a comparatvel represetatve sample ca be obtaed b eumeratg each cluster partall ad dstrbutg the etre sample over more clusters. Ths wll crease the cost of the surve but the proportoate crease cost vs-à-vs cluster samplg wll be less as compared to crease the precso. Ths process of frst selectg clusters ad the further samplg uts wth a cluster s called as two-stage samplg. The clusters a two-stage sample are called as prmar stage uts (psu s ) ad elemets wth a cluster are called as secodar stage uts (ssu s ). A two-stage sample has the advatage that after psu s are selected, the frame of the ssu s s requred for the sampled psu s ol. The procedure allows the flexblt of usg dfferet samplg desg at the dfferet stages of selecto of samplg uts. A two-stage samplg procedure ca be easl geeralzed to mult-stage samplg desgs. Such a samplg desg s commol used large scale surves. It s operatoall coveet, provdes reasoable degree of precso ad s cost-wse effcet. 9. SYSTEMATIC SAMPLIG I the method of sstematc samplg, ol the frst ut s selected at radom whle rest of the uts are selected accordg to a pre-determed patter. Let ad be the populato sze ad sample sze respectvel. Further, =.k, where k s a teger. A radom umber s selected such that < < k. The the sample cotas uts wth seral umber, +k, +k,, +( -)k. Sstematc samplg ca be used stuatos such as selecto of k th strp forest samplg, selecto of cor felds ever k th klometre apart for observato o cdece of borers, or the selecto of ever k th tme terval for observg the umber of fshg crafts ladg at a cetre. The method of sstematc samplg s used o accout of ts low cost ad smplct the selecto of the sample. It makes cotrol of feld work easer. Sce ever k th ut wll be the sample, the method s expected to provde a evel balaced sample. 33

20 : A Overvew of Varous Samplg Schemes.. Whe s ot a exact multple of.e. = k + r, a method of samplg called crcular sstematc samplg (gve b Lahr) s used to select the sample. Ths method cossts of selectg a radom umber from to ad thereafter selectg cclcall ever k th ut utl uts have bee chose the sample. Cclc selecto meas assgg umber (+) to st ut, (+) to d ut ad so o, order to cotue the selecto procedure whe the th ut has bee reached. A drawback of sstematc samplg s that t s ot possble to get a ubased estmate of the varace of the estmator. However, t s possble to get a ubased estmator of the varace based o umber of sstematc samples. 0. VARYIG PROBABILITY SAMPLIG I SRSWOR, the selecto probabltes are equal for all the uts the populato. However, f the samplg uts var sze cosderabl, SRSWOR ma ot be approprate as t does ot take to accout the possble mportace of the larger uts the populato. To gve possble mportace to larger uts, there are varous samplg methods whch ths ca be acheved. A smple method s of assgg uequal probabltes of selecto to the dfferet uts the populato. Thus, whe uts var sze ad the varable uder stud s correlated wth sze, probabltes of selecto ma be assged proporto to the sze of the ut e.g. vllages havg larger geographcal areas are lkel to have larger populatos ad larger areas uder food crops. For estmatg the crop producto, t ma be desrable to adopt a selecto scheme whch vllages are selected wth probabltes proportoal to ther populato szes or to ther geographcal areas. A samplg procedure whch the uts are selected wth probabltes proportoal to some measure of ther sze s kow as samplg wth probablt proportoal to sze (pps). The uts ma be selected wth or wthout replacemet. I samplg wth replacemet, the probablt of drawg a specfed ut at a gve draw s the same. Procedure of selectg a sample wth varg probabltes ) Cumulatve Total Method To draw a sample of sze from a populato of sze wth ppswr, the procedure s as follows: Let x be a teger proportoal to sze of the th ut (=,...,), we make successve cumulatve totals x x x, x x x,..., x x x... x,, 3 3 x ad draw a radom umber R betwee ad from a radom umber Table. 34

21 : A Overvew of Varous Samplg Schemes.. Select the th ut the populato for whch x x x3,..., x R x x x... x 3 x It s clear that ths procedure of selecto gves to the th ut the populato, a probablt of selecto proportoal to x. Ths procedure s to be repeated tmes, f a sample of sze s requred. ) Lahr s Method The ma drawback of cumulatve total method s that t volves wrtg dow the successve cumulatve totals whch s tme cosumg ad tedous especall f the umber of uts the populato s large. Lahr (95) had suggested a alteratve procedure whch avods the ecesst of wrtg dow cumulatve totals. It cossts selectg a par of radom umbers, sa (, j) such that ad j M where M s the maxmum of the szes of the uts the populato. If j x, the th ut s selected; otherwse t s rejected ad aother par of radom umbers s chose. For selectg a sample of uts wth ppswr, the procedure s to be repeated tll uts are selected. It ca be see that the procedure leads to the requred probabltes of selecto.. DETERMIATIO OF SAMPLE SIZE I the plag of a sample surve, determato of sample sze s a mportat decso whch a surve statstca has to take whle decdg the samplg pla. Oe has to be careful whle decdg the sample sze, because too large a sample mples waste of resources, ad too small a sample dmshes the utlt of the results. A effcet samplg pla should eable a optmum utlzato of budgetar resources to provde the best estmators of the populato parameters. As s well kow, effcec of a estmator s ormall measured b verse of mea square error (or varace case of ubased estmators). A desrable proposto would be to mmze the cost as well as varace smultaeousl. But, ufortuatel, t s ot possble. Wth a crease the sample sze, ormall the cost of the surve creases whle the varace decreases, thereb creasg the effcec. Thus for determato of sample sze, a balace s requred to be struck whch s reasoable wth respect to cost as well as effcec. Samplg theor provdes a framework wth whch the problem of determg sample sze ma be tackled reasoabl. We frst cosder the estmato of sample sze case of smple radom samplg. The problem has bee aalzed a ver elegat wa b cosderg hpothetcal example b Cochra (977). We quote the example, A Athropologst s preparg to stud the habtats of some slad. Amog other thgs, he wshes to estmate the percetage of habtats belogg to blood group O. Co-operato has bee secured so that t s feasble to take a smple radom sample. How large should the sample be? Ths s just a tpcal example. I fact, almost all the samplg vestgatos, oe has to face such problems. A aswer to the questo s ot straght forward. Frst of all, oe must be ver clear about the 35

22 : A Overvew of Varous Samplg Schemes.. objectve of the stud. Or atleast, the user must kow to what use ther results are gog to be put, so that he should be able to aswer as to what s the marg of error he s gog to tolerate the results. I the above example, the Athropologst should be able to aswer as to how accuratel does he wsh to kow the percetage of people wth blood group O? I ths case he s reported to be cotet wth a 5% marg the sese that f the sample shows 43% to have blood group O, the percetage for the whole slad s sure to be betwee 38 ad 48. Sce a radom samplg procedure has bee used, ever sample has got some chace of selecto ad the possblt of gettg the estmates lg outsde the above specfed rage caot be ruled out. Aware of ths fact, the Athropologst s prepared to take a 0 chace of gettg a uluck sample wth the estmate lg outsde the above marg. Wth the above formato, gorg fpc ad assumg that the sample proporto p s assumed to be ormall dstrbuted, a rough estmate of ma be obtaed. I techcal terms, p s to le the rage (P 5), except for a 0 chace. Sce p s assumed to be ormall dstrbuted about the populato proporto P, t wll le the rage (P. ) apart from a 0 chace ( 95% cases). p Further, sce the stadard error of p s approxmatel gve b.q / p P where Q = ( - P), we get. p = 5 or.q / P = 5 or = 4 P. Q / 5 At ths pot a dffcult appears that s commo to all problems the estmato of sample sze. A formula for has bee obtaed, but depeds o some propert of the populato that s to be sampled. Here, t s the quatt P that we would lke to measure. We therefore ask the athropologst f he ca gve us some dea of the lkel value of P. He reples that from prevous data o other ethc groups, ad from hs speculatos about the racal hstor of ths slad, he wll be surprsed f P les outsde the rage 30 to 60%. Ths formato s suffcet to provde a usable aswer. For a value of P betwee 30 ad 60, the product P.Q les betwee 00 ad a maxmum of 500 at P = 50. The correspodg les betwee 336 ad 400. To be o the safe sde, 400 s take as the tal estmate of. 36

23 : A Overvew of Varous Samplg Schemes.. PRICIPAL STEPS IVOLVED I THE CHOICE OF A SAMPLE SIZE A statemet about the marg of error to be tolerated the results. Choce of desred cofdece level. Some equato that coects wth the desred precso of the sample should be foud. Ths equato wll cota, as parameters, certa ukow propertes of the populato. Ths must be estmated order to gve specfc results. Usuall a sample surve, more tha oe characterstc s measured. Sometmes, the umber of characterstcs s large. If a desred degree of precso s prescrbed for each characterstc, the calculato leads to a coflctg values of, oe for each characterstc. Some method must be foud for recoclg these values. Fall, the chose value of must be apprased to see whether t s cosstet wth the resources avalable to take the sample. Ths demads a estmato of the cost, labour, tme ad materal requred to obta the proposed sze of sample. It sometmes becomes apparet that wll have to be drastcall reduced. Oe has to choose whether to proceed wth a much smaller sample sze, thus reducg precso, or to abado efforts utl more resources ca be foud. Regardg the choce of a level for tolerable marg of error ad the cofdece level, the user ormall has ol a vague dea ad t s ol through the dscussos ad clarfcatos that a quattatve specfc measures are obtaed. It ma be remarked that these measures are mal subjectve ad deped largel o the judgmet of the user regardg the mportace, applcablt ad vulerablt of the results. Regardg the sample szes case of smple radom samplg, the cases for qualtatve ad quattatve data are preseted below:. QUALITATIVE DATA: ESTIMATIO OF PROPORTIOS The uts are classfed to two classes, C ad C. Some marg of error d the estmated proporto p of uts class C has bee agreed o, ad there s a small rsk that we are wllg to cur that the actual error s larger tha d;.e., we wat Pr ( p - P d ) = Smple radom samplg s assumed, ad p s take as ormall dstrbuted. ow we kow that, 37

24 : A Overvew of Varous Samplg Schemes.. p P. Q Hece the formula that coects wth the desred degree of precso s d t P. Q where t s the abscssa of the ormal curve that cuts off a area of Solvg for, we fd at the tals. t P Q d (.) t P Q d For practcal use, a advace estmate p of P s substtuted ths formula. If s large, a frst approxmato s t p q p q 0 (.) d V where p q V = desred varace of the sample proporto. 0 I practce, we frst calculate 0. If 0 / s eglgble, 0 s a satsfactor approxmato to the of (). If ot, t s apparet o comparso of () ad () that s obtaed as 0 0 ( 0 ) / ( 0 (.3) / ) 38

25 : A Overvew of Varous Samplg Schemes.. EXAMPLE I the hpothetcal blood groups example, we had d = 0.05, p = 0.5, = 0.05, t =. Thus, (4)(0.5)(0.5) (0.005) Let us assume that there are ol 300 people o the slad. The fpc s eeded, ad we fd 0 ( 0 ) / The formula for 0 holds also f d, p ad q are all expressed as percetages stead of proportos. Sce the product pq Icreases as p moves towards /, or 50 %, a coservatve estmate of s obtaed b choosg for p the value earest to / the rage whch p s thought lkel to le. If p seems lkel to le betwee 5 ad 9 %, for stace, we assume 9 % for the estmato of. 3. QUATITATIVE DATA: ESTIMATIO OF POPULATIO MEA Cosder a populato of sze from whch a smple radom sample s to be selected for estmatg the populato mea Y. Suppose, we wsh to cotrol the relatve error r the estmated populato total or mea. As the sample mea estmates the populato mea Y, the marg of error ad cofdece level are specfed as Y Pr r Y Y r P r Pr Y ry where s a small probablt. We assume that s ormall dstrbuted. Its stadard error s = S 39

26 : A Overvew of Varous Samplg Schemes.. Hece ry t. = t. S (3.) Solvg for gves t S ( ) r Y / ( t S ) r Y ote that the populato characterstc o whch depeds s ts coeffcet of varato S / Y. Ths s ofte a more stable quatt ad easer to guess advace tha S tself. As a frst approxmato, we take t S S (3.) r Y C Y 0 ( ) ( ) substtutg a advace estmate of (S / Y ). The quatt C s the desred (cv) sample estmate. of the If 0 / s apprecable, we compute as (.3) 0 (3.3) 0 If stead of the relatve error r, we wsh to cotrol the absolute error d, we take 0 = t S / d = S / V, where V s the desred varace of Y. Sometmes the specfcato error to be tolerated s ol gve terms of desred per cet S.E. of the estmator e.g. the estmate s desred wth a maxmum of sa 5 % S.E. I such cases, s obtaed from the correspodg formulae. I smple radom samplg, f the desred % S.E. s d, the s gve b d ( C ) 40

27 : A Overvew of Varous Samplg Schemes.. where C s the % cv of the populato. METHODOLOGICAL ISSUES RELATIG TO DETERMIATIO OF SAMPLE SIZE The determato of sample sze s geerall based o (a) the avalable facal ad mapower resources ad (b) the requred level of relablt the estmates expected from the sample. Geerall, t would be preferable to start wth the secod cosderato, ad f the budget s a costrat to assess the precso that ca be acheved uder that costrat order to decde whether the achevable precso would be acceptable, ad f ot, whether the budget should be creased. I a sample surve, the samplg error assocated wth a gve sample sze vares from tem to tme. For major tems of frequet occurrece, such as area uder a crop, geerall the samplg error s less that tha the mor tems of frequet occurrece such as use of pestcdes/ sectcdes. Smlarl, for tems whch have greater varablt, the samplg error would be larger tha that for fewer varables. To decde o the sample sze for the surve, t would be ecessar to calculate the sample sze requred for estmatg wth the requste precso a few major tems of terest ad take the largest of the dcated sze requremets as the sample sze for the surve. Estmates from a surve are ofte requred ot ol at the atoal level but also for the ma rego of a coutr ad for certa domas of stud such as rural ad urba sectors ad specfed groups such as rrgated crop ad u-rrgated crop. Geerall, the sample sze requred for a gve precso at the atoal level would ot gve estmates for the regos ad groups wth the same precso because of effectve sample sze for the rego or group. Compromse soluto volvg ths has to be accepted. OVERALL SAMPLE SIZE If a plot surve s udertake for testg questos ad surve procedure before the ma surve s lauched, t ma be possble to estmate roughl the parameters (populato mea ad stadard devato) requred for the determato of sample sze for the varous tems of terest. However, that ma ot be alwas possble, uless the plot surve s take well advace, because estmates of facal resource requred are to be made avalable to the govermet qute advace to make the requste budget. Thus the determato of the sample sze most cases ma have to be doe advace wthout a plot surve. I such cases the surve Statstca has to make use of the formato whch s readl avalable. He ma ofte have to deped o the results of smlar surve coducted the past, preferabl the same coutr or elsewhere the eghborg coutres. If o such results are avalable, the Statstca has to make reasoable guesses of the dfferet parameters whch eter the formula for determato of the sample sze. 4

28 : A Overvew of Varous Samplg Schemes.. If a stratfed mult-stage radom samplg desg s used, whch s usuall the case, the problems are further compouded because formato s requred ot merel o the populato mea ad stadard devato (SD), but also ts compoets of varace betwee prmar stage samplg uts (PSUs) ad wth PSUs. What oe ca do such crcumstaces s to proceed stages b workg out the sample sze requred for a smple radom sample (SRS) ad to make adjustmets to the sample sze to take to accout the effects of mult-stage samplg ad possble stratfcato as llustrated the paragraphs that follow. Geerall the level of precso desred of a estmate s expressed as a percetage of tself or, strctl speakg of the populato parameter. But the subtle dfferece s usuall gored. Let Y be the characterstc uder stud, Y be the populato mea ad be the sample mea. Clearl, s a estmate of Y. The requred samplg precso s prescrbed as a percetage of. For example, samplg precso of should be E per cet of.e., the populato average should le betwee - E x E x ad. 00 l00 Takg 95% cofdece terval, whch s the usual case, ths mples that E x /00 = x SE ( ) SE( ) x 00 E Percetage relatve SE = E If the samplg precso s set at 5% relatve SE, we kow that smple radom samplg SE ( ) Relatve SE ( ) = x 00 Y = Y x 00 = Y x 00 x 4

29 : A Overvew of Varous Samplg Schemes.. = Coeffce t of var ato ( CV) x 00 E CV x 00 = 4 x 0000 x CV E = CV E CV 600 ( CV) 5 If E = 5, = If E = 0, = ( CV) ( CV) 00 Thus to determe the sample szes we requre the value of CV whch ca be estmated o the bass of a prevous surve o the subject or a closel related subject or a surve of a eghborg or smlarl placed coutr. Coeffcet of varato, geerall beg a stable quatt ca also be approxmated o the bass of some related formato. e.g. average household come ca be take to be equal to per capta come x average household sze ad of the kow rage of household 6 come ca be take as a approxmato of the SD o the assumpto of a ormal dstrbuto. However, a better assumpto s that of log-ormal dstrbuto,.e. stead of assumg to be dstrbuted ormall, assume that log e s dstrbuted ormall. (e = 3 =.7) Let us assume that log e s dstrbuted ormall wth mea a ad stadard devato b, the Mea of = e a b ab b Varace of = e e (CV) of = e b 43

30 : A Overvew of Varous Samplg Schemes.. Meda of = e a Mode of = e ab Mea of Meda of b / e Mea Mode of of e 3b / Thus f mea s approxmated (as dcated for household come b per capta come x average household sze) ad the meda or mode s roughl guessed, t s possble to calculate the value of b ad therefore, approxmate C.V. It ma also be oted that a error the estmate of mode effects the estmate of b less tha the same relatve error the estmate of the meda. Also t ma be perhaps less dffcult to make a good guess of the mode tha meda. Further, log ormal dstrbuto has the propert that the proporto of populato wth values less tha or equal to the mea s gve b P(b/), where P(t) s the area to the left of t of a stadard ormal probablt dest fucto. Thus f we guess the proporto of households whose come s less tha or equal to the average, t s possble to obta the value of b b referrg to the correspodg proporto the stadard ormal dstrbuto tables ad thus arrve at a estmate of CV. We shall ow dscuss a smple method that does ot deped upo the estmato of ether the populato mea or the stadard devato, but assumes log-ormal dstrbuto. Take the case of estmato of household come. Based o emprcal data collected from a large umber of coutres, t s reported a techcal stud o Household Icome ad Expedture Surves (Publshed 989 b Statstcs Offce of the Uted atos uder atoal Household Surve Capablt Programme) that earl two-thrds of the populato the case of dstrbuto of come or smlar ecoomc varables le below the average value. Thus wth the propert metoed above para gves CV =.049 It s ot clamed that the observato metoed above s uversal. If the proporto betwee the average s dfferet, CV wll be dfferet as gve below: 44

31 : A Overvew of Varous Samplg Schemes.. Percetage of populato C.V. below average It would be see that as the percetage of populato below the average chages, the value of CV chages ver fast. Thus, rather tha workg wth the assumed proporto of two-thrds, oe ma prefer to err o the safe sde ad take the proporto as 70% ad use CV = or (CV) =. Table 5 (pp80-87) of the U publcato metoed above gves the value of CV ad the proporto of households below the average for some 54 coutres. It ca be see from that Table that a few cases CV exceeded.4. Thus the assumpto of (CV) = s ot urealstc. If (CV) =, we get = 300 ad f (CV) =, as t happes most cases, we get = 600. I the above calculatos, we assumed that fte populato correcto (fpc) factor ca be gored,.e. the populato sze s ver large as compared to sample sze. As a workg rule, whe / < % fpc ca be gored. However, f fpc ca ot be gored, the the sample sze for a smple radom sample wll be case whe fpc ca be gored. =, where s the sample sze for the I large scale sample surves, geerall oe uses a two-stage radom samplg desg. A two-stage desg s geerall less effcet tha SRS of the same ultmate sze ad to acheve the same level of precso as a SRS, a larger umber of ultmate stage samplg uts has to be surveed. Ths s called the desg effect ad the extet of the upward adjustmet to the sample sze depeds o the degree of smlart of secod stage uts wth a PSU, whch s measured b tra-class correlato coeffcet. As a good workg rule, oe ca take the value of for the desg effect as dcated the Hadbook of Household Surves (Revsed Edto), Studes Methods, Seres F.o.3, 984 of the Uted atos. Usg the value for the desg effect, we get = 6400, f (CV) = ad =300, f (CV) =. Here aga the sample sze requred would be less f approprate stratfcato s used at varous stages. SAMPLE SIZE FOR DOMAIS Estmates are geerall requred ot ol at the atoal level but also for certa domas such as geographcal regos, rural ad urba areas. Oe has the to work out the sample sze for each doma ad add them to arrve at the atoal sample sze. We shall assume that doma-wse estmates are requred wth the same precso of 5%. Further, for sake of smplct, we wll deal wth the case of two domas of stud. For sake of llustrato, let us 45

32 : A Overvew of Varous Samplg Schemes.. cosder a Household Icome ad Expedture Surve (HIES) ad rural ad urba areas ma be take as the two domas of stud. Further, let us assume that 80% of households (hh) are rural ad 0% of the hh are urba. Suppose further that the average household come for the urba area s twce the atoal average. Wth the 80 : 0 rato betwee rural ad urba hh, t meas that the average household come the rural areas, s 75% of the atoal average. Let (CV) r ad (CV) u be the CV for rural ad urba areas respectvel. If (CV) =, t ca be show that 0.45 (CV) r ( CV) u =.75 (Please see Aexure) If we assume that (CV) r = (CV) u, we get (CV) r = (CV) u =.4 Thus to obta the same relatve precso of 5% (.e. relatve SE of.5%), the sample sze for each of two sectors wll be.4 x 600 x (desg effect) = 4480 hh Hece, atoal sample sze = 8960 hh. Thus the atoal sample sze s creased b 40% from 6400 to If we do ot have that ma resources, we ca dvde the atoal sample sze of 6400 equall to rural ad urba areas. The effect of allocato of 300 stead of 4480 hh would be that, stead of 5% we shall have 5.9% precso (.95% relatve SE). I the above we have assumed that (CV) r = (CV) u. It s lkel that (CV) u > (CV) r. The table below gves the sample sze requred wth dfferet (CV) u ad (CV) r but satsfg the equato (). (CV) r (CV) u Sample sze ( umber of households ) Rural Urba Total For a other value of (CV) r, all other values ca be determed b lear terpolato. 46

33 : A Overvew of Varous Samplg Schemes.. Aexure Let us use the followg otatos Urba Rural Total Populato sze Mea S.D. We have assumed that x 00 = 0 ad x 00 = 80 We ote that Further we have assumed 0 80, So x Let us ow get the relatoshp amogst, ad. Let x be the observato o the -th ut the urba area, x j be the observato o the jth ut the rural area. We kow that x 47

34 : A Overvew of Varous Samplg Schemes.. 48 x j j j j x x. So = () x x We ote that 6 9 ) 4 3 ( 4,, 5 4, 5, ad o substtuto, () reduces to ( ) ( ) ( ) CV CV x CV x u r = 0.8 ( ) CV u ( ) CV r As (CV) =, so = 0.8( ) CV u ( ) CV r = 0.8 ( ) CV u ( ) CV r

35 : A Overvew of Varous Samplg Schemes.. REFERECES Cochra, W.G. (977): Samplg Techques. Thrd Edto. Joh Wle ad Sos. Des Raj (968): Samplg Theor. TATA McGRAW-HILL Publshg Co. Ltd. Des Raj ad Chadok, P. (998): Sample Surve Theor. arosa Publshg House. Murth, M.. (977): Samplg Theor ad Methods. Statstcal Publshg Socet, Calcutta. Sgh, D. ad Chaudhar, F.S. (986): Theor ad Aalss of Sample Surve Desgs. Wle Easter Lmted. Sgh, D., Sgh, P. ad umar, P. (978): Hadbook of Samplg Methods. I.A.S.R.I., ew Delh. Sgh, R. ad Magat,.S. (996): Elemets of Surve Samplg, luwer Academc Publshers. Sukhatme, P.V. ad Sukhatme, B.V. (970): Samplg Theor of Surves wth Applcato. Secod Edto. Iowa State Uverst Press, USA Sukhatme, P. V., Sukhatme, B.V., Sukhatme, S. ad Asok, C. (984): Samplg Theor of Surves wth Applcatos. Thrd Revsed Edto, Iowa State Uverst Press, USA. 49