3 STRATIFIED SAMPLIG I AGRICULTURAL SURVEYS austav Adtya Ida Agrcultural Statstcs Research Isttute, ew Delh-00 3. ITRODUCTIO The prme objectve of a sample survey 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 survey. The populato may cosst of all the households a vllage / localty, all the felds uder a partcular crop. We may 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 ormally, 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 ways. The frst oe s complete eumerato whch meas collecto of data o the survey characterstcs from each ut of the populato. Ths type 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 survey 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 survey operatos. Some of the advatages of sample surveys as compared to complete eumerato are reducto cost, greater speed, wder scope ad hgher accuracy. 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 surveys s the choce of a proper samplg strategy, whch essetally 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, systematc samplg ad varyg probablty samplg are geerally 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 procedure of stratfed samplg the followg sectos. 3. STRATIFIED RADOM SAMPLIG I radom samplg techques, the selecto s based o radom mechasm. Most of the tme, radom umbers are used for selecto purposes. The smplest method of selecto s smple radom samplg (SRS) whch every sample gets a equal chace of selecto. I SRS, uts are selected wth equal probablty at every draw. We have 3.
see that the precso of a sample estmate of the populato mea depeds ot oly upo the sze of the sample ad the samplg fracto but also o the populato varablty. Selecto of a smple radom sample from the etre populato may be desrable whe we do ot have ay kowledge about the ature of populato, such as, populato varablty etc. However, f t s kow that the populato has got dfferetal behavour regardg varablty, 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 smple radom samplg wthout replacemet, the samplg varace of the sample mea s V ( y ) S Clearly, the varace decreases as the sample sze () creases whle the populato varablty 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 oly way of creasg the precso of a estmate s to devse samplg procedure whch wll effectvely reduce S.e. the heterogeety the populato. I fact, S s a populato parameter ad s heret wth the populato, therefore, t caot be decreased. Istead, the populato may be dvded to umber of groups (called strata), thereby, cotrollg varablty wth each group. Ths procedure s kow as stratfcato. I stratfed samplg, the populato cosstg of uts s frst dvded to sub-populatos of,,, uts respectvely. These sub-populatos are o-overlappg ad together they 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 depedetly 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 mea or total depeds o wth strata varato, the stratfcato of populato s doe such a way that strata are homogeeous wth themselves wth respect to the varable uder study. However, may practcal stuatos t s usually dffcult to stratfy wth respect to the varable uder cosderato especally because of physcal ad cost cosderato. Geerally the stratfcato s doe accordg to admstratve groupgs, geographcal regos ad o the bass of auxlary characters correlated wth the character uder study. Let,,..., deote the sze of strata, such that (total umber of uts the populato). Let y j deotes the j-th observato the -th stratum. Deote Y j y as populato mea. j 3.
Aga j S y Y j S y Y j j j wherey = populato mea square = populato mea square the -th stratum Y j j =.populato mea for the -th stratum 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 j j Y Y PY where P = /. Sce each stratum, the samples have bee draw by smple radom samplg, y ( y sample mea for -th stratum) s a ubased estmator j j of Y ad obvously k k y y Py. st The weghted mea of the strata sample meas wth strata sze as the weghts, wll be a approprate estmator of the populato mea. Clearly y st s a ubased estmator of Y, sce k k k E y E P y PE( y ) PY Y st Sce the sample the -th stratum has bee draw by smple radom samplg wthout replacemet, so V ( y ) S The samplg varace of y st s gve by ( ) ( ) (, ) st j j V y P V y P PCov y y Sce the samples have bee draw depedetly each stratum so Cov ( y, y ) = 0 ad so V( y ) = st j P V ( y ) = P S Sce the sample mea square for the -th stratum, 3.3
s y y ubasedly estmates S j j ad t follows that a ubased estmator of V( y ) s gve by st Vˆ( y st ) = P s From above we see that samplg varace of stratfed sample mea depeds o S 's, varabltes 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. 3.3 EMPIRICAL ILLUSTRATIO The data gve below pertas to the total geographcal area 0 vllages of a block. Treatg ths as populato of 0 uts, stratfy 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 by takg vllages each stratum. Compare the effcecy of stratfed samplg wth correspodg ustratfed smple radom samplg. Vllage Sl.o.: 0 0 03 04 05 06 07 08 09 0 Geographcal : 00 080 050 00 50 070 00 50 0 00 Area ( ha.) Vllage Sl.o.: 3 4 5 6 7 8 9 0 Geographcal : 050 40 080 00 050 030 070 090 00 0 Area ( ha) SOLUTIO : Clearly = 0, = 6 Populato Mea Y = 0 0 80 y 9 ha. 0 Populato Mea Square S = y Y = y Y = 9680 9 3.4 506 ha Samplg Varace of Smple Radom Sample Mea V ( y ) S 590 ha ow stratfy the populato accordg to gve strata szes to three strata
STRATA Sl. o. UITS I (less tha 50 ha) 00 050 00 00 050 00 050 030 II (betwee 50 ad 00 ha) 080 00 070 080 070 090 00 III (more tha 00 ha) 50 50 0 40 0 Clearly = 8, Y = 03.3 ha, = 7, Y = 084.3 ha, 3 = 5, Y = 96.0 ha, 3 S = 070 ha S = 06 ha S = 330 ha 3 From each stratum, a sample of vllages s to be take so = = 3 =. ow V ( y ) P S 67 ha st Obvously the stratfcato has reduced the samplg varace of the sample mea from 590 ha ( case of smple radom samplg) to 67 ha ( case of stratfed samplg).e. a reducto of about 89%. I stratfed samplg, havg decded the strata ad the sample sze, the ext questo whch a survey statstca 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 survey s a fucto of strata sample szes just as the varace. The maer whch cost wll vary wth total sample sze ad wth ts allocato amog the dfferet strata wll deped upo the type of survey. I yeld estmato surveys, the major tem the survey cost cossts of labour charges for harvestg of produce ad as such survey cost s foud to be approxmately proportoal to the umber of crop cuttg expermets (CCE). Cost per CCE may, however, vary dfferet strata depedg upo labour avalablty. Uder such stuatos, the total cost may be represeted by C c Where, c s the cost per CCE the -th stratum. Whe c s same from stratum to stratum, say c, the total cost of a survey s gve by C = c. The cost fucto wll chage form, f travel cost, feld staff salary, statstcal aalyss etc. are to be pad for. 3.5
To fd optmum values of (cost fucto beg C = fucto Φ V( y ) μ C where s some costat. ow = = st V( y ) μ C P S μ c st PS μ c +terms depedet of PS μc + terms depedet of 3.6 c ) where cosder the Clearly, V ( y st ) s mmum for fxed cost C, or cost of a survey s mmum for a fxed value of V ( y st ), whe each of the square terms o rght-had sde of above equato s zero.e. P S = c (=,,, ) or = P S. c From the above, oe ca easly 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 varablty, 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 maxmsg precso for a fxed cost C 0 ca be obtaed by evaluatg the costat of proportoalty as μ ad the total sample sze as = = P S c k C = P S C o c o P S c P S c The allocato of sample sze '' accordg to eq. (A) s kow as optmum allocato. Whe c s the same from stratum to stratum.e. c = c (say ), the cost fucto takes the form C = c., or other words, the cost of survey s proportoal to the sze of the sample, the optmum values of 's are gve by (A)
=. PS PS The allocato of the sample accordg to the above formula s kow as eyma Allocato (eyma, 934). O substtutg for V ( y st ) expresso, we obta V ( y st ) = PS P S where the subscrpt stads for the stratfcato wth eyma 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 above s kow as proportoal allocato ad V ( y st ) ths case becomes V y PS ( ) P st where the subscrpt P dcates the stratfcato wth proportoal allocato. REFERECES Cochra, W.G. (977). Samplg Techques. Thrd Edto. Joh Wley ad Sos. Des Raj (968). Samplg Theory. TATA McGRAW-HILL Publshg Co. Ltd. Des Raj ad Chadok, P. (998). Sample Survey Theory. arosa Publshg House. Murthy, M.. (977). Samplg Theory ad Methods. Statstcal Publshg Socety, Calcutta. eyma, J. (934). O two dfferet aspects of the represetatve method: the method of stratfed samplg ad the method of purposve selecto. Joural of Royal Statstcal Socety, 97, 558-606. Sgh, D. ad Chaudhary, F.S. (986). Theory ad Aalyss of Sample Survey Desgs. Wley 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 Survey Samplg. luwer Academc Publshers. Sukhatme, P.V. ad Sukhatme, B.V. (970). Samplg Theory of Surveys wth Applcato. Secod Edto. Iowa State Uversty Press, USA. Sukhatme, P. V., Sukhatme, B.V., Sukhatme, S. ad Asok, C. (984). Samplg Theory of Surveys wth Applcatos. Thrd Revsed Edto, Iowa State Uversty Press, USA. 3.7