Chapter 11 Systematic Sampling
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1 Chapter stematc amplg The sstematc samplg techue s operatoall more coveet tha the smple radom samplg. It also esures at the same tme that each ut has eual probablt of cluso the sample. I ths method of samplg, the frst ut s selected wth the help of radom umbers ad the remag uts are selected automatcall accordg to a predetermed patter. Ths method s ow as sstematc samplg. uppose the uts the populato are umbered to some order. uppose further that s expressble as a product of two tegers ad, so that. To draw a sample of sze, - select a radom umber betwee ad. - uppose t s. - elect the frst ut whose seral umber s. - elect ever th ut after th ut. - ample wll cota, +,+,..., + ( seral umber uts. o frst ut s selected at radom ad other uts are selected sstematcall. Ths sstematc sample s called th sstematc sample ad s termed as samplg terval. Ths s also ow as lear sstematc samplg. The observatos the sstematc samplg are arraged as the followg table: stematc sample umber ample composto + Probablt 3 ( + + ( ( ( + ample mea 3 amplg Theor Chapter stematc amplg halabh, IIT Kapur Page
2 Example: Let 50 ad 5. o 0. uppose frst selected umber betwee ad 0 s 3. The sstematc sample cossts of uts wth followg seral umber 3, 3, 3, 33, 43. stematc samplg two dmesos: Assume that the uts a populato are arraged the form of m rows ad each row cotas uts. A sample of sze m s reured. The - select a par of radom umbers (, j such that ad j. - elect the (, j th ut,.e., - The the rows to be selected are, +, +,..., + ( m ad colums to be selected are j, j+, j+,..., j+ (. th j ut th row as the frst ut. - The pots at whch the m selected rows ad selected colums tersect determe the posto of m selected uts the sample. uch a sample s called a alged sample. Alteratve approach to select the sample s - depedetl select radom tegers,,..., such that each of them s less tha or eual to. - Idepedetl select m radom tegers j, j,..., j m such that each of them s less tha or eual to. - The uts selected the sample wll have followg coordates: ( + r, jr+,( + r, jr+ +,( 3+ r, jr+ +,...,( + r, jr+ + (. uch a sample s called a ualged sample. Uder certa codtos, a ualged sample s ofte superor to a alged sample as well as a stratfed radom sample. Advatages of sstematc samplg:. It s easer to draw a sample ad ofte easer to execute t wthout mstaes. Ths s more advatageous whe the drawg s doe felds ad offces as there ma be substatal savg tme.. The cost s low ad the selecto of uts s smple. Much less trag s eeded for surveors to collect uts through sstematc samplg. 3. The sstematc sample s spread more evel over the populato. o o large part wll fal to be represeted the sample. The sample s evel spread ad cross secto s better. stematc samplg fals case of too ma blas. amplg Theor Chapter stematc amplg halabh, IIT Kapur Page
3 Relato to the cluster samplg The sstematc sample ca be vewed from the cluster samplg pot of vew. Wth, there are possble sstematc samples. The same populato ca be vewed as f dvded to large samplg uts, each of whch cotas of the orgal uts. The operato of choosg a sstematc sample s euvalet to choosg oe of the large samplg ut at radom whch costtutes the whole sample. A sstematc sample s thus a smple radom sample of oe cluster ut from a populato of cluster uts. Estmato of populato mea : Whe : Let : observato o the ut bearg the seral umber + ( j the populato, j,,...,, j,,...,. uppose the draw radom umber s. ample cossts of th colum ( earler table. Cosder the sample mea gve b s j j as a estmator of the populato mea gve b Y j. Probablt of selectg o E( s Y. j th colum as sstematc sample. Thus s s a ubased estmator of Y. Further, Var( ( Y s. amplg Theor Chapter stematc amplg halabh, IIT Kapur Page 3
4 Cosder where j j ( ( Y ( j + ( Y j ( j ( Y j + ws ( + ( Y ws ( j ( j s the varato amog the uts that le wth the same sstematc sample. Thus ( Var( s ( ws ws Varato Pooled wth as a varato of the whole sstematc sample wth. Ths expresso dcates that whe the wth varato s large, the Var( becomes smaller. Thus hgher heterogeet maes the estmator more effcet ad hgher heterogeet s well expected sstematc sample. Alteratve form of varace: Var( s ( Y j Y j ( j Y j + ( j Y ( j Y( Y j j( + ( ( j Y ( Y. j( amplg Theor Chapter stematc amplg halabh, IIT Kapur Page 4
5 The traclass correlato betwee the pars of uts that are the same sstematc sample s E( Y( Y ρ ; ρ w o substtutg j( j E( j Y ( Var( gves j( ( Y( Y j ( Y ( Y ( ( ρ j w Var( s + w( [ + ρw( ]. [ ρ ] w. Comparso wth RWOR: For a RWOR sample of sze, Var( R. ce Thus Var( s ws Var( Var( + ( ws. s s R s ws - more effcet tha R whe >. ws - less effcet tha R whe <. ws - euall effcet as R whe ws. amplg Theor Chapter stematc amplg halabh, IIT Kapur Page 5
6 Also, the relatve effcec of s relatve to R s Thus Var( R RE Var( s s s w + ρw( [ + ρ ( ] ( ; ρ. ( + ρw( - more effcet tha R whe - less effcet tha R whe - euall effcet as R whe ρ w < ρ w > ρ w. Comparso wth stratfed samplg: The sstematc sample ca also be vewed as f arsg as a stratfed sample. If populato of uts s dvded to strata ad suppose oe ut s radoml draw from each of the strata. The we get a stratfed sample of sze. I dog so, just cosder each row of the followg arragemet as a stratum. stematc sample umber ample composto Probablt 3 + ( + + ( ( ( + ample mea 3 amplg Theor Chapter stematc amplg halabh, IIT Kapur Page 6
7 Recall that case of stratfed samplg wth strata, the stratum mea st j j j s a ubased estmator of populato mea. Cosderg the set up of stratfed sample the set up of sstematc sample, we have - umber of strata - ze of strata (row sze - ample sze to be draw from each stratum ad st becomes st j j j j where Var( st Var( j j usg ( j Var R j. j j j ( j j s the mea sum of suares of uts the th j stratum. j ( j j j ( j s the mea sum of suares wth strata (or rows. The varace of sstematc sample mea s amplg Theor Chapter stematc amplg halabh, IIT Kapur Page 7
8 Var( s ( Y j j j ( j j j j + ( j j ( j j (. j j ow we smplf ad express ths expresso terms of traclass correlato coeffcet. The traclass correlato coeffcet betwee the pars of devatos of uts whch le alog the same row measured from ther stratum meas s defed as o Thus ρ E( j Y( Y E( j Y ( j j ( ( j ( j j j j ( ( j j ( ( Var( s ( ( ( + ρ [ + ( ρ ]. (usg Var( s Var( st ( ρ ad the relatve effcec of sstematc samplg relatve to euvalet stratfed samplg s gve b Var( st RE. Var ( + ( ρ o the sstematc samplg s s - more effcet tha the correspodg euvalet stratfed sample whe ρ > 0. - less effcet tha the correspodg euvalet stratfed sample whe ρ < 0 - euall effcet tha the correspodg euvalet stratfed sample whe ρ 0. amplg Theor Chapter stematc amplg halabh, IIT Kapur Page 8
9 Comparso of sstematc samplg, stratfed samplg ad R wth populato wth lear tred: We assume that the values of uts the populato crease accordg to lear tred. o the values of successve uts the populato crease accordace wth a lear model so that a + b,,,...,. ow we determe the varaces of R, s ad st uder ths lear tred. Uder RWOR V( R. Here + Y a b ( + a+ b + a+ b Y ( + + a b a b b + b + b ( + ( + ( b ( + ( + Var( R b. b ( ( +. amplg Theor Chapter stematc amplg halabh, IIT Kapur Page 9
10 Uder sstematc samplg Earler j deoted the value of stud varable wth the th j ut the th sstematc sample. ow j represets the value of [ ( j ] j j [ ] th + ut of the populato, so a+ b+ ( j,,,..., ; j,,...,. j s Var( s ( Y j a+ b{ + ( j } a+ b + + Y a b a b ( + + b + b + + b b ( + (+ + ( + ( 6 ( Var b b ( s ( (. amplg Theor Chapter stematc amplg halabh, IIT Kapur Page 0
11 Uder stratfed samplg st [ ] a+ b+ ( j,,,...,, j,,..., j Var( where j j ( j j ( st j + a+ b{ + ( j } a b + ( j ( j b + ( j b ( ( ( + b ( + Var( st b b If s large, so that s eglgble, the comparg Var( st, Var( s ad V ( R, Var( st : Var( s : Var( R or or or + + ( + : : : ( ( + : + : : + + : Thus Var( st : Var( s : Var( R :: : : o stratfed samplg s best for learl treded populato. ext best s sstematc samplg. amplg Theor Chapter stematc amplg halabh, IIT Kapur Page
12 Estmato of varace: As such there s ol oe cluster, so varace prcple, caot be estmated. ome approxmatos have bee suggested.. Treat sstematc sample as f t were a radom sample. I ths case, a estmate of varace s Var( s swc where s (. wc + j j 0 Ths estmator uder-estmates the true varace.. Use of successve dffereces of the values gves the estmate of varace as ( Var s + j + ( j+ (. ( j 0 Ths estmator s a based estmator of true varace. 3. Use the balaced dfferece of,,..., to get the estmate of varace as + Var( s + + 5( or Var( s ( 4 4. The terpeetratg subsamples ca be utlzed b dvdg the sample to C groups each of sze. c The the group meas are,,..., c. ow fd c t c t c Var( s ( t. cc ( t amplg Theor Chapter stematc amplg halabh, IIT Kapur Page
13 stematc samplg whe. Whe s ot expressble as the suppose ca be expressed as + p; p <. The cosder the followg sample mea as a estmator of populato mea I ths case + j f p + j j f > p. j s p + E( j + j + j p+ j Y. o s s a based estmator of Y. A ubased estmator of Y s * s j j C where C s the total of values of the th colum. E EC * ( s (. Y C Var( * * s c where * c Y. ow we cosder aother procedure whch s opted whe. [Referece: Theor of ample urves, A.K. Gupta, D.G. Kabe, 0, World cetfc Publshg Co.] amplg Theor Chapter stematc amplg halabh, IIT Kapur Page 3
14 Whe populato sze s ot expressble as the product of ad, the let + r. The tae the samplg terval as f r. + f r > Let M g M deotes the largest teger cotaed. g If * ( or +, the the umber of uts expected sample wth probablt * * * + + wth probablt. * * * If *, the we get * r r r + wth probablt + r r r + + wth probablt. mlarl, f * +, the * r ( r r wth probablt ( r r ( r + + wth probablt. + + ( + Example: Let 7 ad 5. The 3 ad r. ce r<, 3. The sample szes would be * r r r + 5 wth probablt + 3 r r r wth probablt. 3 amplg Theor Chapter stematc amplg halabh, IIT Kapur Page 4
15 Ths ca be verfed from the followg example: stematc sample umber stematc sample Probablt Y, Y4, Y7, Y0, Y3, Y /3 6 3 Y4, Y5, Y8, Y, Y4, Y 7 Y, Y, Y, Y, Y /3 /3 We ow prove the followg theorem whch shows how to obta a ubased estmator of the populato mea whe. Theorem: I sstematc samplg wth samplg terval from a populato wth sze, a ubased estmator of the populato mea Y s gve b Y ˆ ' where stads for the th sstematc sample,,,..., ad ' deotes the sze of th sstematc sample. Proof. Each sstematc sample has probablt. Hece ' ˆ EY (. '. ow, each ut occurs ol oe of the possble sstematc samples. Hece ' whch o substtuto Y, EY ( ˆ proves the theorem. Whe, the sstematc samples are ot of the same sze ad the sample mea s ot a ubased estmator of the populato mea. To overcome these dsadvatages of sstematc samplg whe, crcular sstematc samplg s proposed. Crcular sstematc samplg cossts of selectg a radom umber from to ad the selectg Thereafter ever the earest teger to. the ut correspodg to ths radom umber. th ut a cclcal maer s selected tll a sample of uts s obtaed, beg amplg Theor Chapter stematc amplg halabh, IIT Kapur Page 5
16 I other words, f s a umber selected at radom from to, the the crcular sstematc sample cossts of uts wth seral umbers + j, f j j 0,,,...,(. + j, f j > Ths samplg scheme esures eual probablt of cluso the sample for ever ut. Example: Let 4 ad 5. The, earest teger to 4 3. Let the frst umber selected at radom 5 from to 4 be 7. The, the crcular sstematc sample cossts of uts wth seral umbers 7,0,3, 6-4, Ths procedure s llustrated dagrammatcall followg fgure amplg Theor Chapter stematc amplg halabh, IIT Kapur Page 6
17 Theorem: I crcular sstematc samplg, the sample mea s a ubased estmator of the populato mea. Proof: If s the umber selected at radom, the the crcular sstematc sample mea s where, deotes the total of values the th crcular sstematc sample,,,...,. We ote here that crcular sstematc samplg, there are crcular sstematc samples, each havg probablt of ts selecto. Hece, E( Clearl, each ut of the populato occurs of the possble crcular sstematc sample meas. Hece, Y, whch o substtuto E( proves the theorem. What to do whe Oe of the followg possble procedures ma be adopted whe. ( Drop oe ut at radom f sample has ( + uts. ( Elmate some uts so that. ( Adopt crcular sstematc samplg scheme. (v Roud off the fractoal terval. amplg Theor Chapter stematc amplg halabh, IIT Kapur Page 7
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