Chapter 11 Systematic Sampling

Transcription

1 Chapter Sstematc Samplg 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. Suppose the uts the populato are umbered to some order. Suppose 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. - Suppose t s. - Select the frst ut whose seral umber s. - Select ever th ut after th ut. - Sample wll cota,,,..., ( ) seral umber uts. So 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: Sstematc sample umber Sample composto 3 ( ) Probablt ( ) 3 3 ( ) 3 ( ) Sample mea 3

2 Example: Let 50 ad 5. So 0. Suppose frst selected umber betwee ad 0 s 3. The sstematc sample cossts of uts wth followg seral umber 3, 3, 3, 33, 43. Sstematc 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. - Select 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. Such 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, j ),( r, j ),( r, j ),...,( r, j ( ) ). r r 3 r r Such 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.

3 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. So o large part wll fal to be represeted the sample. The sample s evel spread ad cross secto s better. Sstematc samplg fals case of too ma blas. 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,,...,. Suppose the draw radom umber s. th Sample cossts of 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 j th colum as sstematc sample. 3

4 So E( s ) Y. Thus s s a ubased estmator of Y. Further, Cosder where Var( ) ( Y ) s. j j ( ) S ( Y) ( j ) ( Y) j ( j ) ( Y) j ws ( ) S ( Y) S ws ( j ) ( ) j s the varato amog the uts that le wth the same sstematc sample. Thus ( ) Var( s ) S S ( ) S 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. 4

5 Alteratve form of varace: Var( s ) ( Y ) j Y j ( ) j Y j ( j Y) ( j Y)( Y) j j( ) ( ) S ( j Y )( Y ). j( ) The traclass correlato betwee the pars of uts that are the same sstematc sample s E( j Y)( Y) w ; E( j Y) ( j Y)( Y) ( ) j( ). S So substtutg j( ) Var( ) gves ( Y)( Y) ( )( ) S j w S Var( s ) w( ) S w( ). Comparso wth WOR: For a WOR sample of sze, Var( ) S S S. 5

6 Sce Thus Var( s ) S Sws Var( ) Var( ) S S ( Sws S ). s s s ws - more effcet tha whe S S. ws - less effcet tha whe S S. ws - euall effcet as S S whe ws. Also, the relatve effcec of s relatve to s Var( ) RE Var( ) s S S w( ) w( ) ( ) ;. ( ) w( ) Thus s s - more effcet tha whe - less effcet tha whe - euall effcet as whe w w w. 6

7 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. Sstematc sample umber Sample composto 3 ( ) Probablt ( ) 3 3 ( ) 3 ( ) Sample mea 3 Recall that case of stratfed samplg wth strata, the stratum mea st jj 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 = - Sze of strata = (row sze) - Sample sze to be draw from each stratum = ad st becomes st j j j j 7

8 where Var( st ) Var( ) j j usg ( ) Sj Var S j. j S S S wst j wst S j ( j j) s the mea sum of suares of uts the th j stratum. S wst Sj j ( ) j ( ) s the mea sum of suares wth strata (or rows). j j The varace of sstematc sample mea s 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 8

9 So Thus ( ) j wst j j ( )( ) ( ) ( )( ) ( )( ) S j j j j j wst Var( s ) ( ) S ( )( ) wst wstswst S wst ( ) wst. (usg ) j Var( st) Var( s) ( ) wstswst ad the relatve effcec of sstematc samplg relatve to euvalet stratfed samplg s gve b RE ( ). wst So the sstematc samplg s - more effcet tha the correspodg euvalet stratfed sample whe wst 0. - less effcet tha the correspodg euvalet stratfed sample whe wst 0 - euall effcet tha the correspodg euvalet stratfed sample whe wst 0. Comparso of sstematc samplg, stratfed samplg ad wth populato wth lear tred: We assume that the values of uts the populato crease accordg to lear tred. So the values of successve uts the populato crease accordace wth a lear model so that ab,,,...,. ow we determe the varaces of, ad st uder ths lear tred. s 9

10 Uder WOR V( ) Here Y ab ( ) ab ab S Y S. ( ) abab b b b ( )( ) ( ) 6 4 ( ) b ( ) Var( ) b. b ( )( ). 0

11 Uder sstematc samplg Earler j deoted the value of stud varable wth the th j ut the th sstematc sample. ow j th represets the value of ( j ) ut of the populato, so j j ab ( j),,,..., ; j,,...,. s Var( s ) ( Y ) j a b ( j ) j ab ( Y) ab ab b b b ( )() ( ) b ( ) 6 ( ) Var b b ( s ) ( ) ( ).

12 Uder stratfed samplg where S st ab ( j),,,...,, j,,..., j Var( ) S S st wst wst wst Sj j ( ) j ( ) j ab( j) ab ( j) ( ) j b ( ) b ( ) ( ) b ( ) ( ) Var( st ) b b j j If s large, so that s eglgble, the comparg Var( st ), Var( s ) ad V ( ), Var( st ) : Var( ) : Var( ) s or or or ( ) : : : ( )( ) : : : : Thus Var( st ) : Var( s ) : Var( ) :: : : So stratfed samplg s best for learl treded populato. ext best s sstematc samplg.

13 Estmato of varace: As such there s ol oe cluster, so varace prcple, caot be estmated. Some 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 j0 Ths estmator uder-estmates the true varace.. Use of successve dffereces of the values gves the estmate of varace as Var s j ( j) ( ). ( ) j0 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 4 4 Var( s ) 3. 5( 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 c t t c Var( s ) ( t ). cc ( ) t 3

14 Sstematc 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 j f p j j f p. j s I ths case p E( ) j j j p j Y. So s s a based estmator of Y. A ubased estmator of Y s * s j C j where C s the total of values of the th colum. * ( s ) E( C ) E. Y C Var( ) * * s S c where S * c Y. 4

15 ow we cosder aother procedure whch s opted whe. [Referece: Theor of Sample Surves, A.K. Gupta, D.G. Kabe, 0, World Scetfc Publshg Co.] Whe populato sze s ot expressble as the product of ad, the let r. The tae the samplg terval as f f r. r Let If M g M deotes the largest teger cotaed. g * ( or ), the the umber of uts expected sample wth probablt * * * wth probablt. * * * If *, the we get * Smlarl, f r r r wth probablt r r r wth probablt *, the. * r ( r) r wth probablt ( ) r r ( r) wth probablt. ( ) Example: Let 7 ad 5. The 3 ad r. Sce r, 3. The sample szes would be * r r r 5 wth probablt 3 r r r 6 wth probablt. 3 5

16 Ths ca be verfed from the followg example: Sstematc sample umber Sstematc sample Probablt 3 Y, Y4, Y7, Y0, Y3, Y 6 Y4, Y5, Y8, Y, Y4, Y 7 Y, Y, Y, Y, Y /3 /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 ' ˆ Y 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 6

17 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=, 9-4=5. Ths procedure s llustrated dagrammatcall followg fgure

18 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. 8