Note on the Computation of Sample Size for Ratio Sampling

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Ralph Terry
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1 Not o th Computato of Sampl Sz for ato Samplg alr LMa, Ph.D., PF Forst sourcs Maagmt Uvrst of B.C. acouvr, BC, CANADA Sptmbr, 999 Backgroud ato samplg s commol usd to rduc cofdc trvals for a varabl of trst, basd o th formato obtad for a rlatd varabl dsg dpdt samplg. Usg th formato ad rato samplg, th stmatd total for th varabl s: [] hr N ad ad [] Th probablt that tm appars sampl of obsrvatos s gv b calld th cluso probablt, ad ar masurs for th slctd tm, dcats th Horvtz--Thompso stmator; N s th umbr of tms th ft populato; ad s I modl dpdt samplg, th modl dtrms ho varablt s stmatd, ad th samplg dsg s ot mportat as log as th modl holds. I dsg dpdt samplg, th samplg dsg dtrms ho varablt s stmatd Lohr, 999. Thr s grat dbat o hch approach s bttr for dffrt stuatos. Schrudr t al. 993 ad Lohr 999 prst short dscussos o ths topc ad rfr th radrs to othr dbats prstd ltratur. Lohr 999 plas that th dffrc varacs s du to th dfto of varac. I modl dpdt samplg, th varac s th avrag squard dvato of th stmator from th pctd valu, but ths s avragd ovr all sampls that could b gratd from th populato modl populato s assumd to b ft ad follos th modl gv; Schrudr t al., 993, trmd ths a suprpopulato. I dsg dpdt samplg, th varac s th also th avrag squard dvato of th stmator from th pctd valu, but ths s avragd ovr all sampls that could b obtad usg a gv dsg. E:\BCINENTO\rato samplg\not o th Computato of Sampl Sz for ato Samplg.doc6/0/005

2 th umbr of tms slctd from th populato. Th rato stmator rsults a basd stmat of th total for. Hovr, ths bas dcrass th crasg sampl sz, ad s som cass ll b zro Schrudr, t al If sampls ar tak th rplacmt, th appromat varacs ar: ad + CO, [3] [4] If samplg s thout rplacmt, a addtoal trm must b addd to th appromat varac gv [3], ad ths trm s dpdt upo th samplg dsg usd. Smpl radom samplg For smpl radom samplg SS; samplg th qual probablt, th or thout rplacmt, th stmatd total bcoms: hr N T ad N T N [5] sc N for both th ad thout rplacmt. Th appromat varacs ca b stmatd b: ad N N s + s s [6] [7] s s s, ad hr, ar th usual sampl stmats of th varacs of ad, ad th covarac of ad. Th ft populato corrct factor s rmovd f samplg s th rplacmt. Cochra 977 gvs altratv quatos for th appromat varacs for samplg th rplacmt as: E:\BCINENTO\rato samplg\not o th Computato of Sampl Sz for ato Samplg.doc6/0/005

3 3 N N / N / N hch ar quvalt. / N N Th sampl sz for rato stmato usg smpl radom samplg s: t s t C hr s s + s s AE PE [8] hr AE s th alloabl rror for stmatg th populato ma. Istad of th alloabl rror, th prct rror alloabl rror prssd as a prct of th stmatd ma, hr th stmatd ma s th total gv [] dvdd b N, ad th C stadard dvato prssd as a prct of th stmatd ma ca b usd. For samplg thout rplacmt, th quatos bcom: [9] AE PE + + N t s N t C Uqual probablt samplg th rplacmt For uqual probablt samplg, quatos [] ad [] ar usd to stmat th total ad rato. For samplg th rplacmt, th cluso probablt for tm s: ψ Put s ot th sampl hr ψ s th probablt that ut s slctd th frst dra Lohr, 999. Equato [] ca th b smplfd to Schrudr t al. 983, pag 46: ψ ψ [0] E:\BCINENTO\rato samplg\not o th Computato of Sampl Sz for ato Samplg.doc6/0/005

4 E:\BCINENTO\rato samplg\not o th Computato of Sampl Sz for ato Samplg.doc6/0/005 4 Sc ψ h samplg s th rplacmt. Th follog stmator has b proposd for th appromat varac of th total. [], CO + hr [] ψ ad th quato s smlar for. Hovr, t s ot clar hat th stmator of, CO should b Schrudr t al Schrudr t al. 993, pags 94 ad 95 gv fv varac stmators, basd o ork b Sardal 980 ad 98. For samplg th rplacmt, th frst to varac stmators rsult zro valus Equatos 3.48 ad 3.49 of Schrudr t al.. Th thr rmag stmators ar: [5] [4] mod [3] 5 ' ' 4 3 al t Schrudr s of fcato a s hr hr j j j j j j + + Th thrd stmator s also gv Sardal 984 ad 994 for th spcfc cas of th rato of mas stmator th th varac proportoal to th varabl. Lohr 999, pag 357 gv aothr varac stmator as: L N ad q hr q [6] Lohr also shos ho ths varac stmats dffrs from th modl basd stmator usd b statstcal packags, v h th samplg s basd o SS. All of ths stmators ar basd o th sum of th ghtd rrors, or o a trasformato of th ghtd rrors.

5 5 Th varac of th rato ca th b foud usg: N Istad of usg a fucto to stmat th varac of a uqual probablt rato of mas sampl, ma authors suggst th us of a Talor srs, or rsamplg mthods such as Balacd patd plcato B ad Jackkfg. Bcaus thr s o clar slcto of th varac stmator for uqual probablt samplg, calculatg a appromat sampl sz s dffcult. Altratvs for calculatg th sampl sz ar as follos:. Th dsg ffct s calculatd as th rato of th varac of th uqual probablt dsg, rlatv to th varac of a smpl radom sampl of th sam sz. If th dsg ffct usg uqual probablt samplg stad of qual probablt samplg as ko or could b appromatd, th quato [8] usg smpl radom samplg ould b multpld b th stmatd dsg ffct. Ths s dffcult to appl ulss koldg of th dsg ffct has b obtad through ma survs or through smulatos that ar applcabl to th surv bg coductd. Ksh ad Frakl 974 statd that gral dsg ffcts for compl statstcs ar gratr tha. Thrfor, usg th sampl sz calculato for SS rato samplg ould lkl rsult a udrstmat of th pctd umbr of obsrvatos.. Aothr altratv s to us th Equato [] to obta a stmat of sampl sz basd o uqual probablt samplg th rplacmt for ol as: t ψ AE thout th addtoal formato o th varabl, hr th AE s th alloabl rror for th total of. Ths ould rsult a ovrstmat of th pctd sampl sz, sc ths ould b a lss ffct dsg. Ksh ad Frakl 974 suggst that ths rsults saf ovrstmats. If th alloabl rror s prssd for th ma stad of th total, th quato bcoms: t ψ ψ AE N t [7] AE If th alloabl rror s gv for th rato, ths must b covrtd to a alloabl rror for th total b: E:\BCINENTO\rato samplg\not o th Computato of Sampl Sz for ato Samplg.doc6/0/005

6 6 AE total AE rato Equato [7] ca th b usd. AE rato N AE rato 3. A thrd altratv s to df th dsg ffct as th rato of th varac of th stmatd total usg uqual probablt rato of mas samplg ovr th varac of th total for uqual probablt samplg. Th rsultg dsg ffct ould th b multpld b th sampl sz calculatd for uqual probablt samplg. Usg th quatos gv abov, ths ould b: Eq.3,4,5,6 or rsamplg stmat Eq. Eq.[7] [8] Out of ths thr optos, th thrd altratv appars most fasbl sc a rasoabl stmat of th sampl sz dd could b obtad usg prvousl collctd data for th ara. Hovr, sc thr ar to stmats of th varacs volvd th calculato, th sampl sz ould lkl var gratl for dffrt sts of prvousl collctd data. Uqual probablt samplg thout rplacmt If samplg s thout rplacmt, th calculatos of th cluso probablts ar mor dffcult, sc th probablt of slctg tm a gv dra of a sampl ut dpds o hat happd prvous dras. Also, stmatd varacs ar mor dffcult to calculat. As a altratv, th varac stmators for samplg th rplacmt could b usd. Ths ould rsult a ovrstmat of th varacs, sc samplg th rplacmt s lss ffct tha samplg thout rplacmt Lohr, 999. A smlar approach could b usd for calculatg th sampl sz. Wghtd last squars rgrsso vrsus last squars rgrsso usg samplg ghts A commo approach to corporatg samplg ghts to a lar modl cludg rato of mas samplg s to us ghtd last squars rgrsso stadard statstcal packags. If ghts ar calculatd as th vrs of th cluso probablts as quato [], th statstcal packags that allo th us of ghts for rgrsso ll rsult th sam stmat of th rato slop of th rgrsso as quatos [] ad []. Hovr, th varac stmats for ghtd last squars packag ar ol approprat f th ghts ar qual to th vrs of th varac of th valus Holt t al. 980; Schrudr t al. 993; Lohr 999. Th raso for ths s that th ghtd last squars gv stadard rgrsso packags rsults th follog stmats of th coffcts ad of th varacs: E:\BCINENTO\rato samplg\not o th Computato of Sampl Sz for ato Samplg.doc6/0/005

7 7 T T β W W T ar β σ W hr th ghts ar o th dagoal of th squar W matr, ad th rato s th stmatd slop. Ths s approprat h th ghts ar th vrs of th varacs. If samplg ghts ar ot proportoal to th vrs of th varacs, th coffcts ar stmatd th sam mar, but th varac of thos coffcts s: ar β T T T σ W WW W Kor ad Graubard 995; Lohr 999, pag 36. For rato samplg, both th samplg ghts ad th varac ghts d to b corporatd to th ght matr. Eampl for Uqual Probablt Samplg Sampl data for 05 trs r suppld b Sam Otukol of th Mstr of Forsts, sourcs Ivtor Brach. Th data cludd th actual flld volum ; ANM ad th crusr calld t volum ; CNM. Th lst of data ad all calculatos ar cludd as Appd I. Th data r tall graphd. As pctd, th ANM s vr smlar to th CNM. CNM s foud b calculatg th gross mrchatabl volum usg a tapr fucto b Kozak, ad th subtractg th prct of dca stmatd b th crusr th fld. Sc th tapr fucto s vr prcs, ad th dca prct s commol lo, th CNM valus ar vr smlar to th ANM valus. Basd o th graph, th sampl sz dd to obta a stmat of th total volum for th populato s pctd to b much lor f th CNM formato s usd alog th th flld tr formato. To dffrt valus for th alloabl rror r usd to llustrat th rcommdd approach for calculatg sampl sz tm 3 udr Uqual Probablt Samplg th placmt. Although sampls ar slctd thout rplacmt, th approach for samplg th rplacmt s rcommdd ad usd hr, sc ths mthod s much asr to calculat ad ths rsults a ovrstmat of th sampl sz dd cosrvatv stmat, as otd Uqual Probablt Samplg thout placmt. A 95% probablt t or z appromatl qual to as usd. Th rsults r that , f th CNM formato s ot usd, ad , f th CNM formato s usd. Th stmatd total as,7 both cass. Sttg th alloabl rror for th total as 5% of th stmatd total AE 669, th sampl sz thout th CNM formato as calculatd as 50, ad th th CNM formato as 4 4 flld trs ar dd. Sttg th alloabl rror for th rato as 0% rato th 0.09 uts, th alloabl rror for th total as E:\BCINENTO\rato samplg\not o th Computato of Sampl Sz for ato Samplg.doc6/0/005

8 8 3. Th sampl sz thout th CNM formato as 69, ad th th CNM formato as 3 3 flld trs ar dd. frcs Ctd Cochra, W.G Samplg tchqus, 3 rd dto. Joh Wl & Sos, Toroto. Pp50 to 86. Holt, D., T.M.F. Smth, ad P.D. Wtr grsso aalss of data from compl survs. J.. Statst. Soc. A. 43 Part 4: Kor, E. L. ad B.I. Graubard Aalss of larg halth survs: accoutg for th samplg dsg. J.. Statst. Soc. A 58Part: Lohr, S. L Samplg: dsg ad aalss. Dubur Prss, Toroto. Pp. 59 to 89, 79 to49. Sardal, C.E A to-a classfcato of rgrsso stmato stratgs probablt samplg. Ca. J. Stat. 8: Sardal, C.E. 98. Implcatos of surv dsgs for gralzd rgrsso stmato of lar fuctos. J. Stat. Pla. If. 7: Sardal, C.E Ifrc statstqu t aals ds dos sous ds pla d chatlloag compls. Ls Prsss d L Uvrst d Motral. Pp. 54 to 55. Sardal, C.E. 99. Modl assstd surv samplg. Sprgr-rlag, N ork. 694 pp. Schrudr, H.T., T. G. Grgor, ad G.B. Wood Samplg mthods for multrsourc forst vtor. Joh Wl & Sos, Toroto. Pp 39 to 0 ad 0 to 6. E:\BCINENTO\rato samplg\not o th Computato of Sampl Sz for ato Samplg.doc6/0/005

9 9 Appd I. Eampl for Calculato of Appromat Sampl Sz Usg ato Samplg th Uqual Probablt of Slcto Actual vrsus Crusr olums Actual volum Crusr Estmatd olums Nhat: hat: hat: hat: : hat [] : hat [3] : Dff [3] / [] Eampls for sampl sz stmato usg 95% probablt: AE5% of hat: 669 [Eq. 7]: [Eq. 8]: AE0% of at: covrt to AE for hat: [Eq. 7]: [Eq. 8]: E:\BCINENTO\rato samplg\not o th Computato of Sampl Sz for ato Samplg.doc6/0/005

10 0 CNM * ANM * ph p /ph E E E E E E E E:\BCINENTO\rato samplg\not o th Computato of Sampl Sz for ato Samplg.doc6/0/005

11 E E E E E E E E E E E E:\BCINENTO\rato samplg\not o th Computato of Sampl Sz for ato Samplg.doc6/0/005

12 /ph-hat col.i sq. * *^ -p*col.m E E E:\BCINENTO\rato samplg\not o th Computato of Sampl Sz for ato Samplg.doc6/0/005

13 E E E:\BCINENTO\rato samplg\not o th Computato of Sampl Sz for ato Samplg.doc6/0/005

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