To use adaptive cluster sampling we must first make some definitions of the sampling universe:

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1 8.3 ADAPTIVE SAMPLING Most of the methods dscussed samplg theory are lmted to samplg desgs hch the selecto of the samples ca be doe before the survey, so that oe of the decsos about samplg deped ay ay o hat s observed as oe gathers the data. A e method of samplg that makes use of the data gathered s called adaptve samplg. For example, dog a survey of a rare plat, a botast may feel cled to sample more tesvely a area here oe dvdual s located to see f others occur a clump. The prmary purpose of adaptve samplg desgs s to take advatage of spatal patter the populato to obta more precse measures of populato abudace. I may stuatos adaptve samplg s much more effcet for a gve amout of effort tha the covetoal radom samplg desgs dscussed above. Thompso (99) presets a summary of these methods Adaptve cluster samplg Whe orgasms are rare ad hghly clustered ther geographcal dstrbuto, may radomly selected quadrats ll cota o amals or plats. I these cases t may be useful to cosder samplg clusters a o-radom ay. Adaptve cluster samplg begs the usual ay th a tal sample of quadrats selected by smple radom samplg th replacemet, or smple radom samplg thout replacemet. Whe oe of the selected quadrats cotas the orgasm of terest, addtoal quadrats the vcty of the orgal quadrat are added to the sample. Adaptve cluster samplg s deally suted to populatos hch are hghly clumped. Fgure 8. llustrates a hypothetcal example. To use adaptve cluster samplg e must frst make some deftos of the samplg uverse: codto of selecto of a quadrat: a quadrat s selected f t cotas at least y orgasms (ofte y ). eghborhood of quadrat x: all quadrats havg oe sde commo th quadrat x. edge quadrats: quadrats that do ot satsfy the codto of selecto but are ext to quadrats that do satsfy the codto (.e. empty quadrats). etork: a group of quadrats such that the radom selecto of ay oe of the quadrats ould lead to all of them beg cluded the sample. These deftos are sho more clearly Fgure 8.3, hch s detcal to Fgure 8. except that the etorks ad ther edge quadrats are all sho as shaded. It s clear that e caot smply calculate the mea of the 37 quadrats couted ths example to get a ubased estmate of mea abudace. To estmate the mea abudace from adaptve cluster samplg thout bas e proceed as follos (Thompso 99):

2 Calculate the average abudace of each of the etorks: k k (8.35) m y here Average abudace of the -th etork y k Abudace of the orgasm each of the k-quadrats the -th etork m Number of quadrats the -th etork From these values e obta a estmator of the mea abudace as follos: x (8.36) here x Ubased estmate of mea abudace from adaptve cluster samplg Number of tal samplg uts selected va radom samplg If the tal sample s selected th replacemet, the varace of ths mea s gve by: ( vâr( x) ( ) (8.37) here vâr( x ) estmated varace of mea abudace for samplg th replacemet ad all other terms are defed above. If the tal sample s selected thout replacemet, the varace of the mea s gve by: ( N ) ( âr( x) N( ) v here N total umber of possble sample quadrats the samplg uverse (8.38) We ca llustrate these calculatos th the smple example sho Fgure 8.3. From the tal radom sample of 0 quadrats, three quadrats tersected etorks the loer ad rght sde of the study area. To of these etorks each have plats them ad oe etork has 5 plats. From these data e obta from equato (8.36): x plats per quadrat

3 Sce e ere samplg thout replacemet e use equato (8.38) to estmate the varace of ths mea: ( N ) ( âr( x) N( ) v (400 0) (400)(0)(0 ) We ca obta cofdece lmts from these estmates the usual ay: x ± t α vâr( x ) For ths example th 0, for 95% cofdece lmts lmts become: t α.6 ad the cofdece ( ) ± (.6) ± or from 0.0 to 0.85 plats per quadrat. The cofdece lmts exted belo 0.0 but sce ths s bologcally mpossble, the loer lmt s set to 0. The de cofdece lmts reflect the small sample sze ths hypothetcal example. Whe should oe cosder usg adaptve samplg? Much depeds o the abudace ad the spatal patter of the amals or the plats beg studed. I geeral the more clustered the populato ad the rarer the orgasm, the more effcet t ll be to use adaptve cluster samplg. Thompso (99) shos, for example, from the data Fgure 8. that adaptve samplg s about % more effcet tha smple radom samplg for 0 quadrats ad early 50% more effcet he 30 quadrats. I ay partcular stuato t may ell pay to coduct a plot expermet th smple radom samplg ad adaptve cluster samplg to determe the sze of the resultg varaces. 3

4 Fgure 8. A study area th 400 possble quadrats from hch a radom sample of 0 quadrats (shaded) has bee selected usg smple radom samplg thout replacemet. Of the 0 quadrats, 7 cota o orgasms ad 3 are occuped by oe or more dvduals. Ths hypothetcal populato of 60 plats s hghly clumped. 4

5 Fgure 8.3 The same study area sho Fgure 8. th 400 possble quadrats from hch a radom sample of 0 quadrats has bee selected. All the clusters ad edge quadrats are shaded. The observer ould cout plats all of the 37 shaded quadrats. 5

{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:

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