A Combination of Adaptive and Line Intercept Sampling Applicable in Agricultural and Environmental Studies

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1 ISSN Joural of Statstcs Volume 15, 008, pp Abstract A Combato of Adaptve ad Le Itercept Samplg Applcable Agrcultural ad Evrometal Studes Azmer Kha 1 A adaptve procedure s descrbed for e le tercept samplg. Modfed Hase-Hurwtz ad Horvtz-Thompso estmators are used to fd estamtors for e populato mea, populato desty ad coverage. A example s gve to justfy e meod ad to compare t w ordary le tercept samplg. Keywords Adaptve Cluster Samplg, Le Itercept Samplg, Network Samplg 1. Itroducto I may studes e feld of forestry, rage ad harvest maagemet, ecology etc., t wll be dffcult to apply usual samplg meods to detfy rare characterstcs e populato. For example, suppose e pest cotamato of a crop s of terest. It s uderstood at e cotamato would be rare e wde area uder cultvato of e crop, but e cotamato wll also be expected to be clustered,.e., f a certa plot suffers pest attack, e eghborg plots are also lkely to be attacked. Smlar examples ca be cted for dfferet felds as well. I studes of salty of e sol, smlar complexty ca arse. I sub-coastal zoes sol salty s a rare g but t s of clustered ature as well. Thompso (1990) proposed a adaptve cluster samplg (ACS) desg for e populatos whch are rare ad have cluster tedeces. I hs proposed meod, e study area s frst dvded to several geographcal uts. A prmary sample of uts s draw from e study area usg e smple radom samplg (SRS) 1 Isttute of Statstcal Research ad Trag (ISRT), Uversty of Dhaka, Dhaka-1000, Bagladesh Emal: azmer@srt.ac.bd

2 A combato of Adaptve ad Le Itercept Samplg Applcable 45 Agrcultural ad Evrometal Studes meod. Wheever a ut satsfes a gve codto (for example, possessg a mmum level of e characterstc of terest), e eghborg uts of at selected ut are e added to e sample. Aga some of ese added uts may satsfy e codto, er eghborhood uts are furer added to e sample, ad so o. The group of uts whch are selected e fal sample as a result of selecto of a ut e prmary sample s called a etwork. A ut whch does ot satsfy e codto but selected e prmary sample s called a etwork of sze oe. Thompso (199) obtaed a ubased estmator of e populato mea by modfyg e Hase et al. (1953) estmator. He also obtaed e varace ad e estmated varace of e estmator. Thompso (1991a, 1991b) exteded e ACS procedure where e prmary sample s selected usg e systematc or e strp samplg procedure ad for stratfed samplg. But, for e above metoed stuatos where area uder vestgato s large, e ACS procedure mght ot also be helpful sce e SRS observatos are usually scattered over a very bg rego. Oe of e most frequetly used meods for such felds s le tercept samplg (LIS) (see, for example, Thompso, 199), where a le s draw alog e study area ad e partcles (small rego w evdece of salty or pest attack) tersected by e trasect le are studed (Fg. 1). For e estmato of e total umber of partcles usg e LIS meod e wd of e artcle tercepted by e trasect le s requred to be measured. The LIS procedure ca also be used for estmatg e coverage,.e. how much area of e study rego s covered by e partcles? For coverage estmato, e leg of e tersecto of e partcle parallel to e trasect s measured. Fg. shows e measuremet of e wd of e tercepted partcle ad e leg of e tersecto of e partcle parallel to e trasect. Whe dealg w e populatos of above dscussed type ad f e objects are rare ad have cluster tedeces, a ACS ca be used where e prmary sample s selected by e LIS meod stead of e SRS meod. I stuatos lke e pest attack example, e SRS of plots or oly a LIS procedure mght ot result a substatal peetrato of e patter of e pest attack. Moreover, e coverage of e pest cotamato wll be very dffcult to measure. A samplg procedure, applcable to ese types of stuatos s dscussed s paper ad e meod s termed as adaptve le tercept samplg (ALIS) due to e ature of e meod beg a combato of ACS ad LIS.

3 46 Kha Fg. 1: A le tercept samplg Fg. : The wd of e tercepted partcle le tercept samplg

4 A combato of Adaptve ad Le Itercept Samplg Applcable 47 Agrcultural ad Evrometal Studes. Samplg Procedure I e smple LIS meod, pots are radomly selected alog e basele b ad trasact les of leg l are draw alog e study area whch are perpedcular to e basele. The partcles tersected by e trasect les are observed. But f a rare ad cluster populato s uder study, wheever a partcle s tersected by e trasect le t s lkely at ere are more partcles aroud at tersected partcle. Those eghborg partcles ca be added e sample usg e techque of ACS meod. I e adaptve le tercept samplg (ALIS) procedure, wheever a partcle s tersected by e trasect le, half crcles of radus r (r s arbtrarly chose by e researcher) ceterg e partcle are draw o bo sde of e trasect le, f ay partcle s observed w ay of ose half crcles, e half crcle s exteded w radus r ad e selected half crcles are observed. If ay partcle s observed, half crcle s exteded w radus 3r ad studed. The procedure s cotued as log as ay partcle s observed (Fg. 3). Fg. 3:The wd of e tercepted etwork adaptve le tercept samplg

5 48 Kha As a result of selecto of oe partcle e prmary sample a group of partcles ca be added to e sample, s group of partcles s termed as a etwork. The partcles w a etwork have e property at f ay partcle of s etwork s tersected by e trasect le all oer partcles wll be cluded e sample. 3. Estmators usg ALIS 3.1 Hase-Hurwtz Estmator As e ACS s dealg w etworks, e selecto probablty of etwork s requred. The prmary sample s selected by e LIS meod ad hece e probablty depeds o e wd of e selected partcle ad e wd of e basele (see, for example, Thompso, 199). Suppose at ere are partcles e etwork, y s e value of e varable of terest assocated w etwork. Let w be e wd of e basele from whch e perpedcular les tersect e fal half crcles of bo sdes of e trasect (Fg. 4). So e w probablty of selectg e etwork s p. b Now followg e formulas of e LIS estmator of e populato total τ of e varable of terest ad ts estmated varace are proposed as follows: k 1 y ˆ vl, where vl, (3.1) p l1 1 1 ( v ˆ l ) v( ˆ ). (3.) l1 1 Suppose at e study area s A, e e populato desty per ut τ s D. A Ofte D s requred to be estmated. A estmator of D s as follows: k ˆ ˆ 1 y D, (3.3) A A l1 1 p ad ts estmated varace s gve by ˆ 1 1 ( vl ˆ) v( D) v( ˆ). (3.4) A A 1 l1 K

6 A combato of Adaptve ad Le Itercept Samplg Applcable 49 Agrcultural ad Evrometal Studes 3. HorvtzThompso estmator of ALIS The populato total τ ca also be estmated usg e Horvtz-Thompso estmator as follows: k ~, (3.5) 1 y where 1 (1 p ) s e cluso probablty of etwork ad 1 ( ~ tj j y y j v ) y, 1 (3.6) 1 j1 j j where s e jot cluso probablty of ad j etwork ad tj w w j w j j j 1 1, b w w beg e wd of e basele from whch e perpedcular le tj tersects bo e ad j etworks. The populato desty per ut ca also be estmated as ~ k ~ 1 y D, A A 1 ad w ts estmated varace as yy v( D) v( ˆ ) y t A A 1 1 j1 j j 3.3 Coverage Estmato usg ALIS (3.7) j j t j (3.8) Coverage ca also be estmated by ALIS. Let L be e leg of trasect, be e umber of radomly placed trasect, m be e umber of partcles tersected by e trasect les, M be e umber of partcles e study area A ad a k be e area covered by e partcle k (k 1,,..., M ). To measure e legs of e tersectos of every partcles w e trasects, les parallel to e trasect are draw from radom pots of e partcles. For each partcle, e leg of e tersecto of e j partcle etwork tersected by e trasect s measured, let s leg be deoted by x j.

7 50 Kha The average leg of e tersectos for e follows: K x j etwork ca be obtaed as j1 x. (3.9) K The formula for estmator of e coverage be gve as follows: m j1 C C. (3.10) L l1 x 4. A Example j l To llustrate how oe mght use e estmators gve Secto, smulated data by Dggle (1983) s used. He descrbed a codtoal posso cluster process, where 100 elemets are radomly dstrbuted amog 5 parets. Ths data set s cosdered as our populato s example as t satsfes e assumptos of e ACS procedure beg applcable whe e populato s rare ad has cluster tedecy. The ALIS procedure s appled to s populato to obta a estmate of e populato total ad e populato desty. Here our study area A s 196 sq uts ad e leg of e basele s b 14 ad populato total τ s 100 w a desty of 0.51 per sq ut. Adaptve Le Itercept Samplg To perform ALIS, at frst two trasects of legs l 1 ad l are draw from two radom pots of e basele b, where l1 l 14 ut. Frst trasect le tersects oe elemet ad secod trasect tersects two elemets of e populato (Fg. 4). For e frst trasect, two half crcles of radus 0.4 ut are draw o bo sdes of e trasact ceterg e pot of e tersecto of e elemet ad e trasect ad bo half crcles are observed. Elemets are foud o bo of e half crcles, so e half crcles are exteded to radus r.e. 0.8 ut ad observed. But ow M k1 a k A ca

8 A combato of Adaptve ad Le Itercept Samplg Applcable 51 Agrcultural ad Evrometal Studes o elemet s foud o e left half crcle, so e half crcle s exteded oly o e rght sde of radus 3 r.e. 1. ut ad e area of e half crcle s observed. The procedure s cotued as log as a elemet s observed. Fally, as a result of adaptato, a cluster of elemets s foud ad e wd of e half crcles s measured ad t s foud at w ut. Smlar procedure s followed for e secod trasect ad two clusters are foud ad e wd of e half crcles are measured ad t s foud at w 0. 9 ut ad w ut. Now e probabltes of e selecto of e etworks are calculated usg (3.1) as follows: p1 0.71, p 0.064, p ad e total umber of elemets s estmated usg equato (3.) as follows : ˆ , ad e estmated varace of e above estmate s calculated as follows: v( ˆ ) Ths provdes a two-stadard devato lmt of , , 107. Now e desty of e populato s also calculated usg equato (3.3) as follows: D ˆ 0.43, 196 w estmated varace v(d) ˆ Ths provdes a two-stadard devato lmt for e desty as: , , 0.55.

9 5 Kha Fg. 4: A example of adaptve le tercept samplg 5. Coclusos From e sample obtaed for e example, we ca see at e pot estmate of e populato total possesses eough accuracy (83 as a estmate of 100), also e two stadard devato lmt for e populato total cotas e populato total whch s dcatve of e fact at e estmator ad ts varace bo are of applcable level. Smlar features are observed whle estmatg e desty. The ALIS meod s see to be a practcally applcable meod whch ca be advatageously applcable stuato where e varable uder terest s rare ad has a cluster tedecy. Especally agrcultural ad evrometal studes where e geographc or spatal dsperso, desty, coverage of a characterstc are of mportace, e ALIS meod ca be a approprate choce.

10 A combato of Adaptve ad Le Itercept Samplg Applcable 53 Agrcultural ad Evrometal Studes Refereces 1. Dggle, P. J. (1983). Statstcal Aalyss of Spatal Pot Patters, Academc Press, New York.. Hase, M. H., Hurwtz, W. N. ad Madow, W. G. (1953). Sample Survey Meods, Wley Publcatos, New York. 3. Thompso, S. K. (1990). Adaptve cluster samplg. Joural of e Amerca Statstcal Assocato, 85, Thompso, S.K. (1991a). Adaptve cluster samplg: Desgs w prmary ad secodary uts, Bometrcs, 47, Thompso, S.K. (1991b). Stratfed adaptve cluster samplg, Bometrka, 78, Thompso, S. K. (199). Samplg, Joh Wley & Sos, New York.