SIZE BIAS IN LINE TRANSECT SAMPLING: A FIELD TEST. Mark C. Otto Statistics Research Division, Bureau of the Census Washington, D.C , U.S.A.

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1 SIZE BIAS IN LINE TRANSECT SAMPLING: A FIELD TEST Mark C. Ott Statistics Research Divisin, Bureau f the Census Washingtn, D.C , U.S.A. and Kenneth H. Pllck Department f Statistics, Nrth Carlina State University Bx 8203, Raleigh, Nrth Carlina , U.S.A. SUMMARY, ~" "." llo".' An imprtant prblem in line tranaec\"'ia1iitillin. i. t.ba"t bjects r pint.:..':; ',;,. ":':...;.....;.';::"..~:. :..r..:.:...:,:.,,::..;.:... ';,.:':":::'=:;.: ~,,",. clusters f bjects f different m.fi. have.ditt~req~:jlicllfjng prbabilities. In a recent paper Drummer and McDnsld (1987) develp a bivariate sighting functin. Their functin is dependent.-n 1erPendic~lardistance and bject size. One imprtant special case i:s.~._ex~c!nsin t 'the expnential pwer J series sighting functin first prpsed'-by Pllck (1978).,. In this nte empirical evidence is given fr this model based n-a'field test t line transect sampling thery. Beer cans we1"!l:!j.sed t simulate pint clusters f bjects with 1, 2, 4 and 8 cluster sizes. Key wrds: Clustered ppulatins; Line transect; Size biased sampling; _ Weighted distributins; Expnential pwer series distributins.

2 -2-1. Intrductin An imprtant prblem in line transect sampling is that bjects (r pint clusters f bjects) f different sizes have different sighting prbabilities. This vilates an assumptin f the standard mdel (Burnham et ale 1980) and is an example f size biased sampling (Drummer and McDnald 1987). In a recent paper Drummer and McDnald (1987) present a general mdel fr size bias in line transect sampling. T illustrate, let us cnsider the expnential pwer series sighting functin (Pllck 1978) g(x) =exp [-(x/a) p] (1) where x is the perpendicular distance f the bject f~ the transect line, e expnential sighting functin while if p =2, we have a half nrmal sighting. functin and as p -+ we apprach a unifrji sighting functin. By definitin A is a scale parblleter and p is a shape ~ter. If p = 1, we have an g(x) is the prbability f sighting an bject given it is perpendicular distance x froil the transect line (BurnhmI et a ). A generalized sighting functin dependent n bth perpendicular distance (x) and size (y) wuld be g(x,y) =exp [ - (x/a(y»p(y)] (2) where the scale paraaeter A and the shape parajleter p DOW depend n y. D~r and McDnald (1987) and unpublished wrk by Pllck.(Institute f Statistics, MiJIle Series N. 1669, Nrth Carlina State University, Raleigh, Nrth Carlina, 1985) suggest the JIOdel where p(y) =p, the shape par8lleter des nt depend n y, and the scale parblleter has the simple increasing relatinship A(y) = Ay". (3)

3 -3- Nte that ex is a parameter greater than 0 which defines the degree f size bias. If Gl equals 0 then A(y) =A and there is n size bias. In this paper we present empirical evidence (based n a field test carried ut by Ott) supprting this mdel. 2. Results 2.1 The Field Test In 1982, Ott carried ut a test f the line transect methd in a field near Raleigh, Nrth Carlina. Tw fixed.transects f length 200 m and 160 m were used. Grups f brwn painted beer cans with grup sizes 1, 2, 4 and 8 were used t simulate bjects f differentsisea. The bjects were placed randmly abut each tranaect line t a di8tance f 20 meters. Nine bservers walked alng each transect line and tld Ott which bjects they saw. Frm this Ott was able t recrd the exact perpendicular distance and grup size fr each bject seen by each bserver. The bservers never walked ff the transect line 80 that there was nt the prblem f seeing a secnd bject because f walking t examine an bject. As Ott had a map f every bject it was pssible t estimate the prbability f sighting each bject based n the nine bservers. In Figure 1 the estimated prbability f sighting is pltted against perpendicular distance fr transect 1 fr each grup size separately'. 2.2 The Bxpnential Pwer Sighting Functin The first analysis we carried ut was t fit equatin 2 t the data. We used the prcedure NLIN in SAS (SAS 1982) t fit the expnential pwer sighting functin t each grup size separately. The estimates f A and p are given in Table 1 tgether with their standard errrs. In Figure 1 the

4 -4- estimated sighting functins are pltted. Ntice that there is clear evidence that the scale parameter is a functin f grup size while the pwer parbldeter estimates suggest a cnstant value f p near 2. (The weighted average estimate f p is 2.20 with apprximate standard errr 0.40.) Fr simplicity we decided t fix p = 2 and refit each grup size functin separately using NLIN. The results are given in Table 2. Ntice that the scale parbldeter estimates are almst identical with thse in Table 1. Finally we fitted the relatinship (3) >.(y) = >.yol t ur scale parameter estimates using NLIN and btained i =5.03 (SE = 0.30) and ; = 0.28 (SB = 0.04). The predicted values f icy) are given in Table 2 e and cmpare clsely t the riginal estimates. Nte that ur estimate f ; =0.28 (SE =0.04) shws clearly that there is size bias. (A large saple Z test f H O Ol = 0 VB HI Ol > 0 gives Z = 7 with p value f ) 3. Discussin In this nte we shw strng evidence f the size biased sampling described by Dl'UIIIIIler and McDnald (1987) and we als shw the utility f their bivariate sighting functin In ur case Ol =0.28 (SB =0.04) shwing clear evidence f size bias. We als tried fitting the sblde mdel t the transect 2 data but unfrtunately there was clear evidence f vilatin f a key assumptin f line transect sapling, nbjdely that bjects n the transect line are never missed (Figure 2). BurnhBJD et ale (1980) als nted vilatin f this

5 -5- assumptin in a field study cnducted by Laake n states. We believe this assumptin needs clse scrutiny with real data if it is clearly vilated with artificial data (see als Pllck and Kendall 1987). In retrspect we wish we had nt used a randm distributin f bjects in ur field test. A distributin unifrm within distance classes frm the transect line wuld have been better (by chance we btained sme distance classes with n bjects). This is why in the analyses here we did nt use prgram TRANSECT (Burnham et ale 1980) but rather wrked with the estimated prbabilities f sighting each bject based n the multiple bservers. ACKNOWLEDGEMENTS Ken Burnham prvided helpful ca.enta n the analysis f these data. We thank Luise Lng, Chapel Hill, Ne, fr use f her field. Jim Haefner, Beth Gldwitz, and their bilgy class prvided bservers and helped set up the field experiment. REFERENCES Burnham, K. P., Andersn, D. R. and Laake, J. L. (1980). Estimatin f density frob line transect sampling f bilgical ppulatins. Wildlife Mngraph 72. The Wildlife Sciety, Washingtn, D.C. ~r, T. D. and McDnald, L. L. (1987). Size bias in line transect sampling. Bimetrics 43, Pllck, K. H. (1978). A fa.ily f density estimatrs fr line transect sampling. Bimetrics 34,

6 -6- Pllck, K. H. and Kendall, W. L. (1987). Visibility bias in aerial surveys: a review f estimatin prcedures. Jurnal f Wildlife Management 51, SAS. (1982). NLIN a nn linear regressin prcedure. SAS Users Guide: Statistics Editin. SAS Institute Inc., Cary, Nrth Carlina.

7 Table 1. Fitting the expnential pwer sighting functin t the Ott Beer Can data Transect 1 given in Figure 1. Estimates and (standard errrs) f the scale and shape parameters. Grup Size Scale <i.> Shape (P) (0.44) 2.66 (1.10) (0.72) 1.94 (0.62) (0.95) 1.92 (0.70) (0.54) 3.58 (1. 26) Table 2. Fitting the expnential pwer sighting functin t the Ott Beer Can data Transect 1. Estimates f the scale p&r8lleter and (standard. errrs). ex Predicted values using A(y) = Ay are als presented. The shape parameter is fixed at p = 2. Grup Size Scale (A(y}) Predicted Scale (0.45) (0.69) (0.83) (0.75) 9.00

8 e FIGURE CAPTIONS Figure 1. Ott Beer Can data plt f empirical expnential pwer series sighting functin versus perpendicular distance fr each grup size n Transect 1. The A'S give the prprtin f bservers wh sighted each bject. Figure 2. Ott Beer Can data plt f empirical sighting prbabilities versus perpendicular distance fr grup size 1 OD Transect 2. The A'S give the prprtid f bservers wh sighted each bject.

9 SIGHTING PROBABILITY. SIGHTING PROBABILITY (II :... Clr----,.---r--...,.---,-----,----r--.., , l> l> l> l> l> cr 0 en l> l> > z ",. z > '" (') (') m en m l> ~ J ~'--_---_ J C> :0 (J) N m" SIGHTING PROBABILITY SIGHTING PROBABILITY (II :... O, ,..--,----, "'T'"-_ cr 0 en en > z '" (') m > z (') '" m ca ~ ~' _---I

10 SIGHTING PROBABILITY I I ~ (J'l I I> I [ t e a en -f» (" '"" - z (") m l t -..