1. Estimation equations for strip transect sampling, using notation consistent with that used to

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1 Web-based Supplementary Materials for Line Transect Methods for Plant Surveys by S.T. Buckland, D.L. Borchers, A. Johnston, P.A. Henrys and T.A. Marques Web Appendix A. Introduction In this on-line appendix, we provide:. Estimation equations for strip transect sampling, using notation consistent with that used to develop the line transect methods of the main paper.. Comments on how a full likelihood approach can be implemented, and on why we would not usually wish to do this. 3. A simulation study to explore the performance of the proposed bootstrapping method. 4. Application of the methods to a population of known size.. Strip transect sampling For strip transect sampling, and considering E/W data (indicated by E) only, plant abundance is estimated as AnE Nˆ E = wl E where n E is the total number of plants detected in the E/W strips of half-width w and total length L E, and A is the size of the survey region. We can analyse N/S (N) data similarly, to give Nˆ N. Corresponding variance estimates would typically be obtained by assuming that the systematic sample of strips in each direction is a simple random sample. If there is a strong trend in density through the region, then the

2 stratification methods considered by Fewster et al. (in prep.) may be used to reduce the resulting upward bias in variance estimates. We can combine the two data sets, by including the squares at which two strips intersect just once, along with the number of unique plants detected in these squares. To estimate variance, systematically-spaced sampling units can be defined as shown in Fig. 3 of the main paper, and a simple random sampling variance estimate calculated. The finite population correction is one strip). a / A, where a is the size of the covered region (the total area covered by at least 3. A full likelihood approach Using any of the methods outlined in the main paper or above, we could add a further component to our likelihoods, corresponding to inference about population size N, given the data in our survey strips. For example, if we believed that all plants within the survey strips are detected, and conducted a strip transect analysis based on total number of plants detected within the combined area A + B + C (Fig. ), then the following binomial likelihood allows us to draw inference on abundance N: L ( N n N; n) = P c Pc n ( ) N n where n + = na + nb nc and c P is the (known) proportion of the survey region covered by the survey strips. This approach relies on the assumption that plants are uniformly and independently distributed through the survey region (beyond as well as within the searched strips), and inference is not robust to failures of this assumption, with variances typically underestimated; hence we rely on more robust design-based extrapolation from the covered region to the entire survey region (Borchers and Burnham, 004). An alternative approach to the methods of this paper would be to formulate spatial models for plant density, for example extending the methods of Hedley et al. (004).

3 4. Application to population of known size Yellow split peas were used to simulate a population of plants on a square of mown grass with sides 5m long. They were placed within the survey region in clusters, where cluster sizes were drawn from a Poisson distribution with mean 3. True number of individuals was 4 in 67 clusters. Here, we estimate cluster abundance only. The clusters were distributed through the survey region so as to have a markedly non-uniform distribution with respect to distance from the closest transect line. A crossed design was used, with five lines running in the N/S direction, and five lines E/W. Truncation distance w was set at.5m, so that each strip was 5m wide. Thus strips were contiguous, with five strips spanning the entire survey region. After pooling across each set of five strips, the actual distribution of split peas is shown in Web Fig., together with the distribution of detected peas. A half-normal model was found to provide a good fit to the distance data (Kolmogorov- Smirnov and Cramér-von Mises tests, p > 0.7). AIC indicated that data could be pooled across the two sets of lines (AIC=87.06, compared with summed AIC=87.9 for separate fits; AIC=0.86). For each set of lines, the standard analytic variance was corrected for a finite population correction as described by Buckland et al. (00:87), to allow for the fact that the whole survey region coincided with the covered area. Because for the two sets of lines combined, qe = qn = A/ a =, estimators ˆN and ˆN of the main paper are equivalent. Further, because we assume P = P P say, then we obtain E N = n n Nˆ E + N = and the approximate result (using the Pˆ delta method) [ cv( Nˆ )] Pˆ)] = [ cv( ne + nn )] + [ cv(. To evaluate ( n E nn ) cv +, the variances of n E and n N were again evaluated using a finite population correction. The above analyses gave an estimate of 63 clusters using the data recorded from the N/S lines, with 95% confidence interval (48,83), compared with true abundance of 67. The E/W data 3

4 gave an estimate of 56, with 95% confidence interval (4,76). The above estimator for the pooled data yielded an estimate of 59 clusters with 95% confidence interval (47,74). The distance of objects from the nearest line did not differ significantly from a uniform distribution for either set of lines (Kolmogorov-Smirnov, Cramér-von Mises and χ tests, p > 0.8 in each case). This is despite the fact that the split peas were positioned within strips markedly non-uniformly; pooling of data across five strips in each direction was sufficient to remove evidence of this non-uniformity. Similarly, data were consistent with the assumption that detection on the line was certain ( p ( 0) = ), so results are not given for these cases. 5. Simulation study to assess the performance of the proposed bootstrap method The purpose of this simulation study is to assess the validity of the proposed bootstrapping method. We concentrate therefore on one estimation method: conventional distance sampling as defined in Section.3 of the main paper. We assume that the bootstrap sampling units are as shown in the left-hand side of Fig. 3. These units are systematically spaced through the study region. If objects are uniformly and independently distributed through the region, this should be indistinguishable from simple random sampling, and we explore this scenario first. We then consider the case of a population in which objects occur in clusters, for which the spatial scale of autocorrelation is smaller than the size of a single bootstrap unit. The assumption of independence between units is thus likely to be only slightly compromised, so that a priori, we would still expect the method to do well. The third scenario considers a more extreme case of clustering, with fewer larger clusters, and with the spatial correlation occurring over a greater distance. The final scenario is where there is a linear trend in object density, with zero density along one edge. In this case, systematic sampling gives greater precision than simple random sampling, but by using simple bootstrapping, we fail to exploit this better precision, so that we 4

5 expect the bootstrap variance estimates to be biased high. Fewster et al. (in prep.) explore the issue of variance estimation when transects are laid down according to a systematic design. We thus investigate the following four scenarios, which are designed to be broadly comparable with the Fleecefaulds study area:. A population of N = 000 objects is randomly distributed with constant rate through a square area of side 08m. A systematic grid of lines is positioned over the survey area at random. Line separation is 9m in both the E/W and the N/S directions, and the half-width of each strip centred on each line is w =.5 m. Sampling extends into a bufferzone, extending a distance w =.5 m beyond the survey region boundaries, to avoid bias due to an edge effect (Strindberg et al., 004:00). Defining the Cartesian coordinates of the corners of the study region to be (0,0), (0,08), (08,0) and (08,08), then whatever distribution we give the population of objects, we can ensure a uniform distribution of distances of objects from both the E/W and the N/S lines in a design-based sense by starting the first N/S line at random in the interval (- 4.5,4.5) along the x-axis, and the first E/W line at random in the interval (-4.5,4.5) along the y-axis. Note that we do not achieve this assumption in general if strips are given fixed positions even if we then select a simple random sample of these strips. Detections within the strip are simulated from a half-normal detection function with scale parameter σ = w/ = 0. 75, so that expected sample size from the E/W lines is approximately 00, and similarly for the N/S lines. For each simulated population, the parameter is estimated by maximum likelihood, assuming a truncated halfnormal. (Note that, by generating the position of the first line at random between (- 4.5,4.5), and continuing sampling up to 4.5m outside the opposite boundary, we generate 3 lines. If one of these is more than.5m outside the study region, it is not 5

6 surveyed. If line say is inside the bufferzone, then part of the strip of half-width.5m will be inside the study region, and by surveying this, we exactly compensate for a corresponding part of the strip centred on line 3, which falls outside the study region, given the above set-up. The reason for extending out to 4.5m from the study region boundary is to ensure that partial bootstrap sampling units at the edge are all identified.). As for scenario, except that the population of N = 000 objects is now in 40 clusters, each of 5 objects. The 40 cluster locations are randomly distributed with constant rate through the study region, and the 5 objects associated with each location are given normally distributed x and y coordinates, centred on the cluster location and with standard deviation of m. Objects falling outside the study region are wrapped around to the opposite side. 3. As for scenario, except that there are now just 8 clusters, with 5 objects per cluster, and the standard deviation of each normal distribution is now m. 4. As for scenario, except that the population of N = 000 objects is independently distributed, but with linearly increasing rate with distance along the y-axis, starting at zero density along the x-axis. A single simulated population corresponding to each scenario is shown in Web Fig.. We summarise in Web Table results obtained from 300 simulations of each scenario, using 399 bootstrap replications for each of the 300 populations (Buckland, 984). The following two estimators of population size N were considered: ˆ A n n N = ˆ E + a PE P ˆN N (eqn (3)) and ˆ n n N = 0.5 E + ˆ N qp qpˆ E E N N (eqn (4)). Two implementations of the bootstrap were considered: one in which sampling units were resampled without regard for their size, and one in which the 6

7 resamples were constrained so that each edge configuration appeared in the resample the same number of times as in the original sample. There are two consequences of this second strategy, one beneficial and one detrimental. The beneficial consequence is that the size of the study region, the size of the covered area, and the total line length in each direction are all constant across bootstrap resamples, while the negative consequence is that each of the four corner configurations is unique, so each appears exactly once in each bootstrap resample, which can be expected to generate bias in the variance estimates. Results suggest that the constrained bootstrap performs less well than the unconstrained method for clustered data, and especially poorly for the extreme clustering of scenario 3. Also, estimator ˆN is more precise than ˆN for all but scenario 3, and in that case, the bootstrap variance of ˆN is biased high, so that the improved precision would not be apparent. Using either ˆN or ˆN together with a finite population correction, the means of the unconstrained bootstrap standard errors differ only by small amounts, explicable by Monte Carlo variation, from the standard deviations of estimated abundances for scenarios, and 4, confirming that the proposed bootstrap method works well. For these three cases, incorporating covariance into the variance estimator for ˆN makes little difference. For scenario 3, in which object distribution is very clustered, bootstrap variances are biased high, especially when covariance is allowed for; this possibly reflects instability in the bootstrap estimator for such extreme object distributions (see Web Fig. ), which might be made worse by estimating additional covariance terms. 7

8 References Borchers, D.L. and Burnham, K.P. (004). General formulation for distance sampling. In Advanced Distance Sampling, pp6-30. S.T. Buckland, D.R. Anderson, K.P. Burnham, J.L. Laake, D.L. Borchers and L. Thomas (eds). Oxford University Press, Oxford. Buckland, S.T. (984). Monte Carlo confidence intervals. Biometrics 40, Fewster, R.M., Buckland, S.T., Burnham, K.P., Borchers, D.L., Laake, J.L. and Thomas, L. (in prep.). Estimating the encounter rate variance in distance sampling. Hedley, S.L., Buckland, S.T. and Borchers, D.L. (004). Spatial distance sampling models. In Advanced Distance Sampling, pp S.T. Buckland, D.R. Anderson, K.P. Burnham, J.L. Laake, D.L. Borchers and L. Thomas (eds). Oxford University Press, Oxford. Strindberg, S., Buckland, S.T. and Thomas, L. (004). Design of distance sampling surveys and Geographic Information Systems. In Advanced Distance Sampling, pp90-8. S.T. Buckland, D.R. Anderson, K.P. Burnham, J.L. Laake, D.L. Borchers and L. Thomas (eds). Oxford University Press, Oxford. 8

9 Web Table. Mean bootstrap estimates of standard error for estimators ˆN and ˆN, using both a standard and a constrained bootstrap, averaged across 300 simulations, with a finite population correction ( se ( N ˆ BF ) and se ( N ˆ BF ) ) and without ( se ( N ˆ B ) and se ( N ˆ B ) ). For estimator ˆN, we also show se ˆ BFC ( N ), the bootstrap estimate of standard error with finite population correction and incorporating covariance between n E and n N and between Pˆ E and Pˆ N (see eqn (7)). Also shown is the sample standard deviation of the estimates of N corresponding to each estimator from the 300 simulations of each scenario ( sd( N ˆ,sim ) and sd( N ˆ,sim) ). Standard bootstrap: Scenario Scenario Scenario 3 Scenario 4 se ( N ˆ ) B se ( N ˆ ) BF se ˆ BFC ( N ) sd( N ˆ ) ,sim se ( N ˆ ) B se ˆ BF ( N ) sd( N ˆ ) ,sim Constrained bootstrap: se ( N ˆ ) B se ˆ BF ( N ) sd( N ˆ ) ,sim se ( N ˆ ) B se ˆ BF ( N ) sd( N ˆ ) ,sim 9

10 Web Fig.. Histograms of numbers of split peas by distance from the nearest line. The shaded bars correspond to detected peas. Left-hand-side: N/S lines. Right-hand-side: E/W lines. Frequency Frequency Distance from line Distance from line 0

11 Web Fig.. A single simulated population is shown corresponding to each of scenarios -4. Scenario

12 Scenario

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14 Scenario