A comparison of recapture, removal and resighting designs for population size estimation
Introduction
We consider the problem of estimating the unknown size ν0, of a population based on three competing sampling designs, namely recapture sampling, removal sampling and mark-resighting. Our aim is to quantify the accuracy of the resulting estimators, adjusting for the different sampling efforts involved under the different designs. Throughout we make the following classical assumptions:
(i) There is no immigration or death of individuals in the population during the study period, and the size of the population is constant, apart from possible removals. We say the population is closed.
(ii) Each individual is sampled or sighted according to a Poisson process with the same intensity function for all individuals. We say the population is homogeneous and exhibits no behavioural response.
(iii) The identity of previously sampled individuals is known without error. For instance, if individuals are tagged, then none of these are removed or lost.
In a recapture study, captured individuals have their identity noted, commonly by the attachment of a distinguishing tag. The entire capture history of any individual caught during the experiment is then known. Information about ν0 comes from (i) discovering new untagged individuals, (ii) observing the proportion of marked individuals from subsequent samples.
In a mark-resighting experiment, a known number M0 of marked individuals are deliberately planted in the population. Individuals are then sighted, and the number of marked and unmarked individuals recorded. Information about ν0 now comes solely from the proportion of marked individuals in subsequent samples. The main advantage of this approach is the lower economic cost of sighting compared to capturing individuals. With large animals, such as elks, sightings might be made by helicopter or random ground surveys. This design has been used in elk population studies, see Bear et al. (1989), Minta and Mangel (1989) and Pollock (1981) for further discussion.
In a removal experiment, also known as the catch effort method, sampled individuals are removed from the population rather than being marked and returned. Information about ν0 comes from (i) discovering new individuals, (ii) the rate at which the catch per unit effort declines. The main advantage of this method is that it only involves first captures and so is robust to any behavioural reponse that sampling may induce. Such techniques, firstly used in 1914 for bears in Norway (Hjort and Ottestad, 1933), are now widely used in the study of commerical fishing and small-mammal populations.
The idea of planting a known number of marked individuals prior to the experiment may be combined with the recapture and removal approach, see Mills (1970) and Duran and Wiorkowski (1981), giving rise to plant-recapture and plant-removal experiments. It is of interest to quantify how the number of plants affects the performance of the resulting estimators. Some work in this direction has already been done, see Yip and Fong (1993), Yip (1995).
In this paper, we use martingale theory to derive classes of estimators for ν0 in a continuous time setting. An optimal estimator among the class of estimators for each design is obtained. We compare the efficiencies of the three designs by the information contained in the optimal martingale estimating equations as in Lloyd (1994). Comparison by information is equivalent to comparison by standard deviation for large ν0. Of course, in practice ν0 may not be sufficiently large for the asymptotic results to apply and so the results of a simulation study are reported to confirm the results for finite parameter values.
Section snippets
Optimal estimating equations
We first establish a mathematical framework general enough to include all the three designs described in the introduction, as well as generalisations.
We have a finite homogeneous population of unknown size ν0, whose estimation is the object of the experiment. We initially take a known number M0=θν0 of these, mark them, and reintroduce them into the population as ‘plants’. Subsequently, individuals are sampled (i.e. captured, removed or sighted) over the period [0,τ], according to independent
Asymptotic variance formulae
At time t, let κt be the probability that one of the U0 initially unmarked individuals has been sampled. Let ρt be the probability that one of the ν0 in the initial population is removed. Since it is assumed that individuals behave independently, it follows thatand so Ut,νt satisfy a Law of Large Numbers as ν0 diverges, for fixed t. The stochastic matrix thus converges to the non-stochastic matrixand so the
Optimal planting
Suppose we fix the total cost c of the design. What part of this cost should be spent on planting? Let minimise σ2(c,θ) with respect to θ for fixed c, and denote the minimum value by ? For all three designs, if c>1 then is 1.0 and the variance functions are all zero. This corresponds to initially marking the entire population. Then, for instance, not only asymptotically for large ν0 but for any population size. We henceforth assume that c<1 and note that , since when
Comparison of different designs
We measure relative efficiency of two equal cost experimental designs by the ratio of standard deviations.
Fig. 2 compares recapture and seeding experiments controlled for cost. Three quantities have been plotted, namely,Since ρ3 (dashed) compares experiments with identical seeding, and the sighting experiment is equivalent to the recapture experiment without tagging further individuals, this ratio is
Simulation study
The results of a brief simulation study are reported here, mainly to assess the accuracy of the asymptotic variance formulae on which efficiency comparisons in the previous two sections are based. We also gain a picture of the overall performance of the different estimators. Table 1lists results for ν0=400 for various combinations of c and θ, based on 1000 simulations and 10 000 simulations for the more simply simulated resighting experiment. Each line lists (i) the mean of the estimator, (ii)
Acknowledgements
The work is supported by Hong Kong Research Grant Council. The helpful advice of Professor Anne Chao is also gratefully acknowledged.
References (15)
- et al.
Estimating population size from a removal experiment in discrete time
Statist. Probab. Lett.
(1993) Weak convergence of stochastic integrals related to counting processes
Zeitschrifft fur Wahrverw. Geb.
(1977)- et al.
Evaluation of aerial mark-resighting estimates of elk populations
J. Wildlife Management
(1989) - Duran, J.W., Wiorkowski, J.J., 1981. Capture–recapture sampling for estimating software content. IEEE Trans. Software...
- et al.
Quasi-likelihood and optimal estimation
Int. Statist. Rev.
(1987) - et al.
Initial size estimation for the linear pure death process
Biometrika
(1981) - et al.
The optimum catch Hvalradets skrifter
Oslo.
(1933)
Cited by (5)
Sensitivity-Analysis and Estimating Number-of-Faults in Removal Debugging
1999, IEEE Transactions on ReliabilitySequential procedure for fixed accuracy estimation of the population size in recapture sampling
2003, Australian and New Zealand Journal of StatisticsA unified approach for estimating population size in capture-recapture studies with arbitrary removals
2001, Journal of Agricultural, Biological, and Environmental StatisticsEstimating the population size with a behavioral response in capture-recapture experiment
2000, Environmental and Ecological Statistics