Dung and nest surveys: estimating decay rates

Authors


S. T. Buckland, Centre for Research into Ecological and Environmental Modelling, The Observatory, Buchanan Gardens, St Andrews KY16 9LZ, UK. E-mail steve@mcs.st-and.ac.uk

Summary

  • 1Wildlife managers often require estimates of abundance. Direct methods of estimation are often impractical, especially in closed-forest environments, so indirect methods such as dung or nest surveys are increasingly popular.
  • 2Dung and nest surveys typically have three elements: surveys to estimate abundance of the dung or nests; experiments to estimate the production (defecation or nest construction) rate; and experiments to estimate the decay or disappearance rate. The last of these is usually the most problematic, and was the subject of this study.
  • 3The design of experiments to allow robust estimation of mean time to decay was addressed. In most studies to date, dung or nests have been monitored until they disappear. Instead, we advocate that fresh dung or nests are located, with a single follow-up visit to establish whether the dung or nest is still present or has decayed.
  • 4Logistic regression was used to estimate probability of decay as a function of time, and possibly of other covariates. Mean time to decay was estimated from this function.
  • 5Synthesis and applications. Effective management of mammal populations usually requires reliable abundance estimates. The difficulty in estimating abundance of mammals in forest environments has increasingly led to the use of indirect survey methods, in which abundance of sign, usually dung (e.g. deer, antelope and elephants) or nests (e.g. apes), is estimated. Given estimated rates of sign production and decay, sign abundance estimates can be converted to estimates of animal abundance. Decay rates typically vary according to season, weather, habitat, diet and many other factors, making reliable estimation of mean time to decay of signs present at the time of the survey problematic. We emphasize the need for retrospective rather than prospective rates, propose a strategy for survey design, and provide analysis methods for estimating retrospective rates.

Introduction

To manage wild mammal populations effectively, information is needed on abundance and on factors that affect abundance over time. A wide range of methods exists for direct surveys of animals (Seber 1982; Borchers, Buckland & Zucchini 2002). However, some populations prove particularly problematic, in which case indirect surveys of their signs may be easier. Examples include mammals that are difficult to detect in closed habitats but leave dung piles that are more amenable for survey (e.g. deer, elephants and foxes), and mammals that are too scarce to survey by direct means but leave relatively numerous signs that can be surveyed (e.g. cat scats, otter spraints and ape nests).

Surveys of signs measure usage of the survey area over a period of time, corresponding roughly to the mean time to decay of the signs. In contrast, direct methods usually estimate animal density at the time of the survey, which may be more prone to sample error (Jachmann 1991). Dung methods yield estimates of abundance that are comparable with estimates using direct methods for a range of species (Barnes 2001), and they have been found to yield more precise estimates of elephant abundance than aerial sample surveys (Barnes 2002).

Surveys of dung are typically conducted using quadrat sampling (Bailey & Putman 1981; Putman 1984), strip transect sampling (Plumptre & Harris 1995) or line transect sampling (Barnes et al. 1995; Marques et al. 2001). Surveys of ape nests typically use line transect methods (Plumptre 2000).

To convert estimates of dung or nest density to estimates of animal density, two rates must be estimated: the production (defecation or nest construction) rate and the decay or disappearance rate of the dung or nests. If it is possible to clear the survey plots of signs before the dung or nest surveys, allowing sufficient time for new signs to accumulate but not sufficient time for them to decay, then there is no need to estimate decay rates (see below).

Production rate can be estimated by following animals or animal groups, by monitoring captive animals, or by placing a known number of animals in an enclosure, previously cleared of signs, and estimating the number of signs produced over a fixed time period. There are many practical problems associated with estimating production rate. For example rates may vary seasonally, so care is needed to estimate rates relevant to the surveys of dung or nests; rates may vary between animals, so a representative sample of animals should be monitored; captive animals may exhibit different rates from wild animals; it may be difficult or impossible to follow animal groups. Appropriate methods depend on the study population of interest. In contrast, a generic approach is possible for estimating decay rates, and in this paper we propose such an approach that provides robust estimation of decay rate.

Typically, dung and nest decay rates have been estimated by assuming an exponential rate of decay (McClanahan 1986; Barnes & Jensen 1987), by estimating an ‘instantaneous mortality rate’ of dung (Barnes & Barnes 1992) or by putting down or locating fresh dung and monitoring it until it has decayed (Plumptre & Harris 1995; Marques et al. 2001). In this paper, we develop the suggestions of Marques et al. (2001) and Buckland et al. (2001). They note that, to convert sign density to animal density, it is necessary to estimate the mean time to decay of signs that are present at the time of the survey. A simple way to achieve this is to locate and mark fresh signs on several dates in the lead up to the survey, chosen so that the proportion of signs surviving from the earliest date to the survey is expected to be small, and to return to marked signs just once, at the time of the survey. Data are then binary, recording whether or not the signs survived to the survey. The method of Hiby & Lovell (1991) is also based on this idea. We term the resulting estimates ‘retrospective’ estimates of the mean time to decay, because a time point is identified and the mean time to decay of signs already present is estimated. In contrast, most workers identify or lay down fresh signs at a selected time point, then return regularly to record when the signs disappear. This gives a ‘prospective’ estimate of the mean time to decay; if decay rates vary seasonally, prospective estimates are biased estimates of the required mean time to decay because they do not estimate the mean time to decay of the signs that are present at the time of the survey to estimate sign density. Further, the prospective method requires repeat visits to marked signs. However, this disadvantage is not as great as might be thought, as the design of a retrospective survey often requires repeat visits to identify fresh signs or to allow estimation of decay rate at different times of the year.

We illustrate the methods of this paper using data on red deer Cervus elaphus L. and roe deer Capreolus capreolus L. populations at Abernethy Forest in Scotland, for which the management aim is to maintain deer populations at levels that allow natural regeneration of native Scots pinewood Pinus sylvestris L.

Methods

approaches for estimating mean time to decay

Density of animals D̂a is estimated as:

image(eqn 1)

where D̂s is the estimated density of animal signs in the study area, t̂ is the estimated mean time to decay of the animal signs present when the survey to estimate sign density is conducted, and p̂ is the estimated rate of production of signs per animal during the period preceding the survey. To quantify precision of the estimate of animal density, it is important to estimate the precision of each of the three components of equation 1 (Plumptre 2000). Thus:

[cv(D̂a)]2 ≈ [cv(D̂s)]2 + [cv(t̂)]2 + [cv(p̂)]2(eqn 2)

where cv(t̂) is the coefficient of variation of t̂, defined as its standard error divided by itself, and similarly for other terms. Typically, the contribution of variability in estimated sign density dominates this expression. We assume that estimates D̂s and p̂ are available, together with their standard errors, and consider here the problem of estimating mean time to decay.

The most extensive work on decay rates has been conducted on dung piles of forest elephants in Africa. Short (1983), Merz (1986), McClanahan (1986) and Barnes & Jensen (1987) all estimate prospective rates, assuming that the system is in a steady state throughout the period of the decay experiment. The steady-state assumption states that the number of dung piles being deposited each day equals the number disappearing each day, i.e. the number of dung piles per unit area remains constant from day to day. This assumption means that we can estimate mean time to decay of dung piles present at the survey by the reciprocal of the daily rate of decay. Further, if we assume an exponential rate of decay, then rate of decay is independent of age and we can monitor any dung pile, not just fresh piles, to estimate the rate of decay.

Grimshaw & Foley (1990) and Reuling (1991) found that decay rates were not well modelled by the exponential distribution, with typically slower rates of decay when the dung piles were fresh. Barnes & Barnes (1992) therefore reanalysed their data using six different methods for calculating the mean decay rate from the data. They confirmed that methods that assume a constant exponential rate of decay are substantially biased.

A further major problem with the above methods, however, is that if the steady-state assumption does not hold, bias can again be substantial. Seasonal changes in defecation rates, dung decay rates and elephant distribution all violate the steady-state assumption (McClanahan 1986). Even when methods allow decay rate to vary with the age of the dung pile, the above methods yield a prospective rate, whereas equation 1 requires that we have a retrospective rate. The two rates may differ substantially when the steady-state assumption fails. Hiby & Lovell (1991) derived a method that correctly estimates the retrospective rate, and also provide software (dungsurv) to allow managers to estimate elephant abundance.

Hiby & Lovell (1991) noted that the dung piles visible on a survey represent the remains of the dung piles deposited by elephants in the area over the period preceding the survey, whether or not steady state has been reached. The decay experiment is therefore directed towards estimating the proportion of the dung piles deposited at different times during this period that remain visible on the date of the survey and, similarly, the production rate experiment estimates the defecation rate over this period. If elephant density is varying, they show that equation 1 provides a weighted average of density in the period preceding the dung survey, with greater weight being given to the dates immediately preceding the survey (and to dates on which defecation rates are high, if these vary).

If decay rates are independent of date, and of other factors such as sign size, then the prospective and retrospective rates are the same. However, if seasonal variation occurs, perhaps because of rainfall patterns or diet changes, then prospective rates give biased estimates of animal density. Furthermore, work on deer pellet groups (B.A. Mayle & A.J. Pearce, unpublished data, quoted in Marques et al. 2001) indicates that the average time for a pellet group to decay is a function of the initial number of pellets in a group, so that groups present at the time of the survey are a size-biased selection of groups deposited. The retrospective method of estimating mean time to decay is unaffected by this size-biased selection, but the prospective method generates bias, because too few large, long-lived groups and too many small, short-lived groups are represented in the sample.

design of the decay rate experiment

Given the above considerations, it is advisable to conduct decay experiments prior to every survey, unless sufficient data have been collected to allow a reliable model to be developed that can be shown to predict mean decay times successfully, given data such as rainfall and habitat for the study area (Barnes et al. 1997; Barnes & Dunn 2002). In addition, if surveys to estimate animal density from sign density are carried out over an extended time period, it may be necessary to make repeated visits to marked animal signs so that the mean time to decay can be estimated for different times of the year. Careful survey design may avoid this need. For example, different strata within the study area may be surveyed at different times, and the marked signs within each stratum can be revisited at the time of the survey in that stratum. The dates of the revisits then vary by stratum, but each marked sign is only revisited once.

Decay refers to the disappearance of the animal signs irrespective of the mechanism by which the process occurred. For example, deer pellet groups that have been covered by leaves, that have been spread out over a large area as a result of trampling by the deer, or that have undergone organic decay are all considered to have ‘decayed’ (Marques et al. 2001). It is important that the criterion for determining whether a sign has decayed in the decay rate experiment is the same as that used in the survey to estimate sign density. For deer pellet groups, this is typically taken to be the point at which the number of identifiable pellets falls below some threshold. For dung surveys more generally, it is usually possible to identify stages of decay or disappearance. In that case, the criterion can be chosen to correspond to a change from one stage to the next that is relatively unambiguous.

Following Buckland et al. (2001), we recommend that a time period is estimated over which 90% or more of the signs are expected to have met the criterion that defines ‘decay’. This might be estimated from past data from the study area or from similar studies elsewhere. If a time period that is too long is chosen, field costs will be higher than necessary but estimation of mean time to decay is not compromised. If the time period is too short, bias can be anticipated in estimation of mean time to decay, and hence in estimated animal abundance. Searches for fresh signs should commence this length of time ahead of the sign survey. A criterion will be needed for identifying fresh signs. If, for example, a criterion was shown by experiment to identify a sign as ‘fresh’ if it was up to 4 days old, a fresh sign should be considered to be 2 days old (the average age of signs identified as fresh) for purposes of analysis. Each fresh sign located should be marked to ensure that it can be accurately relocated.

There should be at least five or six visits to the study area to search for fresh signs, roughly evenly spaced in time between the first visit and the subsequent survey from which sign density is estimated. Thus if it was judged that at least 90% of signs decay within 6 months, six monthly searches might be conducted, the last being a month before the sign survey. At each visit, every effort should be made to ensure that a representative sample of fresh signs is located. Ideally, this will involve a designed survey, for example comprising several strip transects, randomly or systematically placed within the study area, that ensures that habitat is sampled in proportion to its occurrence. The length of each transect would then be surveyed, and any fresh sign marked. This ensures that more signs are monitored in areas heavily used by the study species, in proportion to the density of signs, as required for unbiased estimation.

Because variability in estimated decay rate will typically be smaller than variability in estimated density of signs, large sample sizes of fresh signs are not needed. Buckland et al. (2001) suggest a minimum of 50 in the monitoring experiment, and Hiby & Lovell (1991) show by approximate calculation that the contribution of decay rate estimation to overall variability in the animal abundance estimate is small if the number of monitored signs is of the order of 100.

analysis methods

Hiby & Lovell (1991) do not specify how they analyse data from an experiment of this type. Perhaps the most obvious option is logistic regression, and we outline this approach, together with possible transformations to improve model fit, below.

The following methods are based on Buckland et al. (1999), who developed methods for estimating mean willingness to pay for some environmental goal. In that case, a survey is conducted such that a number of respondents must state whether they would be willing to pay a fixed sum of money to achieve a specified level of environmental improvement. Although each respondent is offered a single bid level, different respondents are offered different bids, according to some design. This leads to a binary response of 1 if the bid is accepted or 0 if the bid is refused. If we regard the animal sign as the ‘respondent’, and the length of time between identification of a fresh sign and the subsequent revisit as the ‘bid level’, then the same methods can be used for modelling decay rate. A sign still present at the revisit corresponds to accepting the bid.

For animal sign i, i= 1, … , n, we define the random variable Yi to be 1 if the sign is judged not to have decayed at the revisit, or 0 otherwise, and we denote the time between production of sign i and the revisit by xi, which is therefore the age of the sign at the time of the revisit, if it has survived. Then:

E(Yi | xi) = Pr(Yi = 1 | xi) = p(xi)

where p(xi) is the probability that sign i survives until the revisit, assumed for the moment to depend on xi alone. We assume:

image(eqn 3)

where β0 and β1 are coefficients to be estimated.

Let the random variable X represent the lifetime of an animal sign (i.e. the length of time until the sign is judged to have decayed). The quantity of interest is then the mean time to decay, which we denote µX. Figure 1 shows a diagrammatic representation of the decay experiment.

Figure 1.

Diagrammatic representation of the decay experiment. Pi represents the time at which sign i was produced, and Gi the point in time at which sign i decayed. Sign 1 was therefore still present at the time of the revisit, but sign 2 had decayed.

Estimation of mean time to decay and the corresponding variance are covered in the Appendix. Note that the logistic function of equation 3 is defined over the full range of x, from –∞ to ∞, whereas in the current context x is constrained to be non-negative. One solution is to left-truncate the distribution at x= 0 (see the Appendix). Another solution is to replace x in our model by loge(x), so that the range on x of (0, ∞) transforms to a range on loge(x) of (–∞, ∞), giving a logistic regression over the full range of the real line. The model is then:

image(eqn 4)

Again, results for mean time to decay are deferred to the Appendix.

Although the logarithmic transformation ensures that negative ages are impossible, it can also create problems for some data sets when fitting the upper tail of the logistic curve. In particular the upper tail may be considerably lengthened by the logarithmic transformation, which can result in estimates of mean time to decay that are biased high. A solution to this problem is to identify a transformation that does not alter behaviour of the upper tail. One such transformation is the reciprocal transformation, where x is replaced by w=x–β2/x for some β2. The general logistic equation given by equation 3 is a limiting case of this transformation as β2 → 0. The value of β2 might be fixed arbitrarily, but it is better considered an unknown parameter to be estimated. The regression is carried out as before, but with the additional term β2/x. Equation 3 now becomes:

image(eqn 5)

with β1 < 0 and β2 > 0. See the Appendix for estimating mean time to decay under this model.

Suppose covariates such as habitat type and rainfall are recorded, in addition to age of the sign. The logistic regression equation may be expressed as:

image(eqn 6)

where xij is the value of covariate j for animal sign i, j 1, and βj are coefficients to be estimated, j 0.

If a number of potential covariates are recorded, stepwise methods may be used to reduce the number of covariates in the model. Age xi1 (and/or its transformation where relevant) should always be retained in the model.

The corresponding fitted model may now be expressed as:

image(eqn 7)

Each sign now has a unique estimated decay curve, so that estimation of mean time to decay is less straightforward. The solution proposed by Buckland et al. (1999) is to calculate the prediction ŷi for each animal sign i, using equation 7. These predictions are sign-specific estimates of the probability that the sign has decayed by the time of the revisit. They can be plotted against x1i, the time between production of sign i and the revisit. A logistic curve can be fitted to the plot, assigning a weight 1/{ŷi(1 − ŷi)} to ŷi. [In the willingness-to-pay application of Buckland et al. (1999), the distribution of bid level was discrete whereas that of x1i is continuous, except for the effect of rounding to, say, whole days, so we have modified the proposed method slightly.]

This logistic curve differs from the logistic regressions described earlier, because the logistic regressions assumed that the data were from a binomial distribution and the curves were fitted using iterative reweighted least squares. Here, the logistic curves are fitted using non-linear weighted least squares.

If we denote the logistic curve as:

image(eqn 8)

then this has exactly the same form as equation 3. The fitted logistic curve yields estimates b1 and b2 of β1 and β2, allowing us to estimate the mean time to decay, inline image, and its corresponding variance as previously. The approach is readily extended to allow the logarithmic transformation of x, or the addition of a term in 1/x.

Example: red and roe deer surveys in Scotland

data collection methods

Fifteen plots were established and cleared of dung during January 2000 in Abernethy Forest in Scotland. Each plot was surveyed on average on 10 occasions between January 2000 and May 2001, with intervals between visits of between 6 and 8 weeks. Any new pellet groups were marked and the species of deer recorded. The estimated date of deposit of each pellet group was taken as 24 days before the date that it was marked (i.e. roughly half the time elapsed between the date of marking and the date of the previous visit). For each pellet group, its date of disappearance was recorded. A pellet group was considered to have decayed if less than six identifiable pellets remained. The estimated date of decay was taken as 24 days before the date of the first visit for which the pellet group was judged to have decayed. As a number of pellet groups had not decayed during the period of monitoring, the date when the pellet group was last checked was recorded for these pellet groups.

Line transect surveys of dung were conducted in May 2001. Thus only a single observation was required for each marked sign, whether or not it was still present in May 2001. These observations were readily obtained from the observed dates of decay, which were recorded for research purposes.

data analysis

The survey date chosen as the fixed reference date for the logistic regression analysis of the red and roe deer data sets was 15 May 2001. The status of the pellet group at this reference date was determined: if the estimated date of decay was after the fixed reference date (or if the pellet group was still present at the final visit), then the status of the pellet group was recorded as 1; otherwise, it was recorded as 0.

For each pellet group, habitat was recorded as a factor at three levels. There were in total 54 and 72 observations (pellet groups) for red deer and roe deer, respectively, after deleting those for which habitat had not been recorded.

Three logistic models, each incorporating the habitat covariate, were fitted to the data. Model 1 was the left-truncated logistic model with no transformation of the age variable; model 2 was the logistic model with the log transformation of the age variable; model 3 was the logistic model with the reciprocal transformation of the age variable. The results from fitting each model appear in Table 1 (red deer) and Table 2 (roe deer). In each case, a null model was fitted first, then a model incorporating sign age, and finally a model incorporating habitat as well. For both species, no evidence of an effect of habitat was found, and the left-truncated model without transformation of x proved adequate. Use of loge(x) reduced the error deviance appreciably less for roe deer and slightly more for red deer than use of untransformed x, and the term in 1/x was not significant for either species. Therefore, our favoured option for roe deer was the left-truncated model without the habitat covariate, and for red deer either this model or the model with loge(x) and no habitat covariate appeared satisfactory. We thus dropped the covariate and used the straightforward logistic regression methods for estimating mean time to decay. For comparative purposes, we also show results for the models with a logarithmic and a reciprocal transformation of x, also without the covariate.

Table 1.  Results from fitting the three logistic models to the red deer pellet group data
 ModelResidual devianceChange in devianced.f.Change in d.f.P
  • *

    Significant at the 5% significance level.

No transformationNull68·744 53  
+ age18·86749·877521< 0·0001*
+ habitat15·862 3·184502   0·203
Log transformationNull68·744 53  
+ loge (age)18·16550·579521< 0·0001*
+ habitat15·518 2·647502   0·266
Reciprocal transformationNull68·744 53  
+ age18·86749·877521< 0·0001*
+ 1/age17·931 0·936511   0·333
+ habitat15·181 2·749492   0·253
Table 2.  Results from fitting the three logistic models to the roe deer pellet group data
 ModelResidual devianceChange in devianced.f.Change in d.f.P
  • *

    Significant at the 5% significance level.

No transformationNull98·922 71  
+ age51·05347·870701< 0·0001*
+ habitat48·092 2·960682  0·228
Log transformationNull98·922 71  
+ loge (age)57·25941·663701< 0·0001*
+ habitat55·046 2·213682  0·331
Reciprocal transformationNull98·922 71  
+ age51·05347·870701< 0·0001*
+ 1/age51·024 0·029691  0·866
+ habitat48·082 2·942672  0·230

Figures 2 and 3 show the fitted logistic regression curves under all three models, excluding the habitat covariate, for red deer and roe deer, respectively. The models yielded similar fits to these data. The estimated mean time to decay, inline image, its approximate standard error, SE (inline image) and 95% log-normal confidence interval for all three models are shown in Table 3. Higher variability was evident under model 2, reflecting the greater uncertainty associated with a wider upper tail for the model when age was log-transformed, and high variability was shown by model 3 for red deer only, but generally the different models yielded very similar estimates of mean time to decay for each species.

Figure 3.

The logistic regression curves fitted to the roe deer pellet group data. The open circles show the observed data, which are 1 for pellet groups surviving to 15 May 2001 and 0 otherwise.

Figure 2.

The logistic regression curves fitted to the red deer pellet group data. The open circles show the observed data, which are 1 for pellet groups surviving to 15 May 2001 and 0 otherwise.

Table 3.  Estimated mean time to decay, inline image, its standard error, SE (inline image), and 95% log-normal confidence interval. In model 1, x is left-truncated and untransformed; in model 2, x is log-transformed; in model 3, a term in 1/x is added to model 1
SpeciesModelinline imageSE (inline image)Log-normal 95% confidence interval
Red deer129531240, 362
Red deer228039213, 369
Red deer327542204, 371
Roe deer126025215, 315
Roe deer226034202, 334
Roe deer325226206, 308

Discussion

Methods for estimating the density of signs (usually dung or nests), for example using quadrat sampling, strip transect sampling or line transect sampling, are well developed and understood. There is greater difficulty in estimating the two rates that allow sign density to be converted to animal density: sign production rate and the decay rate (or equivalently its reciprocal, the mean time to decay). In this paper, we address the latter problem. Estimation of the production rate must be addressed on a case-by-case basis, as methods suitable for some populations are not suitable for others. For more difficult populations there would seem to be considerable scope for developing electronic methods to monitor a sample of animals remotely.

The methods developed here can be readily implemented using standard statistical software that provides logistic regression and logistic curve fitting facilities, together with numerical integration. Cameron (1988) adopted a strategy for modelling willingness to pay without including bid level (in our context, age of sign) as a covariate. Instead, Cameron (1988) developed a censored logistic regression approach. However, her more direct approach requires methods that are not available in standard statistical software.

The sign survey methods assumed in this paper are often called ‘standing crop’ methods, because the survey to estimate sign abundance records all detected signs on the survey plots, irrespective of age (unless they are still detectable but are deemed to have decayed). In contrast, ‘clearance plot’ methods avoid the need to have to estimate decay rates. Survey plots are cleared of any signs, and are then revisited before any new signs have had time to decay. The amount of sign deposited per day within the survey region is estimated from the resulting data; this estimate is divided by an estimate of the deposition rate per animal per day, to yield estimated animal abundance. The clearance plot method is generally regarded as efficient only when animal density is high (Buckland 1992). It has the substantial advantage over the standing crop method of not requiring an estimate of decay rate. Thus abundance can be estimated relatively quickly, without the need to monitor signs over a lengthy time period. Its disadvantages over the standing crop method are as follows. Decay rates tend to be highly variable, so that the time period between visits must be short to ensure that new signs do not decay before the site is revisited. This means that many more sampled plots (or larger plots) must be surveyed to allow estimation of sign abundance with comparable precision to that achievable with standing crop methods. [Typically, precision on this estimate dominates precision of the final animal abundance estimate (Plumptre 2000).] Added to this, the sampled plots must be cleared of all signs at the outset, and accurately relocated during the survey of signs, whereas the standing crop method can use distance sampling methods (Buckland et al. 2001) for efficient estimation of dung abundance, without the need to locate all signs on the sampled plots and without the need for marked plots. Except at high densities, the advantage of not having to search for fresh signs from which to estimate decay rates, or to monitor the signs over time, is usually more than offset by these disadvantages, especially as a larger sample of signs is needed to estimate the mean number of signs deposited per day throughout the survey region with comparable precision to that for estimates of mean decay rate. A further factor in favour of the standing crop method is that it may often prove possible to develop a model to predict decay rate so that a decay rate experiment is not needed in every survey site at every time point.

Investigation into how high the density should be for the clearance plot method to be more cost-effective than the standing crop method would be useful. However, conclusions will vary appreciably between studies and species.

Acknowledgements

We thank the Royal Society for the Protection of Birds, who conducted the surveys and funded the development of the survey design. Analyses were conducted for the dissertation element of the first author's MSc, which was funded by BBSRC. We also thank Richard Barnes for his comments on an earlier draft, and two referees for their supportive comments.

Appendix

We need to evaluate µX = E(X). If animal sign i is still present at the time of the survey, then X must be greater than xi, i.e. p(xi) = Pr(X > xi) (see Fig. 1). Hence the cumulative distribution function of X, F(x), is given by:

F(x) = 1 − Pr(X > x) = 1 − p(x)(eqn 9)

The probability density function f(x) is then obtained by differentiating with respect to x:

image(eqn 10)

The mean time to decay, µX = E(X), can then be calculated as:

image(eqn 11)

If we left-truncate the distribution at x= 0, we must rescale f(x) so that it again integrates to unity. This solution to the problem of negative x works well when F(0) for the untruncated distribution is close to 0, and we develop this approach before considering transformations that offer a more satisfactory mathematical solution.

Applying the result of equation 10 to equation 3, we obtain:

image(eqn 12)

where β1 < 0. However:

image(eqn 13)

To ensure that f(x) integrates to 1 after left-truncating at 0, equation 12 must be divided by equation 13. The modified logistic curve is then given by:

image(eqn 14)

from which:

image(eqn 15)

Estimates b0 and b1 obtained from the logistic regression of Y on x are substituted for β0 and β1, respectively, in equation 15. The estimate for µX, inline image, is then obtained by numerical integration of equation 15. The delta method (Seber 1982) yields the following approximate result:

image( eqn 16)

where u = exp{−(b0 + b1x)}

The integrals are evaluated by numerical integration, and standard logistic regression packages give the variances and covariance of the coefficients.

If we replace x in our model by loge(x) (equation 4), then we obtain:

image(eqn 17)

The estimates b0 and b1 obtained from the logistic regression of Y on loge(x) are substituted for β0 and β1, respectively, and inline image is again obtained by numerical integration. The estimated variance is now:

image(eqn 18)

where u = exp{−(b0 + b1 loge(x))}

If instead we add a term β2/x to our model, then:

image(eqn 19)

The estimates b0, b1 and b2 obtained from a logistic regression of Y on the two variables x and 1/x, are substituted for β0, β1 and β2, respectively. The estimate inline image is then obtained by numerical integration of equation 19. The variance of this estimate is estimated as:

image(eqn 20)

Ancillary