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Keywords:

  • blackbird;
  • declining recovery probability;
  • grey heron;
  • logistic models;
  • song thrush;
  • time segmentation;
  • wren

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Simulations
  7. Discussion
  8. Acknowledgement
  9. References
  10. Supporting Information

1. Ring-recovery data can be used to obtain estimates of survival probability which is a key demographic parameter of interest for wild animal populations. Conditional modelling of ring-recovery data is needed when cohort numbers are unavailable or unreliable. It is often necessary to include in such analysis a recovery probability that is declining as a function of time, and failure to do this can result in biased estimates of annual survival.

2. Corresponding estimates of survival probability need to be reliable in order for correct conclusions to be drawn regarding the effects of climate change.

3. We show that standard logistic modelling of a decline in recovery probability is unsatisfactory, and propose and investigate a range of alternative procedures.

4. Methods are illustrated by application to a recovery data set on grey herons. The model selected is a scaled-logistic model, and it is shown to provide a unifying analysis of several data sets collected on different common bird species. The model makes specific predictions, providing potential new insights and avenues for ecological research. The wider performance of this model is evaluated through simulation.

5. In this study, we propose a new scaled-logistic model for the analysis of ring-recovery data without cohort numbers, which incorporates a reporting probability that declines over time. The model is shown to perform well in simulation studies and for both a single real data set and several real data sets in combination. Its use has the potential to reduce bias in estimates of wild animal survival that currently do not incorporate such reporting probabilities. Alternative models are shown to possess undesirable features.


Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Simulations
  7. Discussion
  8. Acknowledgement
  9. References
  10. Supporting Information

Data

In ring-recovery studies, wild birds are ringed and released each year and records kept of reported rings from dead birds; see Balmer et al. (2008). Ring-recovery data can be used to obtain estimates of survival probability which is a key demographic parameter of interest for wild animal populations. Such data are especially important for estimating the survival of birds during the first year of life. We focus on the analysis of data from birds ringed as young, in the nest. However, the modelling can also be applied to data from birds marked as adults. See Greenwood (2009) for relevant historical material. An illustrative recovery data set on grey herons, Ardea cinerea, is provided in Supporting Information Table S1, which is a subset of data available at http://www.tibs.org, analysed by North & Morgan (1979); we analyse the full data set later in the article.

Especially when ringing is carried out by many volunteer ringers at a national level, the number of birds ringed and released in a given year, which we refer to as cohort size, may be unavailable or unreliable. For example, the British Trust for Ornithology (BTO) has recently computerised cohort size numbers for some 20 common bird species, but only since 1985. Computerising records is very time-consuming, and of the order of 15 million remain to be computerised at the BTO. These are entered on paper in ring number order, so selecting out individual species is not practicable, especially for the smaller passerine species, on which the majority of rings are placed. It will take approximately 25 years to complete the task. Most European ringing schemes will have similar data as is also true in North America and Australia.

We use the heron data set to illustrate the work of this study as cohort sizes are known in this case, which allows us to make comparisons between analyses with and without cohort numbers. We refer to data without cohort sizes as conditional data, as when constructing appropriate probabilities we condition on the fact that we only use information from individuals that have been recovered dead. Model parameters are appropriate probabilities of annual survival, probabilities of reporting dead birds and where appropriate regression coefficients, which can be used to incorporate covariate information into the model. Lebreton (2001) provides a useful overview of the use of ring-recovery models to estimate survival and includes a discussion of the conditional ring-recovery models, which are the focus of this study.

Recent years have seen a decline in the reporting probability of many species of wild birds; see Baillie & Green (1987), Robinson, Grantham & Clark (2009), Guillemain et al. (2011) and Mazzetta (2010), and as yet no models for conditional ring-recovery data have been proposed which can incorporate this decline in recovery probability. The motivation for this study is the need for a proper description of large bodies of conditional ring-recovery data which may currently be modelled inappropriately.

Multinomial Models and Conditional Analysis

If we assume that birds behave independently, multinomial models can be constructed for each cohort of ringed birds, parameterised in terms of survival and reporting probabilities. Data from multiple cohorts can be simultaneously analysed by using a product-multinomial model, which dates from Seber (1970, 1971) ; for more recent work, see Freeman & Morgan (1992) and Catchpole & Morgan (1991). Chapter 16 of Williams, Nichols & Conroy (2002) provides a useful overview of modelling this type of data. For illustration we shall assume two age classes for (true) survival, with respective annual survival probabilities φ1(t), for birds in the first year of life at time t, and φa(t), for older birds, for t = 1,…,T. In the simulation studies given later, these probabilities are taken as constant, while for the analyses of real data, each survival probability is logistically regressed on an annual weather covariate, ωt, denoting the number of days below freezing in central England in year t (see Besbeas et al. 2002), to give

  • image
  • image

The form of the logistic regression ensures that the estimates of probabilities are constrained between 0 and 1. We denote the reporting probability in year t by λt. Thus, for example, for cohort c the multinomial cell probability pj, which denotes the probability an individual released at time c is recovered dead at time j, is given by

  • image

for j = c,…,T and inline image. Here, pT+1 denotes the probability of not being observed following marking and S is the probability of being recovered at some point in the study.

We assume that the different cohorts are independent, so that the likelihood is product multinomial over cohorts c = 1,…,C. When cohort sizes are not available, we obtain a conditional analysis, where, for example, for the cohort released at time c, the cell probabilities are given by inline image for j = c,…,T. The cell probability inline image is the probability of being recovered dead in year j for an individual released in cohort c, given that it is recovered within the study.

The widely used computer package MARK (White & Burnham 1999) only allows a constant reporting probability for the conditional analysis of ring-recovery data (model denoted constant in the results below). In the corresponding cell probabilities for the constant model, the recovery probability multiplies terms in both the numerator and denominator and cancels, so that the recovery probability cannot be estimated. This was first observed by Seber (1971). Robinson, Baillie & Crick (2007) state that this is the only possibility for conditional analysis; however, Cole & Morgan (2010) demonstrate that time-dependent recovery probability models for conditional data are parameter redundant, but showed that models incorporating time regressions of the reporting probability are full rank and hence may in principle be fitted in conditional analysis of ring-recovery data. We show in this study that such an approach is not straightforward and offer alternative models for conditional ring-recovery data.

Materials and methods

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Simulations
  7. Discussion
  8. Acknowledgement
  9. References
  10. Supporting Information

We shall compare and contrast the performance of three alternative models for a decline in recovery probability, λt, over time, together with a simpler approach based on time segmentation.

A Scaled-Logistic Model

As we will see in this study, a simple logistic model used to incorporate a decline in recovery probability over time fails to perform well in practice. Therefore, we need to consider a different functional form to model the decline in recovery probability. Here, we propose a scaled-logistic model, in which

  • image
  • image

where τ is a time to be estimated, indicating the start of a temporal decline in reporting probability, corresponding to γs<0. The constant model arises when γs = 0. The parameter κ will not enter conditional analyses owing to cancellation. This means that when the model is fitted to conditional data, we are not able to estimate λt itself, which is not important when the main interest is estimating survival and/or the parameters γs and τ. This contrasts with similar issues in estimating species occurrence probabilities from presence-only data; see Elith et al. (2011). Prior knowledge of κ would allow estimation of λt in conditional analyses, which would also allow estimation of cohort sizes. The alternative models presented in the next section do allow estimation of λt. We note that Phillips & Elith (2011) employ a scaled-logistic model in a different context. See also Richards (1959).

Two Further Logistic Models

The standard logistic model for λt (denoted logistic in what follows) has the form

  • image

We also consider a delayed-logistic model in which the reporting probability has the form

  • image
  • image

We see that the delayed-logistic model reduces to the constant model when γd = 0, and to the logistic model when τ = 0.

Time Segmentation

The models above explicitly incorporate a decline in reporting probability. It is not possible to fit a model with full time-dependent recovery probabilities to a conditional ring-recovery data set because the resulting model is parameter redundant. An alternative approach is to divide time into disjoint continuous segments and then fit a piece-wise linear model for the reporting probability; for instance, we can fit a model with two values, such that λt = ℓ1 for t<τ and λt = ℓ2 otherwise.

Although this model is obviously parameter redundant, it is possible to estimate the ratio of the two recovery probabilities, Λ = ℓ2/ℓ1. An estimate of Λ<1 will provide evidence for a decline in reporting probability, and therefore, the fitting of this simplistic model can be used as a preliminary check that reporting probability decreases with time for any conditional data set. It is possible to generalise this approach to K time segments and values of λ, and in this case, it would be possible to estimate ℓ2/ℓ1,ℓ3/ℓ1,…,ℓK/ℓ1. For some data sets, we have found that it may also be possible to estimate τ when K = 2; however, the parameter redundancy status of this model is affected by the sparseness of a particular data set. Therefore, a natural approach would be to partition data into periods of equal length, and this is how we have estimated Λ for the heron and the combined data sets we analyse below.

Change-point models, such as this time-segmentation approach, are challenging in practice. It is unlikely that the true parameter values exhibit a substantial change in value at a breakpoint; however, they allow us in this case to diagnose whether or not a decline in recovery probability is appropriate for a given conditional data set. Toms & Lesperance (2003) discuss the use of such threshold models for ecological applications.

Computational Aspects

The models of this study are fitted using the method of maximum likelihood. Programs were written in Matlab, and multiple random starts were used for all optimisations to ensure global optima had been reached. Matlab programs are available from McCrea.

Results

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Simulations
  7. Discussion
  8. Acknowledgement
  9. References
  10. Supporting Information

Heron Analysis

Scaled-logistic and constant models

When the constant and scaled-logistic models are fitted to the conditional heron data set, we obtain the results shown in Fig. 1. For comparison, we also present results using the cohort numbers, denoted scaled-logistic (unconditional data). We can see the very good agreement between the conditional and unconditional analyses for the scaled-logistic model, and the lower estimates of survival probability obtained from using the model with a constant reporting probability, compared with the other two analyses. Notable is the effect of the severe winter of 1962 on survival, for all three graphs. Maximum-likelihood parameter estimates are shown in Table 1. We can see that not accounting for time variation in λ affects the survival estimates α1 and αa. For the scaled-logistic model, the precision in these parameters is reduced (a near doubling of standard errors) for conditional data; cf Burnham (1990). In addition, γs and τ are less precisely estimated when using conditional data. A formal Pearson chi-square statistic of the fit of the scaled-logistic model, after arbitrary pooling of cells with small numbers, indicates a significant lack of fit (inline image). This lack of fit could be accounted for through the use of an overdispersion correction in the estimation of standard errors. In this case, inline image. We obtain a better fit to the data by including more than just two probabilities of annual survival (see Besbeas & Morgan 2012), but we adopt the simpler structure here for ease of illustration; a graphical check of goodness-of-fit is provided in the Supporting Information and reveals no systematic defect with the fitted model.

image

Figure 1.  A comparison of (a) first-year survival estimates and (b) adult survival estimates for the constant and scaled-logistic models fitted to the heron data. The latter model is fitted twice, once with cohort numbers (unconditional data) and once without cohort numbers (conditional data). The survival probabilities are modelled through a logistic regression on the temporal covariate, wt, which is the number of days below freezing in central England.

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Table 1.   Maximum-likelihood parameter estimates, with approximate standard errors estimated from the observed Hessian at the maximum likelihood, from fitting the scaled-logistic model, constant model, time-segmentation model with K = 2, logistic model and delayed-logistic model to the heron data. γi denotes γd in the delayed-logistic model, γl in the logistic model and γs in the scaled-logistic model. Similarly, ηi represents ηd and ηl in the respective delayed-logistic and logistic models. We write  log Lmax for the maximised log likelihood. Note that κ, λ, λ1 and λ2 are measured on the logistic scale, and Λ is on the log scale
ParameterScaled-logisticConstantTime segmentationLogisticDelayed logistic
unconditionalconditionalunconditionalconditionalunconditionalconditionalunconditionalconditionalunconditionalconditional
α 1 −0·21 (0·052)−0·22 (0·125)−0·26 (0·051)−0·32 (0·052)−0·21 (0·052)−0·27 (0·055)−0·20 (0·052)−0·33 (–)−0·21 (0·052)−0·10 (0·136)
β 1 −0·03 (0·006)−0·03 (0·006)−0·03 (0·006)−0·03 (0·006)−0·03 (0·006)−0·03 (0·006)−0·03 (0·006)−0·03 (–)−0·03 (0·006)−0·03 (0·006)
α a 0·87 (0·057)0·85 (0·132)0·81 (0·054)0·74 (0·052)0·87 (0·057)0·80 (0·058)0·87 (0·056)0·74 (–)−0·88 (0·057)0·97 (0·147)
β a −0·01 (0·005)−0·01 (0·005)−0·02 (0·005)−0·01 (0·005)−0·02 (0·005)−0·02 (0·005)−0·01 (0·005)−0·01 (–)−0·02 (0·005)−0·02 (0·005)
λ −2·25 (0·026)
1−1·72 (0·042)
2−2·51 (0·034)
κ −1·70 (0·062)
γ i −0·07 (0·006)−0·06 (0·044)−0·04 (0·003)13·83 (–)−0·05 (0·005)−0·08 (0·040)
η i       −1·27 (0·027)−14·37 (–)−1·70 (0·055)−2·46 (2·241)
τ 11·85 (1·866)13·02 (5·682)  13·00 (1·544)13·01 (1·979)
Λ−0·44 (0·164)
− log Lmax7598·62278·77717·802279·77615·12275·97605·12277·77596·02277·4
AIC15211·24569·415445·64567·415242·24561·815222·14567·515206·14568·8

A likelihood-ratio test reveals no strong support for the hypothesis that γs≠0 from the analysis of the conditional data, in line also with the size of the corresponding estimated standard error of inline image in Table 1. However, the analysis of the unconditional data provides very strong evidence that γs≠0. This means that there is no strong evidence that the constant model is inappropriate when using the conditional data; however, when the unconditional data are used, there is very strong evidence that recovery probabilities decline over time.

Time segmentation

We have also fitted the K = 2, K = 3 and K = 18 time-segmentation models to the conditional heron data. The model with the smallest AIC is the K = 2 model (4561·8 compared with 4566·2 and 4573·4 for the K = 3 and K = 18 models, respectively). We observe that the constant model, which is equivalent to a K = 1 time-segmentation model, has an AIC of 4567·4. Therefore, the K = 2 model is more strongly supported by the data. For the K = 2 model, we have obtained an estimate of Λ = 0·64 (0·105). The minimised negative log-likelihood value of this model is 2275·9, which we observe is smaller than the minimised negative log likelihood of the scaled-logistic model. We note that when we fit the time-segmentation model to the unconditional data, the minimised negative log likelihood of the model is 7615·1, which is larger than the corresponding value for the scaled-logistic model. Plots of Λ under the K = 2 and K = 18 time-segmentation models are provided in Fig. 2. We observe from this figure that for K = 2, Λ<1 providing support for a decline in recovery probability over time. Further, the K = 18 estimates detect the main features of the time variation.

image

Figure 2.  (a) A comparison of estimates of the recovery probability, λt, fitted to the unconditional data (left-hand y-axis) and estimates of the ratio of recovery probability values, Λ = ℓk/ℓ1, from the time-segmentation model, for K = 2 and K = 18, fitted to the conditional data (right-hand y-axis); (b) An illustration of the estimates of λt from fitting specified models to the unconditional and conditional heron data set. We observe that the logistic model fitted to the conditional data is not shown owing to the boundary estimate obtained when the model is fitted. We note the near coincidence of the estimates under the scaled-logistic and delayed-logistic models fitted to the unconditional data.

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Logistic and delayed-logistic models

The results from fitting the logistic model and the delayed-logistic model to the heron data are also provided in Table 1. Likelihood-ratio tests applied to the results from fitting models to unconditional data provide strong evidence that τ≠0, which means that the decline in recovery probability only starts part-way through the study period. Additionally, γl≠0 and γd≠0, and there is similar but weaker evidence that γd≠0 when conditional data are used, which means that the decline in recovery probability is significant. A comparison of estimates of the recovery probability λt is provided by Fig. 2. We do not give the plot for the logistic model fitted to conditional data because of the boundary estimates of ηl and γl in Table 1. The near coincidence of the scaled-logistic graph and that for the delayed logistic fitted to the unconditional data are notable. It is clear that time dependence in recovery probability is essential to estimate correctly the survival parameters. A fully time-dependent model can only be fitted to unconditional data; however, Fig. 2 demonstrates how the scaled-logistic model encompasses the key features of the time-dependent recovery probabilities.

There are clear problems with fitting the logistic and the delayed-logistic models to the conditional data. The graph of λt for the delayed-logistic model fitted to the conditional data appears to be too low, compared with the value for the constant model. The logistic model results in boundary parameter estimates, and therefore, we are unable to produce corresponding estimates of standard error. In fact, the estimated Hessians for the logistic and delayed-logistic models at the maximum-likelihood estimates are ill-conditioned. For the logistic model, the eigenvalues of the estimated Hessians range from 0·00 to 47,323·6, and for the delayed logistic, the range is 0·2 to 47,865·8. By contrast, the scaled-logistic model does not have this problem: for the conditional heron data, the eigenvalues range from 4·8 to 47,549·3.

Application to Common Bird Data Sets

For bird species that are likely to share the same value of τ, we can provide an analysis of all the conditional recovery data sets simultaneously. We select ring-recovery data from the BTO for wren, Troglodytes troglodytes, song-thrush, Turdus philomelos, and blackbird, Turdus merula. All are widespread birds of garden, scrub and woodland that coexist. In each case, the data start from 1966, compared with 1960 for the herons. Parameter values are given in Table 2, for all four models. There is very good agreement between the estimates of survival for the delayed-logistic and scaled-logistic models. However, what makes both the delayed-logistic and logistic models untenable are the estimates of ηd and ηl, respectively. They are unrealistically high for the species under study. The ranges of eigenvalues of the estimated Hessians are 2·4 to 17,678·5 for the logistic model, 3·0 to 28,462·9 for the delayed-logistic model and 86 to 30,147·8 for the scaled-logistic model. In this case, we have relatively precise estimates of τ. We note that the decline in recovery probability is larger than for the herons. Likelihood-ratio tests indicate a highly significant improvement in the scaled-logistic model compared with the constant model, while there is an improvement in the delayed-logistic model over the constant and logistic model.

Table 2.   Maximum-likelihood estimates of model parameters when fitting conditional recovery data sets from three bird species simultaneously, with a common value for τ for the scaled-logistic and delayed-logistic models. The subscripts w, s and b refer, respectively, to wren, song thrush and blackbird. γi denotes γd in the delayed-logistic model, γl in the logistic model and γs in the scaled-logistic model. Similarly, ηi represents ηd and ηl in the respective delayed-logistic and logistic models. The time-segmentation estimates correspond to a model with K = 16; estimates of Λ are not provided in the Table but a plot of the estimates can be found in the Web Appendix. We denote the maximised log likelihood by  log Lmax
ParameterConstantLogisticDelayed logisticScaled logisticTime segmentation
φ 1w 0·27 (0·020)0·30 (0·022)0·29 (0·021)0·30 (0·021)0·28 (0·021)
φ a w 0·33 (0·021)0·37 ( 0·025)0·36 (0·024)0·36 (0·024)0·35 (0·024)
φ 1s 0·51 (0·015)0·57 (0·020)0·55 (0·016)0·56 (0·015)0·53 (0·019)
φ a s 0·59 (0·008)0·64 (0·017)0·63 (0·011)0·63 (0·010)0·61 (0·014)
φ 1b 0·53 (0·007)0·60 (0·018)0·58 (0·009)0·59 (0·008)0·56 (0·015)
φ a b 0·64 (0·004)0·70 (0·016)0·69 (0·007)0·69 (0·006)0·67 (0·013)
η i  −0·48 (0·631)0·77 (0·576)  
γ i  −0·17 (0·011)−0·15 (0·015)−0·14 (0·011) 
τ   10·01 (1·048)10·04 (0·705) 
− log Lmax20751·420698·920696·520697·920688·9
AIC41514·841413·841411·041411·841421·8

We have fitted the K = 2, K = 3 and K = 16 time-segmentation models to the conditional common bird data sets. The time-segmentation model with the smallest AIC has K = 16, and the survival estimates resulting from this model are displayed in Table 2. We observe that although the minimised negative log-likelihood value is smaller for the time-segmentation model compared with the scaled-logistic model, the scaled-logistic model has an appreciably smaller AIC (41411·8 compared with 41421·8) owing to the large number of parameters under the K = 16 time-segmentation model. A plot of the estimated Λ ratios can be found in the Web Appendix.

Pearson chi-square statistics have been calculated for each of the component data sets under the scaled-logistic model. Only the blackbird and song thrush ringed-as-young data sets show significant evidence for lack of fit. Therefore, the assumed structure for the recovery probabilities is generally performing well for the scaled-logistic model. A graphical check of goodness-of-fit for the component data sets is provided in the Web Appendix, and no systematic lack of fit is identified from this plot. An estimate of the overdispersion coefficient for the combined data sets is inline image. As with the heron analysis, this could be used to adjust the corresponding standard errors.

Simulations

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Simulations
  7. Discussion
  8. Acknowledgement
  9. References
  10. Supporting Information

Simulation Comparison of Four Models

We present in Fig. 3 a simulation study of the bias that can result from using an incorrect model for the reporting probability in conditional analyses only. The parameter values used in the simulation were φ1 = 0·5, φa = 0·6, τ = 15. Cohort sizes were all 1000, and the length of the study was 30 years. We present kernel density estimates to summarise the estimates from 100 simulations. The model for how the reporting probability declined over time has λ = 0·4 for t<15, while for t ≥ 15, λt = λt−1−0·02. Note that the estimates of parameters ηl, γl, ηd and γd do not correspond directly to values used in the simulation.

image

Figure 3.  Kernel density plots from a simulation study to compare the performance of four alternative models for the reporting probability; mle denotes maximum-likelihood estimates. We note the good performance of the scaled-logistic model.

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Both the scaled-logistic and delayed-logistic models produce approximately unbiased estimates of survival probability. However, we note the possibility that the delayed-logistic model might result in boundary estimates for ηd. As expected, the constant model results in underestimates of survival. Further, we observe that the logistic model, which ignores the period, τ, of constant recovery before a decline commences, results in overestimates of the survival probability; cf Table 2. It is interesting that a naive approach to modelling the decline in reporting probability over time, from using the logistic model, proves to be inadequate, owing to the location of τ. It was this finding that prompted the design of the two alternative logistic models in the study. We also note the superior performance of the scaled-logistic model compared with the delayed-logistic model, with regard to estimation of the slope parameters γs and γd, respectively.

Robustness of the Scaled-Logistic Model

The logistic functions used here all have a lower limit of zero. However, it is possible that a period of decline in recovery probability may be followed by a period of level recovery probability. A simulation study has been run to assess the robustness of the scaled-logistic model when data are simulated from a model such that λt = 0·2 for t ≤ 10, λt = λt−1−0·005 for 10<t ≤ 24 and λt = 0·13 for t > 24. Boxplots of the estimates of φ1 and φa from fitting the scaled-logistic model and the constant model to 100 simulated data sets are displayed in Fig. 4. We observe that the survival probabilities are underestimated; however, the estimates are less biased than those resulting from the constant model. If this structure to reporting probability decline is suspected, then it would be possible to correct the scaled-logistic model appropriately.

image

Figure 4.  Boxplots of the maximum likelihood estimates (mle) on the logistic scale from a simulation study run to examine the robustness of the survival estimates when λt does not follow a logistic decline. SL denotes scaled logistic, and C denotes constant. The horizontal dotted line indicates the true value of the parameter.

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Further simulation results assessing the effect of varying τ can be found in the Web Appendix.

Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Simulations
  7. Discussion
  8. Acknowledgement
  9. References
  10. Supporting Information

The study by Catchpole, Kgosi & Morgan (2001) provided descriptions of near-singular models, which could exhibit poor behaviour when fitted to data if the fitted models were close to parameter-redundant submodels. Fitted to conditional data, the delayed-logistic and logistic models exhibit similar behaviour to near-singular models, as when the parameters that model the slope of the recovery probability decline are near zero it will be hard to estimate the starting value for the recovery probability as the model will be close to the constant model under which the recovery probability parameter cannot be estimated. Indeed we have seen for these models the poor precision in estimating this initial recovery probability in Tables 1 and 2. Checking the entries of the eigenvector of the estimated Hessian at the maximum likelihood, corresponding to the smallest eigenvalues, as proposed by Catchpole et al. (2001), consistently identifies this initial recovery probability as the parameter that is estimated with the least precision, for the analyses of this study and for others not presented here. Essentially both of these models contain too many parameters for estimation with realistic conditional data. By contrast, owing to the cancellation of the parameter κ, the scaled-logistic model avoids this problem.

Time segmentation for the analysis of conditional ring-recovery data is a valuable alternative to fitting the scaled-logistic model. We have found it difficult to provide a clear recommendation with regard to the value of K to use, and this is an area requiring more research. Large values of K may result in overfitting of the data and imprecise estimates. An alternative approach would be to use low-order polynomials or splines – see for example Besbeas & Morgan (2012). These approaches may be employed as preliminary analyses to justify the use of a scaled-logistic model.

For the analysis of conditional ring-recovery data, and with reporting probability that is a declining function of time, we recommend using the scaled-logistic model, unless there is a particular need to estimate the recovery probability as a function of time. In that case, one might also use the delayed-logistic model, with caution. Although we have focussed on conditional ring-recovery models, it is also possible to use a scaled-logistic model for unconditional ring-recovery data.

However in cases where records of cohort numbers are only partially available, then a joint analysis, one for the case of known cohort numbers and one for the case where the numbers are missing, would be possible; cf Lebreton et al. (1995). This would also allow estimation of the parameter κ.

We note the improvement in precision in estimating τ from combining different data sets. The point estimates of τ obtained from the combined analysis and from the heron analysis are compatible, when one considers the different times at which the records began, suggesting a decline in the mid-1970s. The reason for the decline in reporting probability at this time is likely to be a combination of factors. For instance, there has been an increase in scavengers such as corvids and foxes, partly owing to a reduction in game-keepering, and in addition, there is less public engagement in the countryside and reduced use of the postal service resulting in fewer rings being returned to the BTO.

If this modelling approach was used for the simultaneous analysis of many species, it would be possible to incorporate random effects into the model to account for species-specific variability within the recovery probability; see for example Lahoz-Monfort et al. (2010).

A further feature of the fitted scaled-logistic models has been the suggestion that recoveries of different types of bird have declined at different rates. This has not previously been investigated, and the results we have obtained are intuitively sensible.

Detection of the effects of climate change on the annual survival probabilities of wild birds depends essentially upon unbiased estimation of these probabilities; cf Furness & Greenwood (1993). The time periods involved will inevitably require appropriate models for ring-recovery data extending into time periods for which cohort numbers are unavailable. It is then that application of the models of this study will be particularly useful.

Acknowledgement

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Simulations
  7. Discussion
  8. Acknowledgement
  9. References
  10. Supporting Information

The work of Brown was supported by an EPSRC research studentship, through the grant to the National Centre for Statistical Ecology (NCSE), and McCrea is funded by a NERC/EPSRC NCSE grant. The manuscript was appreciably improved following the comments of an Associate Editor and three referees.

References

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Simulations
  7. Discussion
  8. Acknowledgement
  9. References
  10. Supporting Information
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Supporting Information

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Simulations
  7. Discussion
  8. Acknowledgement
  9. References
  10. Supporting Information

Fig. S1. A graphical check of goodness-of-fit for the scaled logistic model for the heron data.

Fig. S2. (a) An illustration of the estimates of λt from fitting the logistic and delayed-logistic models to the combined data. (b) A graphical check of the goodness-of-fit for the scaled-logistic model for the joint data, where “obs” denotes observed values and “exp” denotes expected values.

Fig. S3. Maximum-likelihood estimates of Λ from the K = 16 timesegmentation model fitted to the combined data sets.

Fig. S4. A simulation study to compare the performance of the scaledlogistic model when ô is varied.

Table S1. Recovery data on British grey herons, ringed between 1987 and 1997. Data provided by the British Trust for Ornithology.

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