Age-dependent survival, recovery and recapture E. A. Catchpole1 S. N. Freeman2 B. J. T. Morgan2 M. P. Harris3 1
2
3
School of Mathematics and Statistics University College, UNSW Australian Defence Force Academy Canberra ACT 2600 Australia email:
[email protected] Institute of Mathematics and Statistics University of Kent at Canterbury Canterbury Kent CT2 7NF England email:
[email protected] [email protected] Institute of Terrestrial Ecology Banchory Research Station Banchory Kincardineshire, AB31 4BY. England email:
[email protected] September 1995.
Summary 1
Integrated recovery and recapture data, providing information on the same birds, are essential for the stable tting of realistic stochastic models, in order to estimate annual survival probabilities. Previous work in the area has either concentrated on time-dependent parameters only or, in the age-dependent case, has not provided a suitably exible framework for model re nement, and ultimately model selection. Our motivation in this paper derives from a set of recapture histories on shags (Phalacrocorax aristotelis), whose varying pelagic behaviour requires age-dependence in all of the primary parameters in the model. We derive the likelihood for a perfectly general model, allowing both age- and time-dependent parameters, which we then t to the data by the method of maximum likelihood. The programming is done in MATLAB.
Keywords: Capture-recapture, recovery data, integrated analysis, age-dependence, recapture histories, shags, pelagic behaviour, emigration.
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1. Introduction: Data and background
The starting point for the work described here is the Harris shag data. [***Here we need a full description from Mike of the data collection and the general background***] Data take the form of encounter histories for individual birds, covering c cohorts, and k years of study in total. An illustration is given in Table 1.
Table 1 about here Separate analyses of capture-recapture data (from live birds), and of recoveries data (from dead birds) are now well established { see, respectively, Lebreton et al. (1992) and Freeman and Morgan (1992). The combination of recovery and recapture data into a single analysis when the birds from the two studies are di erent is described by Lebreton et al. (1995). What distinguishes data of the type illustrated in Table 1 is the fact that we frequently have recovery and recapture information on the same individuals. We require a model to t to the entire data set by the method of maximum-likelihood, and which will accurately re ect the prime features of the behaviour of the species studied. Evidently this is not a new problem, but for the kind of data we have here, an analysis has not previously been provided. For birds such as the shags, ringed as young, age-dependence is paramount, particularly with regard to distinguishing a separate survival probability for rstyear birds (which may well also be time-dependent). Pelagic birds such as shags, which spend much of their time at sea until they start breeding, also require models with age-dependent survival of birds older than 1, and almost certainly also age-dependence of recapture and recovery probabilities. This paper is the rst stage in lling the gap in existing methodology. The early paper by Buckland (1980) was signi cant, and integrated recoveries into the Cormack-Jolly-Seber (CJS) model for recapture (Cormack, 1964, Jolly, 1965, Seber, 1965). The original emphasis of the CJS model was on the estimation of population size, and the model (see for example Seber, 1965) resulted in explicit expressions for maximum-likelihood parameter estimates, providing these all lay within permitted bounds. The parameter set therefore included measures of population size, as well as time-dependent measure of (in our context) annual survival, recapture, and reporting (for dead birds), as appropriate. Buckland's integrated approach followed the same procedures as those for the CJS model, and again resulted in an explicit 3
maximum-likelihood estimate (mle), subject to its lying within range. Buckland went on to show how his method could also cope with age-dependent survival probabilities, which is what we need here, { see Buckland (1982) and also Buckland and Bailey (1987). Throughout, this approach is based upon an explicit mle, and involves no numerical optimisation of the likelihood, which immediately becomes necessary once one wishes to constrain model parameters. Previous analyses of the shag data (see Harris et al. 1994) have adopted this approach, based on explicit estimates. The paper by Mardekian and McDonald (1981) also provides an analysis of integrated recapture and recovery data, but at the expense of omitting some of the available information. Burnham (1993) provides a meticulously detailed discussion of integrated analysis possibilities, but only for the case of time-dependent survival, which is not appropriate for our data, being more suited to an analysis of ringed adult birds only. He describes the complexity of a cohort-based analysis, which is possible for time-dependent parameters, in which birds recaptured from one cohort become available for recapture or recovery from the next cohort. As a result, due allowance must also be made for birds recovered dead, that they were not recaptured before dying (thereby switching cohort). As Burnham notes in his nal paragraph, \... additional theoretical models ... should be developed for the joint analysis of recapture-recovery data, in particular extensions to age-speci c models". He also asked: \... is the inclusion of recapture data with recovery data (in what are meant to be ring-recovery studies) worthwhile, and conversely in recapture studies is the inclusion of any recovery data worthwhile?" This last question is also of interest to us in our work; it is a primary focus of Lebreton et al. (1995), where di erent birds were involved in the recapture and recovery studies and also of Catchpole et al. (1993), who consider a simpli ed version of the integrated experimental paradigm considered above, involving a single recapture episode only { see Aebischer et al. (1995). The paper by Szymczak and Rexstad (1991) provides an application of the approach described by Burnham (1993). Burnham (1993) also outlined for his model an analysis based on recapture histories, and this is the way we proceed here. Data can be expected to be presented in recapture history format originally, and initially computerised in this way. However the advantage of a cohort-based analysis, for time-dependent models, is that they may be programmed in SURVIV (White, 1992), and the goodness-of- t of the models is then readily assessed, using standard Pearson X 2 tests. 4
Our aim in this paper is to provide the general framework for an integrated analysis of recovery and recapture data, and to demonstrate how it operates for the shag data. Model notation and model selection are more complex for combined data, as opposed to when just single recapture or recovery data are available, and we shall not be comprehensive in this paper. Full development, including testing for trap-dependence in recapture, and testing goodness-of- t, will be described in the sequel to this paper { Catchpole et al. (1995). Our intention for model-selection is to base this upon score tests, as described by Catchpole and Morgan (1996), for an analysis of recovery data alone. This approach forms the basis of the MATLAB package, EAGLE { see Catchpole (1995) { and MATLAB is the programming environment for our work here. [*** { at some stage we need a full descripition, possibly as an Appendix. We might emphasise how well-suited MATLAB is to the kind of computation needed here. We may note its exibility compared with alternatives, e.g., SURVIV was used by Szymczak and Rexstad (1991), but although this is easily done, since it is by cohort, and multinomial, a clear put-o is the need to front-end SURVIV (as others have done) in order to avoid mistakes and the drudgery of repeated cell speci cation. Possibly make the new program available on WWW? ***] Section 2 of the paper explains how the likelihood is formed. Section 3 provides an analysis of the shag data in terms of mles for a selected model, and a comparison is drawn with the results of Buckland et al. (199 ). The paper ends with a description of additional work to be carried out in the area. [*** NB. here we mention sheep***]
2. Constructing the likelihood
There are k recapture occasions (years in the shag example), denoted by ft g. Let C denote the number of cohorts of marked birds, the cth containing n(c) individuals. Let = Pr(bird from cohort c, alive at t survives until occasion j + 1, 1 j k ? 1, and de ne: 8 > : +1 : : : ?1 c + 1 j k = Pr(bird from cohort c, which dies in (t ; t +1 ), is reported), p = Pr(bird from cohort c, alive at t , is captured at t ), j
cj
j
cj
cc
c;c
c;j
cj
cj
j
j
j
j
5
q =1?p cj
cj
This notation is sucient to incorporate arbitrary time and age dependencies in the model, and is readily extended, as we indicate below, to allow for trap-dependency following recapture. If = Pr(bird from cohort c, having been captured alive at t , is not seen again), then we have immediately the recursion, cj
j
1 ? = (1 ? ) + (1 ? q cj
cj
cj
cj
c;j +1
c;j +1
):
(1)
1 c C; 1 j k ? 1:
Clearly = 1; 1 c C: The recursion of (1) was used by Buckland (1980), but not stated explicitly by him. Throughout we assume we are dealing with a closed population, i.e. that migration can safely be ignored. This is the common assumption on which all recapture analyses are based, and which we shall examine critically for the shag data later. Next, we require the following key 4 matrices: ND(c; j ) = number of birds from cohort c, recovered dead in the interval, (t ?1 ; t ) ck
j
2jk 1 c C; NL(c; j ) = number of birds from cohort c, recaptured at t , j
and not seen again, during the time-course of the study, 2jk 1 c C; (it is natural to de ne NL(c; k) = 0) W (c; j ) =
X
n(c)
W (j ) NL(c; j ) ic
i=1
= number of birds recaptured from cohort c at t 2jk 1 c C: 6
j
j
(Note that many birds are clearly counted in both NL(c; j ) and W (c; j )): Z (c; j ) =
X
n(c)
z (j ) ic
i=1
= number of birds from cohort c not recaptured at but encountered later: (clearly; Z (c; k) = 0)
2jk 1cC
The likelihod is then given by: L=
YY C
k
c=1 j =2
f
c;j
?1(1 ? ?1) ?1 g c;j
c;j
N D (c;j )
f g cj
cj
N L(c;j )
fp
W (c;j ) cj
q
Z (c;j ) cj
g
(2)
The model probability parameters can be made functionally dependent on age and/or time and then each model is tted, by maximum-likelihood, by maximising L. The time-dependent models considered by Buckland (1980) and Burnham (1993) are generalised by the formulation given here. The explicit solutions available for the time-dependent extension of the CJS model provided by Buckland (1980) provide a check of our approach, which was satis ed for a subset of the shag data. The expression for L in (2) has not been given before. It is interesting to observe the structure of L, since the rst component corresponds to the standard model for the recoveries data alone. In a tidy world, at the end of the study no birds would have unknown fates, leading to NL(c; j ) = 0, for all c and j . For such data the likelihood L would factorise simply into recovery and recapture parts, which could be maximised separately. This also provides a useful check of the general maximum likelihood approach. It is the middle components of the terms in L, viz. the f g which are pivotal, linking together the recovery and recapture aspects of the process. We note nally that the last component of L is a simple binomial, focusing on birds alive at t , and considering whether or not they are recaptured at t . It is of interest to proceed from (2) to consider information matrices for particular models, with regard to assessing the relative information contributions from the di erent, recoveryrecapture, parts of the data, using the approach adopted by Lebreton et al. (1995). This is considered in Catchpole et al. (1995). [*** Note: if there is a tie-in with SURGE here too that would be nice, especially if it would require the recoveries part to be transformed, as described in Lebreton et al. (1995). This cj
cj
j
j
7
would add to the coherence of all of the material, and tie in with Biometrics again, emphasise the value of the previous work, and allow SURGE users to do this analysis.***] We do not in this paper consider adding trap-independence to the model. That is done in Catchpole et al. (1995). Note, however, that a simple way of investigating trap-dependence is to suppose that, if it exists, it only a ects the probability of capture at the episode following a recapture, changing p (q ) to p0 (q0 ). De ne the two (c (k ? 1)) matrices R and S as follows: R = fr ; 1 c C; 2 i kg; S = fs ; 1 c C; 2 i kg; and, r = number of birds from cohort c captured at both t ?1 and t , s = number of birds from cohort c captured at t ?1 and not at t , then the likelihood L is modi ed by multiplication by the factor: Y Y p0 ci q0 ci ci
ci
ci
ci
ci
ci
ci
i
ci
i
C
r
k
c=1 i=2
p
q
ci
i
s
ci
ci
ci
i
:
3. Results
[*** at this stage I have only a set of notes for discussion.***]
References
Bailey Buckland, S.T. (1980) A modi ed analysis of the Jolly-Seber capture-recapture model. Biometrics, 36, 419-435. Buckland, S.T. (1982) A mark-recapture survival analysis. J.Animal Ecology, 51, 833-847. Buckland, S.T. and Baillie, S.R. (1987) Estimating bird survival rates from organised mistnetting programmes. Acta Ornithologica, 23, 1, 90-100. Burham, K.P. (1993) A theory for combined analysis of ring-recovery and recapture data. pp.199-213. In J.-D. Lebreton and P.M. North (eds.) Marked Individuals in the Study of Bird Populations. Birkhauser Verlag, Basel/Switzerland. Catchpole, E.A. (1995) MATLAB { an environment for analysing ring-recovery and recapture data. To appear: J.Appl.Statist.. Catchpole, E.A., Freeman, S.N. and Morgan, B.J.T. (1993) On boundary estimation in ring recovery model and the e ect of adding recapture information, pp.215-228. In J.-D. Lebreton and P.M. North (eds.) Marked Individuals in the Study of Bird Populations. Birkhauser 8
Verlag, Basel/Switzerland. Catchpole, E.A., Freeman, S.N. and Morgan, B.J.T. (1995) Integrated recovery/recapture data analysis II: trap dependence and goodness-of- t. In preparation. Catchpole, E.A. and Morgan, B.J.T. (1996) Model selection in ring-recovery models using score tests. To appear: Biometrics. Cormack, R.M. (1964) Estimates of survival from the sighting of marked animals. Biometrika, 51, 429-438. Freeman, S.N. and Morgan, B.J.T. (1992) A modelling strategy for recovery data from birds ringed as nestlings. Biometrics, 48, 217-236. Harris, M.P., Buckland, S.T., Russell, S.M. and Wanless, S. (1994) Year- and age-related variation in the survival of adult European shags over a 24-year period, The Condor, 96, 600-605. Jolly, G.M. (1965) Explicit estimates from capture-recapture data with both death and immigration-stochastic models. Biometrika, 52, 225-247. Lebreton, J.-D., Burnham, K.P., Clobert, J. and Anderson, D.R. (1992) Modeling survival and testing biological hypothesis using marked animals: case studies and recent advances. Ecological Monographs, 62, 67-118. Lebreton, J.-D., Morgan, B.J.T., Pradel, R. and Freeman, S.N. (1995) A simultaneous analysis of dead recovery and bird recapture data. To appear: Biometrics. Mardekian, S.Z. and McDonald, L. (1981) Simultaneious analysis of band recovery and liverecapture data. J.Wildlife Management, 45, 484-488. Seber, G.A.F. (1965) A note on the multiple recapture census. Biometrika, 52, 249-259. Szymczak, M.R. and Rexstad, E.A. (1991) Harvest distribution and survival of a gadwall population. J.Wildlife Management, 55, 592-600. White, G.C. (1992) PC SURVIV User's Manual Version 1.4. Available from Department of Fishery and Wildlife Biology, Colorado State University, Fort Collins, CO 80523, U.S.A.
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Table 1 The available data take the form of individual capture histories, 5 ctitious examples of which are shown below. The ith row refers to bird number i and the j th column contains a 1 if the bird was recaptured on the j th recapture occasion, a 2 if it was recovered dead at some time between the (j ? 1)th and j th occasions, and a zero if neither of these events occurred. The initial capture is marked with a 1, so that the rst 1 in any row indicates the cohort to which the bird belongs. In the example below there are 5 birds from C = 2 cohorts, with n(1) = 2 birds in the rst cohort, and n(2) = 3 birds in the second cohort. In all there are 6 recapture occasions, ft g. i
Recapture occasions cohort 1
cohort 2
t1 t2 t3 t4 t5 t6
1 1
0 1
1 0 0 0
2 1
0 1
0 0 0
1 1 1
2 0 0 0 0 0
0 1 0
0 0 0
10
