To estimate bird survival rates, the shortcomings of methods based on recov.eries. (of dead birds) make it logical to use recapture information (of live birds) in.
THE ESTIMATION OF SURVIVAL IN BIRD POPULATIONS BY RECAPTURES OR SIGHTINGS OF MARKED INDIVIDUALS J. CLOBERT, 1 J.D. LEBRETON, 2 M. CLOBERT-GILLET,l H. COQUILLART2,3 lLaboratoire d'EaoZogie theorique et de Biometrie Universite CathoZique de Louvain Place Croix du Sud 5, 1348 Louvain-la-Neuve Belgique 2Laboratoire de Biom~trie, Universite Lyon I 69622
Villeurbanne-C~dex
France
3Present address: Centre Ornithologique Rh8ne-Alpes 69622
Villeurbanne-C~dex
France
SUMMARY To estimate bird survival rates, the shortcomings of methods based on recov.eries (of dead birds) make it logical to use recapture information (of live birds) in specific statistical models. In this paper, the basic model of Cormack (1964) and some recent developments all owi ng equal i ty constra i nts !letween parameters are reviewed. Applications to Fu1mar, Starling, and Coot data are presented, with the emphasis on model selection. For Swallow data, these models designed to estimate survival are compared to more general recapture models. In conclusion, using the Cormack model and its generalisations in the course of population studies on a local scale is recommended.
Keywords:
1.
CAPTURE-RECAPTURE; SURVIVAL; BIRDS; MAXIMUM LIKELIHOOD
INTRODUCTION
In the management of animal populations and in fundamental research in population dynamics, the emphasis is now on mechanisms - and as a consequence on parameters - of population changes, rather than on changes in numbers by themselves. Survival rates are as such major components of animal population dynamics. To estimate survival rates in birds, ringing provides numerous data that are a real B. J. T. Morgan et al. (eds.), Statistics in Ornithology © Springer-Verlag Berlin Heidelberg 1985
198
J. C10bert et al. challenge to statisticians. Recoveries (i.e. records of time of death by people) have been widely used to estimate survival: good statistical models have been developed (Haldane, 1955; Cormack, 1970; see reviews by Seber, 1973; Brownie et al., 1978) while the statistical shortcomings of some widely used methods have been demonstrated (Anderson et al., 1981). Nevertheless, the number of recoveries available, various sources of bias (e.g. Anderson and Burnham, 1980; Nelson et al., 1980) and problems in the identifiability of the models (Lakhani and Newton, 1983) generally impose severe limitations on the quality of the estimates obtainable. Thus, to estimate survival, it seems logical to use other kinds of information, in particular that based on recaptures of live animals over the years (or sightings when the animals are not actually captured). See also Cormack (1985). Besides general recapture models, some specific approaches have been developed (Chapman and Robson, 1960; Cormack, 1964; Clobert, 1981; Sand1and and Kirkwood, 1981). While Chapman's and Robson's model is based on transversal data (recaptures in the same year of birds ringed for several years), the other models cited use longitudinal data, namely individual recapture histories over the years. The purpose of this paper is to present various longitudinal models available, which seem to provide the most efficient way to estimate survival rates, while being rarely used. We apply the models to the Fu1mar (Fulmarus glacialis) (from data in Cormack, 1964), to the Starling (Sturnus vulgaris), and to the Coot (Fulica atra). Then we compare various recapture models, using data on the Swallow (Hirundo rustica). In conclusion, we discuss the efficiency of the longitudinal models, especially when compared with models using recoveries.
2. 2.1
METHODS General Presentation
The parameters used in all the models are (Fig. 1): the probability of survival from year i to year i+1, si; the probability that a survivor in year i comes back to the study area this year, ni; the probability that a bird present in the study area in year i is captured (prob. of capture), Pi. When there is permanent emigration (n i < 1), the models provide an apparent survival,
199
Estimation of survival in birds by recapture i.e. an estimate of Hisi. The effect of non-permanent absences of birds is included in Pi' From now on, we will suppose Hi = 1, a reasonable assumption in a lot of cases, but which could depend upon the biology of the species studied. The parameters si and Pi are co~non to all individuals, so that, in particular, no agespecific effects are included in the models. The absence of age-specificity seems reasonable when working on breeding populations. On the contrary, the parameters can be time-specific (Cormack, 1964), or sub~ect to various constraints of equality to achieve a greater parsimony (Sandland and Kirkwood, 1981; Clobert, 1981).
YEAR I captured
returning
not captured
surviving
YEAR
emigrating
1-1
dead
Figure
Parameters used in the recapture models to estimate survival. Recoveries from dead birds are not taken into account.
The data can be presented as dichotomous trees of recapture histories (Fig. 2). The various recapture histories in a tree (= the leaves of each tree) are mutually exclusive events: each individual has one history. and only one. Under the hypothesis of independence of the individuals, the numbers in the histories follow a multinomial distribution. Usually. when there are n years of recapture. there are
200
J. Clobert et at.
year
II
R~160 H
/4o0 _________ ;(o
100~NII~(0 600 _____ _______ NR
560 R
H~400 1000 _____ _____ Nil
600
Figure 2
Reaapture trees with surviva~ probabiLities BI=S2=S3=0.5 and probabi~ities of captures PI=P2=P3=O.3: expected numbers in the various recapture historieB. (M=marked, R=recaptured, NR=not recaptured). (1) probabi~ity of thiB event is SIPIB2P2 B3P3' (2) probability of thiB event iB sl{1-PI)B2(1-P2)S3P3'
201
Estimation of survival in birds by recapture n+l years of ringing (Fig. 2) and the likelihood of a whole data set is obtained from a product of multinomial densities. To make the ai' ci (Cormack, bi number ai number in Fig ci number 960 in vi:
likelihood more compact, it is convenient to use the statistics bi ,
1964), and vi defined as follows for i z 1, ... , n: marked in year i (b l = 1000 in Fig. 2); captured in year i+l (a l z 400, a2 = 160 + 40 + 400
= 600
2);
of individuals last seen in year Fi g. 2);
(c 3
= 24 + 96 + 240 + 600 =
i
E(b, - c.,l, number of individuals known to be alive after year i.
j=l J
J
As an example, the data in Fig. 2 are converted to these statistics in Table 1.
With
these notations, the likelihood of a data set reduces to (Statistical Appendix): L = 1Tn
i =1
v1' a. v.-a. p1. 1 ( l-p 1.) 1 1
Si
The Log-likelihood LnL can be maximized under various constraints on the parameters to provide maximum likelihood estimates (see the Statistical Appendix). Table 1 Reduction of the recapture histories of Fi~ 2 to the statistics bi , ai' ci and vi
i
bi
ai
ci
vi
1 2
1000 1000 1000
400 600 700
556 784 960
444 660 700
3
bi ai ci vi
= number = number = number = number
marked in year i; caught in year i+l; caught for the last time in year i; known to be alive after year i.
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J. C10bert et aZ. 2.2 Cormack Model With the full set of parameters, only sl' s2' ... , sn_1' P1' P2' ... , Pn-1 and the product snPn are estimable. Cormack (1964) provides explicit maximum likelihood estimates of these quantities, and estimates of their variances and covariances. This model is noted (Pt,St) by Sand1and and Kirkwood (1981) (2n parameters, 2n-1 estimable) . 2.3 Sand1and-Kirkwood-C1obert l1ode1s Sandland and Kirkwood (1981) and C10bert (1981) proposed independently several kinds of equality constraints between parameters, to enhance parsimony, resulting in (in Sand1and's and Kirkwood's notations): (si = s, for all i) (n+1 parameters) - model (Pt's) - model (p,St) (Pi p, for all i) (n+l parameters) - model (p,s) (Pi = p, si = s, for all i){2 parameters). Sandland and Kirkwood propose also a (pm,st) model with a Markov-type dependence in the probability of capture. C10bert uses also any kind of equality constraints between the survival rates, and between the probabilities of capture, that can be used for example when some years are known as "poor" a priori. In all these cases, numerical methods are required to maximise the likelihood. Likelihood ratio tests help to choose an adequate model for any particular data set (Sahd1and and Kirkwood, 1981; C1obert, 1981), or to compare data sets (males and females within the same population; separate populations of the same species; populations of different species ... ). Numerical estimates of the asymptotic variances and covariances of parameter estimates can be obtained from the second-order derivatives of the likelihood (see statistical appendix).
3. 3.1
APPLICATIONS Fu1mar(FuZmarus gZaaiaZis)
This is Cormack's (1964) original example (Table 2). Comparisons between models (Table 3) do not show any significant variation in survival over the years: when compared with model (Pt,St)' the model (Pt's) is the only model with constraints
203
Estimation of survival in birds by recapture accepted (likelihood ratio test:
x2 = 11.5
s
X2 10(0.05)
Conclusion
model rejected
= 18.3 13
Pt S
2112.1
si =s , for all i
x2=2112.1-2100.6 =11.5< x2 10(0.05)
model not rejected
= 18.3 2
Ps
2235.6
Pi =p ; for all i (under si=s , for aJ 1 i)
x2=2235.6-2112.1 =23. 5>x211 (0.05)
model rejected
= 19.7
(*) in Sandland's and Kirkwood's (1981) notations: - (Pt,St) = probability of capture and survival time-dependent; = probability of capture constant over the years, survival time-dependent; = probability of capture time-dependent, survival constant over the years; - (p,s)
= probability of capture and survival constant over the years.
205
Estimation of survival in birds by recapture 3.2 Starling (Sturnus vUlgaris) C10bert (1981) put 360 nest boxes in a 100 Ha farmland area in Belgium, and studied the breeding population of Starlings for 7 years (n=6 years of recaptures; Table 4). Only the females are studied, by nocturnal capture (the males do not spend the night at the nest during the breeding season). Comparisons between models (Table 5) indicate significant differences between survival rates while the probability of capture can be considered constant: model (p,St)' when compared with model (Pt,St) by a likelihood ratio test, is accepted (X 2=4.0< X2 4(0.05)=9.49). This is obviously related to the experimental design, all the nest-boxes being visited once, and only once, to limit the impact on breeders, each spring. The constraint Pi = p, for all i, in model (p,St) makes it possible to estimate sn (last column of Table 4). The standard deviations of survival rates (Table 4) are slightly smaller in model (p,St) than in model (Pt,St) because of the greater parsimony. The variations in survival are obviously related to weather conditions, the low 53 = 0.37 in 1978-1979 resulting from a particularly cold winter (Fig. 3, and C10bert and Leruth, 1983). Correlations between the estimates si prevent the direct calculation coefficient between a weather variable and survival estimates.
Table 4 Recapture data for Starlings (Clobert, Same notations as in Table J A
Year
i
bi
ai
ci
si
1977 1978 1979 1980 1981 1982
1 '2 3 4 5 6
52 204 102 126 73 43
27
122 74 95 67 48
21 102 151 101 102 75
0.66 0.65 0.37 0.59 0.53
-
si
estimation of si in model (p,St)' a numerica lly.
~(Si)
A
1981)
Pi
s;
0(5 i )
.079 .049 .037 .052 .068
0.79 0.78 0.79 0.73 0.62
-
-
0.67 0.67 0.37 0.58 0.48 0.46
.079 .043 .035 .050 .044 .058
standard-deviation estimated
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J. Clobert et al.
....
Sj
0,8
n
0,7
0,6
~o
0,5
82
0,4
18
61
19
O,J
0,2
0.1
5
6
Fi gure 3 Relationships between average October-March teTlJPerature T. and survival si in a Belgian population of Starling (Sturnul: vulgaris).
207
Estimation of survival in birds by recapture Table 5 ratio tests between reaapture on the Starling data of TabZe 4
Like~ihood
Model
(*)
Number of parameters estimated
- 2LnL
Null hypothesis
PtSt
11
1625.6
Pt=P , for a11 t
P St
7
1629.6
Pt=P , for all t
mode~8
Test
x2 =1629.6-1625.6 = 4.0x\(0.05)
model rejected
= 11.07 (*)
in Sand1and's and Kirkwood's (1981) notations (see Table 3)
However, a linear relationship between survival and an environmental variable can be built into the model (C10bert and Lebreton, in prep.), in the way used by North and Morgan (1979) in a model based on recoveries. 3.3 Coot (FuUaa atra ) The quality of the results in the previous examples may be attributed to a high recapture rate (see Pi in Tables 2 and 4). Data on Coots in the Dombes area (France) (Table 6, Cordonnier, unpub1.) have been obtained with a much lower recapture probability. The model (p,St) is preferable to the model (p,s) for females, but both models are acceptable for males (Table 7). This probably results from fewer data being available for males (Table 6). The comparison between males and females is thus based on model (p,St). The X2 used in the likelihood ratio test is the difference between the likelihood of the sum of the data (1170.77) and the likelihoods obtained separately for males (420.99) and females (745.66) (last column of Table 7). The number of degrees of freedom (5) results from the number of constraints in the null hypothesis (p(01 = p(~); Si(6) = si(~)' i = 1, ... 4). No significant difference exists between males and females (x 2=1170.77-420.99-745.66 = 4.32
