banded at Kaikoura, New Zealand for nine consecutive years. The Fisher Island data provided no evidence of a density dependence relationship between ...
Journal of Applied Statistics, Vol. 29, N os. 1- 4, 2002, 305- 313
M easuring density dependence in survival from mark- recapture data
1
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R IC HARD BARK ER , DAV ID FLETC HER & PAU L SC OFIELD , 1 D epartment of M athematics and Statistics, University of Otago, B ox 56, Dunedin, 2 N ew Zealand and Department of Zoology, University of Otago, B ox 56, D unedin, N ew Zealand
abstract
We discuss the analysis of mark- recapture data when the aim is to quantify density dependence between sur vival rate and abundance. We describe an analysis for a random eþ ects model that includes a linear relationship between abundance and sur vival using an er rors-in-variables regression estimator with analytical adjustment for approximate bias. The analysis is illustrated using data from short-tailed shearwaters banded for 48 consecutive years at Fisher Island, Tasmania, and Hutton’s shearwater banded at Kaikoura, New Zealand for nine consecutive years. The Fisher Island data provided no evidence of a density dependence relationship between abundance and sur vival, and con® dence inter val widths r ule out anything but small density dependent eþ ects. The Hutton’s shearwater data were equivocal with the analysis unable to rule out anything but a ver y strong density dependent relationship between sur vival and abundance.
1 Introduction 1.1 General background M ark- recapture m odels are useful for sum m arizing encounter history data using population param eters such as survival and birth rates or abundance. T he associated theory is well developed w ith m odels suitable for a wide variety of sam pling situations. C om puter software such as program M ARK ( W hite & Burnham, 1999) can be used to ® t these m odels encom passing a large variety of m ark- recapture Correspondence: R. Barker, Departm ent of Mathem atics and Statistics, University of Otago, Box 56, Dunedin, New Zealand. E-m ail: rbarker@m aths.otago.ac.nz ISSN 0266-476 3 print; 1360-053 2 online/02/010305-0 9 DOI: 10.1080 /02664760 12010878 2
© 200 2 Taylor & Francis Ltd
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data types within a com m on analysis fram ework. Studies using these m odels tend to focus on survival rates. M ore recently, m odels have been developed that allow us to focus attention on recruitm ent rates (Pradel, 1996) and probabilities of m oving between states (Brow nie et al., 1993). A statistical m odel can be thought of as a way of sum m arizing data using a sm all num ber of parameters. In an exam ple we discuss below, a m ark- recapture study was carried out w ith 48 sam pling occasions and 1045 unique encounter histories. The full Jolly- Seber m odel has 141 estimable param eters, and although this is considerably fewer than the 1045 encounter histories, it hardly represents a com pact sum m ary of the data. Moreover, the 46 identi® able survival rates in the Jolly- Seber m odel oþ er a sim ple description of the survival process operating over tim e, but provide little insight into the underlying biological processes. W hat is needed in this example is a m odel for the survival probabilities that incorporates key structural features of the biological processes.
1.2 Random eþ ects models To focus our attention on interesting biological processes, we can think of m arkrecapture data as arising through a two-stage process. In stage I, the true abundances, birth rates and sur vival rates can be thought of as being sam pled from som e distribution that depends on a sm all set of param eters. The realized (but unobserved) values for these can be regarded as ® xed parameters at stage II. In stage II, individual encounter histories are obtained by sampling the anim al population that exists during sam pling. There m ay be considerable interest in stage II sam pling, but m atters of real biological interest are in stage I. T his hierarchical sam pling process places us in the realm of `random eþ ects m odels’ , in w hich we envisage a sam pling distribution for the abundances, births and survivals at stage I. We can hypothesize that there is a relationship between som e of these param eters, for exam ple an eþ ect of abundance on sur vival rate. We do not expect that variation in the survival rates over tim e w ill be wholly determ ined by abundance. O ther in¯ uences such as weather are also likely to be im portant. If we have m easured these other in¯ uences then we can specify their eþ ect in the m odel. However, there will inevitably be unexplained sources of variation but if we are prepared to assum e that these unexplained departures from the m odel are random , then their com bined in¯ uence can be expressed by including a random term in the m odel. Random eþ ect m ark- recapture m odels have been developed by Burnham (in press) and im plem ented in program M AR K. These m odels are relatively sim ple and at present restricted to the case w here the vector of param eters is sampled independently from a com m on distribution. T hese m odels are useful where a simple sum m ar y such as an average or trend is sough t, or where the study has been conducted within the context of a sim ple exp erim ental design. The m ethodof-m om ents estimators in MARK are sim ple and allow the user to avoid assuming a particular distribution at stage I. Random -eþ ects m odels and Bayesian inference are closely related and share the sam e m ethodological problem s. In recent years there has been a resurgence of interest in Bayesian m ethods, largely prom pted by the development of new analysis methods such as M arkov chain Monte Carlo (M CM C) (G elfand & Sm ith, 1990; Zeger & K arim , 1991; Sm ith & Gelfand, 1992), that m ake ® tting the necessary m odels feasible.
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1.3 Density dependence D ensity dependence is an im portant ecological concept that has particular relevance to the m anagem ent of exploited populations. O ur interest in density-dependence is m otivated by an exploited `titi’ , or sooty shearwater, (Puý nus g riseus) population in the southern part of the South Island, N ew Zealand. T he Rakiura Maori have harvested titi nestlings for at least 300 years. Recently, a research program m e has been established that aim s to ensure that titi harvest is sustainable. Sim ple population models suggest that population growth rate is likely to be sensitive to changes in survival, age at ® rst breeding and breeding success. However, if these param eters are density-dependent, the titi population m ay be able to com pensate for losses due to harvest. Such a com pensatory m echanism is a central them e of harvest literature (Robson & Youngs, 1971) and the m ain scienti® c justi® cation for wildlife har vest. In trying to m easure the strength of any density dependent relationship between survival and abundance, the form of the relationship will be im portant. For exam ple, a linear relationship between abundance and survival (or suitably transform ed survival) m ight be hypothesized over the range of abundances being considered. Alternatively, density dependence m ight not operate until abundance exceeds a particular level, in w hich case a non-linear relationship would be appropriate. N ote that we are assum ing that abundance of the population under study is a m easure of density. It is possible that a m ore general m easure of density could be m ore appropriate. For titi, the number of breeding birds per square m etre m ight be m ore im portant, or the density of non-breeders. As the term suggests, density dependence im plies that population param eters are a function of anim al density or abundance. Im portantly, if the relationship is between abundance and survival then density-dependence im plies that the density function for survival rate conditional on abundance is not the sam e as the m arginal density for sur vival rate.
1.4 Fitting density dependence models Given long-term data sets w ith relatively constant eþ ort we argue that it is possible to investigate ® ner-scale population processes such as the eþ ect of density on the population. This idea is not new. In their m onograph Lebreton et al. (1992) brie¯ y discuss the idea of using density as a covariate and give som e exam ples. A full likelihood approach to m odelling density dependence requires a distribution to be speci® ed for each of the two stages described above. Let N i denote the abundance at tim e i, S i the survival rate from tim e i until time i + 1, and b i the birth rate at time i. At stage I, we assum e that the abundances, sur vival rates, and birth rates are sam pled from a distribution g({N i }, {S i }, { b i }; l ) with param eters represented by l and cum ulative distribution function G({N i }, {S i }, { b i }; l ). Stage II involves m odelling the mark- recapture data represented by x as being a sam ple from the distribution f (x ½ {N i }, {S i }, { b i }). Estimation is carried out using the distribution h(x; l ) which is found by integration (Link, 1999): h(x; l ) 5
ò
f(x ½ {N i}, {S i}, { b i}) d G({N i}, {S i}, { b i}; l )
(1)
Analytical evaluation of the integral in (1) is problem atic for m ark- recapture m odels (Burnham, in press), although the relatively recent developm ent of M CM C
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m ethods m eans that ® nding h(x; l ) by com puter intensive means is becom ing increasingly feasible. In addition to the structural elem ents specifying m odel density, a distribution m ust also be speci® ed at the ® rst stage of sam pling in order to use M CM C m ethods. In the analysis discussed below we concentrate on density-dependent survival and avoid specifying a full model for stage I sam pling. Instead we only specify the structural form of the relationship between survival and abundance and estim ate m odel param eters using the m ethod of m om ents.
2 M ethods 2.1 Estimating density dependence by linear regression We assum e that from the m ark- recapture analysis we obtain estim ates of SÃ i ½ S i which are random variables with m ean S i and covariance given by Cov(SÃ i ½ S i , 2 SÃ j ½ S j ) 5 x i j . From the stage I process we assume that S 5 (S 1 , . . . , S k ) ¢ (or som e transform ation of S ) is a random vector with m ean X b and variance- covariance m atrix r 2 I, where row i of the m atrix X is given by x i 5 (1 N i ). Unconditionally, YÃ 5 (SÃ 1 ,. . . . , SÃ k ) ¢ is a random vector with m ean vector X b and variance- covariance 2 m atrix D 5 r I + W where
W 5
f
2 11
x x
x
2 12
2 21
x
2 22
2 k1
x
: x
:
¼ ¼
2 2k
x
:
` 2 k2
2 k
x
¼ x
2 kk
g
If we could determ ine N i exactly the obvious estim ator would be: 2 1 2 1 2 1 à 2 b à 5 (X ¢ Dà X) X ¢ Dà Y , obtained by conditioning on values of x à ij from the m arkrecapture study and using an iterative procedure to estim ate b and r 2. There are several problem s with this approach to m odelling a density-dependent relationship à i , which between N i and S i . First, we cannot determ ine N i , but instead must use N à we hope is a good estim ate of N i . If we replace X by X and use the estim ator 2 1 à 2 1 à ¢ Dà 2 1 Yà , we have an errors-in-variable problem . Secondly, in b à 5 (Xà ¢ Dà X) X m ark- recapture problem s there is a sam pling covariance between estim ated abundances and sur vival rates that m ust be taken into account. Thirdly, the elem ents of the m atrix W are not known exactly, but instead are estimated. Provided the m ark- recapture study is based on a large sam ple size, the substitution of x à 2ij for 2 x ij should introduce little bias but it m ay be a problem in studies with sm all sam ple 2 size. A ® nal problem is that the parameter r m ust also be estim ated, but is required à in the matrix D to obtain b .
2.2 B ias-adjusted errors-in-variables estimator It is well known that errors-in-variables attenuate the estim ate of b toward 0, and as a consequence the least-squares estimator will tend to underestim ate the strength of the density-dependence relationship (W ittink, 1988). Fuller (1987, p. 14) provides a bias-adjusted estim ator based on the m ethod-of-m om ents for the case where D 5 r 2I. It is straightforward to extend Fuller’s estim ator to the case w here 2 Ã ½ X and YÃ ½ Y (see D 5 r I + W, and where there are sampling covariances between X the appendix).
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By ® tting the Jolly- Seber m odel to m ark- recapture data we can obtain m axim um à i ½ N i ), Var (Sà i ½ S i ), C ov(Sà i ½ S i , N à i ½ N i ), and likelihood estim ates of N i , S i , Var (N à à Cov(S i 2 1 ½ S i 2 1 , N i ½ N i ) for the sam pling occasions i 5 2, . . . , t 2 2. U sing the invariance property of m axim um likelihood estim ators (M LEs) it is relatively simple to ® nd the M LEs for transform ed abundances and survival probabilities denoted N i¢ and S i¢ . Let
M T¢ T 5
f
0
0 n
0
+
à i¢ ½ N i ) Var (N
i5 1
g
and
M T¢ e 5
f
0 n
+
à i¢ ½ N à i) C ov (Sà i¢ ½ S i , N
i5 1
g
à ¢X à 2 M T ¢ T ) 2 1 (Xà ¢ Y 2 M T ¢ e) is a bias-adjusted estim ator for b (see then b Ä 5 (X à i} and {Sà 2i } we appendix for derivation). Because b Ä is a complicated function of {N Ä Ã Ã found Var ( b ) by jackkni® ng the N i and S i pairs.
2.3 Examples We exam ined density-dependence in shearwater population dynam ics using two data sets. In each case we exam ined the relationship between the logit of the survival probability and untransform ed abundance. Both data sets are from studies of birds closely related to P. griseus: (P. tenuirostris, see Bradley & Wooler, unpublished data; P. huttoni, see C uthbert, unpublished data). T he relationship between survival and abundance in these two species is expected to be sim ilar to that in P. g riseus. Speci® c details of the location and methods used in these studies is available in Bradley et al. (1991) for P. tenuirostris and Cuthbert (1999) for P. huttoni.
3 Results A plot of estim ated survival probabilities against abundance estim ates provides little suggestion of a functional relationship for the short-tailed shearwater population (Fig. 1). Form al analysis of these data provided no evidence of a linear
Estimated Survival
120 110 100 90 80 70 60 50 100
150
200
250
300
350
Estimated Abundance F ig. 1. Survival estimates plotted against abundance estimates from a sam ple of 104 5 short-tailed shearwaters banded at Fisher Island, Tasm ania, between 1947 and 1994. The error bars represent + / 2 one standard error.
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R. B arker et al. Table 1. Estim ates of the slope of the eþ ect of abundance on survival and associated standard errors for density for short-tailed shearwaters banded at Fisher Island 1947 - 1994 and Huttons shearwaters banded at Kaikoura 1991 - 1999 Data
Fisher Island Kaikoura
95% C.I
Slope
SE
Lower
U pper
2 0.002 5
0.005 5 0.067 4
2 0.013 4
0.008 3 0.070 9
2 0.061 2
2 0.193 3
functional relationship between abundance and survival probabilities (Table 1). A con® dence inter val for the odds ratio, indicates that a 20 bird change in the population (about 20% of the average estim ated abundance) would lead to a change in the odds of survival by a factor of between 0.77 (a 23% reduction) and 1.181 (an 18% increase). T hus, although there is no evidence of a density dependent relationship between abundance and survival for short-tailed shearwaters on Fisher Island the data are suý cient to rule out a m oderately strong relationship. T he K aikoura data provided little evidence for density-dependence other than that the lowest survival estim ate corresponds to the highest abundance estim ate (Fig. 2). Even a naõÈ ve estim ate of the relationship between survival and abundance, à i , provides at best weak evidence of a negative ignoring the uncertainty in N relationship ( b à 1 5 2 0.0022, SE 5 0.0012). Our analysis using the bias adjusted estim ator showed no evidence of a density-dependent relationship. Although the point estim ate of the slope of the sur vival-ab undance relationship was negative, the con® dence interval for the odds ratio indicated that a change in abundance of approxim ately 20% would lead to a change in the odds of survival by a factor of between 0.38 (a 62% reduction) and 1.426 (a 43% increase). The correct interpretation of these results is that the K aikoura data set is too sparse to rule out anything but a strong density dependent eþ ect. 140% 130%
Estimated Survival
120% 110% 100% 90% 80% 70% 60% 50% 40% 20
40
60
80
100
120
Estimated Abundance in Previous Year
F ig. 2. Survival estimates plotted against abundance estimates from a sam ple of Hutton’ s Shearwaters banded at Kaikoura on the New Zealand m ainland, 1991 to 1999 . The error bars represent + / 2 one standard error.
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4 D iscussion As emphasised by Link (1999), a particularly im portant role for random eþ ects m odels is in providing a useful sum m ary for a large num ber of param eters. Random eþ ects m odels can also be useful for representing im portant functional relationships between param eters that have biological relevance. T his is particularly relevant to m ark- recapture data sets where relatively lengthy studies lead to a large num ber of param eter estim ates. We have m odelled m ark- recapture data as arising from a 2-stage process in which the m ark- recapture param eters, the abundances survival rates and birth rates are ® rst sam pled from a `hyperdistribution’ . C onceptualizing m ark- recapture param eters as random variables is a natural and logical step in m ark- recapture m odels. D ensity dependent survival im plies the presence of a functional relationship between abundance and survival. However it is unreasonable to believe that survival rates are determ ined wholly by abundance. Obvious environm ental in¯ uences should also be included in the m odel at this stage. In addition to the structural com ponents, the random term in the random eþ ects m odel is particularly im portant as it is needed to account for in¯ uences that we are unable to explain or model explicitly. In order to have a realistic chance of identifying m ajor structural features of the data it is im portant to have a large num ber of observations. It is unreasonable to expect much to be learned from a regression based on a sm all num ber of observations. H owever, the interpretation of `sm all’ m ay be diþ erent for a biologist designing a study than for a statistician analysing the data. This em phasizes the need for biologists to com mit resources to long-term monitoring program mes. In contrast to the development of m ethods for m odelling m ark- recapture data conditional on the param eters, higher-order m odelling is relatively undeveloped and there are few exam ples where such a procedure is followed (for an exception see Pledger, 1999). W ith the exception of Burnham ’ s random eþ ects m odels incorporated in M ARK (W hite & Burnham , 1999), random eþ ects models are not available in accessible software for m ark- recapture data. For biologists to extract the full information from m ark- recapture studies it is im portant that software for random eþ ects m odels is developed and widely distributed. We anticipate that M CM C m ethods for Bayesian analyses will ® gure strongly in this developm ent. O ur results give little support for density dependence in adult survival in both data sets exam ined. In the case of the Fisher Island study, data were suý cient for us to rule out anything but a sm all density dependent relationship. In this case, we would expect to see little com pensation in survival rate as breeding densities were reduced by har vest. The Kaikoura data have too few years of obser vation for us to learn much about density dependence. T he m ethods presented here can also be used to assess density dependence in recruitm ent, which m ight arise as a consequence of intra-speci® c com petition. W ith m ulti-state m odels (Brownie et al. 1993) becom ing m ore prevalent, there is also potential to use them to m easure density dependence in the rates at which birds m iss breeding. In the type of analysis we have considered here, `density’ was m easured by population abundance. However, an inappropriate m easure of density m ay result in density dependence being overlooked when it is present. T his could occur if the abundance of breeder in¯ uences survival or reproduction, or both, but the data do not allow us to discrim inate between breeders and non-breeders.
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Acknowledgem ents We thank Richard Cuthbert (K aikoura), and Ron Wooler and Stuart Bradley (Fisher Island) for perm ission to use their data sets. We thank Christine H unter, Byron M organ, Andrew Royle, and an anonym ous referee for com m ents on the m anuscript
R EF ER EN C ES B radley, J. S., S kira, I. J., & W ooler , R. D. (1991 ) A long-term study of short-tailed shearwaters Puý nus tenuirostris, Ibis, 133, pp. 55 - 61. B rownie , C., H ines , J. E., N ichols , J. D., P ollock , K. H., & H estbeck , J. B. (1993 ) Capture- recapture studies for m ultiple strata including non-M arkovian transitions, B iometrics, 49, pp. 1173 - 1187. C uthbert, R. (1999 ) Huttons Shearwaters in the Kaikoura M ountains. PhD thesis, University of Otago, Dunedin. F uller, W. A. (1987) . Measurement Er ror Models (N ew York, W iley). G elfand , A. E. & S mith , A. F. M. (1990 ) Sam pling-based approaches to calculating marginal densities, Journal of the Am erican Statistical Association, 85, pp. 398 - 409 . L ebreton , J. D., B urnham , K. P., C lobert, J. & A nderson , D. R. (1992 ) M odeling survival and testing biological hypotheses using m arked anim alsÐ a uni® ed approach with case-studies, Ecological M onographs, 62, pp. 67 - 118 . L ink , W. A. (1999 ) M odeling pattern in collections of param eters, Jour nal of W ildlife M anagement, 63, pp. 101 7 - 1027. P ledger , S. (1999 ) Finite mixtures in capture-recapture m odels. PhD thesis, Victoria University, Wellington. P radel , R. (1996 ) U tilization of capture- m ark- recapture for the study of recruitment and population growth rate, B iometr ics, 52, pp. 703 - 709 . R obson , D. S., & Y oungs, W. D. (1971 ) Statistical analysis of reported tag-recaptures in the harvest from an exploited population, Biometrics U nit, Cornell U niversity, Ithaca, New York. BU -369-M . S eber , G. A. F. (1965 ) A note of the m ultiple recapture census, B iometr ika, 52, 249 - 259. W hite , G . C. & B urnham , K. P. (1999 ) Program M ARK: survival estim ation from populations of m arked individuals, B ird Study, 46 (supplement), pp. 120 - 139. W ittink , D. R. (1988 ) The Application of Regression A nalysis (Boston, M A, Allyn and Bacon, Boston). Z eger , S. L., & K arim , M . R. (1991 ) G eneralized linear m odels with random eþ ects: a G ibbs sam pling approach, Jour nal of the American Statistical Association, 86, pp. 79 - 86.
Appendix Let Y 5 X b + e, where Y is an (n 3 1) vector, X is an (n 3 p) matrix of constants, e is an (n 3 1) random vector with m ean zero, and R wz 5 Cov(w, z) is the covariance à 5 X +T m atrix for the random vectors w and z. Suppose we observe Y and X where T is a (n 3 p) random measurem ent error m atrix m ade up of colum n vectors t i (i 5 1, . . . , p) each with m ean 0. From least-squares estim ation theory 2 1 2 1 E[X ¢ X) X ¢ Y ] 5 (X ¢ X ) X ¢ X b 5 b . N ow: à ¢Y] 5 (1) E[X (2) E[T ¢ Y ] 5 tr ( R tp e)) ¢ , à ¢X à ]5 (3) E[X
(4) E[T ¢ T ] 5
E[(X + T ) ¢ Y ] 5 X ¢ X b + E[T ¢ Y ]. E[T ¢ (X b + e)] 5 E[T ¢ X b ] + E[T ¢ e] 5 0 + (tr ( R t1 e), tr ( R t2 e ), . . . , where tr(Z ) is the trace of the m atrix Z. E[(X + T ) ¢ (X + T ) ¢ ] 5 E[X ¢ X + X ¢ T + T ¢ X + T ¢ T ] 5 X ¢ X + E[T ¢ T ]
f
tr ( R
t1t1
) tr ( R
t1 t2
)
tr ( R
t2 t1
) tr ( R
t2 t2
)
: tr ( R
: tpt 1
) tr ( R
¼ ¼
tr ( R
t1 tp
)
tr ( R
t2tp
)
:
` tpt2
)
¼
tr (R
tp tp
)
g
n
where tr ( R
t jt k
)5
+ i5 1
Cov(t i j , t ik ) .
Density dependence in mark- recapture
313
à ¢X à 2 E[T ¢ T ] is a consistent estim ator If the covariances in E[T ¢ T ] are know n, then X ¢ ¢ for X X, and if the covariances in E[T Y ] are also known, then 2 1 b Ä 5 (Xà ¢ Xà 2 E [T ¢ T ]) (Xà ¢ Y 2 E [T ¢ Y ])
should also be a consistent estim ator for b . Assum ing E[T ¢ Y ] 5 0 gives us the m atrix analogue of Feller’ s estimator. N ote that we can generalize the above to à ¢ V 2 1X à )2 1X à ¢ V 2 1 Y however, if the weighted least squares estim ators of the form (X weight m atrix has elem ents that need to be estimated we lack a theory that can be used as a basis for deriving an estim ator.
