Modelling the survival of British Lapwings Vanellus vanellus using ring-recovery data and weather covariates

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1 Bird Study ISSN: (Print) (Online) Journal homepage: Modelling the survival of British Lapwings Vanellus vanellus using ring-recovery data and weather covariates Edward A. Catchpole, Byron J. T. Morgan, Stephen N. Freeman & William J. Peach To cite this article: Edward A. Catchpole, Byron J. T. Morgan, Stephen N. Freeman & William J. Peach (1999) Modelling the survival of British Lapwings Vanellus vanellus using ring-recovery data and weather covariates, Bird Study, 46:sup1, S5-S13, DOI: / To link to this article: Published online: 25 Jun Submit your article to this journal Article views: 387 View related articles Citing articles: 34 View citing articles Full Terms & Conditions of access and use can be found at Download by: [ ] Date: 04 December 2017, At: 15:00

2 Bird Study (1999) 46 (suppl.), S5-13 Modelling the survival of British Lapwings Vanellus vanellus using ring-recovery data and weather covariates EDWARD A. CATCHPOLE 1*, BYRON J.T. MORGAN', STEPHEN N. FREEMAN2 and WILLIAM J. PEACH 3 'School of Mathematics & Statistics, University College, University of New South Wales, Australian Defence Force Academy, Canberra ACT 2600, Australia; 2Institute of Mathematics & Statistics, University of Kent at Canterbury, Canterbury, Kent CT2 7NF, UK; 3British Trust for Ornithology, The Nunnery, Nunnery Place, Thetford, Norfolk IP24 2PU, UK Two sets of data are considered. One describes recovery data from Lapwings ringed as young in Britain in The other provides several weather covariates for the same period, and our objective is to try to relate these two data sets. We use score tests, provided by the computer package Eagle, to build a simple model which describes the data well. We first establish a need for time variation in the three basic elements of the model: first-year annual survival probability, adult annual survival probability and reporting probability. We then investigate the direct incorporation of covariates, using score tests, to give a limited set of models to compare. The final choice between these models is based on the Akaike information criterion (Arc). We find that adult annual survival is negatively related to measures of winter weather severity. In contrast to previous work, we have found a pronounced decline of reporting rate with time. Our methods produce appreciably higher estimates of adult survival than those previously obtained using methods which assumed reporting rates to be constant over time. The aim of this paper is twofold. We provide a new analysis of recovery data from British Lapwings Vanellus vanellus ringed as chicks. Unlike the work of Peach et al.,' this analysis incorporates weather covariates directly into the models. We thereby demonstrate new facilities of the Eagle computer package, 2 which is particularly well-suited to the incorporation of covariate information. The recovery data are for birds ringed as chicks during the years (Table 1). This period was chosen to avoid the very severe 1962 / 63 winter, whose influence otherwise overwhelms the other effects. In contrast to the earlier analysis, these data include birds found dead within 30 days of ringing, which changes *Correspondence author. e-catchpole@adfa.edu.au the estimates of first-year survival. Years are always measured as starting from 1 May. For the complete background, see the paper by Peach et al.' which deals with recovery data up to 1990; in particular they note that ring wear and loss are unlikely to be problems during the years we are concerned with. Five weather variables are available: mean air temperature (Temp), mean soil temperature (Soil), number of days when the ground grass temperature falls below 0 C (Frost), number of days with lying snow (Snow) and number of days with more than 0.2 mm of rain (Rain). Monthly mean values were calculated over a range of weather stationsl and these have been further amalgamated to cover the three main periods: the breeding season, April June (which we term period 1); the autumn season, period 2, which covers July October; and period 3, covering the C) 1999 British Trust for Ornithology

3 S6 E.A. Catchpole et al. Table 1. Recovery data for British Lapwings ringed as chicks during the years Ringing took place before 1 May in each year. For example, of the 1147 birds ringed in 1963,14 were found dead during the period 1 May 1963 to 30 April 1964, four were found dead during the period 1 May 1964 to 30 April 1965, etc. Year of Number Year of recovery ringing ringed winter months, November February. As there are no snow days in periods 1 or 2, we have a total of 13 weather variables. In addition we use calendar year (Year) as a covariate, to allow for the smooth changing of parameters over time. The correlation matrix for the set of 13 weather variables and Year is shown in Table 2. The large absolute correlations, for example the positive ones between Soil and Temp, and the negative ones between Rain and Temp, are unsurprising. Most recoveries of first-year birds occur during June August, which spans periods 1 and 2, and during December February (period 3). Most adult recoveries occur in December- May, 1 which corresponds roughly to period 3, and to some extent period 1. The previous analysis of the recovery datal assumed a constant reporting rate for dead birds, allowing no variation with respect to age or time. We allow full time dependence of reporting rate. This is important, since if the assumption of constant reporting rate is made incorrectly, then the time variation may manifest itself through estimates of parameters elsewhere in the model. We assume initially that there is no age variation in reporting rate. This is plausible, as Peach et al.' found that the proportion of recoveries for various major causes of death does not differ between adults and firstyear birds. A formal test of this assumption could be made if we also analysed recovery data for birds ringed as adults, but very few such data exist. However, as we discuss later, estimation of the adult annual survival probability is robust with respect to the violation of this assumption. Peach et al.' concluded that both first-year and adult annual survival probabilities were temperature dependent. Their analysis was done in two stages, by first obtain British Trust for Ornithology, Bird Study, 46 (suppl.), S5-13

4 Modelling Lapwing survival S7 Table 2. Correlations of weather covariates. Tempi Temp2 Temp3 Soul Soil2 Soil3 Frost1 Frost2 Frost3 Stiow3 Rainl Rain2 Rain3 Year Tempi Ternp2 Tern p3 Soi11 Soil2 Soil3 Frostl Frost2 Frost3 Snow3 Rain1 Rain ing the survival estimates for each year and then relating these to covariates using multiple regression. It is preferable to perform an integrated analysis in which survival is modelled explicitly as a function of the covariates, and we do this via the Eagle computer package. 2 Eagle currently allows for complete time dependence in survival and reporting rates, ' and allows age dependence in survival but not in reporting rate. However, Catchpole 2 showed how any desired model can be fitted in the Matlab environment of Eagle. See also North & Morgan3 for relevant work and discussion. MODELS, SCORE TESTS AND DATA REDUCTION Models We adopt the x/y/z notation 2,4 for model specification. Here x describes the type of first-year annual survival assumed in the model, either C (constant) or T (time-dependent, i.e. varying from year to year); y denotes the type of adult annual survival assumed in the model (C, T or An, where An indicates the degree of age variation in adult survival probability: for example, A2 corresponds to a separate annual survival probability for second-year birds, and then a different, constant annual survival probability for older birds); and z describes the reporting probability, which may be C or T. Time-dependent covariates may be introduced into all three model components, denoted for example by V(Temp). All regressions on covariates are logistic regressions. Note that the model C/T/T cannot be fitted to the data, because, if k denotes the length of the study in years, the components, 0, and A.k, the adult survival and reporting probabilities, respectively, for the last year of the study, occur only together as the product (1 - O k )A,k. They are confounded, so cannot be estimated individually. We use the notation C/ T*/T to denote a model with equal adult survival probabilities for the last two years of the study, which allows all individual parameters to be estimated. Model selection Variable selection in regression-type models is a difficult problem with a long history, and is the subject of continuing research. 5 Our situation is more complicated, as each of the x/y /z components can be modelled, which results in many possible models for the data. We adopt an exploratory approach, based on score tests, 4,6 which form the basis of model selection in the Eagle package. Score tests are asymptotically equivalent to likelihood-ratio tests, but require the fitting of only the simpler of the two models being compared. Our approach is a modified step-up procedure. For example, we first fit the C / C/C model, and then examine whether the C /C/ T model is better. (If we were using a likelihood-ratio test then we would need to fit the C /C/ T model and compare maximized likelihoods. When a score test is used, only model C/C /C need be fitted.) In Catchpole & Morgan, 4 a simple step British Trust for Ornithology, Bird Study, 46 (suppl.), S5-13

5 S8 E.A. Catch pole et al. up procedure was followed. Score tests were applied in sequence, and at each stage in the sequence the model which produced the largest score statistic was fitted. This model then formed the basis for the next stage in the score test procedure. Care must be taken that variation in one part of the model does not contaminate other parts of the mode1. 4,7 This is especially true when score tests are used for variable selection. For example, if the C / C/ C model is significantly improved by a C/C/V model, it is potentially dangerous to use a C / C/ V model as a basis for further variable selection. This is because time variation which may be necessary for describing survival probabilities might result in an inappropriate variable being selected in a C/C/V model. Instead, we adopt the following conservative strategy. In terms of variable selection, if we are considering which variables may be important for adult survival, we may fit the T/C/ T model and then perform score tests of this model vs. the 14 alternatives, resulting from replacing C by the covariate Year and by each of the 13 weather variables in turn. This approach has two attractions. Firstly, using score tests means that all of these comparisons can be made from fitting just the one model, T/C/ T, saving much computing time. Secondly, the step-up approach means that a suitable model is built up without unnecessary models having to be considered, saving time and possibly confusion. This is a conservative approach, because, with T / C/T, for example, as the basic model, much of the variation in the data is already described by the time-dependent parameters for first-year survival and reporting probability. We found no evidence to suggest age variation in adult survival. We simplify the description of the model selection procedure by omitting details of this step. Thus we proceed as follows: (i) The models T/T/C, T/C/T and C/T*/T are fitted to the data. (ii) For each of the models of stage (i), score tests are carried out, comparing C to V for Year and each of the weather covariates in turn. These covariates are included singly and then, if necessary, multiply. (iii)if stage (ii) suggests covariates for inclusion in the model, then relevant models are fitted and compared using the AIC, which is analogous to the use of Mallow's C p index in multiple-regression problems (Atkinson, ref. 8, p. 62). The utility of AIC for model selection is well established. 9 (iv)finally, goodness-of-fit is checked, both by means of a formal test and by examination of residuals. RESULTS Score tests The results of the stage (ii) score tests are presented in Table 3. We show only those covariates with a score statistic significant at the 5% level. For adult survival the score tests select just three weather variables, all from period 3 and highly correlated. Of these, we choose Snow3, as it is the most significant. Score tests of T/V(Snow3)/T vs. T /V(Snow3, cov)/t (not shown in Table 3) are not significant for any choice of additional covariate, so there is no need for any further modelling of adult sur- Table 3. Score tests for choosing weather covariates, for adult survival, first-year survival and reporting probability. In all cases the statistics are referred to x2 tables on one degree of freedom. T/C/T vs. T/V/T C/T*/T vs. V/T*/T T/T/C vs. T/T/V V Statistic V Statistic V Statistic Snow Frost Year Tern p Rain Frostl Frost Year 4.32 Rainl Rainl 4.00 Soil Tern p Tern p Soil British Trust for Ornithology, Bird Study, 46 (suppl.), S5-13

6 Modelling Lapwing survival S9 Table 4. AIC values for selected models. C/C/C 1184 C/C/T 1088 C/T/C 1147 T/C/C 1183 T/C/T 1104 T/T/C 1137 CI T*IT 1117 T/T/V(Year) 1106 T/T/V(Year, Paint) 1097 T /V(Snow3)/T 1101 V(Frost2)/T" /T 1111 T /V(Snow3)/V(Year, Rainl) 1082 V(Frost2)/T/V(Year, Rainl) 1116 V(Frost2)/V(Snow3)/T 1080 V(Frost2)/V(Snow3)/V(Year) 1089 V(Frost2)/V(Snow3)/V(Year, Rainl) 1089 V(Frost2)/V(Snow3)/V(Year, Year 2 ) 1081 vival. This is also true if Snow3 is replaced by Tern p3 or Frost3, so any one of these can be used to model adult survival, resulting in a model which is biologically plausible. The score tests for first-year survival select Year and weather variables from periods 1 and 2. The results for Frost2 and Rain2 are largely due to single outliers. We choose the model based on Frost2 to illustrate the approach, but other variables result in a comparable fit to the data (see Discussion). After Frost2 is included in the model, no other variables contribute significantly to the annual variation in first-year survival. To some extent, the reporting probability can be regarded as a nuisance parameter, and it is arguable whether its variation needs to be modelled. However the score tests show strong evidence of a trend in reporting rate over time. After Year has been fitted, only Rainl contributes significantly to the annual variation in reporting probability. We now proceed to stage (iii) of the variable selection. Table 4 presents AIC values for a range of models suggested by the screening procedure of (ii). The last stage of model selection reveals a case for adding a quadratic term to the logistic regression of reporting probability on Year, and such a model is also shown in Table 4. Sole use of the AIC selection procedure for these data would involve fitting a large number of models, but our selection procedure has kept the total number of models fitted to 17, only 3 of which needed to be fitted in the variable screening of stage (ii). Relationships between adult annual survival and the number of winter snow days, and between reporting rate and Year, are shown in Fig. 1. The models V(Frost2)/V(Snow3)/T and V(Frost2)/V(Snow3)/V(Year, Year,) have the lowest AIC values of the models considered, and we select these models to provide parsimonious descriptions of the data. (We do not select the model T /V(Snow3)/V(Year, Rainl), as we are interested in modelling survival. See also the Discussion.) The parameter estimates, their estimated standard errors and the correlations between the estimators for the second of the chosen models are presented in Table 5. The goodness-of-fit of the Table 5. Maximum-likelihood parameter estimates and estimated correlation matrix, from fitting the model V(Frost2)/V(Snow3)/V(Year, Year 2 ). Note that all covariates are mean-centred. Thus the logistic regressions are all of the basic form p = (1 + expha + fl(x - i )1D -1, where p denotes the appropriate probability, x the appropriate covariate and f its sample mean. The coefficient for Year2 is 7. Also shown is the Wald statistic W for the regression coefficients, and the corresponding p-value. Variable Parameter Coefficient Estimate W p Correlation matrix (standard errors on diagonal) Frost2 Juvenile a survival p Snow3 Adult a survival Year Reporting a rate C 1999 British Trust for Ornithology, Bird Study, 46 (suppl.), S5-13

7 S10 E.A. Catchpole et al a Tz 0.8.> MME o o < Snow c Year Figure 1. (a) Adult survival versus Snow3. The point estimates, marked by asterisks, are from the model V(Frost2)/ T/ T. The line shows the prediction of model V(Frost2)/V (Snow3)/T. (b) Reporting probability varying with time. The point estimates, marked by asterisks, are from the model V(Frost2)/V (Snow3)/T.---, Predictions of the model V(Frost2)/V (Snow3)/V(Year);, predictions of the model V(Frost2)/V (Snow3)/V(Year, Year 2 ). first of the models is demonstrated in Fig. 2. The residuals from the two models reveal no striking patterns. The goodness-of-fit can in principle be tested by comparing the deviance of the selected model with its asymptotic X2 distribution. However, with such sparse data matrices as that shown in Table 1, this asymptotic assumption is unjustified. A better procedure is a Monte Carlo test. Accordingly, we generate 99 artificial data sets, assuming that the selected model V(Frost2)/V(Snow3)/V(Year, Year2) is the correct CD 1999 British Trust for Ornithology, Bird Study, 46 (suppl.), S5-13

8 Modelling Lapwing survival Sll Observed numbers of recoveries Figure 2. Expected numbers of recoveries, from each cohort for each year, for the model V(Frost2)/V (Snow3)/T, plotted against the observed numbers recovered. model, and using the parameter values from Table 5. We refit the model to each data set in turn, and compute the deviance. This gives an approximation to the distribution of the deviance if the selected model is correct (Fig. 3) Comparing the deviance of the observed data with this distribution gives a p-value for goodness-of-fit of approximately (Assuming the asymptotic distribution, which is x2 on 458 degrees of freedom, leads to a p-value of 0.94, Deviance Figure 3. The Monte-Carlo distribution of the deviance from the model V(Frost2)/V (Snow3)/V(Year, Year2). The deviance from the observed data is marked with an asterisk British Trust for Ornithology, Bird Study, 46 (suppl.), S5-13

9 S12 E.A. Catchpole et al. which considerably overstates the goodnessof-fit.) DISCUSSION The model of this paper has obvious qualitative similarities with the multiple regression results of Peach et al) However, our model is simpler, and has the advantage of being the result of an integrated process in which the regressions on the weather covariates take place as part of the model-fitting process, and not afterwards, as a separate least-squares regression on covariates. A separate regression is inappropriate, since standard regression assumptions are violated: the variables are correlated with unequal variances, and these correlations and variances cannot be estimated. Our work differs from the previous analysis in having a reporting rate which decreases with time, in keeping with the noted declines in the UK, over the time period of this study, in the reporting rate of several other bird species, 1031 and the decrease in shooting of Lapwings. 1 Especially for a hunted species, we need to consider whether allowing different reporting rates for adult and first-year birds would change the conclusions. While such a model can be fitted to recovery data, estimators of parameters may be very imprecise. 12 Instead we test the sensitivity of our estimates by assuming that, instead of being equal, first-year and adult reporting rates differ by a factor of 10. This results in a substantial change to the estimates of first-year survival, which has obvious implications for studies of population dynamics, but the estimates of adult survival are almost unchanged. The model V(Rain1)/V(Snow3)/V(Year,Year 2), for which the AIC value is 1085, also fits the data well, and provides an alternative description of the data in which first-year survival is positively related to rainfall during April-June, which agrees with the findings of Beintema. 13 Note that the covariate Rainl is included in the description of reporting probability only when first-year survival is fully time-dependent. When first-year survival is modelled, Rain1 no longer contributes significantly to the regression. This is a feature of these models that requires further investigation. Our analyses were repeated for the same data set as used by Peach et al) For the period British Trust for Ornithology, Bird Study, 46 (suppl.), S , which included the severe winter of 1962 / 63, the constant adult survival model of Peach et al. (not fully reported in that paper) estimated average annual adult survival as (0.011). Using our method which allows for temporal variation in reporting rate and first-year survival, we obtain a comparable estimate of (0.011) from fitting the T / C / T model. This higher survival estimate is equivalent to a 13% increase in average adult life expectancy. Fitting the T / C / T model to our current data set gives an adult survival estimate of (0.011). In further work we plan to incorporate our estimates of annual survival into a population model for British Lapwings. ACKNOWLEDGEMENTS These analyses have only been possible because of the labour and dedication of the many ringers who have ringed Lapwings since ringing began in Britain. The Ringing Scheme is funded as part of a partnership of the British Trust for Ornithology and the Joint Nature Conservation Committee (on behalf of English Nature, Scottish Natural Heritage and also on behalf of the Environment and Heritage Service in Northern Ireland) and also by the ringers themselves. REFERENCES 1. Peach, W.J., Thompson, P.S. & Coulson, J.C. (1994) Annual and long-term variation in the survival rates of British Lapwings Vanellus vanellus. J. Anim. Ecol., 63, Catchpole, E.A. (1995) Matlab an environment for analysing ring-recovery and recapture data. J. App!. Stat., 22, North, P.M. & Morgan, B.J.T. (1979) Modelling heron survival using weather data. Biometrics, 35, Catchpole, E.A. & Morgan, B.J.T. (1996) Model selection in ring-recovery models using score tests. Biometrics, 52, Shao, J. (1993) Linear model selection by crossvalidation. J. Am. Stat. Assoc., 88, Catchpole E.A., Morgan B.J.T. & Boucher, M. (1997) Score tests.the Ring, 19, Boucher, M. (1996) A comparison of the score test with the likelihood-ratio test for capture-recapture models. MSc dissertation, University of Kent at Canterbury, UK. 8. Atkinson, A.C. (1985) Plots, Transformations and Regressions. Oxford University Press, Oxford.

10 Modelling Lapwing survival S13 9. Burnham, K.P., White G.C. & Anderson, D.R. the survival rates and movements of British Cor- (1995) Model selection strategy in the analysis morants. Bird Study, 46 (suppl.), of capture-recapture data. Biometrics, 51, 12. Catchpole E.A., Freeman, S.N. & Morgan B.J.T (1995). Modelling age variation in survival and 10. Baillie, S.R. & Green, R.E. (1987) The importance reporting rates for recovery models. J. App!. Stat., f variation in recovery rates when estimating 22, survival rates from ringing recoveries. Acta 13. Beintema, A. J. (1994) Condition indices for wader Ornithol., 23, chicks derived from body-weight and bill-length. 11. Wernham, C. V. & Peach, W. J. (1999) Changes in Bird Study, 41, British Trust for Ornithology, Bird Study, 46 (suppl.), S5-13