Estimating age-specific survival rates from historical ring-recovery data Diana J. Cole and Stephen N. Freeman Mallard Dawn Balmer (BTO) Sandwich Tern Jill Pakenham (BTO)

2 of 15 Introduction (Robinson, 2010, Ibis) Prior to 2000 BTO ringing data were submitted on paper forms which have not yet been computerised. Free-flying birds can be categorised as: – Juveniles (birds in their first year of life) – “Adults” (birds over a year) There are more than paper records listed by ringing number rather than species. Each record will indicate whether a bird was a juvenile or an adult at ringing. Recovered birds can be looked up and assigned to their age- class at ringing. However the totals in each category cannot easily be tabulated. There is also separate pulli data (birds ringed in nest), where totals are known.

3 of 15 Introduction Example ring-recovery data (simulated data) Total Ringed Ringed as JuvenilesRinged as Adults Year

4 of 15 Introduction Robinson (2010, Ibis) use Sandwich Terns (Sterna sandvicensis) historical data as a case study. In Robinson (2010) a fixed proportion in each age class is assumed. For the Sandwich Terns this is 38% juvenile birds. This is based on the average proportion for computerised data where the totals in each age class are known (range 25-47%) Using parameter redundancy theory we show that this proportion can actually be estimated as an additional parameter.

5 of 15 Historic Model Assume there were n 1 year of ringing, n 2 years of recovery We know N i,t,1 and N i,t,a - the number of juvenile and adult birds ringed in year i who were recovered dead in year t. We only know T i - the total number of birds ringed in each year i. Parameters: – p t is the proportion of birds ringed as juveniles at time t, with (1 – p t ) ringed as adults; –  1,t is the annual probability of survival for a bird in its 1st year of life in year t; –  a,t is the annual probability of survival for an adult bird in year t; – 1,t is the recovery probability for a bird in its 1st year of life in year t. – a,t is the recovery probability for an adult bird in year t.

6 of 15 Historical Model The probability that a juvenile bird ringed in year i is recovered in year t The probability that an adult bird ringed in year i is recovered in year t Likelihood: (number of birds never seen again)

7 of 15 Parameter Redundancy Methods Symbolic algebra is used to determine the rank of a derivative matrix (Catchpole and Morgan, 1997, Catchpole et al,1998 and Cole et al, 2010a). Rank = number of parameters that can be estimated Parameter redundant models: rank < no. of parameters Full rank model: rank = no. of parameters Example: Constant survival in 2 age classes, constant recovery, constant proportion juvenile, n 1 = 2 years of ringing, n 2 = 2 years of recovery Parameters: Exhaustive summary: Age class 1 (ringed in first year)Age class 2 (ringed as adults)

8 of 15 Methods Derivative matrix: rank = 4 = no. parameter, model full rank Extend result to general n 1 and n 2 using the extension theorem (Catchpole and Morgan, 1997 and Cole et al, 2010a)

9 of 15 Results – constant p Model parametersRankDeficiency Deficiency of standard model  1,  a,, p 400  1,  a, t, p n  1,  a, 1, a, p 410  1,  a, 1,t, a,t, p n 1 + n  1,t,  a,t,, p n 1 + n  1,t,  a,t, t, p (n 2 = n 1 ) 3n13n1 11  1,t,  a,t, 1, a, p n 1 + n  1,t,  a,t, 1,t, a,t, p (n 2 = n 1 ) 4n 1 – 232  1,t,  a,, p n  1,t,  a, t, p n 1 + n  1,t,  a, 1, a, p n  1,t,  a, 1,t, a,t, p (n 2 = n 1 ) 3n13n1 21  1,  a,t,, p n  1,  a,t, t, p (n 2 = n 1 ) 2n  1,  a,t, 1, a, p n  1,  a,t, 1,t, a,t, p (n 2 = n 1 ) 3n13n1 21

10 of 15 Results – time dependent p Model parametersRankDeficiency Deficiency of standard model  1,  a,, p t n  1,  a, t, p t 2n  1,  a, 1, a, p t n  1,  a, 1,t, a,t, p t n 1 + 2n 2 20  1,t,  a,t,, p t n 1 + 2n  1,t,  a,t, t, p t (n 2 = n 1 ) 4n 1 – 111  1,t,  a,t, 1, a, p t n 1 + 2n  1,t,  a,t, 1,t, a,t, p t (n 2 = n 1 ) 5n 1 – 442  1,t,  a,, p t n 1 + n  1,t,  a, t, p t n 1 + 2n  1,t,  a, 1, a, p t n 1 + n  1,t,  a, 1,t, a,t, p t (n 2 = n 1 ) 4n 1 – 231  1,  a,t,, p t 2n  1,  a,t, t, p t 3n  1,  a,t, 1, a, p t 2n  1,  a,t, 1,t, a,t, p t (n 2 = n 1 ) 4n 1 – 231

11 of 15 Simulation True Value Standard ModelHistorical Model ParameterMeanStdevMSEMeanStdevMSE 1 aa p Data simulated from  1,  a,, p model with n 1 = 5 and n 2 = 5 Results from 1000 simulations

12 of 15 Mallard Data Mallard data ( ). Two data sets: – ringed as juveniles – ringed as adults of unknown age We pretend to not know the total in each age class - historical data model. Compare to the standard ring-recovery model, where totals are known. All the full rank models in the previous tables were fitted to Mallard data. Standard model with lowest AIC:  1,  a,t, t followed by  1,  a, 1,t, a,t (  AIC = 6.7) Historic model with lowest AIC:  1,  a,t, t, p followed by  1,  a,t, t, p t (  AIC = 4.8)

13 of 15 Mallard Data - Models with smallest AIC -  1,  a,t, t

14 of 15 Discussion Recommended that symbolic methods are used to detect parameter redundancy before fitting new models. In this example we have shown that the historic model is mostly full rank if standard model is full rank. The historic model is nearly as good as the standard model at estimating parameters, when the historic model is full rank. Some problems with first or last time points for time dependent parameters, particularly as p gets closer to 1 (1963 for Mallard data). Mallard adult data is of unknown age. McCrea et al (2010) fit an age-dependent mixture model to this data. Such a model fitted to the adult data alone is parameter redundant, but can estimate adult survival parameter. If combined with juvenile data most models are no longer parameter redundant. Robinson (2010) Sandwich Terns model has separate survival parameters for 1 st year, 2 nd and 3 rd year, older birds. Ideal model: – standard model for the pulli data – a historical model for free-flying birds with an age-mixture model for the ‘adult’ part.

15 of 15 References Catchpole, E. A. and Morgan, B. J. T (1997) Detecting parameter redundancy. Biometrika, 84, Catchpole, E. A., Morgan, B. J. T and Freeman, S. N. (1998) Estimation in parameter redundant models Biometrika, 85, Cole, D. J., Morgan, B. J. T and Titterington, D. M. (2010) Determining the Parametric Structure of Non-Linear Models. Mathematical Biosciences. 228, 16–30. McCrea, R. S., Morgan, B. J. T and Cole, D. J. (2010) Age- dependent models for recovery data on animals marked at unknown age. Technical report UKC/SMSAS/10/020 Paper available at Robinson, R. A. (2010) Estimating age-specific survival rates from historical data. Ibis, 152, 651–653.