Estimating age-specific survival rates from historical ring-recovery data

1. 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. 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 700 000 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. Introduction • Example ring-recovery data (simulated data)

4. 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 2000-2007 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. Historic Model • Assume there were n1 year of ringing, n2 years of recovery • We know Ni,t,1 and Ni,t,a - the number of juvenile and adult birds ringed in year iwho were recovered dead in year t. • We only know Ti - the total number of birds ringed in each year i. • Parameters: • ptis the proportion of birds ringed as juveniles at time t, with (1 – pt) ringed as adults; • 1,tis 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,tis 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. 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. 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, n1 = 2 years of ringing, n2 = 2 years of recovery Parameters: Exhaustive summary: Age class 1 (ringed in first year) Age class 2 (ringed as adults)

8. Methods Derivative matrix: rank = 4 = no. parameter, model full rank Extend result to general n1 and n2 using the extension theorem (Catchpole and Morgan, 1997 and Cole et al, 2010a)

9. Results – constant p

10. Results – time dependent p

11. Simulation Data simulated from 1, a, , p model with n1 = 5 and n2 = 5 Results from 1000 simulations

12. Mallard Data • Mallard data (1964-1971). 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, pt (AIC = 4.8)

13. Mallard Data -Models with smallest AIC - 1,a,t, t

14. 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 1st year, 2nd and 3rd 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. References • Catchpole, E. A. and Morgan, B. J. T (1997) Detecting parameter redundancy. Biometrika, 84, 187-196. • Catchpole, E. A., Morgan, B. J. T and Freeman, S. N. (1998) Estimation in parameter redundant models Biometrika, 85, 462-468. • 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 http://www.kent.ac.uk/ims/personal/djc24/McCreaetal2011.pdf • Robinson, R. A. (2010) Estimating age-specific survival rates from historical data. Ibis, 152, 651–653.