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

Diana J. Cole and Stephen N. Freeman

MallardDawn Balmer (BTO)

Sandwich TernJill Pakenham (BTO)

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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.

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Introduction

• Example ring-recovery data (simulated data)

TotalRinged

Ringed as Juveniles Ringed as AdultsYear 1996 1997 1998 1999 1996 1997 1998 1999

1996 300 15 3 6 1 14 11 3 5

1997 300 13 1 3 13 9 8

1998 300 27 2 11 7

1999 300 19 4

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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.

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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 i who were recovered dead in year t.• We only know Ti - the total number of birds ringed in each year i.• Parameters:

– pt is the proportion of birds ringed as juveniles at time t, with (1 – pt) 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.

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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)

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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 recoveryParameters:Exhaustive summary:

Age class 1 (ringed in first year) Age class 2 (ringed as adults)

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Methods

Derivative matrix:

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

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Results – constant p

Model parameters Rank DeficiencyDeficiency of

standard model1, a, , p 4 0 01, a, t , p n2 + 3 0 0

1, a, 1, a, p 4 1 01, a, 1,t, a,t, p n1 + n2 + 2 1 0

1,t, a,t, , p n1 + n2 + 2 0 01,t, a,t,t, p (n2 = n1) 3n1 1 11,t, a,t, 1, a, p n1 + n2 + 3 0 0

1,t, a,t, 1,t, a,t, p (n2 = n1) 4n1 – 2 3 21,t, a, , p n1 + 3 0 01,t, a,t, p n1 + n2 + 2 0 0

1,t, a, 1, a, p n1 + 4 0 01,t, a, 1,t, a,t, p (n2 = n1) 3n1 2 1

1, a,t, , p n2 + 3 0 01, a,t,t, p (n2 = n1) 2n1 + 2 0 01, a,t, 1, a, p n2 + 3 1 0

1, a,t, 1,t, a,t, p (n2 = n1) 3n1 2 1

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Results – time dependent pModel parameters Rank Deficiency

Deficiency ofstandard model

1, a, , pt n2 + 3 0 01, a, t , pt 2n2 + 2 0 0

1, a, 1, a, pt n2 + 3 0 01, a, 1,t, a,t, pt n1 + 2n2 2 0

1,t, a,t, , pt n1 + 2n2 + 1 0 01,t, a,t,t, pt (n2 = n1) 4n1 – 1 1 11,t, a,t, 1, a, pt n1 + 2n2 + 2 0 0

1,t, a,t, 1,t, a,t, pt (n2 = n1) 5n1 – 4 4 21,t, a, , pt n1 + n2 + 2 0 01,t, a,t, pt n1 + 2n2 + 2 0 0

1,t, a, 1, a, pt n1 + n2 + 3 0 01,t, a, 1,t, a,t, pt (n2 = n1) 4n1 – 2 3 1

1, a,t, , pt 2n2 + 2 0 01, a,t,t, pt 3n2 + 1 0 0

1, a,t, 1, a, pt 2n2 + 3 0 01, a,t, 1,t, a,t, pt (n2 = n1) 4n1 – 2 3 1

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Simulation

True Value

Standard Model Historical Model

Parameter Mean Stdev MSE Mean Stdev MSE

1 0.4 0.3984 0.0517 0.00267 0.3979 0.0517 0.00268a 0.6 0.6013 0.0554 0.00307 0.6015 0.0582 0.00338 0.3 0.3027 0.0238 0.00057 0.3031 0.0241 0.00059p 0.6 0.5972 0.0336 0.00114

Data simulated from 1, a, , p model with n1 = 5 and n2 = 5Results from 1000 simulations

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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 by1, a,t,t, pt (AIC = 4.8)

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Mallard Data - Models with smallest AIC - 1, a,t, t

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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.

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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.