parameter redundancy in ecological models

Parameter Redundancy in Ecological ModelsDiana Cole, University of Kent

Byron Morgan, University of KentRachel McCrea, University of KentBen Hubbard, University of Kent

Stephen Freeman, Centre for Ecology and HydrologyRemi Choquet, Centre d'Ecologie Fonctionnelle et Evolutive

Mike Titterington, University of GlasgowTed Catchpole, University of New South Wales

Wandering Albatross Striped Sea Bass

Mallard(Dawn Balmer, BTO)

Sandwich Tern(Jill Pakenham, BTO)

Herring Gulls

Introduction

• If a model is parameter redundant you cannot estimate all the parameters in the model.

• Parameter redundancy is equivalent to non-identifiability of the parameters.

• A model that is not parameter redundant will be identifiable somewhere (could be globally or locally identifiable).

• Parameter redundancy can be detected by symbolic algebra.• Ecological models and models in other areas are getting more

complex – then computers cannot do the symbolic algebra and numerical methods are used instead.

• In this talk we show some of the tools that can be used to overcome this problem using ecological examples.

Introductory Example Cormack Jolly Seber (CJS) ModelCapture-Recature

Herring Gulls (Larus argentatus) capture-recapture data for 1983 to 1986 (Lebreton, et al 1995)

Numbers Ringed:

Numbers Recaptured:

111

123

78

R

9100

31030

2467

N

83

84

85Ringing yr

83

84

85

Recapture yr

84 85 86

James McCrea James McCrea

Introductory Example CJS Model

i – probability a bird survives from occasion i to i+1

pi – probability a bird is recaptured on occasion i

= [1, 2, 3, p2, p3, p4 ]

recapture probabilities

Can only ever estimate 3 p4 - model is parameter redundant or non-identifiable.

etc1

00

0

22

43

433232

433221322121

pp

p

ppp

pppppp

Q

9100

31030

2467

N

r cn

ij ijicr c

ij

n

i

NRn

ij ij

n

i

n

ij

Nij QQL

11

1

111

123

78

R

Derivative Method (Catchpole and Morgan, 1997)

Calculate the derivative matrix D

rank(D) = 5 rank(D) = 5 Number estimable parameters = rank(D). Deficiency = p – rank(D) no. est. pars = 5, deficiency = 6 – 5 = 1

T

p

pp

p

ppp

pp

p

)ln(

)ln(

)ln(

)ln(

)ln(

)ln(

43

4332

32

433221

3221

21

κ

432321 ppp

j

i

D

T

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pppR

ppR

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)1( 4332213221211

4332211

32211

211

μ

i

j

D

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433211321111

432211

4323113211

432321322121

00

0

00

0

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ppRpR

ppRpRR

pppR

pppRppR

pppRppRpR

14

14

14

13

13

13

13

12

12

12

13

13

13

12

12

12

12

11

11

11

000

00

000

000

00

000

ppp

pppp

ppp

Derivative or Jacobian Rank Test

• Jacobian is the transpose of the derivative matrix, so the two are interchangeable.

• Uses of rank test:

– Catchpole and Morgan (1997) exponential family models, mostly used in ecological statistics.

– Rothenberg (1971) original general use, examples econometrics.

– Goodman (1974) latent class models.

– Sharpio (1986) non-linear regression models.

– Pohjanpalo (1982) first use for compartment models.

Derivative or Jacobian Rank Test• The key to the symbolic method for detecting parameter

redundancy is to find a derivative matrix and its rank.• Models are getting more complex.• The derivative matrix is therefore structurally more complex.• Maple runs out of memory calculating the rank.• Examples: Hunter and Caswell (2009), Jiang et al (2007)

• How do you proceed?– Numerically – but only valid for specific value of parameters.

But can’t find combinations of parameters you can estimate. Not possible to generalise results.

– Symbolically – involves extending the theory, again it involves a derivative matrix and its rank, but the derivative matrix is structurally simpler.

Wandering AlbatrossMulti-state models for sea birds

Striped Sea BassAge-dependent tag-return

models for fish

Exhaustive SummariesCole, Morgan and Titterington (2010, Mathematical Biosciences)• An exhaustive summary, , is a vector that uniquely defines

the model (Walter and Lecoutier, 1982).

• Derivative matrix

• r = Rank(D) is the number of estimable parameters in a model.• p parameters; d = p – r is the deficiency of the model (how

many parameters you cannot estimate). If d = 0 model is full rank (not parameter redundant, identifiable somewhere) . If d > 0 model is parameter redundant (non-identifiable).

• More than one exhaustive summary exists for a model• CJS Example:

i

j

D

T

p

pp

p

ppp

pp

p

)ln(

)ln(

)ln(

)ln(

)ln(

)ln(

43

4332

32

433221

3221

21

κ

T

ppppppR

pppR

ppR

pR

)1( 4332213221211

4332211

32211

211

μ

• Choosing a simpler exhaustive summary will simplify the derivative matrix.

• CJS Example:

• Computer packages, such as Maple can find the symbolic rank of the derivative matrix if it is structurally simple.

• Exhaustive summaries can be simplified by any one-one transformation such as multiplying by a constant, taking logs, and removing repeated terms.

• A simpler exhaustive summary can also be found using reparameterisation.

323211

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433211321111

432211

4323113211

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00

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ppRpR

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D

14

14

14

13

13

13

13

12

12

12

13

13

13

12

12

12

12

11

11

11

000

00

000

000

00

000

ppp

pppp

ppp

D

Exhaustive SummariesCole, Morgan and Titterington (2010, Mathematical Biosciences)

Methods For Use With Exhaustive SummariesWhat can you estimate in a parameter redundant model?

Exponential family models: Catchpole and Morgan (1998)Compartment models: Chappell and Gunn (1998) and Evans and Chappell (2000)

Exhaustive Summaries: Cole, Morgan and Titterington (2010, Mathematical Biosciences)

• A model: p parameters, rank r, deficiency d = p – r

• There will be d non-zero solutions to TD = 0.

• Zeros in s indicate estimable parameters.

• Example: CJS, regardless of which exhaustive summary is used

• Solve PDEs to find full set of estimable pars.

• Example: CJS, PDE:

Can estimate: 1, 2, p2, p3 and 3p4

djfp

i iij ,...,10

1

10000

4

3

pT

α

0434

3

p

ff

p

432321 ppp

Methods For Use With Exhaustive SummariesExtension Theorem

Catchpole and Morgan, 1997 Extended to exhaustive summaries in Cole, Morgan and Titterington (2010, Mathematical Biosciences)

• Suppose a model has exhaustive summary 1 and parameters 1.

• Now extend that model by adding extra exhaustive summary terms 2, and extra parameters 2 (eg. add more years of ringing/recovery). New model’s exhaustive summary is = [1 2]T and parameters are = [1 2]T.

• If D1 is full rank and D2 is full rank, the extended model will be full rank. The result can be further generalised by induction.

• Method can also be used for parameter redundant models by first rewriting the model in terms of its estimable set of parameters.

i

j

,1

,11

D

2

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,1

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00 D

DD

i

j

i

j

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j

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Methods For Use With Exhaustive SummariesExtension Theorem

Catchpole and Morgan, 1997 Extended to exhaustive summaries in Cole, Morgan and Titterington (2010, Mathematical Biosciences)

• Example: Ring-Recovery Mallards• Birds are ringed and then recovered dead.• Parameters:

•

• D1 = [1/1] is of full rank 6.• Adding an extra year of recovery adds:

• D2 = [2/2] has rank 1 – not full rank• Adding an extra year of ringing and recovery simultaneously adds:• • D2 = [2/2] is of full rank 2. In general r = 2n1 d = 0 if n1 = n2

Methods For Use With Exhaustive Summaries The PLUR decomposition

Cole, Morgan and Titterington (2010, Mathematical Biosciences)• Write derivative matrix which is full rank r as D = PLUR (P is a square

permutation matrix , L is a lower diagonal square matrix, with 1s on the diagonal, U is an upper triangular square matrix, R is a matrix in reduced echelon form).

• If Det(U) = 0 at any point, model is parameter redundant at that point (as long as R is defined). The deficiency of U evaluated at that point is the deficiency of that nested model.

• Example: Ring-recovery model:

Rank(D) = 5

10000

01000

00100

00010

00001

R)1)(1(

)(Det2,11,112,11,1

2,11,1

aa

U

0)(DetIf 2,11,1 U

Therefore nested model is parameter redundant with deficiency 1.

4)(Rank2,11,1

U

aa 12,11,1

aa

aaaaa

2,112,1

1,11,111,1

0Q

Finding simpler exhaustive summaries Cole, Morgan and Titterington (2010, Mathematical Biosciences)

1. Choose a reparameterisation, s, that simplifies the model structure.

CJS Model (revisited):

2. Reparameterise the exhaustive summary. Rewrite the exhaustive summary, (), in terms of the reparameterisation, (s).

43

32

32

21

21

5

4

3

2

1

p

p

p

p

p

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sss

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s

s

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433232

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ppp

pppppp

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3. Calculate the derivative matrix Ds.

4. The no. of estimable parameters = rank(Ds)

rank(Ds) = 5, no. est. pars = 5

5. If Ds is full rank ( Rank(Ds) = Dim(s) ) s = sre is a reduced-form exhaustive summary. If Ds is not full rank solve set of PDE to find a reduced-form exhaustive summary, sre.

There are 5 si and the Rank(Ds) = 5, so Ds is full rank. s is a reduced-form exhaustive summary.

15

15

15

14

14

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i

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.)(DimRankif

si

js


6. Use sre as an exhaustive summary.

A reduced-form exhaustive summary is

Rank(D2) = 5; 5 estimable parameters.Solve PDEs: estimable parameters are 1, 2, p2, p3 and 3p4

43

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ReparameterisationMulti-state Example

Cole, Morgan and Titterington (2010)

• Hunter and Caswell (2009) examine parameter redundancy of multi-state mark-recapture models, but cannot evaluate the symbolic rank of the derivative matrix (developed numerical method).

• 4 state breeding success model:

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1

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1 1

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psurvival breeding given survival successful breeding recapture

Wandering Albatross

1 3

2 4

1 success

2 = failure

3 post-success

4 = post-failure



1. Choose a reparameterisation, s, that simplifies the model structure.

2. Rewrite the exhaustive summary, (), in terms of the reparameterisation - (s).

2

1

333

222

111

14

13

3

2

1

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145

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sss

ss

ss

ss

ss

s

3. Calculate the derivative matrix Ds.

4. The no. of estimable parameters =rank(Ds)

rank(Ds) = 12, no. est. pars = 12, deficiency = 14 – 12 = 2

5. If Ds is full rank s = sre is a reduced-form exhaustive summary. If Ds is not full rank solve set of PDE to find a reduced-form exhaustive summary, sre. Tre sssssssssssssssss 104934837141312116521 //

139

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6. Use sre as an exhaustive summary.T

re pp

2244411333

4

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Breeding Constraint

Survival Constraint

1= 2=

3= 4

1= 3,

2= 4

1= 2,

3= 4

1, 2,

3,4

1= 2= 3= 4 0 (8) 0 (9) 1 (9) 1 (11)

1= 3 ,2= 4 0 (9) 0 (10) 0 (10) 2 (12)

1= 2, 3= 4 0 (9) 0 (10) 1 (10) 1 (12)

1,2,3,4 0 (11) 0 (12) 0 (12) 2 (14)



Age-dependent tag return models for estimating fishing mortality (Cole and Morgan, 2010, JABES)

Jiang et al (2007, JABES) developed an age-dependent fisheries model:

F – fishing mortality, M natural mortality, Sela selectivity, - reporting.Numerical method applied to N = 3 years of data, with K = 3 age classes found the general model parameter redundant with rank 15 and deficiency 1. Assumed this meant full model was parameter redundant.Actually for N > 3, K > 1 model is full rank, with rank = 3N + 3K – 2Problem of near redundancy.

All parameters constant (Sela = 1) rank = 2, deficiency = 1.

Striped Sea Bass

Parameter redundancy with covariatesCole and Morgan (2010, Biometrika)

• One approach to removing parameter redundancy is to include covariates in a model.

• Suppose that the model with covariates has pc parameters, and in the equivalent model without covariates the rank = q.

• The rank of the model with covariates = min(pc,q)

• Example: Conditional ring-recovery, 1, a, t

• For 4 years of ringing and 4 years of recovery:

• Model without covariates: p = 6, r = 4

• Model with covariates: t = 1/{1 + exp( + t)} pc = 4

r = min(4,4) = 4, d = 0.

Parameter redundancy in mark-recovery modelsCole, Morgan, Catchpole, Hubbard (submitted)

• The probability of an animal being ringed in year i and recovered in year j is

with survival probability and recovery probability .

• Likelihood:

Model rank deficiencyC/C 2 0C/T 1 + n2 0C/A n2 1

C/A,T E 1T/C 1 + n2 0T/T n1 + n2 - 1 n2 – n1+ 1

T/A 2n2 0

• Model notation y/z: y represents survival probability and z represents reporting probability, which can be constant (C) or dependent on age (A), time (T) or (A,T).• The rank of any ring-recovery model is limited by the number of terms in the exhaustive summary.

E = n1n2 – ½n12 + ½n1

• How does the data effect parameter redundancy?• a main diagonals of data; Ni,j = 0 if j – i + 1 > a

• An exhaustive summary consists of Pi,j with Ni,j 0 and the probabilities of never being seen again:

• Reparameterisation method is used to find general results.• Now a maximum rank of Ea = E – ½ (n2 – a)(n2 – a +1)


Without Missing Valuesunchanged

With Missing ValuesModel rank deficiency rank deficiency

C/C 2 0 alwaysC/T 1 + n2 0 alwaysC/A n2 1 always

C/A,T E 1 - Ea E – Ea + 1

T/C 1 + n2 0 alwaysT/T n1 + n2 - 1 n2 – n1+ 1 alwaysT/A 2n2 0 a 2 Ea = n2 + n1 – 1 n2 – n1 + 1 (a=1)

T/A,T E n2 - Ea E – Ea + n2

A/C n2 1 alwaysA/T 2n2 – 1 1 a 2 Ea = n2 + n1 – 1 n2 – n1 + 1 (a=1)

A/A n2 n2 alwaysA/A,T E n2 - Ea E – Ea + n2

A,T/C E 1 - Ea E – Ea + 1

A,T/T E n2 - Ea E – Ea + n2

A,T/A E n2 - Ea E – Ea + n2

A,T/A,T E E - Ea 2E – Ea

• Similar tables of results are also available for x/y/z models, where x represents 1st year survival, y represents adult survival and z represents reporting probability.

• There are 24 models – 3 of which remain unchanged for a 1 – 10 of which remain unchanged for a 2– 3 of which remain unchanged for a 3– 8 are limited by E/Ea

• A lot of data values can be zero and the number of estimable parameters remains unchanged.


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

(Joint work with Stephen Freeman)

• 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.



• Example ring-recovery data (simulated data)

TotalRinged

Ringed as Juveniles Ringed as Adults

Year 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

• 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.



• 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)



Constant p

Model parameters RankDeficiency of Historic model

Deficiency of standard model

1, a, , p 4 0 0

1, a, t , p n2 + 3 0 0

1, a, 1, a, p 4 1 0

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

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

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

1,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 2

1,t, a, , p n1 + 3 0 0

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

1,t, a, 1, a, p n1 + 4 0 0

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

1, a,t, , p n2 + 3 0 0

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

1, a,t, 1, a, p n2 + 3 1 0

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

True Value

Standard Model Historical ModelParameter Mean Stdev MSE Mean Stdev MSE

1 0.4 0.3984 0.0517 0.00267 0.3979 0.0517 0.00268

a 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

Historic model is almost as good as the standard model.



Age-dependent mixture models for animals marked at unknown age

McCrea, Morgan and Cole (invited revision, Applied Statistics)

• Mallard data: ringed as first years, ringed as adults of known age.

• If model data sets together only allows one age-class adult survival.

• t(a) - survival probability of an individual aged a.

• t(a) - reporting probability of an individual aged a.

• t(a) - proportion of individuals marked that are age a.

• i,t(a) - probability an individual is marked at time i at age a and recovered dead at time t, i,t(u) - probability an individual of unknown age is marked at time i and recovered dead at time t.

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Age-dependent mixture models for animals marked at unknown age

McCrea, Morgan and Cole (invited revision, Applied Statistics)

• If only adult-marked data are available, the model is always parameter redundant.

• However it is possible to estimate adult annual survival.• If we combine 1st year data with adult data we can estimate all

parameters for many models.• Notation: X/Y/Z/W 1st year survival/adult survival/reporting/

q parameters, J+1 age classes starting at age J0, I years of ringing years of recovery

Which is the best method to use?

• Symbolic method is now possible in structurally complex models using reparameterisation, but method is not automatic.

• Numerical methods can be automatic, but can be inaccurate.

• Develop general simpler exhaustive summaries, eg multi-state models.

• Hybrid Symbolic-Numerical.

Multi-state mark–recapture models for sea birds

Cole (to appear in Journal of Ornithology)

State 1: Breeding site 1

State 2: Breeding site 2

State 3: Non-breeding,

Unobservable in state 3

- survival

- breeding

- breeding site 1

1 – - breeding site 2

Wandering Albatross


Cole (to appear in Journal of Ornithology)

• General Multistate-model has S states, with the last U states unobservable with N years of data.

• Survival probabilities released in year r captured in year c:

• t is an SS matrix of transition probabilities at time t with transition probabilities i,j(t) = ai,j(t).

• Pt is an SS diagonal matrix of probabilities of capture pt

• pt = 0 for an unobservable state

Wandering Albatross

r = 10N – 17

d = N + 3


Cole (to appear in Journal of Ornithology)Wandering Albatross

A Hybrid Symbolic-Numerical Method for Determining Model Structure

Choquet and Cole (invited review Mathematical Biosciences)

• Derivative matrix evaluated symbolically, rank is determined at 5 random points. The model rank is equal to the maximum rank of the 5 points.

• (Can also determine which parameters can be estimated in parameter redundant models).

• Example: Additive trap-dependence

t = logit-1(t), p*t = logit-1(t+3 + m) pt = logit-1(t+3) t =1,..,4

Gimenez et al (2003) found rank = 9, problem with Maple not simplifying logit functions.

Other and future work• Random effect models (joint with Remi Choquet).

• Joint mark-recapture-recovery models (Ben Hubbard).

• Pledger et al (2009)'s stopover models (Eleni Matechou).

• (Discrete) state-space models for census data.

.

Sandpiper (Stop-over models)

eg. Jose Lahoz-Monfort’s Integrated Population Model

Atlantic puffinJose Lahoz-Monfort

Parameters Fj confounded. a estimable.

Conclusion• Exhaustive summaries offer a more general framework for

symbolic detection of parameter redundancy.• Parameter redundancy can be investigated symbolically by

examining a derivative matrix and its rank.• In the symbolic method we can find the estimable parameter

combinations (via PDEs).• The symbolic method can easily be generalised using the

extension theorem.• Parameter redundant nested models can be found using a

PLUR decomposition of any full rank derivative matrix.• The use of reparameterisation allows us to produce structurally

much simpler exhaustive summaries, allowing us to examine parameter redundancy of much more complex models symbolically.

• Methods are general and can in theory be applied to any parametric model.

References– Catchpole, E. A. and Morgan, B. J. T. (1997) Biometrika, 84, 187-196– Catchpole, E. A., Morgan, B. J. T. and Freeman, S. N. (1998) Biometrika, 85, 462-468

– Chappell, M. J. and Gunn, R. N. (1998) Mathematical Biosciences, 148, 21-41.

– Choquet, R. and Cole D. J. (2010) A Hybrid Symbolic-Numerical Method for Determining Model Structure. University of Kent Technical Report UKC/SMSAS/10/016.

– 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.

– Cole, D. J. (2010) Determining parameter redundancy of multi-state mark-recapture models for sea birds. To appear in Journal of Ornithology.

– Cole, D. J. and Morgan, B. J. T. (2010a) A note on determining parameter redundancy in age-dependent tag return models for estimating fishing mortality, natural mortality and selectivity, JABES, 15, 431-434.

– Cole D. J. and Morgan B. J. T. (2010b) Parameter redundancy with covariates. Biometrika, 97, 1002-1005.

– O. Gimenez, R. Choquet, J. D. Lebreton (2003), Parameter redundancy in multi-state capture-recapture models, Biometrical Journal, 45, 704-722.

– 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

– Robinson, R. A. (2010) Estimating age-specific survival rates from historical data. Ibis, 152, 651–653.

– Evans, N. D. and Chappell, M. J. (2000) Mathematical Biosciences, 168, 137-159.– Goodman, L. A. (1974) Biometrika, 61, 215-231.– Hunter, C.M. and Caswell, H. (2009). Ecological and Environmental Statistics, 3, 797-825– Jiang, H. Pollock, K. H., Brownie, C., et al (2007) JABES, 12, 177-194– Lebreton, J. Morgan, B. J. T., Pradel R. and Freeman, S. N. (1995). Biometrics, 51, 1418-1428.– Pledger, S., Efford, M. Pollock, K., Collazo, J. and Lyons, J. (2009) Ecological and Environmental Statistics Series:

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