Determining Parameter Redundancy of Multi-state Mark-Recapture Models for Sea Birds

Diana Cole

University of Kent

IntroductionCJS Example

• Consider the Cormack-Jolly-Seber Model with time dependent annual survival probabilities, i, and time dependents annual recapture probabilities, pi.

• For 3 years of ringing and 3 subsequent years of recapture the probabilities that a bird marked in year i is next recaptured in year j + 1 are:

• Can only ever estimate 3 p4 - model is parameter redundant

etc1

00

0 22

43

433232

433221322121

pp

p

ppp

pppppp

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IntroductionParameter Redundancy at Euring

• Euring 2003: Gimenez et al (2004) Methods for investigating parameter redundancy– Compare different methods for determining parameter

redundancy (profile likelihood, hessian, simulation, symbolic method)

– Conclusion: symbolic method more reliable, provides estimable parameter combinations and can be extended.

• Euring 2007: Hunter and Caswell (2009) examined multi-state mark-recapture models for seabirds. It was not possible to evaluate the algebra of symbolic method. Developed a better numerical based method instead.

IntroductionSymbolic Method

• A model is parameter redundant (or non-identifiable) if you cannot estimate all the parameters

• Can determine whether a model is parameter redundant by forming a derivative matrix.

• The rank, r, of the derivative matrix is equal to the number of estimable parameters. If there are p parameters and r < p the model is parameter redundant (Catchpole and Morgan, 1997).

• In a parameter redundant model estimable parameter combinations can be found by solving a set of partial differential equations (PDEs).

• All symbolic algebra can be executed in Maple.

• In complex models calculating the rank becomes impossible.

• This talk will show how it is now possible to use the symbolic method instead and to find general rules.

Multi-state Mark-Recapture Framework(Hunter and Caswell, 2009)

• S different states. U of which are unobservable.

• N different sampling occasions (ringing in years 1 to N – 1 and recapture in years 2 to N).

• Transition matrix t. S by S matrix with entries i,j(t), the probability of transition from state j at time t to state i at time t + 1.

• Recapture matrix Pt. Diagonal matrix of size S, with diagonal elements pi,i, the probability of recapturing an animal in state i at time t.

• The p-array:

in an unobservable i state i,j(r,c) = 0 for all j and pi,i = 0

1

1

1211

1),(

rc

rcT

rrcccc

Trrcr

ΦPIΦPIΦP

ΦPΨ

Multi-state Mark-Recapture Framework 3 state time-invariant model (Hunter and Caswell, 2009)

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00

00

)1()1()1(

)1()1()1(

2

1

332211

333222111

333222111

p

p

P

(Fig 1 Hunter and Caswell )

survivalBreeding given survival

Breeding at location 1

recapture

Probability of moving from state 3 to state 1.

Breeding location 1

Breeding location 2

Non-Breeding

Exhaustive Summary Framework(Cole and Morgan, 2009)

• To be able to calculate the rank a structurally simple derivative matrix is required.

• Hunter and Caswell (2009) differentiate the p-array wrt the parameters to form the derivative matrix. This is an example of an exhaustive summary.

• An exhaustive summary is a vector of parameter combinations that uniquely defines the model.

• Different exhaustive summaries will result in different derivative matrices. But the rank (and PDEs) will remain the same.

• Structurally simpler exhaustive summaries result in structurally simpler derivative matrices. Therefore are able to calculate the rank.

• Simpler exhaustive summaries can be found using reparameterisation (Cole and Morgan, 2009)

A Simpler Exhaustive Summary for Multi-State Capture Recapture Models

• Consider a multi-state model with S states, U 0 of which are unobservable, with states 1 to S – U observable and states S–U +1 to S unobservable.

pi = 0 if state unobservable

• If there are more than one observable state, and N is large enough exhaustive summary is given by table on next slide

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A Simpler Exhaustive Summary for Multi-State Capture Recapture Models

Exhaustive Summary Terms

Range No. of Terms

pi(t+1)ai,j(t)t = 1,...,N – 1 i = 1,...,S – U j = 1,...,S – U

(N – 1)(S – U)2

pi(t)t = 2,...,N – 1 i = 1,...,S – U (N – 1)(S – U)

pi(t+1)ai,j(t) aj,1(t)t = 2,...,N – 1 i = 1,...,S – U

j = S – U + 1,...,S U(N – 2)(S – U)

t = 2,...,N – 1 i = 2,...,S – U

j = S – U + 1,...,S U(N – 2)(S – U – 1)

t = 3,...,N – 1 i = S – U + 1,...,S j = S – U + 1,...,S

U2(N – 3)

)1(

)1(

1,

,

ta

ta

j

ij

)1(

)1()1(

1,

1,,

ta

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A Simpler Exhaustive Summary for Multi-State Capture Recapture Models 3 state time-invariant model (N=4)

> A:=Matrix(1..3,1..3): A[1,1]:=sigma[1]*beta[1]*g[1]:

A[1,2]:=sigma[2]*beta[2]*g[2]: A[1,3]:=sigma[3]*beta[3]*g[3]:

A[2,1]:=sigma[1]*beta[1]*(1-g[1]):A[2,2]:=sigma[2]*beta[2]*(1-g[2]):

A[2,3]:=sigma[3]*beta[3]*(1-g[3]):A[3,1]:=sigma[1]*(1-beta[1]):

A[3,2]:=sigma[2]*(1-beta[2]):

A[3,3]:=sigma[3]*(1-beta[3]):

> P := <<p[1]|0|0)>,<0|p[2]|0>,<0|0|0>>:

> pars:=<sigma[1],sigma[2],sigma[3],beta[1],beta[2],

beta[3],g[1],g[2],g[3],p[1],p[2]>:

> kappa:=simexsum(A,P,4):

> DD:=Dmat(kappa,pars):

> r:=Rank(DD);

r:=10

> Estpars(DD,pars);

tΦenter

tPenter

parametersenter

simexsum(,P,N) procedure for finding simple exhaustive summary.

Dmat(kappa,pars) procedure for finding the derivative matrix.

Estpars(DD,pars) procedure for finding

the estimable parameter combinations.

3-state time varying model

Full Model Hunter and Caswell Constraints

Alternative Constraints

N r d p r d p r d p

4 23 7 30 10 1 11 23 0 23

5 33 8 41 21 1 22 33 0 33

6 43 9 52 31 2 33 43 0 43

7 53 10 63 41 3 44 53 0 53

N 10N-17 N+3 11N-14 10N-29 N-4 11N-33 10N-17 0 10N-17

000

00

00

)1()1()1(

)1()1()1( ,2

,1

,3,3,2,2,1,1

,3,3,3,2,2,2,1,1,1

,3,3,3,2,2,2,1,1,1

t

t

t

tttttt

ttttttttt

ttttttttt

t p

p

PΦ

Hunter and Caswell Constraint: First two and last two time points equal for all parsAlternative Constraints: 2,t = 1,t , j,N-1 = j,N-2 , pi,N = pi,N-1.

Length of exhaustive summary 10N – 17

4-state Time Varying Breeding Success and Failure Model

Full Model Hunter and Caswell Constraints Alternative Constraints

N r d p r d p r d p

7 62 16 78 48 8 56 62 0 62

8 74 18 92 60 10 70 74 0 74

9 86 20 106 72 12 84 86 0 86

N 12N-22 2N+2 14N-20 12N-36 2N-6 14N-42 12N-22 0 12N-22

)1(0)1(0

0)1(0)1(

)1()1()1()1(

,4,4,2,2

,3,3,1,1

,4,4,4,3,3,3,2,2,2,1,1,1

,4,4,4,3,3,3,2,2,2,1,1,1

tttt

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t

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survival breeding given survival successful breeding recapture

1 3

2 4

1 success

2 = failure

3 post-success

4 = post-failure

• Seabirds with delayed maturity tend to be only be observable when they are young or breeding

• k = 4 age at first recruitment. y = 5 recruitment years. • state y + k = 9 breeding state

• pk+y,t = pt (p1,t = 1, pi,t = 0 otherwise)

• Only 2 out of 9 states observable. Transition matrix has lots of 0s.• 9-state example required N 40 to be able to use simpler exhaustive

summary• Instead a general exhaustive summary for the n – state recruitment model is

developed.

Recruitment Example

(Fig 3 Hunter and Caswell, 2009. = survival, = recruitment)

1 – 1st year

9 – breeding

Recruitment Example

• k+y,t is estimable for t = 1,...,N – 2

• pt is estimable for t = 2,...,N – 1

• Last time point only pNN-1 is estimable

• The parameters 1,t to k,t with k,t are always confounded

• Even without time dependence, full age-dependence would not be estimable.

Exhaustive Summary Terms Range No. Terms

pt+1k+y,tt = 1,...,N – 1 N – 1

ptt = 2,...,N – 1 N – 2

t = k,....,N – 1 N – k

t = i,...,N – 1i = k + 1,....,y + k – 1

N(y – 1) – ½ (y2 – y) – yk + k

1

1,,,1

k

jkjtjtktktp

1,1

1,1,,1 )1(

ti

titititp

Recruitment Example with Constraints

Time Dep. Constraints Deficiency

no k-1 = ... = 2 = 1 y+1

no k-1 = ... = 2 = 1

logit(i) = a + bi2

nok-1 = ... = 2 = 1

logit(i) = a + bilogit(i) = a + bi

max(0,7-y)

yes pN = pN-1

k-1 = ... = 2 = 1

yN – ½ (y2 – y) – yk + 1

yes

pN = pN-1

k-1 = ... = 2 = 1

logit(i) = ai, + bi,xt

logit(i) = ai, + bi,xt

k i y + k

mostly 0

The number of estimable parameters is equal to the minimum of number of estimable parameters in the equivalent model without covariates and the number of parameters in the covariate model (Cole and Morgan, 2007).

DiscussionNumerical v Symbolic Methods

Numeric Method Symbolic Method

Ease of use Fairly Easy Requires some algebra to find a simple exhaustive summary.

Then relatively easy to use.

Computation Could be added to any computer program

Needs a symbolic algebra package such as Maple

Estimable parameter

combinations

Trial and error only Can be found using a Maple procedure

AccuracyNot always, although

Hunter and Caswell’s work improves this

Finds the actual redundancy

Near Redundancy Is not distinguishable from actual redundancy

Can be detected using PLUR decompositions (Cole and Morgan, 2009 extending work of Gimenez et al, 2003)

General Rules Not possible to prove Can be found using extension theorems (Catchpole and Morgan, 1997)

Discussion

• Based on these advantages and disadvantages:

– if interest lies in whether a particular model for a specific data set is parameter redundant then a numerical method would be sufficient.

– However if interest lies in the redundancy of a model in general or a particular class of models, general rules can be found using the symbolic method.

• It is now possible to use the symbolic method to determine parameter redundancy in complex models.

Other / future work:

– Only one observable state: Developed a simple exhaustive summary for the case S = 2 and U = 1, in particular examining a two-state model for breeding and non-breeding of Great Crested Newts (McCrea and Cole work in progress).

– Parameter redundancy in Pledger et al (2009)'s stopover models (Matechou and Cole unpublished work).

– Rouan et al (2009)'s memory models

– MacKenzie et al (2009)'s multi-site occupancy models.

References

• Recent Advances in Symbolic Approach:– Cole, D. J. and Morgan, B. J. T (2009) Determining the Parametric Structure of Non-Linear

Models IMSAS, University of Kent Technical report UKC/IMS/09/005 – Cole, D.J. and Morgan, B.J.M (2007) Detecting Parameter Redundancy in Covariate Models.

IMSAS, University of Kent Technical report UKC/IMS/07/007,– See http://www.kent.ac.uk/ims/personal/djc24/parameterredundancy.htm for papers and Maple

code• Other 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. – Gimenez, O., Choquet, R. and Lebreton, J. (2003) Parameter Redundancy in Multistate Capture-Recapture Models

Biometrical Journal 45, 704–722– Gimenez, O., Viallefont, A., Catchpole, E. A., Choquet, R. & Morgan, B. J. T., (2004) Methods for investigating

parameter redundancy. Animal Biodiversity and Conservation, 27. 1-12– Lebreton, J. Morgan, B. J. T., Pradel R. and Freeman, S. N. (1995) A simultaneous survival rate analysis of dead

recovery and live recapture data. Biometrics, 51, 1418-1428.– Pledger, S., Efford, M. Pollock, K., Collazo, J. and Lyons, J. (2009) Stopover duration analysis with departure

probability dependent on unknown time since arrival. Ecological and Environmental Statistics Series: Volume 3. – Hunter, C. M. and Caswell, H. (2009) Rank and redundancy of multi-state mark- recapture models for seabird

populations with unobservable states. In Environmental and Ecological Statistics Series : Volume 3. – Mackenzie, D.I., Nichols, J.D., Seamans, M.E, and Gutierrez, R.J. (2009) Modelling species occurrence dynamics

with multiple states and imperfect detection. Ecology, 90, 823-835.– Rouan, L., Choquet R. and Pradel, R. (2009) A General Framework for Modelling Memory in Capture-Recapture

Data To appear in JABES