Determining Parameter Redundancy of Multi-state Mark- Recapture Models for Sea Birds Diana Cole University of Kent

Introduction CJS Example Consider the Cormack-Jolly-Seber Model with time dependent annual survival probabilities,  i, and time dependents annual recapture probabilities, p i. 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 p 4 - model is parameter redundant

Introduction Parameter 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.

Introduction Symbolic 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 P t. Diagonal matrix of size S, with diagonal elements p i,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 p i,i = 0

Multi-state Mark-Recapture Framework 3 state time-invariant model (Hunter and Caswell, 2009) (Fig 1 Hunter and Caswell ) survival Breeding 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. p i = 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

A Simpler Exhaustive Summary for Multi- State Capture Recapture Models Exhaustive Summary Terms RangeNo. of Terms p i (t+1)a i,j (t) t = 1,...,N – 1 i = 1,...,S – U j = 1,...,S – U (N – 1)(S – U) 2 pi(t)pi(t) t = 2,...,N – 1 i = 1,...,S – U(N – 1)(S – U) p i (t+1)a i,j (t) a j,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 U 2 (N – 3)

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 :=,, >: > 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);  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 ModelHunter and Caswell Constraints Alternative Constraints Nrdprdprdp N10N-17N+311N-1410N-29N-411N-3310N-170 Hunter and Caswell Constraint: First two and last two time points equal for all pars Alternative Constraints:  2,t =  1,t,  j,N-1 =  j,N-2, p i,N = p i,N-1. Length of exhaustive summary 10N – 17

4-state Time Varying Breeding Success and Failure Model Full ModelHunter and Caswell ConstraintsAlternative Constraints Nrdprdprdp N12N-222N+214N-2012N-362N-614N-4212N-220 survivalbreeding given survivalsuccessful breedingrecapture 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 p k+y,t = p t (p 1,t = 1, p i,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,  = survival,  = recruitment) 1 – 1 st year 9 – breeding

Recruitment Example  k+y,t is estimable for t = 1,...,N – 2 p t is estimable for t = 2,...,N – 1 Last time point only p N  N-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 TermsRangeNo. Terms p t+1  k+y,t t = 1,...,N – 1N – 1 ptpt t = 2,...,N – 1N – 2 t = k,....,N – 1N – k t = i,...,N – 1 i = k + 1,....,y + k – 1 N(y – 1) – ½ (y 2 – y) – yk + k

Recruitment Example with Constraints Time Dep.ConstraintsDeficiency no  k-1 =... =  2 =  1 y+1 no  k-1 =... =  2 =  1 logit(  i ) = a  + b  i 2 no  k-1 =... =  2 =  1 logit(  i ) = a  + b  i logit(  i ) = a  + b  i max(0,7-y) yes p N = p N-1  k-1 =... =  2 =  1 yN – ½ (y 2 – y) – yk + 1 yes p N = p N-1  k-1 =... =  2 =  1 logit(  i ) = a i,  + b i,  x t logit(  i ) = a i,  + b i,  x t 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).

Discussion Numerical v Symbolic Methods Numeric MethodSymbolic Method Ease of useFairly 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 onlyCan be found using a Maple procedure Accuracy Not 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 RulesNot 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 for papers and Maple codehttp:// Other 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, – 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, – 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, – 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, – Rouan, L., Choquet R. and Pradel, R. (2009) A General Framework for Modelling Memory in Capture- Recapture Data To appear in JABES