1. Adding individual random effects results in models that are no longer parameter redundant Diana Cole, University of Kent Rémi Choquet, Centre d'Ecologie Fonctionnelle et Evolutive

2. 2/12 Cormack Jolly Seber (CJS) Model A model is parameter redundant (or non-identifiable) if you cannot estimate all the parameters. Consider the CJS model with time dependent survival probabilities,  t, and time dependent recapture probabilities, p t. This model is parameter redundant.

3. 3/12 Detecting Parameter Redundancy Symbolic Method (see for example Cole et al, 2010): Let  denote the parameters. Let  denote an exhaustive summary for a model, where an exhaustive summary is a vector that provides a unique representation of the model, for example the probabilities of life- histories. Form a derivative matrix Rank D gives the number of parameters in a model. Rank D < no. of parameters model is parameter redundant (or non- identifiable). Rank D = no. of parameters model is full rank. Extensions to more complex models using reparameterisation. Hybrid-Symbolic-Numerical Method (Choquet and Cole, 2012) Derivative matrix formed symbolically, rank evaluated at about 5 random points.

4. 4/12 CJS Model with random effects Consider adding individual random effects to the time dependent CJS model. Survival parameter: where As b i appears in all  i,t there is now separation between  T-1 and p T Is the model still parameter redundant? How do we investigate parameter redundancy in models with random effects? Statistics are defined by a one dimensional integral. Impossible to manage them exactly. => approximation.

5. 5/12 Theory for detecting parameter redundancy in models with individual random effects Exhaustive summary,  : where probability of history i evaluated at. Easiest to use with hybrid-symbolic method and with results for specific data sets. Simpler exhaustive Summary,  : Can be used with symbolic method to get general results. (Results generalise for multiple random effects.)

6. 6/12 European Dipper Example has rank 13, so can theoretically estimate all the parameters. (Can also show using the simpler exhaustive summary that the CJS model with individual random effects is always full rank.)

7. 7/12 European Dipper Example EstimateStandard Error  1 = logit -1 (  1 ) 0.720.16  2 = logit -1 (  2 ) 0.430.07  3 = logit -1 (  3 ) 0.480.06  4 = logit -1 (  4 ) 0.630.06  5 = logit -1 (  5 ) 0.60.06  6 = logit -1 (  6 ) 0.5730.1 p2p2 0.70.17 p3p3 0.920.07 p4p4 0.910.06 p5p5 0.90.05 p6p6 0.930.05 p7p7 0.9449.8  1.32  10 -6 0.0045 lowest Eigen value 8.49  10 -8

8. 8/12 Near Redundancy A near redundant model behaves similarly to a parameter redundant model because it is close to a model that is parameter redundant (Catchpole, et al, 2001). In a near redundant model the smallest Eigen value of the hessian matrix will be close to zero (Catchpole, et al, 2001). Potential near redundancy can be found using a PLUR decomposition of D. Det(U) = 0 indicates parameter redundancy and near redundancy (Cole, et al, 2010). Dipper example: Will be parameter redundant if  = 0. Will be near redundant if  is close to 0.

9. 9/12 Simulation Example (100 simulations) TrueMeanStd 11 0.70.710.07 22 0.50.510.05 33 0.60.600.05 44 0.55 0.06 55 0.650.660.08 66 0.50.540.24 p2p2 0.80.800.07 p3p3 0.75 0.06 p4p4 0.85 0.04 p5p5 0.750.740.05 p6p6 0.650.640.06 p7p7 0.70.720.22  11.050.30 lowest Eigen value 0.510.21

10. 10/12 Near Redundancy Simulation

11. 11/12 Conclusion The time dependent CJS model is no longer parameter redundant when there are individual random effects, but there are potential problems with near redundancy. Smallest Eigen value of Hessian can be used to detect near redundancy (if the Hessian is reasonably well approximated). Theory is applicable to any model with individual random effects, if the life-history probabilities can be written down.

12. 12/12 References This talk: Cole, D.J. and Choquet, R. (2012) Parameter Redundancy in Models with Individual Random Effects, University of Kent technical report. References: Catchpole, E. A., Kgosi, P. M. and Morgan,B. J. T. (2001) On the Near-Singularity of Models for Animal Recovery Data. Biometrics, 57, 720-726. Choquet, R. and Cole, D.J. (2012) A Hybrid Symbolic- Numerical Method for Determining Model Structure. Mathematical Biosciences, 236,117-125. Cole, D.J. and Morgan, B.J.T. and Titterington, D.M (2010) Determining the parametric structure of models. Mathematical Biosciences, 228,16-30.