Parameter Redundancy in Mark-Recapture and Ring-Recovery Models with Missing Data Diana Cole University of Kent

2/12 A model is parameter redundant (or non-identifiable) if you cannot estimate all the parameters. – Caused by the model itself (intrinsic parameter redundancy). – Caused by the data (extrinsic parameter redundancy). CJS example: Consider the Cormack-Jolly-Seber Model with time dependent annual survival probabilities,  i, and time dependent annual recapture probabilities, p i. For 3 years of marking and 3 subsequent years of recapture the probabilities that an animal 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

3/12 Introduction – Detecting Parameter Redundancy Can determine whether a model is parameter redundant symbolically, which involves 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). Matrix algebra executed in Maple. CJS example: Rank D is 5. There are 5 out of 6 estimable parameters, so the model is parameter redundant.

4/12 Introduction - Reparameterisation A model reparameterised in terms of s, will have the same rank as the original parameterisation (Cole et al, 2010). Reparameterisation can be used in structurally complex models (eg Cole and Morgan, 2010 or Cole, 2010) when the standard derivative method fails to calculate the rank. Reparameterisation is also useful for proving general results in simpler models. CJS example: Rank(D s ) = 5

5/12 Missing Values – CJS Model So far we have considered parameter redundancy results that are based on having perfect data (intrinsic parameter redundancy). In reality there may be some marking recapture combinations which never occur. For example: How does missing values effect parameter redundancy? (extrinsic parameter redundancy). In forming our derivative matrix we also need to include the probabilities of being marked and never seen again. We also exclude any entries which correspond to missing values.

6/12 Missing Values – CJS Model Rank still 5. Rank is now 4.

7/12 Missing Values – Ring Recovery Models The corresponding probability of an animal being ringed in year i and recovered in year j is with survival probability  and recovery probability. 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 age and time (A,T). The rank and deficiency can be determined for any model combination, using the derivative method. The rank of any ring-recovery model is limited by the number of terms in P. There are E = n 1 n 2 – ½n ½n 1 terms. For example the full model (A,T/A,T) has rank E but has 2E parameters, therefore has deficiency E.

8/12 Missing Values – Ring Recovery Models a main diagonals of data; N i,j = 0 if j – i + 1 > a A derivative matrix is formed from the probabilities with associated non-zero N i,j values and the probabilities of never being seen again. Reparameterisation method is used to find general results. There will now be a maximum rank of E a =an 2 + n 1 – ½a 2 – ½a.

Without Missing Values unchanged With Missing Values Modelrankdeficiencyrankdeficiency C/C20 a  1 C/T1 + n 2 0 a  1 C/An2n2 1 a  1 C/A,TE1-EaEa E – E a + 1 T/C1 + n 2 0 a  1 T/Tn 1 + n 2 - 1n 2 – n a  1 T/A2n22n2 0 a  2 E a = n 2 + n 1 – 1n 2 – n (a=1) T/A,TEn2n2 -EaEa E – E a + n 2 A/Cn2n2 1 a  1 A/T2n 2 – 11 a  2 E a = n 2 + n 1 – 1n 2 – n (a=1) A/An2n2 n2n2 a  1 A/A,TEn2n2 -EaEa E – E a + n 2 A,T/CE1-EaEa E – E a + 1 A,T/TEn2n2 -EaEa E – E a + n 2 A,T/AEn2n2 -EaEa E – E a + n 2 A,T/A,TEE-EaEa 2E – E a

10/12 Missing Values – Ring Recovery Models Similar tables of results are also available for x/y/z models, where x represents 1 st 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/E a A lot of data can be missing and the number of estimable parameters remains unchanged. Generally the number of estimable parameters, r I, only changes if there are less than r I data points.

11/12 Conclusion General results can be obtained for capture-recapture and ring- recovery. For many models a lot of data can be missing and the number of estimable parameters in a model does not change. Other parameterisations of the model are possible, for example the recovery rate j at time j can be reparameterised as f j = (1 –  j ) j (eg Hoenig et al, 2005). By the reparameterisation theorem of Cole et al (2010) the number of estimable parameters will be the same regardless of the parameterisation used.

12/12 References See for papers and Maple code. Cole, D. J., Morgan, B. J. T. and Catchpole, E. A. (2010) Parameter Redundancy in Ring-Recovery Models, University of Kent Technical report UKC/SMSAS/10/010 Cole, D. J. and Morgan, B. J. T (2010) 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. T. (2010) A note on determining parameter redundancy in age-dependent tag return models for estimating fishing mortality, natural mortality and selectivity. Jabes DOI: /s Cole, D.J. (2010) Determining Parameter Redundancy of Multi-state Mark- recapture Models for Sea Birds. Presented at Euring 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,