Detecting Parameter Redundancy in Integrated

Population Models

Diana Cole and Rachel McCrea National Centre for Statistical Ecology, School of

Mathematics, Statistics and Actuarial Science, University of Kent

Lapwing Example

• Lapwing (Vanellus vanellus) census data consists of a yearly index of abundance derived from counts of adult Lapwings.

• Let denote the number of 1 year old birds (unobserved) and number of adults (observed). Besbeas et al (2002) considered the following state-space model

juvenile survival probability; adult survival; productivity; and error processes.

• The two parameters and only ever appear as a product. It will only ever be possible to estimate the product and never the two parameters separately.

• This is an example of parameter redundancy.

Lapwing Example• Ring-recovery data on Lapwings 1963-

• Probabilities of being ringed in year i and recovered in year j, Pij,

• Ring-recovery model alone is not parameter redundant (Cole et al, 2012).• Integrated state-space model and ring-recovery model is not parameter redundant.• What happens if reporting probability is dependent on age-class? Ring-recovery model alone is parameter redundant. Is the integrated model parameter redundant?

𝑷=[ ( 1−𝜙1 )𝜆1 𝜙1 ( 1−𝜙𝑎 )𝜆𝑎 …

¿ (1−𝜙1 )𝜆1 ¿ ¿⋱ ]

Symbolic Method for Detecting Parameter Redundancy

• In some models it is not possible to estimate all the parameters and the model can be written in terms of a smaller set of parameters. This is termed parameter redundancy.• Symbolic methods can be used to detect parameter redundancy in less obvious

cases (see for example Catchpole and Morgan, 1997, Cole et al, 2010).• Firstly an exhaustive summary is required, , for each data set. An exhaustive

summary is a vector of parameter combinations that uniquely define the model. • An exhaustive summary for ring-recovery data are the Pij. • There are p parameters, .• We then form a derivative matrix, , where • Then calculate the rank, r, of .• When r = p, model is full rank; we can estimate all parameters.• When r < p, model is parameter redundant, and we can also find a set of r

estimable parameter combinations by solving a set of PDEs (see Catchpole et al, 1998 or Cole et al, 2010 for details).

Parameter Redundancy in State-Space Models• Linear state-space model format: observation process, state equation, measurement matrix, transition matrix, and are error processes.• An exhaustive summary can be obtained from the • (Alternative exhaustive summary when parameters depend on time / non-linear) Lapwing Example:

Rank(D2) = 2, but there are 3 parameters, therefore model is parameter redundant. (Solving a set of PDEs shows estimable parameter combinations are and .)

Lapwing ExampleIntegrated Model

• Probabilities of being ringed in year i and recovered in year j, Pij, form an exhaustive summary for the ring-recovery data (Cole et al, 2012).

• Integrated model is not parameter redundant, so in theory it is possible to estimate all the parameters.

Symbolic method for more complicated models• In more complex models it may not be possible to find Rank.• Suppose there are data sets in the integrated model with exhaustive summaries and, and

parameters and of lengths and .• Rather than considering we can use the following result:

Step 1: Calculate .Step 2: If let If let be the estimable parameter combinations. Write as a function of s with extra parameters to given .Step 3: Calculate .Step 4: Rank of the integrated model is .

• Advantage is this requires calculating rank of two structurally simpler derivative matrices.

Lapwing ExampleIntegrated Model with Age Dependence in Recovery Parameter

Step 2: Estimable parameters: The state-space model:

Step 4: Rank of integrated model is , but 5 parameters so model is parameter redundant. Estimable parameter combinations:

Common Guillemots Example

• Reynolds et al (2009) examine four data sets on common guillemots.

• Parameters:

• Survival: (1st year), (2nd – 4th year), (adult)• Recapture, Resight and Reporting: • Tag loss: Emigration:, Variance:

• Data set 1 (): Productivity data, number of successful fledged chicks.

• Data set 2: (, ): Capture-recapture of adults.

• Data set 3: ( , ) Mark-resight-recovery of chicks.

• Data set 4: () Count of adults.

• With all 4 data sets, all parameters can be estimated.

• Do we need to continue collecting all types of data (Lahoz-Monfort et al, 2014)? Are all data sets needed? Can MRR be removed?

Common Guillemots Which Parameters Can be Estimated?

(1st Year Survival) (2nd – 4th Year Survival) (Adult Year Survival) (Productivity)

Count No No No No

Productivity No No No Yes

Adult CR No No ALT No

Chick MRR Yes Yes Yes No

Count + Productivity No No No Yes

Count + Adult CR No No ALT No

Count + Chick MRR Yes Yes Yes No

Productivity + Adult CR No No ALT Yes

Productivity + Chick MRR Yes Yes Yes Yes

Adult CR + Chick MRR Yes Yes Yes No

Count + Productivity + Adult CR No No ALT Yes

Count + Productivity + Chick MRR Yes Yes Yes Yes

Count + Adult CR + Chick MRR Yes Yes Yes Yes

Productivity + Adult CR + Chick MRR Yes Yes Yes Yes

ALT = All except last time point

References• Besbeas, P., Freeman, S. N., Morgan, B. J. T. and Catchpole, E. A. (2002) Integrating Mark-

Recapture-Recovery and Census Data to Estimate Animal Abundance and Demographic Parameters. Biometrics, 58, 540-547.

• 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.

• Cole, D. J., Morgan, B. J. T. and Titterington, D. M. (2010) Determining the parametric structure of models. Mathematical Biosciences, 228, 16-30.

• Cole, D. J. and McCrea, R. S. (2014) Parameter Redundancy in Discrete State-Space and Integrated Models. Invited Revision Biometrical Journal.

• Cole, D.J., Morgan, B.J.T., Catchpole, E.A. and Hubbard, B. A. (2012) Parameter Redundancy in Mark-Recovery Models. Biometrical Journal, 54, 507-523.

• Lahoz-Monfort, J. J., Harris, M. P., Morgan, B. J. M., Freeman, S. N. and Wanless, .S (2014) Exploring the consequences of reducing survey effort for detecting individual and temporal variability in survival. Journal of Applied Ecology, 51, 534-543.

• Reynolds, T. J., King, R., Harwood, J., Frederikesen, M., Harris, M. P. and Wanless, S. (2009) Integrated Data Analyses in the Presence of Emigration and Tag-loss. JABES, 14, 411-431.