Adding individual random effects results in models that are no longer parameter redundant

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30 Νοε 2013 (πριν από 3 χρόνια και 6 μήνες)

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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
/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
/12

Detecting Parameter Redundancy

Symbolic Method (see for example Cole
et al
, 2010):


Let


摥湯瑥 瑨攠灡牡m整敲献


Let


摥d潴攠慮

數e慵a瑩癥v獵sm慲礠景爠愠m潤o氬⁷桥牥 慮a
數e慵a瑩癥v獵sm慲礠楳 愠癥v瑯爠瑨慴 灲潶o摥d 愠畮煵q
牥灲敳敮e慴楯渠潦o瑨攠e潤o氬⁦潲⁥硡x灬攠瑨e 灲潢o扩汩瑩敳 潦楦i
-
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
/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
/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
/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
/12

European Dipper Example



Estimate

Standard

Error


1

= logit
-
1
(

1
)

0.72

0.16


2

= logit
-
1
(

2
)

0.43

0.07


3

= logit
-
1
(

3
)

0.48

0.06


4

= logit
-
1
(

4
)

0.63

0.06


5

= logit
-
1
(

5
)

0.6

0.06


6

= logit
-
1
(

6
)

0.57

30.1

p
2

0.7

0.17

p
3

0.92

0.07

p
4

0.91

0.06

p
5

0.9

0.05

p
6

0.93

0.05

p
7

0.94

49.8

s

1.32

10
-
6

0.0045

lowest Eigen value

8.49

10
-
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
s

= 0.


Will be near redundant if
s

is close to 0.

9
/12

Simulation Example
(100 simulations)



True

Mean

Std


1

0.7

0.71

0.07


2

0.5

0.51

0.05


3

0.6

0.60

0.05


4

0.55

0.55

0.06


5

0.65

0.66

0.08


6

0.5

0.54

0.24

p
2

0.8

0.80

0.07

p
3

0.75

0.75

0.06

p
4

0.85

0.85

0.04

p
5

0.75

0.74

0.05

p
6

0.65

0.64

0.06

p
7

0.7

0.72

0.22

s

1

1.05

0.30


lowest
Eigen
value



0.51

0.21

10
/12

Near Redundancy Simulation

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

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.