Mark-recapture analysis of natural populations

Mark-recapture analysis of
natural populations
Practice: Recaptures only models
After the MARK “book”
“Program MARK: a gentle introduction” by Cooch & White
Available at http://www.phidot.org/
Le Galliard J.-F., CNRS, Paris

Aims of mark-recapture analysis
Unbiased estimates of crucial demographic parameters
population density
survival, recruitment, migration …
Testing biological hypothesis in ecology
temporal and age variation
effects of population density, climate …
effects of individual covariates (size …)
Input for demographic analyses
time-series
population viability analysis, matrix population models

MARK : a platform for MCR models

MARK : a bunch of models!

“Recaptures only” models: data type

Encounter history file
/* European Dipper Data, Live Recaptures,
7 occasions, 2 groups
Group 1=Males Group 2=Females */
1111110 1 0 ;
1111100 0 1 ;
1111000 1 0 ;
1111000 0 1 ;
1101110 0 1 ;
1100000 1 0 ;
1100000 1 0 ;
1100000 1 0 ;
1100000 1 0 ;
1100000 0 1 ;
1100000 0 1 ;
1010000 1 0 ;
1010000 0 1 ;
1000000 1 0 ;
1000000 1 0 ;
Encounter history (1 = seen, 0 = not seen)
Encounter session
first column = initial session
second column = recapture sessions
Group covariates (dummy variables)
Here 2 group covariates indicating males
and females
Individual covariates
(raw/standardized data)

Start a project: “recaptures only”

Fitting a CJS model on dipper data
Step 1: Check input data summary
Step 2: Use the parameter index chart
Step 3: Run the default model (the “time*sex” model)
* link function
* variance estimation
* options
Step 4: Check the output of the model

Number of parameters
(important for model selection)
Number of intrinsically identifiable parameters
the number of survival and capture parameters than can be
estimated given the model
How to calculate np_i ? Check the model
Number of extrinsically identifiable parameters
the number of survival and capture parameters than can be
estimated given the model and the data
How to calculate np_e ? Check the estimates, check
the sensitivity of deviance to problematic parameters or use a
numerical approach (implemented in another program, called
M-SURGE)

Number of parameters
The time*sex dipper model
Number of intrinsically identifiable parameters
Survival: 6 par, 2 sexes = 12 par
Capture: 6 par, 2 sexes = 12 par
Time model in each sex
Last survival and capture estimates not
identifiable in this model, only their
product can be estimated
Hence, 5 surv + 5 capt + 1 capt*surv
estimate in each sex == 22 np_i

Number of parameters
The time*sex dipper model
Number of extrinsically identifiable parameters
European dipper
Real Function Parameters of {phit(t*sex)p(t*sex)}
95% Confidence Interval
Parameter Estimate Standard Error Lower Upper
------------------------- -------------- -------------- -------------- --------------
1:Phi 0.6969698 0.2049831 0.2555230 0.9390714
2:Phi 0.4230769 0.0968907 0.2519587 0.6148826
3:Phi 0.5052879 0.0874935 0.3396458 0.6697773
4:Phi 0.6094017 0.0838412 0.4389146 0.7567907
5:Phi 0.5708180 0.0776952 0.4166816 0.7123428
6:Phi 0.7637587 0.0000000 0.7637587 0.7637587
7:Phi 0.7428572 0.2372217 0.2021058 0.9705431
8:Phi 0.4468410 0.0982829 0.2703613 0.6378177
9:Phi 0.4538126 0.0815350 0.3036420 0.6128857
10:Phi 0.6404243 0.0832336 0.4672411 0.7834076
11:Phi 0.6280454 0.0810516 0.4610285 0.7692151
12:Phi 0.6928203 0.0000000 0.6928203 0.6928203
13:p 0.7173914 0.2238537 0.2257356 0.9567136
14:p 1.0000000 0.2616558E-07 0.9999999 1.0000001
15:p 0.9093023 0.0855744 0.5674378 0.9871171
16:p 0.9274193 0.0693437 0.6291491 0.9897162
17:p 0.9358289 0.0616791 0.6607893 0.9909235
18:p 0.7637665 0.0000000 0.7637665 0.7637665
19:p 0.6730771 0.2452306 0.1881302 0.9481657
20:p 0.8600683 0.1263738 0.4397367 0.9796466
21:p 0.9164970 0.0792066 0.5907658 0.9881583
22:p 0.8788732 0.0794409 0.6269547 0.9690646
23:p 0.9283667 0.0684261 0.6330113 0.9898349
24:p 0.6928203 0.0000000 0.6928203 0.6928203
Check this parameter by fixing it at 0.5
Check new model deviance

Survival estimates
The time*sex dipper model

Capture estimates
The time*sex dipper model

Goodness-of-fit tests (GOF tests)
CJS model assumptions
(1) Every animal of the same group present at time t has the
same probability of being captured at time t
(2) All animals marked at time t and present in the population
have the same probability of surviving until time t+1
(3) Marks are not lost, missed …
(4) Sampling is instantaneous compared to time intervals

Assessing GOF with RELEASE
RELEASE procedure (contingency tables)
(1) TEST.1: tests overall difference between groups (not useful here)
(2) TEST.2.C: tests difference in time until next capture between animals
seen at time t and animals not seen at time t among all animals known to
be alive between t and t+1
(3) TEST.3.SR: tests difference in “seen again” probability between
animals marked at time t and animals marked before time t and known to
be alive at time t
(4) TEST.3.Sm: tests in “when seen again” between animals marked at
time t and animals marked before time t and known to be alive at time t

Assessing GOF with RELEASE
TEST2.C
Pooled in a2 by 2 matrix
TEST3.SR
TEST3.Sm
Pooled in a 2 by 2 matrix

GOF-test for the “time*sex” model
Summary of TEST 3 (Goodness of fit) Results
Group Component Chi-square df P-level Sufficient Data
----- --------- ---------- ---- ------- ---------------
1 3.SR2 0.1771 1 0.6739 Yes
1 3.SR3 1.0950 1 0.2953 Yes
1 3.SR4 3.5740 1 0.0586 Yes
1 3.SR5 0.0881 1 0.7666 Yes
1 3.SR6 0.3416 1 0.5589 Yes
Group 1 3.SR 5.2759 5 0.3831
1 3.Sm2 0.0000 0 1.0000 No
1 3.Sm3 0.0000 1 1.0000 No
1 3.Sm4 0.0000 1 1.0000 No
1 3.Sm5 0.0000 0 1.0000 No
Group 1 3.Sm 0.0000 2 1.0000
Summary of TEST 2 (Goodness of fit) Results
Group 1 TEST 3 5.2759 7 0.6263
2 3.SR2 0.2359 1 0.6272 Yes
Group Component Chi-square df P-level Sufficient Data
2 3.SR3 2.7551 1 0.0969 Yes
2 3.SR4 0.3764 1 0.5396 Yes
2 3.SR5 0.0891 1 0.7654 Yes
2 3.SR6 0.0000 1 1.0000 Yes
Group 2 3.SR 3.4565 5 0.6300
2 3.Sm2 1.5426 1 0.2143 No
2 3.Sm3 0.0000 1 1.0000 No
2 3.Sm4 0.4986 1 0.4800 No
2 3.Sm5 0.0000 0 1.0000 No
Group 2 3.Sm 2.0412 3 0.5639
Group 2 TEST 3 5.4977 8 0.7033
All Groups TEST 3 10.7735 15 0.7685
----- --------- ---------- ---- ------- ---------------
1 2.C2 0.0000 0 1.0000 No
1 2.C3 0.0000 0 1.0000 No
1 2.C4 0.0000 1 1.0000 No
1 2.C5 4.2839 1 0.0385 No
Group 1 TEST 2 4.2839 2 0.1174
2 2.C2 0.0000 1 1.0000 No
2 2.C3 0.0000 1 1.0000 No
2 2.C4 0.0000 1 1.0000 No
2 2.C5 3.2503 1 0.0714 No
Group 2 TEST 2 3.2503 4 0.5168
All Groups TEST 2 7.5342 6 0.2743

GOF-test for the “time*sex” model
TEST 3.SR4: Animals captured on occasion 4
+------+------+
O| 7 | 10 | 17
E| 10.0| 7.0|
C| 0.9| 1.3|
+------+------+
O| 16 | 6 | 22
E| 13.0| 9.0|
C| 0.7| 1.0|
+------+------+
23 16 39
Chi-square=3.9456 (df=1) P=0.0470
Fisher's Exact Test P=0.0586
Seen before
Newly marked
Seen and released
at occasion 4
Not seen again
Seen again

Bootstrap GOF-test with MARK
Bootstrap procedure
(1) Retrieve the model to be tested (full model)
(2) The bootstrap procedure runs simulations of a “similar”
encounter history file (same number of animals released at
each occasion) based on estimates of the full model and then
runs the full model on the simulated data
(3) The procedure stores the deviance … of the simulated
model and can repeat n times (n ≈ 500-1000)
(4) GOF-test based on deviances of the simulated model versus
deviance of the full model based on true data

Bootstrap GOF with the dipper data
Deviance of the full model based on true data
deviance = 71.47, df_deviance = 19
observed c-hat = 3.76
Deviance of simulated models
mean deviance = 55.41, mean c-hat = 3.275
% deviance above = 92/1000 = 0.098 (slight over-dispersion, but not significant)
Computation of model c-hat
based on mean deviance = 1.40
based on mean c-hat = 1.15

Model selection
(1) Define an a priori set of model corresponding to a set of
biological hypotheses; select capture terms first, then survival
terms (Lebreton et al. 1992)
(2) Run the models, check parameters’ identifiability and
compute the model deviance
(3) Calculate an information-based criterion for each model
(4) Select the “best” model or the best subset of models; this
model can be used to make the “best predictions”

Information-based criterion
Aikake Information Criterion (and relatives)
AIC = DEV + 2*np_e
Small sample size (N = effective sample size)
AICc = DEV + 2*np_e + 2*np_e*(np_e+1)/(N-np_e-1)
Small sample size and over-dispersion in the data
AICc = DEV/c-hat + 2*np_e + 2*np_e*(np_e+1)/(N-np_e-1)

Model selection for the dipper data (1)
European dipper
-------------------------------------------------------------------------------------------
Delta AICc Model
Model AICc AICc Weight Likelihood #Par Deviance
-------------------------------------------------------------------------------------------
{phi(t*sexe)p(cte)} 685.124 0.00 0.68431 1.0000 13.000 75.764
{phit(t*sex)p(sex)} 686.922 1.80 0.27862 0.4072 14.000 75.422
{phi(t*sexe)p(t)} 690.973 5.85 0.03674 0.0537 17.000 72.996
{phit(t*sex)p(t*sex)} 700.462 15.34 0.00032 0.0005 22.000 71.474
-------------------------------------------------------------------------------------------
European dipper
Reduced Model General Model Chi-sq. df Prob.
------------------------- ------------------------- ---------- --- ------
{phi(t*sexe)p(cte)} {phit(t*sex)p(sex)} 0.341 1 0.5591

Model selection for the dipper data (2)
European dipper
-------------------------------------------------------------------------------------------
Delta AICc Model
Model AICc AICc Weight Likelihood #Par Deviance
-------------------------------------------------------------------------------------------
{phi(cte)p(cte)} 670.866 0.00 0.62389 1.0000 2.0000 84.361
{phi(sexe)p(cte)} 672.733 1.87 0.24529 0.3932 3.0000 84.199
{phi(t)p(cte)} 673.998 3.13 0.13032 0.2089 7.0000 77.253
{phi(t*sexe)p(cte)} 685.124 14.26 0.00050 0.0008 13.000 75.764
-------------------------------------------------------------------------------------------
European dipper
Reduced Model General Model Chi-sq. df Prob.
------------------------- ------------------------- ---------- --- ------
{phi(cte)p(cte)} {phi(sexe)p(cte)} 0.161 1 0.6878

Model selection for the dipper data
New set of a priori hypotheses:
We expect temporal variation in survival due to flooding during
the second and third interval. We wish to test the hypotheses
that flooding affected survival in male and female dippers
To do this, we need to use the design matrix!

Identity design matrix for dipper data
Model time*sex

Constrained design matrix
Model flood*sex

Model selection for the dipper data (3)
European dipper
-------------------------------------------------------------------------------------------
Delta AICc Model
Model AICc AICc Weight Likelihood #Par Deviance
-------------------------------------------------------------------------------------------
{phi(flood)p(cte)} 666.160 0.00 0.60043 1.0000 3.0000 77.626
{phi(flood+sexe)p(cte)} 668.116 1.96 0.22578 0.3760 4.0000 77.544
{phi(flood*sexe)p(cte)} 670.134 3.97 0.08231 0.1371 5.0000 77.514
{phi(cte)p(cte)} 670.866 4.71 0.05708 0.0951 2.0000 84.361
{phi(sexe)p(cte)} 672.733 6.57 0.02244 0.0374 3.0000 84.199
{phi(t)p(cte)} 673.998 7.84 0.01192 0.0199 7.0000 77.253
{phi(t*sexe)p(cte)} 685.124 18.96 0.00005 0.0001 13.000 75.764
-------------------------------------------------------------------------------------------
European dipper
Reduced Model General Model Chi-sq. df Prob.
------------------------- ------------------------- ---------- --- ------
{phi(cte)p(cte)} {phi(flood)p(cte)} 6.735 1 0.0095

Model results: estimates and CI

Work out example: Apus apus colony
Encounter history file : aa.inp
8 years of captures
2 colonies (1 column = P colony, 2 column = G colony)
Fit the initial model phi(time*colony) p(time)
Check GOF of the initial model
Select the best model among the following models
phi(time*colony) p(cte) phi(colony) p(time)
phi(time+colony) p(cte) phi(time) p(cte)
phi(time) p(time)
phi(colony) p(cte)