Mark-Recapture Maximum Likelihood Estimators adapted from Seber et al. 1982 (page
200) and Burnham et al. 1987 (page 114) for CJS models.
R. Perry
Updated: 2/22/2008
Definition of symbols
i
= detection site, i = 1, 2, …k, numbered from upstream to downstream where k is the
maximum number of detection sites.
mi
= number of fish detected at site i.
ri
= number of fish detected downstream of site i (i.e., at sites i+1…k), of those detected at
site i.
zi
= number of fish not detected at site i, but detected downstream of site i (i.e., at sites
i+1…k).
R
= total number of fish released.
Note: (ri + zi) = the total number of fish detected downstream of site i and therefore, known to be
alive at site i.
Estimated parameters
ˆi
= estimated survival probability from site i-1 to site i
pˆ i
Mˆ
= estimated probability of detection at site i, conditional on the fish being alive at site i.
̂
i
= estimated number of tagged fish alive at site i
= k pk = estimated joint probability of surviving the last reach and being detected at the
kth detection site.
Maximum Likelihood Estimators
The maximum likelihood estimate for detection probability, pˆ i , is:
r
pˆ i  i
ri  zi
and the standard error of pˆ i is:
SE( pˆ i ) =
 pˆ 1  pˆ  
i
i
2
1 1 
  
 ri zi 
1
The maximum likelihood estimator of the survival probability ˆi is:
ˆi 
Mˆ i
Mˆ i 1
m r  z 
where Mˆ i  i i i
ri
Mˆ
if i = 1 then ˆ1  1
R
 Mˆ  mi
and Var ˆi  ˆi2  i
ˆ
 M i
 
 1 1  Mˆ i 1  mi 1  1
1  1  ˆi 


  


Mˆ i 1  ri 1 mi 1  Mˆ i 
 ri mi 
when i =1 the variance formula reduces to:
 Mˆ  m1  1 1  1  ˆi 
Var ˆ1  ˆi2  1
   ˆ 
ˆ
M
 r1 m1  M i 
1

 
The maximum likelihood estimator of ̂ is:
ˆ 
rk 1
mk 1
CJS Example
I generated capture histories for a 2-occasion CJS model with true parameter values 1 =
0.85, p1 = 0.93, and  = 0.80. Capture histories were generated for R = 378 fish by calculating
the expected value of each capture history frequency as R*P where P is the probability of
observing each capture history:
Capture History (i)
111
101
100
110
ni
239
18
61
60
Statistics:
R = 378
m1 = n111 + n110 = 299
r1 = n111 = 239
z1 = n101 = 18
Parameter estimates:
r
239
pˆ1  1 
 0.93
r1  z1 239  18
m  r  z  299  239  18 
Mˆ 1  1 1 1 
 321.3
r1
239
2
Mˆ 1 321.3

 0.85
R
239
 Mˆ  m1  1 1  1  ˆ1 
 321.3  299  1
1  1  0.85 
Var ˆ1  ˆ12  1
 
 0.852 





  0.0003796
ˆ
ˆ
r
m
321.3
239
299
321.3
M
M




1
1


1
1


ˆ1 
 
 
 
SE ˆ1  Var ˆ1  0.01945
ˆ 
rk 1
r
239
 1 
 0.80
mk 1 m1 299
Note that the parameter estimates from the maximum likelihood estimators are the same as the
true values used to generate the capture histories.
Derivation of Maximum Likelihood Estimators for double-array detection probabilities
The double-array model for calculating detection probabilities follows a multinomial model
conditional of the fish observed at either detection array. The likelihood can be specified as:

p1 1  p2 
 N 
L  p1 , p2 | n10 , n01 , N   

 
 n10 , n01   1  1  p1 1  p2  
n10


p2 1  p1 


 1  1  p1 1  p2  
n01


p1 p2


 1  1  p1 1  p2  
Where p1 and p2 are detection probabilities for each independent detection array, n10 is the
number of fish detected at the first array but not the second, n01 is the number of fish detected at
the second array but not the first, and N is the total number of unique fish detected at both arrays.
The MLEs are found by taking partial derivatives of the log-likelihood with respect to each
parameter, setting to zero, and solving for the simultaneous solution of the parameters.
To simplify derivatives, the likelihood can be expressed as:
 N 
1
n
n
L  p1 , p2 | n10 , n01 , N   
p1N n01 p2N n10 1  p2  10 1  p1  10

N
 n10 , n01  1  1  p 1  p  
1
2
The log-likelihood is then
 N 
log L  log 
  N log 1  1  p1 1  p2     N  n01  log p1   N  n10  log p2
 n10 , n01 
 n01 log 1  p1   n10 log 1  p2 
3
N  n10  n01
Taking derivatives
N  p2  1
n
 log L N  n01

 01 
0
p1
p1
1  p1 1  1  p1 1  p2 
N  p1  1
n
 log L N  n10

 10 
0
p1
p2
1  p2 1  1  p1 1  p2 
Solving this system of equations for p1 and p2 yields the following MLEs:
pˆ1 
N  n10  n01
N  n10
pˆ 2 
N  n10  n01
N  n01
Note: Since N = n10+n01+n11 where n11 are fish detected at both arrays, the MLE’s can also be
expressed as:
pˆ1 
n11
n01  n11
pˆ 2 
n11
n10  n11
The overall detection probability is a function of p1 and p2:
p = 1-(1-p1)(1-p2)
References
Burnham, K. P., D. R. Anderson, G. C. White, C. Brownie, and K. H. Pollock. 1987. Design and
Analysis Methods for Fish Survival Experiments Based on Release-Recapture, American
Fisheries Society Monograph 5, Bethesda, Maryland.
Seber, G.A.F. 1982. The estimation of animal abundance and related parameters. The Blackwell
Press, Caldwell, New Jersey
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