Determining Parameter Redundancy of Multi-state Mark-Recapture Models for Sea Birds
Mark recapture lecture 2: Jolly-Seber Confidence intervalssampath/Bio404/Bio404/Lecture_Notes… ·...
Transcript of Mark recapture lecture 2: Jolly-Seber Confidence intervalssampath/Bio404/Bio404/Lecture_Notes… ·...
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Mark recapture lecture 2:
• Jolly-Seber• Confidence intervals
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Jolly-Seber
• For an OPEN population
• Repeatedly sampled
• Information on when an individual was last marked
Year
LPB
Col
ony
size
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Open populationsIndividuals enter or leave the
population between surveys
Survey 1 Survey 2
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Catch nt animals
Check if each animal is marked
Total unmarked (ut ) Total marked (mt )
Mark all withcode for this time
period
Release St (equals nt if no handling mortality)
NO YES
Question: What is formula for proportion marked?
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Jolly-Seber
Remember Petersen (biased):
N= C MR
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Problem: We don’t know how many marked in population (M)
Sample 1: mark 21 animalsSample 2: mark 41 animalsSample 3: mark 46 animals
How many marked at beginning of sample 4?
Not 21+41+46=108, as some will have died or emigrated
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A
A
A
Time 1
AB
A
BB
Time 2
ACC
B
A
B
Time 3
Mark 3, but1 of these emigrates
Mark 3 more, but 1 marked animal dies
Mark 2 more, no loss of marked animals
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Time 4
Marked animals in sample 4 (m4 ) = 3
+ Marked animals not in sample 4
=Total number of marked animals in population
How many marked animals are alive and present in the population at time 4?
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D
DD
DD
Time 4
Marked animals in sample 4 (m4 ) = 3
+ Marked animals not in sample 4
=Total number of marked animals in population
D
6 marked at end of time 4 (S4 )
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D
DD
DD
Time 4
Marked animals in sample 4 (m4 ) = 3
+ Marked animals not in sample 4
=Total number of marked animals in population
DDD
Time 5
D D
6 marked at end of time 4 (S4 )
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Time 4
Marked animals in sample 4 (m4 ) = 3
+ Marked animals not in sample 4 (> 1)
=Total number of marked animals in population
Time 5
6 marked at time 4 (S4 ), recaptured (R4 )=1
D
DD
DD D
DD
D D
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Time 4
Marked animals in sample 4 (m4 ) = 3
+ Marked animals not in sample 4 (> 1)
=Total number of marked animals in population
Time 5
6 marked at time 4 (S4 ), recaptured (R4 )=1
D
DD
DD
E
D
E
ED
D D
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Time 4
Marked animals in sample 4 (m4 ) = 3
+ Marked animals not in sample 4 (> 1)
=Total number of marked animals in population
Time 5
E
D
E
ED
Time 6
D
6 marked at time 4 (S4 ), recaptured (R4 )=1+1
D
DD
DD
E
D
E
ED
D D
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Marked animals in sample 4 (m4 ) = 3
+ Marked animals not in sample 4 (> 1)
=Total number of marked animals in population
6 marked at time 4 (S4 ), recaptured (R4 )=1+1
Marked animals alive but not found in sample 4
= Recaptures after sample 4 (Z4 =1)x
factor accounting for animals missed or lost from population (S4 / R4 ) = 6/2 = 3
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Marked animals in sample 4 (m4 ) = 3
+ Marked animals not in sample 4 (=3)
=Total number of marked animals in population (M4 = 6)
Marked animals alive but not found in sample 4
= Z4 * S4 = 1* 6 = 3R4 2
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Mt = mt + Zt * StRt
Biased formula for number of marked animals in population:
Mt = mt + Zt * (St + 1)(Rt + 1)
Unbiased formula for number of marked animals in population:
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Jolly-Seber
Remember Petersen (biased):
N= C MR
Rearrange to:
N = M(R/C)
Number marked in population
Proportion markedin sample
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Jolly-Seber
Nt = Mt(?)
Number marked in population (t)
Proportion markedin sample t
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Jolly-Seber
Nt = Mt(?)
Number marked in population
Proportion markedin sample
mtnt
? = mt +1nt + 1? (unbiased) =
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Question:
m5 = 21S5 = 9R5 = 4Z5 = 10n5 = 43
What is N?
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Confidence intervals
( )
A range around the estimate of a parameter which
-if repeated –
would include the true value of the parameter a certain percentage of the time
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( )( )
( )
( )( )
( )( )
( )( )
( )
( )( )
( )( )
( )
( )
( )
( )
( )
( )95% Confidence interval:
19/20 of these confidence intervals contain the true value
true value
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Example of 95% confidence intervals:
501 British Columbians: “Which party would you vote for in the next provincial election?”:
CI = ± 4.5%
Sept 2004 Dec 2004 Feb 2005 Feb 2009
BC Liberals 43% 40% 46% 52%
NDP 37% 43% 40% 36%
Green 10% 8% 10% 12%
Reform 4% 2% 2% --
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Difference between confidence interval and variance:
Variance: know distribution of MANY data points around estimate (mean)
Eg. We measured height of 500 British Columbians (1.4 m + 0.2 m)
Confidence interval: only have ONE parameter estimate, have to guess what the distribution of repeated measurements might look like
Eg. We obtained percentage of British Columbians who “disapprove of Campbell’s performance”, and estimated CI (51% + 4.5%)
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Is the ratio ofR/C > 0.10?
Is the number of recaptures, R > 50?
Poisson
Normal
BinomialPetersen
Schnabel
Schumacher-Eschmeyer
Jolly-Seber: complex lognormal assumed,See Krebs p 47
Step 1: Make an educated guess as to the distribution (p 22 Krebs)
Y
Y
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Step 2: Calculate CI for either R or R/C (as appropriate)
-see formulae in Krebs
Step 3: Insert upper and lower bound for R or R/C into the formula for estimating population size to obtain CI
For example, if CI for R/C is (0.083, 0.177), to calculate CI for N by Petersen:
N=M/ 0.083 (upper bound) N=M/ 0.177 (lower bound)