Transcript 1-p 2
BRIEF INTRODUCTION TO
CLOSED CAPTURE-RECAPTURE
METHODS
Workshop objectives
Basic understanding of capture-recapture
Estimators
Sample designs
Uses and assumptions
Detectability
and abundance estimation
N = true abundance
C = catch
p = probability of capture
E(C)= pN
Incomplete capture: Inference
Inferences about N require inferences about p
ˆ
N
C
pˆ
Estimating abundance with capture
probability known = 0.5 (or 50%)
2
ˆ
N
4
0 .5
• If you ignore p then C =2 is biased
• Usually we have to collect other data to
estimate p!
Closed Population Estimation
Parameters
• Abundance
• Capture probability
Population closed
• No gains or losses in the study area
Replicate samples used to estimate N, p
Commonly Used Estimators:
Lincoln-Petersen/Schnabel/etc.
Design
• Animals caught
• Unmarked animals in sample given (or have) unique marks
• Marks on any marked animals recorded
• Release marked animals into population
• Resample at subsequent occasions
• Minimum two sampling periods (capture and recapture)
• (Ideally) a relatively short interval between periods
Not during migration, harvest period, other period with
significant gains, losses, movement
• Must be long enough to generate recaptures
Closed Population Estimators
Key Assumptions
• Population is closed
(no birth/death/immigration/emigration)
• Animal captures are independent
• All animals are available for capture
• Marks are not lost or overlooked
• L-P and Schnabel
• assume equal p (never ever possible)
• Probability of recapture not affected by previous
capture
Violations of Assumptions
Closure violation
• Mortality or emigration during sampling
Unbiased estimate of N at first sample time
• Immigration or birth
Unbiased estimate of N at last sample time
• Both
Valid inferences not possible
Violations of Assumptions
All animals are not available for capture
- underestimate N
- overestimate p
Violations of Assumptions
Equal capture probability (when assumed)
• Differences (heterogeneity) among individuals
Underestimate abundance
• Trap response: “trap-shy”
Overestimate N
Underestimate p
•“Trap happy”
Underestimate N
Overestimate p
Potential Violations of Assumptions
Tag loss
• Lost between sampling periods
Underestimate p
Overestimate N
• Overlooked or incorrectly recorded
Underestimate p
Overestimate N
Effect can be eliminated or minimized by double-tagging
Variance of abundance estimate
Depends on
Variance in true N
Capture probability
Variance in estimated p
Affected by sample size
Sample size
Number of marked animals
Number of capture occasions
Rule of thumb
Number of animals captured each occasion
(C) determines precision of estimates of N
If capture probabilities low or true
abundance low:
More effort in fewer occasions
Increases occasion specific p
Increases C
Closed population estimators
Definitions
pt = probability of first capture sampling
occasion t
ct = probability of recapture sampling
occasion t+1 (don’t confuse with big
C)
N = population size
Note: there are t-1 estimates possible for c
Closed population estimators
Definitions
If there is no effect of first capture on
recapture probability
- no trap happy
- no trap shy, etc.
pt+1 = ct
Capture (encounter) histories
H1 = 101
Verbal description: individual was captured on
first and third sample occasion, not captured on
second occasion
Mathematical depiction:
P(H1 = 101) = p1(1-c1)c2
Capture (encounter) histories
H1 = 111
Verbal description: individual was captured on
all three occasions
Mathematical depiction:
P(H1 = 111) = p1c1c2
Capture (encounter) histories
H1 = 001
Verbal description: individual was captured on
first and third sample occasion, not captured on
second occasion
Mathematical depiction:
P(H1 = 001) = (1-p1)(1-p2)p3
Capture (encounter) histories
100
p1(1-c1)(1-c2)
010
(1-p1)p2(1-c2)
001
(1-p1)(1-p2)p3
110
p1c1(1-c2)
101
p1(1-c1)c2
011
(1-p1)p2c2
111
p1c1c2
Capture (encounter) histories
Capture and recapture
equal differ in time
Capture and recapture equal
across time
p(1-p)2
H
100
010
001
110
p1(1-c1)(1-c2)
(1-p1)p2(1-c2)
(1-p1)(1-p2)p3
p1c1(1-c2)
p1(1-p2)(1-p3)
(1-p1)p2(1-p3)
(1-p1)(1-p2)p3
p1p2(1-p3)
(1-p)p(1-p) or p(1-p)2
(1-p)2 p
p2(1-p)
101
011
111
p1(1-c1)c2
(1-p1)p2c2
p1c1c2
p1(1-p2)p3
(1-p1)p2p3
p1p2p3
p(1-p)p or p2(1-p)
(1-p)p2
p3
Huggins version of CR
estimator
Why Covariates?
Capture probability known to be related to:
species, body size, habitat characteristics
More efficient means of accounting for heterogeneity
e.g., assume p varies through time (5 time periods) due to
differences in stream discharge
Number of parameters time varying model = 5
Number parameters p in f(discharge) = 2
Effects model selection: AIC = -2LogL + 2*K
Danger of over parameterization (more parameters than data)
Frequently encountered problem
I don’t have enough marked and/or recaptured
individuals
Make sure closure assumption not violated
Include data from other years/locations to
estimate p for poor recapture year (Huggins)
Bayesian hierarchical approaches
p?
p1
p2
Frequently encountered problem
Lake Sturgeon in Muskegon River, MI
Year
Catch Statistic
1
2
3
4
Total Gill Net
Hours
3030
2250
1247
1852
Total marked
adults
13
10
8
15
Recaptured adults
8
5
1
2
Schnabel Estimate
(95% CL) each
year seperate
24
(12-74)
15
(9-45)
---
---
22
(16-45)
16
(12-37)
45
(14-247)
18
(16-39)
Estimate (95% CL)
modeled together
f(soak time, size)
Double Sampling
Disadvantages of capture recapture approaches: Can be labor/time
intensive!!
But….double sampling can reduce effort:
Capture recapture
Estimate p
and adjust
data
Normal sampling
Mark-resight
(will not cover in this course)
Estimate population size
Resighting marked and unmarked individuals
Requires known number of marks
But version available if marks unknown (not recommended)
Used terrestrial applications but potential fish uses
snorkeling: if marks detectable
weir or trap where unmarked fish returned unmarked
Marks
Batch marked
Individually identifiable
Open and closed versions
BREAK!
then
ON TO MARK