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Mixture models for
estimating population size
with closed models
Shirley Pledger
Victoria University of Wellington
New Zealand
IWMC December 2003
Acknowledgements
•
•
•
•
•
•
Gary White
Richard Barker
Ken Pollock
Murray Efford
David Fletcher
Bryan Manly
2
Background
• Closed populations - no birth / death /
migration
• Short time frame, K samples
• Estimate abundance, N
• Capture probability p – model?
• Otis et al. (1978) framework
3
M(tbh)
M(tb)
M(th)
M(bh)
M(t)
M(b)
M(h)
M(0)
4
Models for p
• M(0), null model, p constant.
• M(t), Darroch model, p varies over time
• M(b), Zippin model, behavioural response
to first capture, move from p to c
• M(h), heterogeneity, p varies by animal
• M(tb), M(th), M(bh) and M(tbh),
combinations of these effects
5
Likelihood-based models
• M(0), M(t) and M(b) in CAPTURE, MARK
• M(tb) – need to assume connection,
e.g. c and p series additive on logit scale
• M(h) and M(bh), Norris and Pollock (1996)
• M(th) and M(tbh), Pledger (2000)
• Heterogeneous models use finite mixtures
6
M(h)
C animal classes, unknown membership.
Animal i from class c with probability pc.
p1
Class
1
Capture probability p1
Animal
i
p2
Class
2
Capture probability p2
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M(h2) parameters
•
•
•
•
•
N
p1 and p2
p1 and p2
Only four independent, as p1 + p2 = 1
Can extend to M(h3), M(h4), etc.
8
M(th) parameters
•
•
•
N
p1 and p2 (if C = 2)
p matrix, C by K, pcj is capture probability
for class c at sample j
• Two versions:
1. Interactive, M(txh), different profiles
2. Additive (on logit scale), M(t+h).
9
Capture probability
M(t x h), interactive
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
Class 1
Class 2
• Different classes of
animals have different
profiles for p
• Species richness
applications
1 2 3 4 5
Sample
10
Capture probability
M(t+h), additive (on logit scale)
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
Class 1
Class 2
1 2 3 4 5
Sample
•
•
•
•
•
For Class 1,
log(pj/(1-pj)) = m + tj
For Class 2,
log(pj/(1-pj)) = m + tj + h2
Parameter h2 adjust p
up or down for class 2
• Similar to Chao M(th)
• Example – Duvaucel’s
geckos
11
M(bh) parameters
• N
• p1 . . . pC (C classes, Sp = 1)
• p1 . . . pC for first capture
• c1 . . . cC for recapture
• Two versions:
1.Interactive, M(bxh), different profiles
2.Additive (on logit scale), M(b+h).
12
M(b x h), interactive
Capture probability
0.6
0.5
0.4
Class 1
Class 2
0.3
0.2
0.1
• Different size of trapshy response
• One class bold for
first capture, large
trap response
• Second class timid at
first, slight trap
response.
0
First Recap
13
M(b + h), additive (logit scale)
Capture probability
0.6
0.5
0.4
Class 1
Class 2
0.3
0.2
0.1
0
First Recap
• Parallel lines on logit
scale
• For Class 1,
log(p/(1-p)) = m + h1
log(c/(1-c)) = m + h1 + b
• For Class 2,
log(p/(1-p)) = m + h2
log(c/(1-c)) = m + h2 + b
• Common b adjusts for
behaviour effect
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M(tbh)
• Parameters N and p1 . . . pC (C classes)
• Interactive version – each class has a p
series and a c series, all non-parallel.
• Fully additive version – on logit scale,
have a basic sequence for p over time,
use b to adjust for recapture and h to
adjust for different classes.
• There are also other intermediate models,
partially additive.
15
M(t x b x h)
For class c, sample j, Logit(pjc) =
m + tj + b + hc + (tb)j + (th)jc + (bh)c + (tbh)jc
where b is a 0/1 dummy variable, value 1 for
a recapture. (Constraints occur.)
16
Other Models
•
•
•
•
•
•
•
M(t+b+h) – omit interaction terms
M(t x h) – omit terms with b
M(t + h) – also omit (th) interaction term
M(b x h) – omit t terms
M(0) has m only.
17
M(t x b)
• Can’t do M(t x b) – too many parameters
for the minimal sufficient statistics.
• Can do M(t+b) using logit. Similar to
Burnham’s power series model in
CAPTURE.
• Why can we do M(t x b x h) (which has
more parameters), but not M(t x b)?
18
Now have these
models:
M(txbxh)
M(t+b+h)
M(txb)
M(txh)
M(bxh)
M(t+b)
M(t+h)
M(b+h)
M(t)
M(b)
M(h)
M(0)
19
Example - skinks
•
•
•
•
•
•
•
Polly Phillpot, unpublished M.Sc. thesis
Spotted skink, Oligosoma lineoocellatum
North Brother Island, Cook Strait, 1999
Pitfall traps
April: 8 days, 171 adults, 285 captures
Daily captures varied from 2 to 99 (av<40)
November: 7 days, 168 adults, 517
captures (20 to 110 daily, av>70)
20
21
22
April:
Rel(AICc)
npar
M(t + b + h)
M(t x h)
0.00
8.82
12
18
M(t x b x h)
M(t + h)
M(t)
9.79
26.65
63.43
26
10
9
M(t + b)
M(b x h)
M(b + h)
65.25
200.56
205.06
10
6
5
M(b)
M(h)
M(0)
267.15
289.81
328.53
3
4
2
23
November: Rel(AICc) npar
M(t x b x h)
M(t x h)
0.00
4.65
22
16
M(t + b + h)
M(t + h)
M(t + b)
7.82
8.24
145.08
11
10
9
M(t)
M(b + h)
174.76
190.50
8
5
M(h)
M(b x h)
M(0)
M(b)
200.44
219.41
323.76
325.60
4
6
2
3
24
Abundance Estimates
• Used model averaging
• April, N estimate = 206 (s.e. = 33.0)
95% CI (141,270).
• November, N estimate = 227 (s.e. = 38.7)
95% CI (151,302).
25
Using MARK
• Data entry – as usual, e.g. 00101 5; for 5
animals with encounter history 00101.
• Select “Full closed Captures with Het.”
• Select input data file, name data base,
give number of occasions, choose number
of classes, click OK.
• Starting model is M(t x b x h)
• Following example has 2 classes, 5
sampling occasions.
26
Parameters for M(t x b x h)
p1
1
p for class 1
2
3
4
5
6
p for class 2
7
8
9
10
11
c for class 1
12
13
14
15
c for class 2
16
17
18
19
N
20
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M(t x h): set p=c
p1
1
p for class 1
2
3
4
5
6
p for class 2
7
8
9
10
11
c for class 1
3
4
5
6
c for class 2
8
9
10
11
N
12
28
M(b x h): constant over time
p1
1
p for class 1
2
2
2
2
2
p for class 2
3
3
3
3
3
c for class 1
4
4
4
4
c for class 2
5
5
5
5
N
6
29
M(t)
p1
1 (fix)
p for class 1
2
3
4
5
6
p for class 2
2
3
4
5
6
c for class 1
3
4
5
6
c for class 2
3
4
5
6
N
7
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M(b)
p1
1 (fix)
p for class 1
2
2
2
2
2
p for class 2
2
2
2
2
2
c for class 1
3
3
3
3
c for class 2
3
3
3
3
N
4
31
M(0)
p1
1 (fix)
p for class 1
2
2
2
2
2
p for class 2
2
2
2
2
2
c for class 1
2
2
2
2
c for class 2
2
2
2
2
N
3
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M(t + h): use M(t x h) parameters (as
below), plus a design matrix
p1
1
p for class 1
2
3
4
5
6
p for class 2
7
8
9
10
11
c for class 1
3
4
5
6
c for class 2
8
9
10
11
N
12
33
Design matrix for M(t + h). Use logit link.
B1
p1
p class 1
B2
B3
B4
B5
1
b7 is h2
1
p class 1
p class 2
p class 2
N
Adjusts
for class 2
1
p class 1
p class 2
B8
1
p class 1
p class 2
B7
1
p class 1
p class 2
B6
1
1
1
1
1
1
1
1
1
1
1
1
34
M(b + h)
Start with M(b x h) and use this
design matrix, with logit link
B1
p1
p class 1
B2
c class 2
N
B4
B5
1
1
p class 2
c class 1
B3
1
1
1
1
1
1
35
M(t + b + h)
• Start with M(t x b x h)
• Use one b to adjust for recapture
• For each class above 1 use another b for
the class adjustment.
36
Time Covariates
• Time effect could be weather, search effort
• Logistic regression: in logit(p), replace tj with
linear response e.g. gxj + dwj where xj is search
effort and wj is a weather variable (temperature,
say) at sample j
• Logistic factors: use dummy variables to code
for (say) different searchers, or low and high
rainfall.
• Skinks: maximum daily temperature gave good
models, but not as good as full time effect.
37
Multiple Groups
• Compare – same capture probabilities?
• If equal-sized grids, different locations, N
indexes density – compare densities in
different habitats.
• Cielle Stephens, M.Sc. (in progress) –
skinks. Good design - eight equal grids,
two in each of four different habitat types.
Between and within habitat density
comparisons. Temporary marks.
38
Discussion
• Advantages of maximum likelihood
estimation – AICc, LRTs, PLIs.
• Working well for model comparison.
• Two classes enough? Try three or more
classes, look at estimates.
39
• If heterogeneity is detected, models
including h have higher N and s.e.(N).
• If heterogeneity is not supported by AICc,
the heterogeneous models may fail to fit.
See the parameter estimates.
• M(t x b x h) often fails to fit – see
parameter estimates (watch for zero s.e.,
p or c at 0 or 1).
40
• Alternative M(h) – use Beta distribution for p
(infinite mixture). Which performs better? depends on region of parameter space chosen
by the data. Often similar N estimates.
• Don’t believe in the classes or the Beta
distribution. Just a trick to allow p to vary and
hence reduce bias in N.
41
• All models poor if not enough recaptures.
Warning signals needed.
• Finite mixtures, one class with very low p.
• Beta distribution, first parameter estimate < 1.
• Often with finite mixtures, estimates of p and p
are imprecise, but N estimates are good.
42
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