Download Mixture models for estimating population size with closed models

Mixture models for
estimating population size
with closed models
Shirley Pledger
Victoria University of Wellington
New Zealand
IWMC December 2003
Acknowledgements
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Gary White
Richard Barker
Ken Pollock
Murray Efford
David Fletcher
Bryan Manly
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Background
• Closed populations - no birth / death /
migration
• Short time frame, K samples
• Estimate abundance, N
• Capture probability p – model?
• Otis et al. (1978) framework
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M(tbh)
M(tb)
M(th)
M(bh)
M(t)
M(b)
M(h)
M(0)
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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
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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
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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
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•
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N
p1 and p2
p1 and p2
Only four independent, as p1 + p2 = 1
Can extend to M(h3), M(h4), etc.
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M(th) parameters
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•
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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).
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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
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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
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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
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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).
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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
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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.
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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.)
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Other Models
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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.
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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)?
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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)
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Example - skinks
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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)
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April:
Rel(AICc)
npar
M(t + b + h)
M(t x h)
0.00
8.82
12
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M(t x b x h)
M(t + h)
M(t)
9.79
26.65
63.43
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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
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November: Rel(AICc) npar
M(t x b x h)
M(t x h)
0.00
4.65
22
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M(t + b + h)
M(t + h)
M(t + b)
7.82
8.24
145.08
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10
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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
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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).
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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.
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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
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c for class 1
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c for class 2
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N
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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
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c for class 1
3
4
5
6
c for class 2
8
9
10
11
N
12
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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
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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
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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
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c for class 1
3
4
5
6
c for class 2
8
9
10
11
N
12
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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
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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
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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.
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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.
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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.
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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.
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• 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).
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• 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.
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• 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.
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