Mixture models for estimating population size
with closed models
Shirley PledgerVictoria University of Wellington
New ZealandIWMC December 2003
2
Acknowledgements
• Gary White
• Richard Barker
• Ken Pollock
• Murray Efford
• David Fletcher
• Bryan Manly
3
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 c.
Animal
i
Class1
Class2
Capture probability p1
Capture probability p2
1
2
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M(h2) parameters
• N
• 1 and 2
• p1 and p2
• Only four independent, as 1 + 2 = 1
• Can extend to M(h3), M(h4), etc.
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M(th) parameters
• N
• 1 and 2 (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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M(t x h), interactive
00.050.1
0.150.2
0.250.3
0.350.4
0.450.5
1 2 3 4 5
Sample
Cap
ture
pro
bab
ilit
y
Class 1
Class 2
• Different classes of animals have different profiles for p
• Species richness applications
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M(t+h), additive (on logit scale)
00.050.1
0.150.2
0.250.3
0.350.4
0.450.5
1 2 3 4 5
Sample
Cap
ture
pro
bab
ilit
y
Class 1
Class 2
• For Class 1,
• log(pj/(1-pj)) = j
• For Class 2,
• log(pj/(1-pj)) = j 2
• Parameter2 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
• 1 C (C classes, )
• 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
0
0.1
0.2
0.3
0.4
0.5
0.6
First Recap
Cap
ture
pro
bab
ilit
y
Class 1
Class 2
• Different size of trap-shy response
• One class bold for first capture, large trap response
• Second class timid at first, slight trap response.
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M(b + h), additive (logit scale)
0
0.1
0.2
0.3
0.4
0.5
0.6
First Recap
Cap
ture
pro
bab
ilit
y
Class 1
Class 2
• Parallel lines on logit scale
• For Class 1, log(p/(1-p)) = 1
log(c/(1-c)) = 1 • For Class 2,
log(p/(1-p)) = 2
log(c/(1-c)) = 2 • Common adjusts for
behaviour effect
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M(tbh)
• Parameters N and 1 . . . C (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 to adjust for recapture and 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) =
j ++c+j+jc+c+jc
where is a 0/1 dummy variable, value 1 for a recapture. (Constraints occur.)
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Other Models
• M(t+b+h) – omit interaction terms
• M(t x h) – omit terms with • M(t + h) – also omit () interaction term
• M(b x h) – omit terms
•
•
• M(0) has 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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M(txbxh)
M(txb) M(txh) M(bxh)
M(t) M(b) M(h)
M(0)
M(t+b) M(t+h) M(b+h)
M(t+b+h)
Now have thesemodels:
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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)
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22
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April: Rel(AICc) npar
M(t + b + h) 0.00 12
M(t x h) 8.82 18
M(t x b x h) 9.79 26
M(t + h) 26.65 10
M(t) 63.43 9
M(t + b) 65.25 10
M(b x h) 200.56 6
M(b + h) 205.06 5
M(b) 267.15 3
M(h) 289.81 4
M(0) 328.53 2
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November: Rel(AICc) npar
M(t x b x h) 0.00 22
M(t x h) 4.65 16
M(t + b + h) 7.82 11
M(t + h) 8.24 10
M(t + b) 145.08 9
M(t) 174.76 8
M(b + h) 190.50 5
M(h) 200.44 4
M(b x h) 219.41 6
M(0) 323.76 2
M(b) 325.60 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)
1 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
1 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
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M(b x h): constant over time
1 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)
1 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)
1 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)
1 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
1 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
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Design matrix for M(t + h). Use logit link.
B1 B2 B3 B4 B5 B6 B7 B8
1 1
p class 1 1
p class 1 1
p class 1 1
p class 1 1
p class 1 1
p class 2 1 1
p class 2 1 1
p class 2 1 1
p class 2 1 1
p class 2 1 1
N 1
7 is
Adjustsfor class 2
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M(b + h) Start with M(b x h) and use this design matrix, with logit link
B1 B2 B3 B4 B5
1 1
p class 1 1
p class 2 1
c class 1 1 1
c class 2 1 1
N 1
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M(t + b + h)
• Start with M(t x b x h)
• Use one to adjust for recapture
• For each class above 1 use another for the class adjustment.
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Time Covariates
• Time effect could be weather, search effort• Logistic regression: in logit(p), replace j with
linear response e.g. xj + wj 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 and p
are imprecise, but N estimates are good.
Recommended
