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.