### David Borchers

```Survival models without
mortality
Casting closed-population wildlife
survey models as survival- or
recurrent event models
David Borchers
Roland Langrock, Greg Distiller,
Ben Stevenson, Darren Kidney, Martin Cox
Closed-Population Methods
1.
2.
3.
4.
Removal methods
Distance Sampling Methods
Capture-Recapture Methods
Occupancy Methods
The Removal Method
40
t
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30
20
10
Cumulative catch
50
N - å nt
(an example with mortality)
+
N×F(t)
å nt
t
0
+
2
t
4
6
8
10
Occasion
(t) unknown number of
• This is a discrete survival model
with
censored subjects (N - å nt)
t
“Survivor function”
• pdf of time of death: f (t; p) = S(t-1)p
S(t-1) = (1- p)t-1
The Removal Method
40
30
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20
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+
10
Cumulative catch
50
(an example with mortality)
0
+
2
t
4
6
8
10
Occasion
(t)
• Continuous survival model with
unknown
number of censored
subjects
h is mortality hazard
(per unit time)
• pdf of time of death: f (t;h) = S(t)h
Removal models are survival models with
unknown number of censored subjects.
{
}
S(t) = exp - ò h¶u
t
0
Diagramatically:
Mortality hazard h
t
Time
0
T
Survival Model:
{
Survivor function
}
S(t) = exp - ò h du
f(t ;h) = S(t) h
Detection hazard
t
0
The Removal Method
(an example with mortality)
Continuous time likelihood
(with Poisson rather than Binomial/multinomial)
n
L(D, p) = Po(n; D)Õ f (ti ;h | detected)
i=1
n
S(ti )h
= Po(n; D)Õ
i=1 [1- S(T )]
pr(detect)
The Removal Method
(an example with mortality)
Continuous time likelihood with individual random effect
(with Poisson rather than Binomial/multinomial)
Hazard depends
on x
n
L(D, p) = Po(n; D)Õ p (xi | detected) f (ti ;h(xi ) | detected)
i=1
Random
distribution,
p (xeffect
i )p(xi )
conditional on detection
p (xi )p(xwith
… and hazard thatò changes
i )¶x time:
n
Hazard depends
on x and t
L(D, p) = Po(n; D)Õ p (xi | detected) f (ti ;h(ti xi ) | detected)
i=1
Diagramatic Removal Model, for a given x:
Mortality hazard at x: h(t |x)
x
t
Time
0
T
Survival Model:
f(t |x) = S(t |x)h(t |x)
pr(detect |x) = 1-S(T |x)
{
S(t | x) = exp - ò h(t | x)dx
t
0
}
Diagramatic
LineRemoval
Transect Model, for a given x:
Diagramatic
Detection hazard at x: h(t |x)
x
t
Time
0
T
Survival Model:
f(t |x) = S(t |x)h(t |x)
pr(detect
p(x)
|x) = 1-S(T |x)
{
S(t | x) = exp - ò h(t | x)dx
t
0
}
Line Transect models are survival models with unknown number
of censored subjects, and individual random effects.
Line Transect Models
Continuous time likelihood with individual random effect
(with Poisson rather than Binomial/multinomial)
Perpendicular distance distribution,
conditional on detection
n
L(D, p) = Po(n; D)Õ p (xi | detected) f (ti ;h(ti , xi ) | detected)
i=1
p (xi )p(xi )
S(ti | xi )p(ti | xi )
p(xi )
ò p (xi )p(xi )¶x
p(xi )
p(xi )
Q: Why is this ignored??
=
ò p(xi )¶x m
A: Hayes and Buckland (1983) are to “blame”
Hayes and Buckland are to blame
• Prior to Hayes & Buckland (1983), various models for 2-D
distribution of detection functions were proposed.
• Some fitted the data in some situations, but none was
robust (i.e. fitted in many situations).
• H&B (1983) proposed a hazard-rate formulation
(effectively a survival model) and showed that
marginalising over t resulted in robust forms for p(x), i.e.
forms that fitted many cases.
• Distance sampling has been 1-D ever since.
Is time-to-detection any use?
• Fewster & Jupp (2013) showed that additional data
improves asymptotic efficiency, even when it involves
estimating additional nuisance parameters.
• There are other benefits too…
1. Removal method does not require p(0)=1
2. Removal method does not require known random effect
distribution (π(x); uniform for line transects)
3. Can accommodate stochastic availability (i.e. overcome
“availability bias”)
30
20
0
5
15
0.05
0.10
Proportion of population
that is missed
0.00
f(t)
10
Time (t)
0.15
0.20
0
10
P(catch
by t)
N×F(t)
40
50
1. p(0)<1
0
5
10
15
1. Time-to-detection enables you to estimate p(0)
f(t|x=0): pdf of detection times for animals at peprendicular distance zero
Proportion of population
at x=0 that is missed:
= 1 – p(0)
Observer
1. Time-to-detection enables you to estimate p(0)
(Sometimes not so well)
f(t|x=0): pdf of detection times for animals at peprendicular distance zero
Removal method poor
unless large fraction of
population is removed.
Þ
LT p(0) estimation from
time-to-detection data
is poor when p(0) is not
“close” to 1.
Observer
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+ ++ ++
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+ + ++ + + + +++
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+ ++
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+ + ++
++
+++
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++ +++++ +
+++ ++ + + +
+++
+ ++ +++
+
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+ +
+ + + +++
+
+ +
+++
+ + + + + + ++
++ +++
+
+
+ ++ + +
++
+
+
+
-0.5
0.0
0.5
++
-1.0
Perpendicular distance
1.0
2. Time-to-detection enables you to estimate π(x)
0.0
0.5
1.0
Forward distance
1.5
+
+
+
2.0
0.8
0.6
0.4
p(x)=1-S(T|x)
1.0
2. Time-to-detection enables you to estimate π(x)
0.0
0.2
0.4
0.6
Perpendicular distance
0.8
1.0
1.0
0.5
0.0
f(y|x) at x=0
1.5
2. Time-to-detection
Forward distance enables you to estimate π(x)
0.0
0.5
1.0
Forward distance
1.5
2.0
10 20 30 40
0
Frequency
Perp dist dbn
0.0
0.2
0.4
0.6
0.8
1.0
Perpendicular distance
30
20
10
0
Frequency
40
Forward dist dbn
0.0
0.5
1.0
Forward distance
1.5
2.0
2. Forward distance enables you to estimate π(x)
Fitted curves (grey=true)
2.0
Estimated
1.0
0.5
0.0
pdf
1.5
f(x)
p(x)
pi(x)
0.0
0.2
0.4
0.6
perpendicular distance (x)
0.8
1.0
3. Stochastic availability
Detection
Detection
hazard
hazard
at x,t
at x,
given
given
availability:
availability:
h(x,t)
h(t |x)
0
t
2-State Markov-modulated Poisson Process (MMPP) in which
State 1 = shallow diving:
(high Poisson event rate)
State 0 = deep diving:
(low Poisson event rate)
T
Time
3. Stochastic availability: Bowhead aerial survey
Recap: Is time-to-detection any use?
• Fewster & Jupp (2013) showed that additional data
improves asymptotic efficiency, even when it involves
estimating additional nuisance parameters.
• There are other benefits too…
1. Removal method does not require p(0)=1 (if p(0) not too small)
2. Removal method does not require known random effect
distribution (π(x); uniform for line transects)
3. Can accommodate stochastic availability (i.e. overcome
“availability bias”) 
Distance
Sampling
Methods
Line Transect
Method
n
L(D, p) = Po(n; D)Õ p (xi | detected) f (ti ;h(ti | xi ) | detected)
i=1
p (xi )p(xi )
S(ti | xi )p(ti | xi )
p(xi )
ò p (xi )p(xi )¶x
Capture-recapture with camera traps
Trap k
Spatially Explicit Capture-Recapture (SECR)
1
x
2
3
t21 t22
t11
1
.........
2
Detection function
parameters
Density model
parameters
R
Time
Number of times animal
i is detected on camera k
on occasion r
Poisson
Location of animal
i’s activity center
Continuous-time Spatially Explicit Capture-Recapture
• Each animal can be detected multiple times, so not a “survival”
model.
• Detection times modelled as Non-homogeneous Poisson Process
(NHPP), with rate hk(t|x;θ) for trap k, given activity center at x
• For generality, allow detection hazard to depend on time
• Ignoring occasion for simplicity:
Number of times animal
i is detected on camera k
Continuous-time SECR
Discrete-time Model
Continuous-time SECR
Discrete-time Model
Time-to-detection is NOT informative about density IF
hk (xi ;q )
hk (xi ;q )
1
=
=
=C
(a) hk (tikj | xi ;q ) = hk (xi ;q ) so that
ò hk (xi ;q )dt hk (xi ;q ) ò 1dt ò 1dt
(b) D(x; f ) = D (then D factorises out of integral and product)
ELSE time-to-detection IS informative about denstiy
Aside: In case (a) above, continuous-time model is identical to discrete-time Poisson count model.
Continuous-time SECR
Discrete-time Model
wik
Sk (T | x;q )Õ hk (tikj | x;q )
j=1
Sk (Tr | x;q )
Continuous-time SECR
wik
Sk (T | x;q )Õ hk (tikj | x;q )
j=1
Notes:
1. Mark-recapture distance sampling is a special case of SECR
2.
Aside from independence issues (ask a question if you don’t now
what I mean), there is no reason to impose occasions when you
have detectors that sample continuously.
Time-to-detection in Occupancy Models
F(t)
F(t)
From Bischoff et al. (2014): Prob(detect | Presence)
Kaplan-Meier
Constant-hazard
This is just the continuous-time removal method again
(constant hazard, no individual random effects).
Time-to-detection in Occupancy Models:
incorporating availability
Guillera-Arroita et al. (2012): tiger pugmarks along a transect
Constant hazard of detection, given pugmark
Distance
l
0
2-State Markov-modulated Poisson Process (MMPP) : 3 different animals
Does this look familiar?
L
Recall: Line Transect with Stochastic availability
Detection
Detection
probability
hazard at x,t
x,
given
given
availability:
availability:
h(x,t)
h(t |x)
0
t
Time
T
Line Transect with stochastic availability and multiple detections
2-State Markov-modulated Poisson Process (MMPP) in which
State 1 = shallow diving:
(high Poisson rate)
State 0 = deep diving:
(low Poisson rate)
Time-to-detection in Occupancy Models:
incorporating availability
Gurutzeta et al. (2012): tiger pugmarks along a transect
Constant hazard of detection, given pugmark
Distance
l
0
Same as Line Transect with stochastic availability, except:
1. Distance, not time and pugmarks, not whale blows
2. Constant detection hazard
3. Can’t distinguish between individuals (this adds lots of complication!)
4. Estimating presence, not abundance (this makes things little simpler)
L
Summary
• Time-to-detection is informative about
density/occupancy
• Removal, Distance Sampling, SECR and Occupancy
models share common underlying theory
• Fertile ground for further method development, each
method borrowing from the other.
My email: [email protected]
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