Survival and Associated Risk Factors for Mortality Among ...
Survival models without mortality
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Survival models without mortality
Casting closed-population wildlife survey models as survival- or
recurrent event models
David BorchersRoland Langrock, Greg Distiller,
Ben Stevenson, Darren Kidney, Martin Cox
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Closed-Population Methods
1. Removal methods2. Distance Sampling Methods3. Capture-Recapture Methods4. Occupancy Methods
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• This is a discrete survival model with unknown number of censored subjects ( )
• pdf of time of death:
The Removal Method(an example with mortality)
“Survivor function”
t
N×F(t)
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The Removal Method(an example with mortality)
t
h is mortality hazard (per unit time)
• Continuous survival model with unknown number of censored subjects
• pdf of time of death:
Removal models are survival models with unknown number of censored subjects.
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Diagramatically:
Timet
T0
Mortality hazard h
f(t ;h) = S(t) h
Survivor function
Survival Model:
Detection hazard
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The Removal Method(an example with mortality)
Continuous time likelihood (with Poisson rather than Binomial/multinomial)
pr(detect)
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The Removal Method(an example with mortality)
Continuous time likelihood with individual random effect (with Poisson rather than Binomial/multinomial)
Random effect distribution, conditional on detection
Hazard dependson x
… and hazard that changes with time: Hazard dependson x and t
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Diagramatic Removal Model, for a given x:
Timet
x
T0pr(detect |x) = 1-S(T |x)
Mortality hazard at x: h(t |x)
f(t |x) = S(t |x)h(t |x)
Survival Model:
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Diagramatic Removal Model, for a given x:
Timet
0
x
Tpr(detect |x) = 1-S(T |x)
Detection hazard at x: h(t |x)
p(x)
Line Transect models are survival models with unknown number of censored subjects, and individual random effects.
f(t |x) = S(t |x)h(t |x)
Survival Model:
Diagramatic Line Transect
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Line Transect Models
Continuous time likelihood with individual random effect (with Poisson rather than Binomial/multinomial)
Perpendicular distance distribution, conditional on detection
A: Hayes and Buckland (1983) are to “blame”Q: Why is this ignored??
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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.
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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)=12. 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”)
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Proportion of population that is missed
N
×F(t
)1. p(0)<1
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1. Time-to-detection enables you to estimate p(0)
Observer
Proportion of population at x=0 that is missed: = 1 – p(0)
f(t|x=0): pdf of detection times for animals at peprendicular distance zero
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1. Time-to-detection enables you to estimate p(0)
Observer
(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.
LT p(0) estimation from time-to-detection data is poor when p(0) is not “close” to 1.
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2. Time-to-detection enables you to estimate π(x)
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2. Time-to-detection enables you to estimate π(x)
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2. Time-to-detection enables you to estimate π(x)Forward distance
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2. Forward distance enables you to estimate π(x)
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3. Stochastic availability
Timet T0
Detection hazard at x,
given availability: h(t |x)Detection hazard at x,t
given availability: h(x,t)
2-State Markov-modulated Poisson Process (MMPP) in whichState 1 = shallow diving: (high Poisson event
rate)State 0 = deep diving: (low Poisson event
rate)
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3. Stochastic availability: Bowhead aerial survey
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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)=12. 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”)
(if p(0) not too small)
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Line Transect MethodDistance Sampling Methods
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Capture-recapture with camera traps
Trap k
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Spatially Explicit Capture-Recapture (SECR)
Time1 2 R. . . . . . . . . t11
t21 t22
1
2
3
x
Poisson Location of animal i’s activity center
Number of times animali is detected on camera k on occasion r
Detection functionparameters
Density model parameters
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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 animali is detected on camera k
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Continuous-time SECRDiscrete-time Model
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Continuous-time SECRDiscrete-time Model
Time-to-detection is NOT informative about density IF
(a) so that
(b) (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.
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Continuous-time SECRDiscrete-time Model
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Continuous-time SECR
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.
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Time-to-detection in Occupancy Models
F(t)
F(t)
(t) (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).
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Time-to-detection in Occupancy Models:incorporating availability
2-State Markov-modulated Poisson Process (MMPP) : 3 different animals
Constant hazard of detection, given pugmark
Guillera-Arroita et al. (2012): tiger pugmarks along a transect
Does this look familiar?
l L
0
Distance
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Recall: Line Transect with Stochastic availability
Timet T0
Detection hazard at x,
given availability: h(t |x)Detection probability at x,t
given availability: h(x,t)
2-State Markov-modulated Poisson Process (MMPP) in whichState 1 = shallow diving: (high Poisson rate)State 0 = deep diving: (low Poisson rate)
Line Transect with stochastic availability and multiple detections
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Time-to-detection in Occupancy Models:incorporating availability
Constant hazard of detection, given pugmark
Gurutzeta et al. (2012): tiger pugmarks along a transect
Same as Line Transect with stochastic availability, except:1. Distance, not time and pugmarks, not whale blows2. Constant detection hazard3. Can’t distinguish between individuals (this adds lots of complication!)4. Estimating presence, not abundance (this makes things little simpler)
l L
0
Distance
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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]