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

Closed-Population Methods

1. Removal methods2. Distance Sampling Methods3. Capture-Recapture Methods4. Occupancy Methods

• 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)

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.

Diagramatically:

Timet

T0

Mortality hazard h

f(t ;h) = S(t) h

Survivor function

Survival Model:

Detection hazard

The Removal Method(an example with mortality)

Continuous time likelihood (with Poisson rather than Binomial/multinomial)

pr(detect)

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

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:

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

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??

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)=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”)

Proportion of population that is missed

N

×F(t

)1. p(0)<1

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

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.

2. Time-to-detection enables you to estimate π(x)

2. Time-to-detection enables you to estimate π(x)

2. Time-to-detection enables you to estimate π(x)Forward distance

2. Forward distance enables you to estimate π(x)

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)

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)=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)

Line Transect MethodDistance Sampling Methods

Capture-recapture with camera traps

Trap k

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

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

Continuous-time SECRDiscrete-time Model

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.

Continuous-time SECRDiscrete-time Model

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.

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).

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

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

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

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: dlb@st-andrews.ac.uk