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.Removal methods 2.Distance Sampling Methods 3.Capture-Recapture Methods 4.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: Time t T 0 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 depends on x … and hazard that changes with time: Hazard depends on x and t

Diagramatic Removal Model, for a given x : Time t x T 0 pr(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 : Time t 0 x T pr(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)=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”)

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)

Forward distance

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

3. Stochastic availability Time t T 0 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 which State 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)=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”)  (if p(0) not too small)  

Line Transect Method Distance Sampling Methods

Capture-recapture with camera traps Trap k

Spatially Explicit Capture-Recapture (SECR) Time 1 2 R t 11 t 21 t x Poisson Location of animal i’s activity center Number of times animal i is detected on camera k on occasion r Detection function parameters 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 h k (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 (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 SECR Discrete-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) (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 Time t T 0 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 which State 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 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 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