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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 + + + + + + + + + 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 + + + + + + + + 20 + + 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 + + + + + ++ + + + + ++ + + + ++ +++ + + + + + + ++ ++ +++ + ++ + + + + + + +++ + + + + + + + + + + + + ++ + + + +++ + + + + + + + + + + + + + + + + + ++ ++ + + + + + ++ + + ++ + + + + + + ++ ++ +++ + + + + + + + + + + + ++ +++++ + +++ ++ + + + +++ + ++ +++ + + + + + + + + +++ + + + +++ + + + + + + ++ ++ +++ + + + ++ + + ++ + + + -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]