Multiple covariate distance sampling (MCDS)

Transcription

1 Multiple covariate distance sampling (MCDS) Aim: Model the effect of additional covariates on detection probability, in addition to distance, while assuming probability of detection at zero distance is 1 References: Marques (F) and Buckland (2004) Covariate models for the detection function. Chapter 3 in Buckland et al. (eds). Advanced Distance Sampling. Marques (T) et al. (2007) Improving estimates of bird density using multiple covariate distance sampling. The Auk 127: Section 5.3 of Buckland et al. (2015) Distance Sampling: Methods and Applications

2 Contents Why additional covariates? Multiple covariate models Estimating abundance MCDS in Distance 2 nd Lecture: Complications Clusteredpopulations Adjustm entterm s S tratification MCDS analysis guidelines

3 Why additional covariates? In conventional distance sampling (CDS) analysis all factors affecting detectability, except distance, are ignored In reality, many factors may affect detectability g(x) g(x) x Sources of heterogeneity: Object : species, sex, cluster size Effort: observer, habitat, weather x

4 Examples of heterogeneity 1 Effect of time of day on Rufous Fantail birds in Micronesia (point transects). Ramsey et. al Biometrics 43:1-11 g(x) x g(x) x

5 Examples of heterogeneity 2 Effect of sea state (and other covariates) on sea turtles in the Eastern Tropical Pacific (shipboard line transects). Beavers and Ramsey, 1998, J. Wildl. Manage. 63:

6 Examples of heterogeneity 3 Effect of cluster size on beer cans. Otto and Pollock, 1990, Biometrics 46: Gp Size=4 SIGHTING PROBABILITY SIGHTING PROBABILITY Gp Size=1 DISTANCE DISTANCE Gp Size=5 SIGHTING PROBABILITY SIGHTING PROBABILITY Gp Size=2 DISTANCE DISTANCE

7 Why worry about heterogeneity? In CDS, we use models that are pooling robust, so why worry about heterogeneity? Pooling robustness works for all but extreme levels of heterogeneity Potential bias if density is estimated at a lower level than detection function (e.g. density by geographic region, detection function global) Could potentially increase precision of detection function estimate Interest in sources of heterogeneity in their own right (e.g. group size)

8 Dealing with heterogeneity Stratification Requires estimating separate detection function parameters for each stratum, so often not possible due to lack of data Model as covariates in detection function Allows a more parsimonious approach: - can model effect of numerical covariates ~ can share information about detection function shapebetween covariate levels ~ 320 ~ 140

9 Multiple covariate models Recap of CDS models g(x) = Pr[animal at distance x is detected] m k(x) 1 ajpj(xs) /c j 1 Key function jth series adjustment term Scaling constant to ensure g(0) = 1

10 CDS models continued Key functions Shape parameter Series adjustments Hazard rate x b k(x) 1 exp Cosine cos(jπxs) Half-normal x k(x) exp 2 2 Hermite poly. Hj(xs) Neg. exp. x k(x) exp Uniform k(x) 1 2 Scale parameter Polynomial xsj xsare scaled distances (see later)

11 Modelling with covariates ignoring adjustments terms (for now) g(x, z) = Pr[animal at distance x and covariates z is detected] Assume the covariates affect the scaleof the key function, not its shape.so choose key functions with a scale parameter J Let ( z) exp 0 jzj j 1 e.g. Hazard rate x b k( x, z) 1 exp ( z) x2 Half normal k( x, z) exp 2 ( z)2 kis used here to denote the key function

12 Modelling with covariates Example: Dolphin tuna vessel data Model: half-normal, with no adjustments Covariate: cluster size, s x2 g(x,s) exp 2 (s)2 (s) exp 0 1 s exp( 0 ).exp( 1 s) A1.exp(A 2 s) From distance output Â1 = Â2 = ~ 680 ~ 320 ~ 140

13 Estimating abundance without covariates using Horvitz-Thompson estimator n 1 1 na Nˆ anim alincluded i 1 2 L ˆ 2 L ˆ i 1 P r A n Recall that f(x)= pdf of observed x s g(x) g(x) g(x)dx Because g(0)=1 by assumption, then f(0) = 1/μ na ˆ So Nˆ f(0 ) 2L

14 Estimating abundance with covariates n 1 1 A Nˆ anim alincluded i 1 2 L ˆ(zi) 2 L i 1 P r A g(x,z) g(x,z) Now f(x z) g(x,z)dx (z) n n i 1 Because g(0,z)=1 by assumption, then f(0 z) 1 (z) So A Nˆ 2L n fˆ(0 z ) i 1 i Note similarity to CDS estimator 1 ˆ(zi)

15 MCDS in Distance In Model Definition, choose MCDS analysis engine See Chapter 9 of online Users Guide Covariate type: Factor covariates classify the data into distinct classes or levels. Can be numerical or text. One parameter per factor level. Non-factor (i.e., continuous) covariates must be numerical (integer or decimal). One parameter per covariate + 1 for the intercept.