CREEM. Centre for Research into Ecological and Environmental Modelling. University of St Andrews INTERNATIONAL WORKSHOPS

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1 CREEM Centre for Research into Ecological and Environmental Modelling University of St Andrews INTERNATIONAL WORKSHOPS Advanced Techniques and Recent Developments in Distance Sampling St Andrews 25 th 27 th August 2008 CREEM 2008

2 Advanced Techniques and Recent Developments in Distance Sampling St Andrews 2008 INDEX Lectures Page 1 Distance sampling overview 1 2 Advanced overview of Distance 15 3 Multiple covariate distance sampling (MCDS) 27 4 Estimation with incomplete detection at distance zero g(0) 41 4a Mark-recapture distance sampling (MRDS) in Distance Automated survey design 61 6 Stratification 87 7 Adaptive distance sampling 97 8 Temporal inferences 105 Exercises 0 Introduction to Distance Multiple covariate distance sampling Double platform analysis Automated survey design Complex stratification 137 Solutions 1 Multiple covariate distance sampling Double platform analysis Automated survey design Complex stratification 147 Formulas and notation 149

3 Schedule Advanced Techniques and Recent Developments in Distance Sampling 25 th 27 th August 2008 Monday August 25 th 08:30 Registration (in Observatory) 09:00 Welcome Introduction to new developments, plus quick recap of main ideas of distance sampling Introduction to Distance 5 s advanced features 10:45 Coffee/tea 11:00 Multiple covariate distance sampling (MCDS) Computer session: MCDS 12:45 Lunch 14:15 Computer session: MCDS (continued) Double-platform (mark-recapture distance sampling MRDS) 16:00 Coffee/tea 16:15 Computer session: MRDS 18:00 Adjourn Tuesday August 26 th 09:00 Automated survey design 10:45 Coffee/tea 11:00 Computer session: Automated survey design 12:45 Lunch 14:15 Advanced stratification Adaptive sampling Participants data sets or practical exercises 16:00 Coffee/tea 16:15 Temporal inferences Participants' data sets or practical exercises 18:00 Adjourn

4 Wednesday August 27 th 09:00 Current research directions 10:45 Coffee/tea 11:00 Special topic discussion groups / Computer session / Further material by request 12:45 Lunch 14:15 Special topic discussion groups / Computer session / Further material by request 16:00 Coffee/tea 16:30 Summary and discussion: future research directions 17:00 Workshop ends

5 Distance Sampling Overview 1. Some Common Issues A. Design- & Model-based inference: B. Using Models: 2. Overview of Advances Reasons animals are missed: Because they were not seen although they were in covered strips: detection function g(0)=1 g(x) perpendicular distance x w Because they were not in covered strips: design Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 5

6 A. Design & Model Strip Transects Covered 25% (shaded strips) Every animal equally likely to be covered Hence, for every animal: Pr(include animal) = 0.25, and N ˆ = n = n = Pr( include) 0.25 = Pa Line Transects Covered 25% Detect only μ =0.5 W of those in strip Effectivelycover 0.25x0.5 = of region Hence for every animal: Nˆ = n = n = 43 = 344 Pr( include) a μ A w Pr(include animal) = 0.25 x 0.5 = 0.125, and Page 6 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

7 Horvitz-Thompson For individual animals Nˆ 1 = animals Pr( animal included) If Pr(animal included) equal for all animals: N ˆ = Pr( animal n included) For animals in groups group size Nˆ = groups Pr( group included) Nˆ = OR, we often use groups 1 Eˆ[ s] Pr( group included) Formula reminders μˆ W N ˆ = n = Pr( include) n 2Lμˆ A W na na = = fˆ(0) Eˆ [ s ] 2Lμˆ 2L Clustered populations ρˆ ˆ n n N = = Pr( include) 2 kπ ˆρ A na = na 2 = h ˆ(0 ) kπ ˆρ 2kπ Eˆ [ s ] Clustered populations Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 7

8 Design & Model with Line Transects P a = 0.25 is design-based μ =0.5 estimate is model-based W (based on detection function model) Inclusion probability estimate 0.25x0.5 = has designbased and model-based parts Conventional distance sampling variance: Variance of n/l usually design-based Variance of estimated μ usually model-based Variance of estimated E[s] usually model-based B. Models a. Detection function b. Mean cluster size c. Encounter rate & Density Page 8 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

9 B.i. Detection Function Hazard rate model has 2 parameters = 2 df Hazard rate model + 1 adjustment term has 3 parameters = 3 df g(x) W Model Selection Issues Is model plausible? Does it fit well? Look at residual plots, GOF tests Does it have a good trade-off between GOF and model df (number of model parameters)? AIC, GCV, F-ratio,... Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 9

10 Multiple Covariate Detection Functions Extra dimensions: distance and cluster size, and visibility, and... New modelling decision: which variables to include? B.ii. Mean Cluster Size, E[s] Linear model (flat line: 1 parameter) (sloped line: 2 parameters) ln(s) g(x) Page 10 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

11 Model Selection Issues Is model plausible? Does it fit well? Look at residual plots, GOF tests Does it have a good trade-off between GOF and model df (number of model parameters)? AIC, GCV, F-ratio,... B.iii. Encounter rate and Density Density Stratum 1 Stratum 2 Stratum 3 longitude Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 11

12 B.iii. Encounter rate and Density (ctd.) 3 Stratum Factor levels 3 model df Density Stratum 1 Stratum 2 Stratum 3 longitude B.iii. Encounter rate and Density (ctd.) GLM with log link & 1 parameter 1 model df Density Stratum 1 Stratum 2 Stratum 3 longitude Page 12 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

13 B.iii. Encounter rate and Density (ctd.) Generalized Additive Model 4 model df Density Stratum 1 Stratum 2 Stratum 3 longitude Model Selection Issues Is model plausible? Does it fit well? Look at residual plots, χ 2 GOF statistic Does it have a good trade-off between GOF and model df (number of model parameters)? AIC, GCV, F-ratio,... Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 13

14 ln(s) Linear model (2 parameters) Some Model Types g(x) Parametric Nonlinear model (hazard-rate: 2 parameters) Density g(x) Piecewise linear model (3 parameters) Generalized linear model (2 parameters) Density w Stratum 1 Stratum 2 Stratum 3 longitude Density Generalized additive model (4 parameters) Stratum 1 Stratum 2 Stratum 3 longitude Stratum 1 Stratum 2 Stratum 3 longitude Stratification of Density using a model Example with factor 1 (3df): D ~ stratum Unstratified continuous variable 1 (1df): D ~ longitude Density Density Stratum 1 Stratum 2 Stratum 3 longitude Stratum 1 Stratum 2 Stratum 3 longitude 1 Data has variables stratum (with values 1,2,3) and longitude Page 14 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

15 In 3-D: Example with factor 1 (3df): D ~ stratum Density Latitude Longitude Unstratified continuous variable 1 (1df): D ~ longitude Density Latitude Longitude 1 Data has variables stratum (with values 1,2,3) and longitude 2. Overview of Advances A. Related to Design: i. Automated survey design ii. Adaptive Sampling B. Related to Detection Function: i. Covariates in detection function ii. g(0)<1 Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 15

16 A.i. Automated survey design Nˆ = i 1 Pr( animal i included) Now Pr(animal i included) depends on where animal i is: coverage probability changes with location. Can deal with difficult design problems without introducing bias A.ii. Adaptive sampling Nˆ = i 1 Pr( animal i included) Now Pr(animal i included) depends on where animal i is relative to other detected animals: coverage probability changes with density. Can increase efficiency Page 16 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

17 B.i. Covariates in Detection Function Nˆ = i 1 Pr( animal i included) Now Pr(animal i included) depends on covariates of animal i. Can improve precision Alternative to stratification Can examine effects of covariates B.ii. g(0)<1 Nˆ = i 1 Pr( animal i included) Again Pr(animal i included) depends on covariates of animal i and g(0) can be <1. Remove/reduce bias due to g(0)<1 Can examine effects of covariates Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 17

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19 Overview of Distance Advanced Workshop Introduction Distance projects Data Analysis Survey design Introduction History of Distance Distance 4.x Distance 5.0 Pre-history TRANSECT DISTANCE 1-3 Distance 3.5 Distance AD Time 2008 AD Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 19

20 Introduction Distance Manuals Users Guide pdf version Distance help see also Advanced Distance Sampling book Introduction Support distance-sampling list To join, send the following message to join distance-sampling Your Name stop Searchable archived messages are at or in Distance, choose Help Distance on the web distance-sampling list archive Distance home page or in Distance, choose Help Distance on the web Distance Home Page Page 20 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

21 Distance Projects Project File MyProject.dst Designs Surveys Analyses Data Filters Model Definitions Data Folder MyProject.dat Data File DistData.mdb Shapefiles.shp.shx.dbf.prj R folder R Distance Projects To create a New Project: New Project Setup Wizard (File New Project) To open a Project: File Open Project To check Project Settings: Project Properties (File Project properties) Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 21

22 Distance Projects Project File + Project Folder Zip archive file To export a project to zip archive: File Export Project Save as type zip archive files *.zip To unpack a project from zip archive, and open: File Open Files of type zip archive files *.zip Survey data in Distance Data Layers Main layer types: Global Stratum SubStratum 1-5 Sample SubSample 1-5 Observation Coverage Page 22 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

23 Survey data in Distance Data Fields Field name Field type (Integer, Decimal, Text, ID, Label, LinkID, ParentID) Units Data source (Internal, External, Geographic) Geographic data in Distance Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 23

24 Geographic data in Distance Getting data into Distance Data Entry Wizard Data Explorer Data Import Wizard Import from Distance 3.5, 4.x Linking (editing DistData.mdb) Page 24 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

25 Getting geographic data into Distance Manually Shape Properties window Copy and paste vertex coordinates into Shape Properties window Via a GIS Rename shapefiles and put in project s data folder Link to shapefiles by editing DistData.mdb Analysis in Distance Analysis browser (Analyses tab of Project Browser) Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 25

26 Analysis in Distance Survey Analysis details Data filter Model definition Analysis in Distance Survey Survey methods Data layers Data fields Page 26 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

27 Analysis in Distance Data Filter Some advanced features: Powerful data selection Units conversion Analysis in Distance Model Definition Some advanced features: Multiple covariates analysis engine User-defined sample layer Mark-recapture distance sampling analysis engine Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 27

28 Automated Survey Design Design browser (Analyses tab of Project Browser) Automated Survey Design Design Details Design Properties Page 28 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

29 Automated Survey Design Automated Survey Design Design results Survey results Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 29

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31 Multiple covariate distance sampling (MCDS) Aim of MCDS Model the effect of additional covariates on detection probability, in addition to distance, while assuming probability of detection at zero distance is 1 i.e. model f(0) as a function of covariates Based on PhD thesis work by Fernanda Marques Chapter 3 of Advanced book (Covariate models for the detection function, by Marques and Buckland) Marques, T. A., Thomas, L., Fancy, S. G., and Buckland, S. T Improving estimates of bird density using multiple covariate distance sampling. The Auk 124: Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 31

32 Contents Why additional covariates? Multiple covariate models Estimating abundance MCDS in Distance Complications Clustered populations Adjustment terms Stratification MCDS analysis guidelines 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 x Sources of heterogeneity: Object : species, sex, cluster size Effort: observer, habitat, weather Page 32 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

33 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 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: Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 33

34 Examples of heterogeneity 3 Effect of cluster size on beer cans. Otto and Pollock, 1990, Biometrics 46: Why worry about heterogeneity? In CDS, we use models that are pooling robust, so why worry about heterogeneity? Pooling robustness only works for moderate 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) Page 34 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

35 Stratification Dealing with heterogeneity 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 shape between covariate levels ~ 680 ~ 320 ~ 140 Multiple covariate models Recap of CDS models g(x) = Pr[animal at distance x is detected] Key function m = k( x) 1 + a j p j ( xs ) / c j= 1 j th series adjustment term Scaling constant to ensure g(0) = 1 Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 35

36 CDS models continued Key functions Hazard rate Half-normal Neg. exp. Uniform x k( x) = 1 exp σ 2 x k( x) = exp 2 2σ x k( x) = exp λ k( x) = 1 Shape parameter b Scale parameter Series adjustments Cosine cos(jπx s ) Polynomial x j s Hermite poly. H j (x s ) x s are scaled distances (see later) 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 scale of the key function, not its shape. So choose key functions with a scale parameter J Let σ( z ) = exp β + β z 0 j j j= 1 e.g. Hazard rate Half normal x k( x, z) = 1 exp σ( z) k( x, z) 2 x exp 2σ( z) = 2 b Page 36 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

37 Modelling with covariates Example: Dolphin tuna vessel data Model: half-normal, with no adjustments Covariate: cluster size, s 2 x g( x, s) = exp 2 2 ( s) σ σ( s) = exp( β0 + β1s) = exp( β0).exp( β1s) = A1.exp( A2s) From distance output Â1 = Â2 = ~ 680 ~ 320 ~ 140 Recap of estimating abundance without covariates n n 1 Nˆ = = i Pr i= 1 [ animal included] 2Lμˆ 2 μ = 1 L ˆ Recall that f(x) = pdf of observed x s Because g(0)=1 by assumption, then f(0) = 1/μ 1 A = na g( x) g( x) = = g( x) dx μ So N ˆ = na 2L fˆ(0) Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 37

38 Estimating abundance with covariates Nˆ = Now n 1 = [ ] = animal included 2Lμˆ ( z ) 2 i= 1 Pr i= 1 i i= 1 μ zi ) g( x, z) g( x, z) f ( x z) = = g( x, z) dx μ( z) Because g(0,z)=1 by assumption, then f ( 0 z) = 1 μ( z) n 1 A A L n 1 ˆ ( So Nˆ = A 2L n i= 1 fˆ(0 z ) i Note similarity to CDS estimator In Model Definition, choose MCDS analysis engine MCDS in Distance See Chapter 9 of online Users Guide Covariate type: Factor covariates classify the data into distinct classes or levels. Can be numeric or text. One parameter per factor level. Non-factor (ie. continuous) covariates must be numeric. One parameter per covariate + 1 for the intercept. Page 38 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

39 Complications 1. Clustered populations From introductory talk: There are two approaches to estimating number of individuals when objects are in clusters: n 1 (1) Nˆ = Eˆ[ s] (2) Pr i= 1 A = Eˆ( s) 2L [ group included] n i= 1 fˆ(0 z ) i Nˆ n group size = i= 1 Pr n A = si fˆ(0 zi ) 2L i= 1 [ group included] When cluster size is not a covariate, we use (1); when it is a covariate, we use (2) Clustered populations (contd.) To tell Distance that a covariate represents cluster size, tick the box: When cluster size is a covariate: Distance does not estimate variance using analytic methods: the bootstrap must be used (Reflected in the Variance tab) There is no need for size bias regression methods (Cluster size tab changes) No stratification allowed (Estimate tab) Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 39

40 With adjustments: Complications 2. Adjustment terms g ( x, z ) = k ( x, z ) 1 + a m j =1 j p j ( x s ) c Adjustment terms use scaled distances, x s cosine adjustment of order 2: cos(2πx s ) simple polynomial of order 4: x 4 s Why scale? Avoid numerical problems Limits cosine adjustment to a small number of wiggles How to scale? Adjustment terms (contd.) Scenario 1: Scale distances by w, the right truncation distance x s = x/w Then covariates affect the scale of the key function, but adjustment terms are unaffected by covariates, so the overall shape varies with covariate value: e.g. half-normal with 1 cosine adjustment of order 2 Note: no monotonicity constraint 2 x g x z) exp 2 2σ( z) 1 + a ( 2 2πx cos w Page 40 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

41 Adjustment terms (contd.) Scenario 2: Scale distances by σ(z), the estimated scale parameter x s = x/σ(z) Then covariates affect the scale of the key function, and the scale of the adjustment terms, so only the scale and not the shape of the overall function is affected: e.g. half-normal with 1 cosine adjustment of order 2 2 x g x z) exp 2σ( z) 1 + a ( 2 2 2πx cos σ( z) Adjustment terms (contd.) The previous was an extreme example, to illustrate the difference between scaling factors. Generally: start with no adjustment terms and introduce covariate terms one by one check the fit with adjustments looks reasonable consider whether to scale by w or σ you may need fewer adjustment terms with MCDS than CDS analyses Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 41

42 Complications 3. Stratification If we want stratum-level estimates of density/abundance we can fit the detection function with covariates globally, and estimate f(0 z) by stratum: Tick both boxes If estimating density by sample, could estimate f(0 z) by stratum Global variance estimate for density/abundance must be calculated via the bootstrap MCDS analysis guidelines Choose covariates that are: independent of distance not strongly correlated with each other Specifying the model: factor covariates generally harder to fit avoid or limit automatic selection of adjustment terms if using adjustments, consider whether to scale by w or σ check convergence and monotonicity add only one covariate at a time where necessary, use starting values and bounds for parameters consider reducing the truncation distance, w, if more than 5% of the P a (z i ) are <0.2, or if any are less than 0.1 Page 42 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

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45 Estimation with incomplete detection at distance zero g(0)<1 Chapter 6 of Advanced book (Methods for incomplete detection at distance zero by Laake and Borchers) Borchers, D., Laake, J., Southwell, C., and Paxton, C Accommodating unmodeled heterogeneity in doubleobserver distance sampling surveys. Biometrics 62: Conventional Distance sampling estimates are biased if g(0)<1: D* = D g(0) where D is the true density and D* is the density obtained if you assume g(0)=1. g(0)<1 when there is Availability Bias Perception Bias at distance 0 Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 45

46 Availability Bias : When animals are unavailable for detection. Perception Bias : When observers fail to detect animals at distance 0 although they are available Animals available for Missed detection Seen Animals UNavailable for detection Availability Bias : When animals are unavailable for detection. Perception Bias : When observers fail to detect animals on the transect although they are available Availability Bias Perception Bias Page 46 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

47 Visual Mark-Recapture Seen by 2 = marked Obs 2 = trapping occasion Obs 1 = trapping occasion Visual Mark-Recapture Seen by 2 = marked Seen by 2 = marked Seen by 1 = success Obs 2 = trapping occasion Obs 1 = trapping occasion Passes unseen by 1 = failure Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 47

48 Visual Mark-Recapture Seen by 2 = marked Seen by 2 = marked Passes unseen by 1 = failure We know 2 animals passed (because Obs 2 saw them) Seen by 1 = success Of these, Obs 1 saw 1 So estimate: Pr(Obs 1 sees) = p = 1 = n ˆ1 2 n 12 2 = number duplicates number seen by 2 Note: In this section, we use p, not g for the detection function If the 2 animals were at distance 0: Availability Bias p (0) ˆ1 = 1 2 Perception Bias Page 48 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

49 Sources of Heterogeneity The animals themselves (distance, size, availability,...) Group size The environment (sea state, ground cover,...) The kind of survey effort (the observers, their platforms,...) Observer Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 49

50 Observer 1 detections Problem? Unmodelled Heterogeneity here Proportion of Observer 2 detections seen by Observer 1 Full Independence (FI) Model Proportion of Observer 2 detections seen by Observer p 1 (0) 1 Detection function Page 50 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

51 Point Independence (PI) Model Independence at 0 only Observer 1 detections Detection function p 1 (0) Proportion of Observer 2 detections seen by Observer 1 Conditional detection function (given detection by Observer 2) Yet more recent advances Buckland, S.T., Laake, J.L., and Borchers, D.L. submitted. Double observer line transect methods: levels of independence. (see poster in foyer) Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 51

52 Conditional Detection Function estimated by binary regression can include any observable explanatory variables in the detection function; can easily be done with standard statistical packages; provides model selection tools Example: Should p be stratified? Fit stratified detection function p(x,size,stratum), giving AIC strat Fit pooled detection function p(x,size), giving AIC pooled Don t stratifiy if AIC pooled < AIC strat Page 52 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

53 Binary Regression Types Cue-based methods: Binary regression on cues (not animals). Getting adequate estimates of cue generation process can be difficult. Able to incorporate heterogeneity due to availability process. Animal-based methods: We focus on these; in some applications cue-based methods perform better Binary Regression on animals Don t need to estimate availability/cue-ing process. Less able to incorporate heterogeneity due to availability process. Configuration: Trial-Observer Observer 2 sets up trials for Observer 1 to estimate p 1 The Observer at the end of an arrow must be independent of the Observer at the start of the arrow Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 53

54 Configuration: Independent Observer to estimate p 2 Observer 2 sets up trials for p. = p 1 + p 2 - ( p 1 p 2 ) Observer 1 to estimate p 1 The Observer at the end of an arrow must be independent of the Observer at the start of the arrow Abundance Estimation Trial-Observer Nˆ = all i (note) ˆ 1 seen by 1 p ( x, ) 1 i Independent Observer Nˆ = (note) all i seen 1 pˆ ( x, ) i Page 54 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

55 Design to deal with availability bias Use enough effort for certain detection at x=0 May not be possible Use cue-based methods need to estimate availability process Use different capture methods Separate search areas of the observers Use different types of observers (e.g. visual and acoustic; visual and radio-tag) Example: Visual & Acoustic Observers Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 55

56 Duplicate Identification Duplicate Identification Field methods Use a dedicated duplicate identifier Record measure of confidence in duplicate identification. Record positions and times as precisely as possible Record ancillary data Have at least one observer track animals Page 56 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

57 Duplicate Identification Analysis methods Bracket "best" estimate by two extremes Rule-based duplicate identification after the survey. (e.g. Schweder et al., 1996) Probabilitistic duplicate identification after the survey. (e.g. Hiby & Lovell, 1998) Schweder, T., Hagen, G., Helgeland, J. and Koppervik, I Abundance estimation of northeastern Atlantic minke whales. Rep. Int. Whal. Commn. 46: Hiby, A. and Lovell, P Using aircraft in tandem formation to estimate abundance of harbour porpoise. Biometrics 54: Critical Assumptions of Mark Recapture Line Transect Have the required independence between observers No unmodelled heterogeneity Duplicates (resightings) known (else need to include uncertainty in duplicate status in estimated variance) Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 57

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59 Mark-recapture distance sampling (MRDS) in Distance 5.0 Setting up Distance for MRDS Setting up a Distance project for MRDS Data requirements MRDS analyses Setting up Distance You need a copy of R installed on your computer Currently supported version is R Check: Distance automatically installs mrds R library when you run an MRDS analysis Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 59

60 Project setup Choose Double observer in New project Setup Wizard Choose appropriate observer configuration Project setup This causes 3 extra fields to be added to the Observation layer And their roles defined in the default Survey object Page 60 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

61 Data requirements Observation data must have: 2 rows per object one for Observer 1 and one for Observer 2 Fields for: object ID observer (1 or 2) detected (1=yes, 0=no) Additional covariate data can go in fields at the appropriate level Example: (golf tee project) the 3 new required fields observation-level covariates fields created during data import MRDS analyses Select MRDS engine in Model Definition Estimate tab Stratification options as for CDS/MCDS engines but no post-stratification for now Quantities to estimate Can choose not to estimate density (saves time during model selection) Can choose to estimate a detection function, or to use a fitted function from a previous analysis. Useful to apply a detection function estimated with all data to a subset of the data See manual for details. Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 61

62 Detection function tab 5 methods at present ds CDS and MCDS (but no adjustment terms) IO (independent observer) both point and full independence Trial both point and full independence Choice of method determines which model you need DS model = distance sampling model. half-normal or hazard rate, optionally with covariates in the scale parameter MR model = mark recapture model GLM with logit link Model formulae Type in variable names joined by + (main effect), : (interaction), * (main effect + interaction) Note that some fields get renamed: distance, size, object, observer, detected any field from layers above the observation layer Tip look in Analysis Details log to see what new names are Page 62 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

63 Factors Need to specify which variables in the formulae are factors Tip: type in all possible factors in the first Model Definition, and this will be used as the basis of all subsequent definitions Results Produces diagnostics (qq plots, detection function plots, goodness-of-fit tests) parameter estimates, and estimated density and abundance Can customize plots (in Preferences) Plots stored as graphics files in a folder R within project data folder Results optionally stored in an.rdata file in the R folder, so if you know R software you can access them Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 63

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65 Survey Design Laura Marshall University of St. Andrews Automated Survey Design Aim: Use geographic information system (GIS) within Distance to aid survey design and evaluate properties of different designs Based on PhD thesis work by Samantha Strindberg See Chapter 7 of Advanced book (Design of distance sampling surveys and Geographic Information Systems by Strindberg, Buckland and Thomas) Strindberg, S. and Buckland, S. T Zigzag survey designs in line transect sampling. Journal of Agricultural, Biological, and Environmental Statistics 9: Thomas, L., Williams, R., and Sandilands, D Designing line transect surveys for complex survey regions. Journal of Cetacean Research and Management 9:1-13 Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 65

66 Contents Background and Terminology Point Transect Designs Line Transect Designs Design-based Abundance Estimates Survey Design in Distance ArcGIS Pen and Paper Background Page 66 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

67 Automated survey design using GIS technology Easily generate surveys based on randomised designs Print out maps or download to GPS Evaluate properties of different designs optimise for any situation Terminology Sampler a sample unit Strip (line transect) Circle (point transect) Design an algorithm for laying out samplers Survey a single realisation of a design Sampling strategy design & estimator Coverage probability probability for a given design that a point within the survey region will be sampled Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 67

68 Example: Coverage Probability P Uniform coverage probability, π = 1/3 P Survey Region Uniform coverage probability, π = 1/3 Uneven coverage for any given realisation Which Design? Uniformity of coverage probability Even-ness of coverage within any given realisation Overlap of samplers Cost of travel between samplers Efficiency when density varies within the region Edge effects Page 68 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

69 Point Transect Designs Simple Random versus Systematic Grid Comparison Uniform coverage both have uniform coverage probability Systematic has more even coverage for any given realisation Can have overlap of samplers in simple random design Cost of travel is similar If this is important a cluster sampling design can be used Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 69

70 Density Variation Survey Region Animals Systematic generally more reliable If variation in density is predictable Consider stratification Or unequal coverage probability design If not predictable Adaptive sampling Consider modelling the density surface, (i.e. a model-based estimate) Stratification Example showing complex nested strata: a nested grid Effort allocation set using formulae in section of Introduction to Distance Sampling (For more about this example, see Central Africa Pilot Project at ) Page 70 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

71 Edge Effects A problem if study area is small or narrow relative to w 2w Issues Coverage probability close to the edge Animals detected outside the region boundary Coverage probability is lower within w of the edge Assumption Animal density within w of the survey region boundary is the same as for > w For data collection and analysis options, see 6.7 of Introduction to Distance Sampling Minus Sampling Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 71

72 Minus Sampling Observed Distribution Availability Detection Function distance distance distance Minus Sampling Survey up to distance w outside region boundary Assumption Animal density is similar either side of the survey region boundary Page 72 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

73 Plus Sampling Sample all points within a buffer w around the survey region w 2w Record only animals within the survey region Analysis: 0 s and 1 s Proportions (GIS) Other Point Transect Designs Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 73

74 Line Transect Designs - Full Length Transects Parallel Random Systematic Often used in aerial (and sometimes shipboard) surveys Full Length Line Transects - considerations Coverage for a given realisation is more critical as there tend to be fewer lines than points lines are more expensive Transit (off-effort) time can be considerable Other full-width line transect designs include random line orientation, nonoverlapping random parallel, etc. Page 74 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

75 Line Transects - considerations Edge Effects: Surveyed area decreases with distance from transect Conventional analysis can give valid density estimate. Coverage probability lower at edge See, Introduction to Distance Sampling, 2001 chapter 6.7 Line Transects - considerations Edge Effects: Extrapolate lines beyond boundary recording only animals within survey region See, Introduction to Distance Sampling, 2001 chapter 6.7 Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 75

76 Segmented Line Transect Designs -Fixed Length Transects Systematic segmented trackline Systematic segmented grid Edge Effects Not a problem if your willing to survey incomplete segments Otherwise you could reflect back incomplete segments already more than ½ inside This gives even coverage probability but is hard to automate reliably Page 76 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

77 Edge Effects (contd.) Or, could push segments back in if they are already more than ½ inside Systematic segmented trackline Systematic segmented grid Edge Effects (contd.) but this leads to uneven coverage probability near the edge Systematic segmented trackline Systematic segmented grid N.B. Both use random orientation of transects in the northern stratum Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 77

78 Fixed Length Line Transects - considerations Systematic segmented grid seems superior Consider random orientation of lines, (in Distance, type -1 under angle in Effort Allocation tab) Random orientation of each segment may be even better, (not yet in Distance) Other designs (such as circuit samplers) are worth considering, (again not yet in Distance) Zig-zag Line Transect Designs Used commonly in shipboard surveys Advantage (over systematic parallel) Improved efficiency Disadvantages Design is difficult in complex regions Coverage probability may be uneven Page 78 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

79 Design Difficulties non-convex survey region Convex hull Survey Region Minimum bounding rectangle Dealing with Complex Survey Regions Example: Antarctic shipboard survey Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 79

80 Dealing with Complex Survey Regions Example: Antarctic shipboard survey, (contd.) Study region divided into suitable strata to increase efficiency Efficiency Example: SCANS II ship survey Cross survey region twice Page 80 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

81 Effort Allocation Example: SCANS II aerial survey Distance outputs total track length for survey Considerations: Total effort available Required transit effort Rest periods Spare survey Stratification Example: SCANS II aerial survey Stratification based on prior knowledge of animal density Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 81

82 Coverage probability for zig-zag designs Equal angle zig-zag Equal spaced zig-zag Coverage probability for zig-zag designs (contd.) Adjusted angle zig-zag Even coverage probability parallel to the design axis In practice, approximate curved path with a series of straight lines Page 82 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

83 Comparison of Coverage Probability for Zig-zag Designs coverage probability Design-based estimates of abundance Recap- Horvitz-Thompson Nˆ = n 1 Pr[included i= 1 ] = n i= 1 1 Pr[in covered area] Pr[seen in covered area] When coverage probability is even, and we estimate a global detection function: Nˆ = n 1 2Lw μ A w i= 1 ˆ na = 2Lμˆ Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 83

84 Horvitz-Thompson Variable (but known) coverage probability Let π i = Pr[in covered area] Nˆ = n i= 1 1 Pr[in covered area] Pr[seen in covered area] = n 1 μ w i= 1 ˆ π i w = ˆ μ If you estimate probability of detection by animal (e.g., MCDS), then n 1 n Not yet in Distance Nˆ = 1 = w i= 1 ˆ μi π i= 1 π i ˆ μi i Variance will be via w the bootstrap n i= 1 1 π i Automated survey design in Distance Import data from GIS (or type it in!) Create coverage grid Create design Generate example surveys from design (run 2 nd option) Assess even-ness of coverage probability via simulation (run 1 st option) Finally can export GIS data, map or sampler coordinates Page 84 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

85 Main Points Line transects are preferable Parallel designs give uniform coverage Systematic designs give more even coverage for any survey Zig-zag designs usually more practical Lines should be placed parallel to density gradient Otherwise should be placed to maximise number of samplers Arc ToolBox Ensure Geographic Coordinate System is defined Project data on to a flat surface Eg. Albers Equal Area Conical Projection Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 85

86 RIGHT CLICK - Attribute Table Page 86 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

87 Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 87

88 Next Steps Create a new distance project Add correct number of strata Close project Replace the files created in the Distance project.dat folder with those manipulated in ArcView Things to Remember! Arc ToolBox to define coordinate system and project shapefile Cut Polygon Features in ArcView can be used to divide area into strata Define LinkID in attributes Distance help Importing Existing GIS Data Page 88 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

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91 Dealing with complex stratification Stratification components nf E s Dˆ ˆ(0) ˆ( ) = 2L Estimating different components of N or D separately for different parts of the survey or components of the population Why? Reduce bias when estimating N or D for these components of the population Increase precision We need a larger sample size for estimating f(0) than for n or E(s), so we typically estimate f(0) at a higher level Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 91

92 Types of stratification I Area/Geographic Dˆ = A A v Dv Effort (e.g., survey, observer, etc) L Dˆ = v Dˆ v L Components of the population (species, sex, etc) Dˆ = Dv ˆ ˆ Nested Types of stratification II Crossed Page 92 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

93 Nested stratification Example: Global Observer f ˆ (0) ˆ (0) o 1 f o2 Geographic region n s r1 r1 n s r2 r2 n s r3 r3 Crossed stratification Example: Global Observer f ˆ (0) ˆ (0) o 1 f o2 Geographic region n s r1 r1 n s r2 r2 n s r3 r3 both observers survey the same set of transects in geographic region 2 Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 93

94 What can Distance do? Distance will give estimates of N and/or D, plus analytic and bootstrap variances if there is one level of nested stratification In other situations, multiple runs of Distance are required, and some hand calculations For crossed stratification, you ll need to program a bootstrap in other software (see later) Estimating N and/or D Start at the lowest level and write down the standard distance sampling formula for N or D (depending on your target) Obtain expressions for N or D at the higher levels for which you want estimates, in terms of weighted sums or averages of the lower level estimates Factor out common terms and simplify where possible Page 94 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

95 Start at the lowest level: nr f (0) ˆ 1 o s n 1 r1 D r = D ˆ = 1 2L r1 Nested stratification example f r2 o2 r2 2 (0) s L r2 r2 n D ˆ = f r3 o2 r3 2 (0) s L r3 r3 Combine: ˆ A1 ˆ A2 ˆ A3 D = Dr + D 1 r + 2 A A A Dˆ Factor out common terms and simplify A Dˆ 1 = Dˆ r + fo (0) M 1 2 A A where 2 A n 3 r s 2 M = M 2 + M 3 and M 2 = A A 2L r 3 r2 r2 n s M 3 = 2L r3 r3 r3 Estimating variance of N and/or D for Nested Stratification Start with your formula for N or D and use the following rules: variance of a weighted sum of independent random variables: if 2 then var( Y ) = c i var( Y i ) variance of a product of independent random variables (delta method): if Y Y then so Y = c Y i i 2 { CV Y) } { CV( Y )} = i var( Y ) 2 [ ] ( i 2 var( Yi ) Y 2 Yi Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 95

96 Nested stratification example A Dˆ 1 = Dˆ r + fo (0) M 1 2 A 2 so A Dˆ 1 var( ) = var( Dˆ r ) + var fo (0) M 1 2 A ( ) ( f (0)) 2 A 2 var = ˆ o + ( ) var( M 2 var( Dr ) f (0) o M 2 2 A f (0) M o2 1 ) Note var( D ˆ r ) and var( f can be obtained from Distance. 1 o2 (0)) What about var(m)? A2 A3 Recall that M = M 2 + M 3 A A so 2 A2 A3 var( M ) = var( M 2) + var( M 3) A A 2 Nested stratification example (contd.) But now we need var(m 2 ) and var(m 3 ). Recall so nr s 2 r2 M 2 = 2Lr 2 var( n 2 r ) var( s var( M = 2) M nr s 2 r2 2 r ) 2 similarly var( M 3 var( n 2 r ) var( s ) = M nr s 3 r3 3 r ) 3 Page 96 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

97 Nested stratification example (contd.) Suggested setup for getting required quantities from Distance: Use geographic region as stratum, but have observer as a field in the stratum layer Do an analysis with density, encounter rate, detection function, and cluster size by stratum to obtain Dr, n and their variances 1 r, s 2 r, n 2 r, s 3 r3 Do an analysis post-stratified by observer to estimate f o 2 (0) and var( f o (0)) 2 Other examples of nested stratification Simple example in Intro. to Distance Sampling, section Complex example in section 8.5 Here stratification was performed by geographic block, also by group size of fin whales as well as Beaufort sea state As an exercise, try the crossed stratification example, but treating encounter rate and cluster size in region 2 separately for each observer (this is then no longer crossed, but is nested) Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 97

98 Crossed stratification The only way I know to estimate variance and confidence limits is using a bootstrap Using a separate programming language, program a resampling routine that resamples according to the design (for example if the same transects were surveyed in each time period then the bootstrap routine should do the same) For each bootstrap resample, the program would call the Distance analysis engine, MCDS.exe, pass in the data, wait for the engine to finish, and then read and store the results For more information, see the Distance help Running the analysis engine as a stand-alone program, search the distance-sampling list archives (e.g., for stand alone ), look at the technical report Rexstad (2007b) on your thumb drive, or Visit the website of Tiago Marques ( Page 98 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

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101 ADAPTIVE DISTANCE SAMPLING Adaptive point transect sampling Adaptive line transect sampling Advantages/Disadvantages Chapter 8 in Advanced book Adaptive point transect sampling Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 101

102 Neighbourhood Points Initial Sample Point Neighbourhood Points Initial Sample Point Neighbourhood Points Initial Sample Point Simulated example Page 102 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

103 Results for conventional and adaptive simulations on a population of 722 objects Parameter # points sampled, k # detections, n ĥ (0) V ˆ ( h ˆ(0) ) E ˆ( n ) V ˆ( n ) Dˆ Conventional Adaptive V ˆ( D ˆ ) 95% CI of Dˆ , , Adaptive line transect sampling Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 103

104 Fixed-effort adaptive line transect sampling Actual Effort Nominal Effort Page 104 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

105 Experimental harbour porpoise surveys in the Gulf of Maine/Bay of Fundy Effort completed and number of groups sighted by search mode Conventional Adaptive Efficiency = CV(conventional method) CV(adaptive method) Total effort (nm) Nominal effort (nm) # of sightings Nominal # of sightings Parameter f(0) Encounter rate Group size Group density Animal density Efficiency Fixed-effort adaptive line transect sampling Pros and cons relative to conventional line transect sampling + Increases sample size for fitting detection function + Improves efficiency for estimating density + If effort is lost, eg. through bad weather, adaptive effort is reduced, thus allowing complete coverage of survey effort + More animals recorded and hence, more information obtained on study species - Complicates field protocol and analysis - Improvement in efficiency for estimating D is modest - Choice of trigger function especially for multi-species surveys Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 105

106 Fixed-effort adaptive line transect sampling Pros and cons relative to standard adaptive line transect sampling + Trackline can be continuous, which allows ship time to be fully utilized, without off-effort time between legs + Effort is known in advance essential for shipboard surveys + Coverage throughout the study region can be (almost) guaranteed - The method is not design unbiased but bias is minimal - Field protocol is more complex Page 106 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

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109 Temporal inferences from distance sampling surveys Aims: Estimate population change/trend over time. Design effective monitoring programmes. For more information, and references, see Chapter 5 of Advanced Distance Sampling (ADS) book. Contents Analysis Short-term: difference between two time points Long-term: trend analysis Concepts Trend estimation from global abundance estimates Linear models Smooths Other approaches Survey design issues Repeating transects Sample size Planning long-term studies Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 109

110 Analysis Short term: Difference between two time points Two abundance estimates ˆN 1 and ˆN 2. If degrees of freedom (df) for both are large (>30), then Nˆ ˆ 1 N 2 z = vâr Nˆ Nˆ is distributed as Normal (0,1) ( ) If ˆN 1 and ˆN 2 are independent, then vâr N ˆ Nˆ = vâr Nˆ + vâr Nˆ where vâr ( ) ( ) ( ) 1 2 ( N ˆ ) = [ Nˆ CV( Nˆ )] 2 1 If df not large, or the N s share components of estimation (e.g. f(0)), see Introduction to Distance Sampling, Section Analysis Long term: Trend analysis Concepts: 1. Components of variation 2. Sampling covariance 3. Empirical and process models 4. Trend 5. Abundance as a fixed vs. random variable Page 110 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

111 Mallard numbers 1 Concepts 1 Components of variation T=49 years Mallard (millions) Year Total variation var(nˆ ) Population variation or + Process variation var(n ) Sampling variation var( Nˆ N ) If estimates are independent T 1 var( Nˆ t N t ) T t= 1 Can estimate population variation e.g. for Mallard: ˆ v âr( N ) = v âr( Nˆ N ) = v âr( N ) = vâr( Nˆ ) vâr( Nˆ N ) = From North American Waterfowl Breeding Population and Habitat Survey Concepts 2 Sampling covariance ˆ ˆ 1,..., The estimates N N T will not be independent when They share components of estimation, e.g. f(0) So far as possible, want to estimate detection function separately in each year (to avoid confounding change in f(0) with change in N) The same transects are re-used in successive years In this case, estimating sampling variation var( N N ) requires estimates of both sampling variances and covariances Bootstrap by transect (see ADS Ch5) ˆ Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 111

112 Concepts 3 Empirical vs Process models Empirical model a statistical model for the components of population variation (e.g., linear trend + independent errors about the trend line) + can be easy to implement estimates (e.g. trend) can be biologically implausible unreliable to extrapolate/predict Process model a biological of population dynamics + more biologically realistic + better predictions: constrained by biology requires strong assumptions requires long time series (and often additional data) can be difficult to implement Concepts 4 Trend Trend is: smooth and slow movement over the long term long-term change in the mean level Try drawing your own trend line for the mallard data below. How would you summarize the trend (say for a written report)? Mallard (millions) Year Page 112 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

113 Concepts 4 Trend Trend is: smooth and slow movement over the long term long-term change in the mean level But what is long term? years (df=1) years (df=3) Conclusion: trend definition is subjective years (df=5) years (df=10) Mallard (millions) years (df=15) years (df=25) Year Linear trend models Concepts 4 Trend + 1-number summary: easy to present and interpret inadequate for longer time series e.g., recent rapid declines hard to detect Smooth trend models + more realistic for longer time series harder to summarize trend have to select amount of smoothing Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 113

114 Concepts 5 Abundance as a fixed or random quantity Abundance fixed Given a model of trend, the only source of uncertainty in the trend estimate comes from sampling variation e.g., imagine if we could repeat the survey and there was no sampling variation: Mallard (millions) Year Abundance random Abundances are random realizations of a stochastic process Uncertainty comes both from sampling variation and from randomness in the true abundances e.g., repeated surveys, again with no sampling variation: Mallard (millions) Year Concepts 5 Abundance as a fixed or random quantity Which to use? Abundance fixed Inferences restricted to the study area and time period of the survey More precise as it ignores non-trend population variation. Useful as a first-pass analysis, to see if a more complex random-abundance method is justified. Abundance random Essential for making inferences about other times / places (predictions); effect of management actions, etc. Sources of randomness Demographic variation vs environmental variation Can only be separated using a biological process model Page 114 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

115 Trend estimation from global abundance estimates Graphical methods Mallard (millions) Year Essential first step May be all that is needed for short time series Trend estimation from global abundance estimates Linear trend models t = 0 + β1t trend sampling error True abundances assumed random Nˆ β + δ + ε ( ) ( ) t t process error 2 If assume δ t + ε t ~ iid Normal 0,σ 2 then can use simple linear regression to estimate β, β σ 0 1, Can estimate confidence limits on parameters, confidence band for the trend line, and do the usual tests, e.g.: ˆ β1 ~ t(0,1) with T 2 df vâr( ˆ β1) 2 ( ˆ ˆ σ where vâr β1) = T 2 t t t = 1 ( ) Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 115

116 Trend estimation from global abundance estimates Linear trend models trend sampling error True abundances assumed fixed ε t Now are fixed, not random Assume ~ iid Normal 0, var( Nˆ N ) δ t Nˆ t = 0 + β1t β + δ + ε ( ) Trend estimate, βˆ1 N, is the same as when abundances assumed random, but variance excludes process error: var( Nˆ N ) vâr( ˆ β1 N ) = T 2 = ( t t ) t 1 ˆ β1 also ~ Normal(0,1) vâr( ˆ can use Normal rather than t β1) distribution t t process error Trend estimation from global abundance estimates Linear trend models Mallard (millions) Year True abundance assumed random 4 4 β1 = %CI ( , ˆ 4 True abundance assumed fixed 4 4 β1 N = %CI ( , ˆ 3 ) ) 95% pointwise confidence bands true abundance fixed true abundance random Note: for the random-abundance analysis, do the errors look independent? Page 116 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

117 Trend estimation from global abundance estimates Other linear trend models Log-linear regression log ( Nˆ t ) = β 0 + β1t + δt + ε t Corresponds with biological model of exponential growth t E( N t ) = N 0 λ where N 0 = exp( β 0 ) λ = exp( β 1 ) Corresponds with usual assumption in distance sampling that sampling errors are log-normally distributed Generalized linear model (GLM), e.g.: Nˆ ( β 0 + β t) + δt + ε t t = exp 1 log link function; gamma errors Trend estimation from global abundance estimates Smooth trend models smooth trend Nˆ = s t ( t) + δ t + ε t sampling error process error e.g., smoothing spline, local regression, kernel methods smoothness / wiggliness specified by degrees of freedom or log-linear model log ( Nˆ t ) = s( t) + δ t + ε t or Generalized Additive Model (GAM), e.g. Nˆ = exp log link function; gamma errors t ( s( t) ) + δ t + ε t Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 117

118 Trend estimation from global abundance estimates Smooth trend models 95% pointwise df=8.35 Mallard (millions) Year confidence bands true abundance fixed Generated with function gam in free software R automatic selection of degrees of freedom can generate pointwise confidence bands when abundance assumed random the above bands generated via a resampling scheme (see ADS Ch5) true abundance random Trend estimation from global abundance estimates Sampling covariance When samples covary δ t no longer independently distributed the previous methods will underestimate variance of the trend estimate Solutions true abundances assumed random variance components methods (also called random effects or mixed models ) true abundances assumed fixed bootstrap by transect (see ADS Ch5) Page 118 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

119 Other approaches Spatio-temporal trend models analyze data at the level of the transect or individual observation Biological process models Time series methods Spectral analysis Markov methods (ARIMA) Quality control methods Survey Design Repeating transects For best precision If the goal is to estimate mean abundance over several years, select new transects each year If the goal is to estimate change in abundance over time, keep permanent transects Why? var ( N ˆ 1 + Nˆ 2 ) = var ( Nˆ 1 ) + var ( Nˆ 2 ) + 2cov ( Nˆ ˆ 1, N 2 ) var( Nˆ Nˆ ) = var( Nˆ ) + var( Nˆ ) 2cov( Nˆ, Nˆ ) 1 2 Compromise panel designs Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 119

120 Survey Design Sample size Can frame calculations in terms of statistical power of a test for trend E.g., for log-linear regression model (See ADS Ch5 for formulae; R code at Number of years required for power of 0.8 to detect trend λ for given error CV. (2-tailed test, α = 0.05) For specifying error CV, can assume N s are fixed (only sampling error) or random (sampling error + process error) λ λ Number of years required for power of 0.8 Relative size of the population after this many years ( λ t ) CV CV See ADS Section Survey Design Planning long-term studies Our intuition, analysis methods and organizational support structures are mainly suitable for short-term small-scale studies. These do not carry over well to long-term large-scale monitoring programmes. Issues (inter-related): Goals, design Maintaining integrity over time technical advisory board publications periodic peer review Ability to adapt New technologies Range shifts Correcting deficiencies Quality control Minimizing personnel turnover Documentation Page 120 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

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123 Advanced Techniques and Recent Developments in Distance Sampling Computer exercise 0: Introduction to Distance 5 This exercise is intended as an introduction for those who have had little or no experience with Distance 5. It takes you through opening a distance project, looking at data, exploring survey properties and analyses components. If you are already familiar with Distance, then go straight to the Multiple covariate distance sampling exercises. Opening a Distance project Start by opening Distance 5 (under Start Programs Distance). The data that will be used for the MCDS exercises is stored as a zip archive called BeerCan.zip. You will need to extract and open the project. To do this click on File, Open Project and change the Files of type to zip archive files (*.zip). Select the BeerCan.zip archive and click Open. Distance then prompts you for the directory to extract the archive into the directory currently specified is fine, so click OK. Distance then extracts the project and creates the project file BeerCan.dst and the project folder BeerCan.dat, and opens the project. Next time you want to open the project, you can just select the project file BeerCan.dst. Looking at the Data When the project is open, a Project Browser window appears. Click on the Data tab of the Project Browser called the Data Explorer and have a look at the beer can data. Click on Region there is one survey region of 0.8 hectares. Click on Line transect data from 9 observers has been pooled and so there is one transect with a length of 1800 metres. Finally, click on Observations - there are several fields in the observation layer; Perpendicular distance, Cluster size, Observer, Can (number of can) and Side (indicates on which side of the transect line the can was detected). More details of the data are given in the MCDS exercises. About Survey objects Now, click on the Surveys tab to show the Survey Browser. Only one survey has been defined. Highlight it, and click on the Show Details button (3rd along on toolbar after the word Survey). The survey object has two functions: the first is in automated survey design you can generate instances of a survey from a design (we'll cover this in a separate session); the second is in analyses the survey object tells an analysis what type of survey it is (point, line, etc) and which data layers and fields to get the data from. When you clicked Show Details, a Survey Details window popped up. On the Inputs tab of the Survey Details, click Properties..., and the properties window for this survey is displayed. The first tab of the Survey Properties contains information about the survey methods. You can have more than one Survey object in the same project file, so you can keep point and line transects, clustered and unclustered data etc in the same project file. The second tab (Data layers) tells Distance which data layers contain the data for this survey. This is necessary because you can have more than one layer of each type in a Distance 5 project (e.g. multiple observation layers for multiple survey years). In the last tab (Data fields) you tell distance which fields are the Distance, Angle, Cluster size, etc fields. In Distance 5 you can define the roles of each field separately for different surveys. For example, if you are running a simulation exercise on objects where the distance from the object to the line is known, and was also estimated by the observers, you could have two Distance fields: KnownDistance and EstimatedDistance. You could define one Survey object to use the KnownDistance and another Survey object to use the EstimatedDistance. You could run the same models through both surveys and see what difference it made to the final density estimate to use estimated distances. Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 123

124 Analyses in Distance 5 Close the Survey Properties window, and the Survey details, and click on the Analyses tab of the Project Browser to show the Analysis Browser. Highlight the analysis called New Analysis and click on Show details (third button along after Analysis: ) to see the inputs for this new analysis in the Inputs tab of the Analysis Details window. Analyses in Distance 5 are made up of 3 components, the Survey, which we have met already, the Data Filter and the Model Definition. The list of data filters and model definitions (called Analysis Components ) is available on the inputs tab. In addition to the inputs tab, information about data filters and model definitions are also available in its own window: the Analysis Components manager. Access this by clicking the View Analysis Components Manager button, which is 5th button along on the main toolbar at the top of the Distance window. The first button on the Analysis Components Manager shows you the list of the Data Filters, while the second button shows you a list of the Model Definitions. In contrast with the Inputs tab of the Analysis Details windows, the Analysis Components Manager allows you to see more data filters and model definitions at one time, making it easier to keep track of them (you can re-size the window) see whether a component is being used by any analyses (the Used column is Y if it is being used; you can also double-click on the used column to see which analyses use which components) re-arrange the order they display (the up and down buttons) delete components Once you are familiar with the Analysis components, close the window to go back to the Analyses tab and move onto the MCDS exercises. Page 124 CREEM, Univ. of St. Andrews August 2008 Advanced distance sampling workshop

125 Advanced Techniques and Recent Developments in Distance Sampling Computer exercise 1: Multiple covariate distance sampling This exercise consists of three datasets, of increasing difficulty. Everybody should work through the first dataset, and the other datasets can be examined later if you wish, or you may work on them when you complete the first analysis. The first analysis will show you the rudiments of conducting an analysis, while the remaining analyses take you deeper into the heart of understanding multiple covariates. 1 A whale of a dataset Rather than relaxing here in the serenity and tranquility of the Scottish coast, image instead that you are a research biologist collecting distance sampling data during December on gray whales as they migrated through the Aleutian chain near Unimak Pass en route to their wintering grounds off Baja California (some luckier, more senior researcher, got the job of data collection on their wintering grounds). These data will now be the focus of your attention for this exercise examining the potential utility of covariates in explaining variation in animal detectability. Detections were of individuals (not groups), and you chose to record not only distance, but also time of observation (at this latitude at this time of year, the crew was restricted to making observations between 1000 and 1500 during the day). However, because of the low sun angles during much of this time, there was some reason to believe that time of day might play a role in whale detectability. [In what manner might you wish to incorporate this covariate?] Under extreme weather conditions, observer motion sickness can influence the performance of the observers. An additional covariate, "motion sickness tablet effective dosage at time of observation (MSTEDTO)" was recorded each time a whale was detected. The data are available for your inspection in the Distance project adv_practical_1.dst. Notice the extreme precision with which the perpendicular distances were measured (how do you suppose this could happen on a rolling ship in the Bering Sea?). Describe your candidate model set (what models did you construct) and your rationale for the final estimates you provide. You may also comment upon the use of time of observation as a measure of glare from oblique sun angles. If you have been successful in performing the analysis of this dataset (which can now be revealed to have been simulated), you can continue to sharpen your skills in using covariates in your analysis of distance sampling data by exploring two other data sets, that are considerably more elaborate. Advanced distance sampling workshop August 2008 CREEM, Univ. of St. Andrews Page 125