1. 5.2 Distance Sampling Estimating abundance from counts when detection is imperfect

2. Basic Scenario Assume we want to estimate density Treatment 1 Treatment 2 2

3. Estimate Density 3

4. Strip transect 4

5. Transects Line Transects –k strip transects, –Each of length L i and width w –Walk down center of strip, Note detected individuals 5

6. Transects Length of all transects L Width = 2*w Density is: 6

7. Strip transect 7

8. Detection decreases with Distance Census: Out to distance w, all objects seen, p = 1 –unrealistic Distance sampling uses data to estimate p ̂ 8

9. Transects Line Transects –k strip transects, –Each of length L i –Walk down center of strip, When see an individual i, note either: 1. Perpendicular distance from line, x i, 2. Radial distance from observer (r i ) and angle (θ i ). xixi riri 9

10. Data SiteTransectLength (m)Distance (m) NorthA10001.2 NorthA10003.4 NorthA10005.0 NorthB8000.5 NorthB8001.7 ………… 10

11. Duck Nests 11

12. Kangaroos 12

13. Detection Function g(x) = Pr(detection | distance x) g(x) = the probability of detecting an object, given that it is at distance x from the transect 0 ≤ g(x) ≤ 1 13

14. Average Detection 14

15. Effective strip width 15

16. Estimate Density 16

17. Basic Assumptions: Transects are located randomly with respect to distribution of animals. Animals do not need to be randomly distributed Necessary for inference Roadside is bad Animals are identified correctly No double counting Sightings are independent. But cluster analysis possible. 17

18. Key Assumptions Objects on line are detected with certainty –g(0) = 1 –Survey protocols –“its importance cannot be over-emphasized” Buckland et al. 2001 18

19. Key Assumptions Objects detected at initial location –Random movement ok –Avoidance movement will bias D low –Attractive movement will bias D high AvoidanceAttraction g(x)g(x) g(x)g(x) 19

20. Key Assumptions Distance measurements are correct –Biased estimates –Heaping Round distances Round angles –Training and equipment 20

21. Distance Function 21

22. Distance Function f(x) ?  ? g(x) Pr(x|d) ?  ? Pr(d|x) 22

23. Distance Function f(x) ?  ? g(x) Pr(x|d) ?  ? Pr(d|x) Bayes formula Pr(x|d) = Pr(d|x) * Pr(x) / Pr(d) 23

24. Distance Function Pr(x|d) = Pr(d|x) * Pr(x) / Pr(d) ? ? 24

25. Distance Function Pr(x|d) = Pr(d|x) * Pr(x) / Pr(d) ? ? 25

26. Distance Function Pr(x|d) = Pr(d|x) * Pr(x) / Pr(d) ? ? Assume random sample U(0,w) =1/w 26

27. Distance Function Pr(x|d) = Pr(d|x) * Pr(x) / Pr(d) f(x) = g(x) * 1/w * w/u f(x) = g(x) /u g(0) = ? 27

28. Distance Function Pr(x|d) = Pr(d|x) * Pr(x) / Pr(d) f(x) = g(x) * 1/w * w/u f(x) = g(x) /u g(0) = ? f(0) = 1/u 28

29. Number of Duck Nests, by 1 foot intervals along a 1600 mi transect (Anderson and Pospahala 1970) 29

30. So, how do we estimate f(0)? Historically, ad-hoc approaches –Anderson and Pospahala Duck Nest Transect Parametric models –Chuck Gates and Program LINETRAN “Semi-parametric models” –TRANSECT –DISTANCE 30

31. % of Duck Nests, by 1 foot intervals along a 1600 mi transect Intercept ~ 14.2% 31

32. Plug into the equation 32

33. Parametric Models for g(x) Gates et al. 1968. Negative exponential –g(x) = exp(-λx) Pollock (1977) Generalized exponential (Weibull) –g(x)=exp(-(x/β) α Negative exponential (α = 1, β = 1/λ) half-normal (α = 2, β = sqrt(2)*σ) uniform (α =α, x < β). 33

34. Criteria for g(x) Robustness –Function fits a variety of reasonable shapes Shape criterion –g ’ (0) = 0 –Shoulder, not a peak Efficient –Precise, i.e., small variance Buckland, Borchers, Laake, Anderson… 34

35. Hazard Rate Good: Uniform Half-normal Hazard Rate Bad: Exponential Quadratic 35

36. Key functions for g(x) Key function –Uniform = 1/w, (0 parameters) –Half-normal (1 parameter) –Hazard rate (2 parameter) 36

37. Field Work g(0)=1 g(0) has a shoulder Good survey methods! –Train observers– Don’t rush –Use multiple observers– Rest –Dedicated note taker– Pilot studies –Use a camera– Don’t guard the center 37

38. Field Work No movement Good survey methods! –Look ahead –use binoculars –use a wait period –use a quiet vehicle –fly at a greater altitude –don’t talk 38

39. Field Work Accurate measurement –Tools Measuring tape, range finder Angle board, compass –Guides Flag distances, sight-guide on aircraft –Pilot study Decoys, dummies, stakes –Train observers Avoid heaping 39

40. Field Work Random placement –Needed to achieve Pr(x) as uniform –Not roadside surveys Non-overlapping –Avoid double counting –Calculate area Useful practice –Regular, parallel transects with random start –Along a known gradient 40

41. Field Work Random start 41

42. Pilot Study! 42

43. Data Review 43

44. Data Review Outlier Truncate Distant observations have less effect on f(0) Rules of thumb Drop 5-10% of distances Drop if g(x)<0.15 Drop when gaps appear 44

45. Data Review Evasive movement Visibility under an aircraft Left truncate 45

46. Data Review Spiked/peaked Heaping to zero Attractive movement 46

47. Outliers Heaping Group data 47

48. Data Collected Ideal –ungrouped data (actual distances to the line or point) –{x 1, x 2,..., x n } If poor shape (heaping) –group data into distance categories grouping can be by design (data are recorded by distance categories in the field) grouping can be a posteriori (distances are recorded in the field and later grouped for analysis) –{n 1, n 2,..., n m } frequencies of occurrences for m categories 48

49. Data Review Guarding the centerline 49

50. Data Review Cluster 50

51. Clustered populations Count clusters and cluster size Distance is to center of cluster –If x(center) > w, do not count any –If x(center) < w, count all Estimate n and f(0) of clusters (D c ), Average cluster size (E(s)). 51

52. Clustered populations Size-biased sampling: detection a function of cluster size. –If small clusters undetected E(s) too big D too high Solutions –Truncate cluster size data where g(x) ~ 0.6 to 0.8 –Regression: s ~ x Use s ̂ at x=0 as E(s) –Stratify by cluster size Reduces sample size –Use cluster size as covariate 52

53. Series Expansion DISTANCE models have 2 parts: g(y) = key(y)[1+series(y)] Key Functions –Uniform –Half-normal –Hazard rate Series expansion (added to improve overall fit) –Cosine series –Simple polynomial –Hermite polynomial 53

54. Series Expansion Uniform with three cosine adjustments 54

55. Series Expansion Uniform with two polynomial adjustments 55

56. Series Expansion Four recommended models Uniform + cosine Half normal + cosine Half normal + Hermite Hazard + polynomial –Thomas et al. 2010 56

57. Goodness of Fit Compare the model to the distance histogram 57

58. Goodness of Fit Review the q-q plot 58

59. Goodness of Fit Chi-square test 59

60. Model Selection Akaike's Information Criterion (AIC): –AIC = -2 Ln L + 2*(number of parameters) –L = value of likelihood at its maximum for the model. –Select model with smallest AIC. 60

61. Estimating Variance Two sources of sampling variance. –n, uncertainty regarding the average number of individuals per transect –f(0), uncertainty regarding the detection function –If cluster size estimated another variance component needed 61

62. Estimating Variance Estimate variance of f(0) from the maximum likelihood information matrix OR Estimate variance in density using a non- parametric bootstrap estimator –Resample with replacement –Repeat analysis –CI obtained from distribution of density estimates 62

63. Point counts Point transects –Go to k points, –At each point measure distances to birds seen. 63

64. Point counts 64

65. Point count distances Random Placement implies Transect: Pr(x) = U(0,w) = 1/w Point: Pr(r) = ? 65

66. Point count distances Random Placement implies –Pr(r)= 2π r /π w 2 = 2r /w 2 Transect: Pr(x) = U(0,w) = 1/w Point: Pr(r) = ? 66

67. Because fewer animals occur near the observer in point transects, distribution of observed animals differs in point and line transects Line TransectPoint Transect True animal distribution Observed distribution g(x) Observed distribution 67

68. Density Transect: Point: 68

69. Sample Size: Rules of thumb Line transects –n > 60 – 80 –Clustered populations require larger samples –Not good if cluster size is variable Point transects –Need more samples (observations further from observer) –n > 75-100 69

70. Sample Size: Pilot Study 70

71. Combining with other studies 71

72. Some relative advantages of line and point transect sampling Line transects: –usually greater proportionate coverage of the target area for a given level of effort –often higher rates of encounter of individuals along transect lines –more effective for low density species –less time spent moving among transects, more time dedicated to observations –lower potential for disturbance of animals Point transects: – at the sampling location, attention can be concentrated on observing objects rather than walking along a transect line –patchy habitats can be sampled more easily with point transects –easier to estimate distances from object to observer than distances from object to transect line –known distances from a sampling point can be flagged, as aids in distance estimation –can be more efficient for high density organisms that exhibit little responsive movement 72