1. Single Season Occupancy Modeling 5.13 UF-2015

2. 2 Occupancy Modeling State variable is proportion of patches that is occupied by a species of interest In surveys, target species is detected or not Advantages over abundance Practical, esp. for large scale Metapopulations

3. 3 Occupancy Modeling

4. 4

5. Alldredge et al. 2007 Playback experiment with skilled observers Detection ranged from 0.19 to 0.65 of individual songs 5

6. 6 Occupancy Modeling Due to imperfect detection, naïve occupancy < occupancy 3 problems Distribution underestimated Covariate relationships underestimated Factors affecting detection generate spurious covariate relationships

7. Covariates Underestimated X Occupancy 0 1True p < 1

8. Covariates Underestimated 8 Tyre et al. 2003, “Improving Precision and Reducing Bias in Biological Surveys: Estimating False-Negative Error Rates”

9. Spurious Relationships 9

10. 10 Single Season Model Various approaches to account for imperfect detection. (e.g., Geissler and Fuller 1987, Azuma et al. 1990, MacKenzie et al. 2002, Tyre et al. 2003, Wintle et al. 2004 and Stauffer et al. 2004) MacKenzie et al. (2002) provide most general treatment of the problem. State space approach that accounts for both observation process and unobserved system state

11. 11 Single Season Model Consider the data as 2 ‘layers’ 1. True presence/absence of the species. 2. Observed data which is conditional upon species distribution. Knowledge about the first layer is imperfect. Must account for the observation process to make reliable inferences about occurrence.

12. 12 Single Season Model  = probability a unit is occupied. p j = probability species is detected at a unit in survey j, given presence.

13. 13 Basic Field Situation From a population of S sampling units, s are selected and surveyed. Units are repeatedly surveyed within a season. Provides information about detection Units are closed to changes in occupancy during a common ‘season’.

14. Data Summary: Detection Histories A detection history (h i ) for each visited site or sample unit (i) 1 denotes detection 0 denotes nondetection Example detection history: h i = 1 0 0 1 Denotes 4 visits to site Detection at visits 1 and 4

15. Detection histories 15 111 110 101 011 100 010 001 000

16. Detection histories 16 111 110 101 011 100 010 001 000 High detection Low detection

17. Detection histories 17 111 110 101 011 100 010 001 000 High detection Low detection

18. 18 Model Development 101 000 001 000 111 000 010 000 110 000 Biological Reality Field Observations

19. 19 Likelihood Approach Consider all possible stochastic processes that may have resulted in observed detection histories. Take verbal description of the observed data and translate it into a mathematical equation.

20. 20 Likelihood Approach For example, Verbal description: species is present at the unit, was detected in first and third survey, not detected in second survey. Mathematical translation:

21. 21 Likelihood Approach For example, Verbal description: species is absent, OR species is present at the unit and was never detected. Mathematical translation:

22. 22 Likelihood Approach Biological Reality Present Absent 1 0 0 0 0 1 1 1 0 0 0 1 1 1 Observations 111 110 101 100 011 010 001 000 0 0 0

23. 23 Likelihood Approach Biological Reality Present Absent 1 0 0 0 0 1 1 1 0 0 0 1 1 1 Observations 101 0 0 0

24. 24 Likelihood Approach Biological Reality Present Absent 1 0 0 0 0 1 1 1 0 0 0 1 1 1 Observations 000 0 0 0

25. 25 Likelihood Approach Model likelihood is the product of the probability statements. Likelihood can be maximized to obtain MLE’s

26. Likelihood Approach 26 Naive occupancy estimate = 0.4211 Number of parameters = 2 Number of function calls = 62 -2log(likelihood) = 129.0606 AIC = 133.0606 LikeNRSig=6 eps=0.01 ETA=1e-013 Individual Site estimates of Site estimate Std.err 95% conf. interval psi 1 0 : 0.5928 0.1482 0.3042 - 0.8290 ============================================================ Individual Site estimates of Site estimate Std.err 95% conf. interval P[1] 1 0 : 0.2664 0.0729 0.1487 - 0.4301 ============================================================ DERIVED parameter - Psi-conditional = [Pr(occ | detection history)] Site psi-cond Std.err 95% conf. interval 1 0 1.0000 0.0000 1.0000 - 1.0000 8 0 1.0000 0.0000 1.0000 - 1.0000 9 1 0.2966 0.1925 0.0646 - 0.7201 10 0 1.0000 0.0000 1.0000 - 1.0000 13 1 1.0000 0.0000 1.0000 - 1.0000 14 0 0.2966 0.1925 0.0646 - 0.7201 15 0 0.2966 0.1925 0.0646 - 0.7201

27. 27 Pr(occupied|nondetection) “Given the species was not detected, is the unit occupied?” is sometimes of interest. Can be derived from modelling results using Bayes theorem.

28. 28 Pr(occupied|nondetection) Standard errors can be derived with the delta method.

29. Likelihood Approach 29 Naive occupancy estimate = 0.4211 Number of parameters = 2 Number of function calls = 62 -2log(likelihood) = 129.0606 AIC = 133.0606 LikeNRSig=6 eps=0.01 ETA=1e-013 Individual Site estimates of Site estimate Std.err 95% conf. interval psi 1 0 : 0.5928 0.1482 0.3042 - 0.8290 ============================================================ Individual Site estimates of Site estimate Std.err 95% conf. interval P[1] 1 0 : 0.2664 0.0729 0.1487 - 0.4301 ============================================================ DERIVED parameter - Psi-conditional = [Pr(occ | detection history)] Site psi-cond Std.err 95% conf. interval 1 0 1.0000 0.0000 1.0000 - 1.0000 8 0 1.0000 0.0000 1.0000 - 1.0000 9 1 0.2966 0.1925 0.0646 - 0.7201 10 0 1.0000 0.0000 1.0000 - 1.0000 13 1 1.0000 0.0000 1.0000 - 1.0000 14 0 0.2966 0.1925 0.0646 - 0.7201 15 0 0.2966 0.1925 0.0646 - 0.7201

30. Hierarchical Bayesian Approach Goal is to estimate posterior probability distribution for probability of occupancy (and detection) at sampling units. Assume a model for species occurrence. Assume prior probabilities. Bayes formula

31. 31 Hierarchical Bayesian Approach Implemented using software such as WinBUGS. → posterior distributions of model parameters and conditional occupancy state for each unit.

32. 32 Hierarchical Bayesian Approach: WinBUGS Code Occupancy state Det. History (data) Prob occ Conditional det model { ### Priors psi ~ dunif(0,1) p0 ~ dunif(0,1) ### Likelihood for (i in 1:s) { z[i] ~ dbern(psi) for (j in 1:k) { h[i,j] ~ dbern(p.eff[i,j]) p.eff[i,j] <- z[i]*p0 }

33. 33 Hierarchical Bayesian Approach If sample size large and priors vague, results identical to MLE

34. 34 Hierarchical Bayesian Approach Advantages Can be used for finite/small populations. Easier to implement complicated models (e.g., spatially explicit models). Derived parameters Disadvantages Model comparisons may be difficult Run time is greater

35. 35 Covariates Site-specific e.g., habitat type, patch size Survey-specific may vary between surveys. e.g., local environmental conditions, observers

36. 36 Covariates Occupancy and detection probabilities may be functions of site-specific covariates (via logit link, say).

37. 37 Covariates Detection probabilities may also be a function of survey-specific covariates.

38. 38 Covariates

39. 39 Covariates Categorical covariates with m categories should be represented with m – 1 indicator (dummy) variables. e.g., if 4 habitat types, use 3 indicator variables; HabA, HabB, HabC, with habitat D considered the ‘standard’. However, suggest all m indicator variables be included in data file.

40. 40 ‘Missing’ Observations Implicit assumption that j th survey of all units are conducted at (approximately) the same time; possibly unlikely in practice. Weather or breakdowns may result in some units not being surveyed. Equal sampling effort may not be possible with available resources.

41. 41 ‘Missing’ Observations Day Unit12345 1101-0 2-0-11

42. 42 ‘Missing’ Observations Survey-specific covariates can only be missing if associated detection survey is also missing. Site-specific covariates cannot be missing.

43. 43 Model Assumptions Closure. Surveys are independent. No unmodelled heterogeneity. Species identified correctly (no false detections)

44. 44 What if Occupancy Changes? If species physically occupies units at random within a season, ‘occupancy’ parameter relates to probability a unit is used by the species. Immigration/emigration only; ‘occupancy’ parameter relates to probability that species is present at a unit at end/beginning of season respectively. should allow detection probability to vary within a season. should pool some survey occasions.

45. 45 What if Occupancy Changes? Closure implies 2 nd component = 1. When changes are random, ‘detection probability’ is the product of 2 nd and 3 rd components

46. 46 What if Occupancy Changes? Some designs (2 observers at each survey, j) permit Direct estimation of component 2

47. 47 What if Occupancy Changes? Other non-random changes may cause biases. Is a ‘season’ defined appropriately? Is time between surveys appropriate? Area of active research.

48. Staggered entry Staggered entry models (Kendall et al. 2013) Permits 1 entry and 1 exit in season Migratory spp 48

49. Staggered entry 2 new parameters e ij = Pr(species enters site i after survey j | site i is not used at survey j) d ij = Pr(species departs site i after survey j | site i is used at survey j) 49

50. Staggered entry 50

51. 51 Lack of Independence Surveys are not independent if the outcome of survey A is dependent upon the outcome of survey B. Some forms of dependence may be accommodated with good designs or modelling. In some instances, parameter estimates may be OK, but standard errors too small.

52. 52 Lack of Independence To account for ‘trap response’; define a site- and survey-specific covariate (x ij ) that equals 1 for all surveys after first detection at a site, 0 otherwise. X ij h1234 01010011 11010111 00010000 00000000

53. 53 Accounting for Lack of Closure/Independence Correlation of spatial replicates (Hines et al. 2010, Ecol. Appl.) Motivating example: surveying tigers on trails in India Detection of signs in one spatial replicate indicates high probability of presence of signs on an adjacent spatial replicate

54. 54 Accounting for Lack of Closure/Independence

55. 55 Accounting for Lack of Closure/Independence

56. 56 Accounting for Lack of Closure/Independence

57. 57 Accounting for Lack of Closure/Independence Hines et al. (2010) approach could also possibly be applied to account for temporal correlation or non-random changes in occupancy within a season.

58. 58 Unmodelled Heterogeneity In occupancy probabilities parameter estimates should still be valid as average values across the sites surveyed. In detection probabilities occupancy will be underestimated. covariates may account for some sources of variation.

59. 59 Incorporating Heterogeneity Finite Mixtures Assume occupied sites consist of G groups. Each group has a different p. Group membership is unknown hence a site may belong to any of the G groups.

60. 60 Incorporating Heterogeneity Random Effects Assume p i is a random value from a continuous probability distribution (e.g., beta distribution, logit-normal). Implement using WinBUGS.

61. 61 Abundance Induced Heterogeneity in Detection Differences in the local abundance of the species between sites may induce heterogeneity in detection probability. Royle and Nichols (2003) suggested an extension of occupancy models to accommodate this.

62. 62 Abundance Induced Heterogeneity Local abundance is unknown, but a spatial distribution could be assumed (e.g., Poisson).

63. 63 Abundance Induced Heterogeneity

64. 64 Abundance Induced Heterogeneity Occupancy is now a derived parameter: (1–e –  ). Implicit assumption that the number of animals at a unit is constant. “several hundred sites may be required”

65. 65 Abundance Induced Heterogeneity A good approach for incorporating heterogeneity in detection for occupancy estimation, less reliable if inferences about abundance are desired. Can also use a 0-inflated Poisson.

66. 66 Species Misidentification If species is falsely detected then occupancy could be overestimated. Can account for false detections if some surveys are free of misidentification (Miller et al. 2011). More later…

67. 67 Assessing Model Fit MacKenzie and Bailey (2004) suggest a test based on the observed and expected number of sites with each possible detection history. The expected number is predicted by the model, which may include covariates.

68. 68 Assessing Model Fit For example, for the history 101 : Site 10.300.760.042 20.350.740.050 30.400.720.058 40.450.700.066 Total0.216

69. 69 Assessing Model Fit Use parametric bootstrap to assess the evidence for lack of fit. used to adjust SE’s and AIC values.

70. 70 Assessing Model Fit Test has been shown to perform well to identify poor model structure w.r.t. detection probabilities, but not occupancy probabilities. Found to have generally low-power, especially for the sample sizes expected in many applications ( s <100). Recommend ≥10,000 bootstraps.

71. 71 Assessing Model Fit Test should be conducted on the most complicated model under consideration (the global model) and results applied to all models in candidate set. e.g., if from global model, this value is used to adjust AIC values and SE’s for all models.

72. 72 Finite Population In some circumstances, the sample of s units may constitute a large fraction of the population of interest. Strictly speaking, the methods above estimate the probability of occupancy. an underlying characteristic of the population. The proportion of units occupied is a realization of this process.

73. 73 Finite Population Distinction between them is of little practical consequence for ‘infinite’ (large) populations, but may be for ‘finite’ populations. SE’s will be too large if not accounted for.

74. 74 Finite Population The proportion of occupied sites could be calculated as: SE derived from delta method

75. 75 Finite Population Alternatively, easily implemented using the data augmentation approach. Presence/absence of the species is predicted for units where species not detected. Occupancy state can also be predicted for units that were never surveyed.

76. 76 Finite Population Post. Distn.  mean = 0.56, sd = 0.10 {2.5, 50, 97.5} %iles = {0.37, 0.56, 0.78} Post. Distn.  * mean = 0.56, sd = 0.08 {2.5, 50, 97.5} %iles = {0.46, 0.54, 0.74}

77. 77 Spatial Correlation Probability of occupancy at a unit may depend upon whether a ‘neighbouring’ unit is also occupied. Some forms of clustering may be well explained by covariates. May not be important to account for if interest is in an overall measure of occupancy. May be more important if interest is in maps or predictions of unit-level occupancy.

78. 78 Spatial Correlation What fraction of cells is occupied? What would be the appropriate SE? Random sampling with standard occupancy estimators should work

79. 79 Spatial Correlation Alternatively, could model occupancy with the autologistic function. essentially the logit link with an ‘effect’ related to the number of occupied neighboring units. g i is a function of the occupied ‘neighboring’ units But the exact occupancy state of units will often be unknown…

80. 80 Spatial Correlation Predict the occupancy state for units where it is unknown (either due to nondetection or unsurveyed) and use the data augmentation approach.

81. 81

82. 82 Summary Historically, investigating patterns in occupancy has been the main focus of such studies. Useful for assessing a snapshot of a population, but not for understanding the underlying dynamics. A suite of flexible methods are now available that account for: detectability covariates unequal sampling effort heterogeneity finite populations spatial correlation