Estimation of Animal Abundance and Density Miscellaneous Observation- Based Estimation Methods 5.2 Basic concept (canonical form) To estimate animal abundance (N): 1. Count Animals 2. How many did we miss? (p) To estimate p, we need replicated counts Possible levels of replications: -Time: Several surveys -Observers: Several observers / survey Several ways of doing this: Multiple observers: Multiple observers: Dependent observers Independent observers Multiple surveys: Multiple surveys: Temporal removal modeling Marked subpopulation Sighting probability modeling Repeat counts/occupancy (N-mixture) Defining population and Decomposing the Detection Process N * =number of individuals whose ranges overlap the large area of interest (A) N * =number of individuals whose ranges overlap the large area of interest (A) Pr (a particular member of N * is detected in a survey) Pr (a particular member of N * is detected in a survey) p s =Pr (individuals home range overlaps a selected sample unit) p p =Pr (individual is physically present in sample unit at time of survey) p a = Pr (individual is available for detection at some time during the survey) p d = Pr (individual is detected) Above probabilities can vary by individual, possibly in association with covariates Above probabilities can vary by individual, possibly in association with covariates Components of Detection p s =Pr (individuals home range overlaps a selected sample unit) p s =Pr (individuals home range overlaps a selected sample unit) p p =Pr (individual is physically present in sample unit at time of survey) p p =Pr (individual is physically present in sample unit at time of survey) Might vary among individuals Population ? Components of Detection p a = Pr (individual is available for detection at some time during the survey) p a = Pr (individual is available for detection at some time during the survey) p d = Pr (individual is detected) p d = Pr (individual is detected) Might vary among individuals Photo by John Weller Available for detection ?? p d = Pr (individual is detected) Conceptual Framework Geographic closure Geographic closure Most abundance estimation methods based on ideas of fixed N in known area A Observation-Based Methods Multiple dependent observers Multiple dependent observers Multiple independent observers Multiple independent observers Temporal removal modeling Temporal removal modeling Marked subpopulation Marked subpopulation Sighting probability modeling Sighting probability modeling Repeat counts/occupancy (N-mixture) Repeat counts/occupancy (N-mixture) 1. Double-observer approaches Double-observer Approach 2 observers conduct counts at same time 2 observers conduct counts at same time Broad applications, e.g.: Broad applications, e.g.: Aerial surveys Avian point counts Amphibian egg mass counts In a nutshell: Use information on animals observed or missed by each observer to estimate proportion missed 2 Ways to Implement Double- Observer Approach Dependent Observers Dependent Observers Apply removal model thinking to observations by the 2 observers Independent Observers Independent Observers Apply capture-recapture thinking to observations by the 2 observers = independent replicates Dependent Observers e.g., Grant et al. 2005 Dependent Observers Primary and Secondary roles Primary and Secondary roles Primary observer communicates birds seen/heard to secondary observer Secondary observer records birds detected by primary observer & additional birds (s)he detects Observers alternate the roles Observers alternate the roles Dependent Observers: Expected Values of Sufficient Statistics N = true abundance (# birds in area sampled) N = true abundance (# birds in area sampled) p i = detection probability for observer i p i = detection probability for observer i x i1 = birds detected by observer i in primary role x i1 = birds detected by observer i in primary role x i2 = additional birds detected by observer i in role of secondary observer x i2 = additional birds detected by observer i in role of secondary observer E(x 11 ) = Np 1 E(x 21 ) = Np 2 E(x 11 ) = Np 1 E(x 21 ) = Np 2 E(x 12 ) = N(1-p 2 )p 1 E(x 22 ) = N(1-p 1 )p 2 E(x 12 ) = N(1-p 2 )p 1 E(x 22 ) = N(1-p 1 )p 2 Estimating detectability Estimate of overall detectability: Estimate of overall detectability: Estimate of population size: Estimate of population size: Total count Overall detect. Dependent Multiple Observers: Assumptions All birds have same probability of being detected All birds have same probability of being detected Observer detection probability does not change with role Observer detection probability does not change with role Detection and recording by secondary observer do not influence detections by primary observer Population is closed for survey duration Population is closed for survey duration Independent Observer Approach 2 observers conduct survey independently at point (no communication) 2 observers conduct survey independently at point (no communication) Map bird locations by species Map bird locations by species After survey, review data and decide which individuals were detected by both observers, and which were missed by an observer After survey, review data and decide which individuals were detected by both observers, and which were missed by an observer After each stop After each stop Look at data sheets, conduct summary on the spot Independent Observer Approach Independent Observer Approach x 11 = birds detected by both observers x 11 = birds detected by both observers x 10 = additional birds detected by observer 1 but not by observer 2 x 10 = additional birds detected by observer 1 but not by observer 2 x 01 = additional birds detected by observer 2 but not by observer 1 x 01 = additional birds detected by observer 2 but not by observer 1 Independent Double-Observer: Expected Values detected by obs 2 but not by obs 1 detected by obs 1 but not by obs 2 detected by both Independent Double-Observer: Expected Values detected by obs 2 but not by obs 1 detected by obs 1 but not by obs 2 detected by both Independent Double-Observer: Sufficient statistic p i = detection probability for observer i N = abundance (# birds in sampled area) = (# detected by obs. 1) / (Pr[detec.] for obs. 1) Independent Double-Observer Likelihood Function (Unconditional) Independent Double-Observer Likelihood Function (Conditional) Independent Double Observers: Assumptions All birds have same detection probability All birds have same detection probability Independence of detections among observers Independence of detections among observers There are no matching errors of individual birds (no double counting) (= ability to ID birds detected by both observers) There are no matching errors of individual birds (no double counting) (= ability to ID birds detected by both observers) Population is closed for survey duration Population is closed for survey duration More than 2 Observers: Multiple-observer approach >2 observers permits use of heterogeneity models (finite mixture) that permit to model different detection probabilities for different individual birds (Alldredge et al. 2006) = allows relaxing assumption 1 >2 observers permits use of heterogeneity models (finite mixture) that permit to model different detection probabilities for different individual birds (Alldredge et al. 2006) = allows relaxing assumption 1 5 observers (Simons, Smoky Mountains) 5 observers (Simons, Smoky Mountains) Found evidence of heterogeneous detection probabilities for many species Found evidence of heterogeneous detection probabilities for many species Multiple Observers Key issue: Key issue: ability to identify birds detected by multiple observers Elaborate bird radio study by Simons, Alldredge, Pollock (Alldredge et al. 2007) suggests that this is more difficult than expected, especially when based detections are aural Elaborate bird radio study by Simons, Alldredge, Pollock (Alldredge et al. 2007) suggests that this is more difficult than expected, especially when based detections are aural Multiple Observers Combined with Other Approaches Multiple observers plus distance sampling Multiple observers plus distance sampling Each observer records distance to each detected bird Multiple observers plus time-at-detection Multiple observers plus time-at-detection Each observer records time at which each bird is detected (=> time intervals) Relax p a =1 (birds can be unavailable = not singing) Key references: series of papers by Alldredge, Pollock, Simons (2006, 2007, 2008) Key references: series of papers by Alldredge, Pollock, Simons (2006, 2007, 2008) 2. Temporal removal approaches Temporal Removal Modeling In a nutshell: Use information of new animals found at each survey (= were missed before) to estimate the proportion missed over all surveys Count number of animals detected at time t1 Count number of animals detected at time t1 Count again at time t2, and record how many new animals are detected (same for t3, t4, ) Count again at time t2, and record how many new animals are detected (same for t3, t4, ) => Works well when trapped animals are not released between 2 surveys (removed) or if you can mark caught animals Example: Avian Point Counts Often most detections are by sound Often most detections are by sound Forested habitats Cryptic species Birds can only be detected if they sing Birds can only be detected if they sing Detection probability is product of Detection probability is product of Probability bird sings during interval (p a ) and Probability song is heard by observer (p d ) Temporal Removal: Simple Case Point-count is divided into 2 equal intervals Point-count is divided into 2 equal intervals x 1 is number of birds counted in t 1 x 2 is number of birds first counted in t 2 (not in t 1 ) Assume detection prob. is same for t 1 & t 2 Assume detection prob. is same for t 1 & t 2 E(x 1 ) = Np E(x 2 ) = N(1-p)p N = pop. within ear-shot p = detection probability Temporal Removal Approach: Estimators for p and N Per interval probability of detection Total number of birds available to be detected (Zippen 1958) Temporal Removal: Assumptions All birds have same detection probability (relax with finite mixture models) All birds have same detection probability (relax with finite mixture models) Birds exhibit constant, non-Markovian (=random) probability of vocalizing at each time interval Birds exhibit constant, non-Markovian (=random) probability of vocalizing at each time interval No errors in recognizing and counting individual birds that are detected No errors in recognizing and counting individual birds that are detected Population is closed for survey duration Population is closed for survey duration Birds are correctly assigned as inside or outside of sample unit (e.g., fixed radius point counts) Birds are correctly assigned as inside or outside of sample unit (e.g., fixed radius point counts) Temporal Removal Approach: Extensions Unequal time intervals (Farnsworth et al. 2002) Unequal time intervals (Farnsworth et al. 2002) Heterogeneous detection probabilities, finite mixture approach (Farnsworth et al. 2002) Heterogeneous detection probabilities, finite mixture approach (Farnsworth et al. 2002) Combine approach with other methods (e.g., multiple observers, distance sampling) to increase precision and deal with non-Markovian vocalization Combine approach with other methods (e.g., multiple observers, distance sampling) to increase precision and deal with non-Markovian vocalization Use entire detection history (Alldredge et al. 2007) e.g.: (=better for modeling heterogeneity) Use entire detection history (Alldredge et al. 2007) e.g.: (=better for modeling heterogeneity) 1. Mark-recapture approaches (Marked subpopulation) Marked Subpopulation Sampling design: Sampling design: Mark and release animals at time 1 Sample animals via observations a short time later at time 2, recording the number of animals observed with and without marks In a nutshell: Use information on individual animals observed or missed at each survey occasion to estimate the proportion missed overall Marked Subpopulation Sampling design: Sampling design: Mark and release animals at time 1 Sample animals via observations a short time later at time 2, recording the number of animals observed with and without marks Obtain 3 statistics: Obtain 3 statistics: n 1 = number animals marked and released in 1 n 2 = number of animals observed in period 2 m = number of marked animals observed in 2 Marked Subpopulation: Main Assumptions Animals marked and released in period 1 have similar probabilities of detection in period 2 as animals not marked Animals marked and released in period 1 have similar probabilities of detection in period 2 as animals not marked No losses of marked animals between times 1 and 2 No losses of marked animals between times 1 and 2 Short time period between 1 and 2 or Place radios on marked animals so losses are known Marked Subpopulation: Estimation Detection probability: Detection probability: Abundance: Abundance: This approach can be extended to multiple observation periods Examples: White and Garrott (1990) White (1993) Canonical form = Lincoln-Peterson estimator (can also use ML) 1. Sighting Probability Model Sighting Probability Model In a nutshell: Two steps: 1. Develop a model (covariates) for detection (radiomarked animals: observed & missed) 2. Conduct wide survey (observed animal only) Use model for p to estimate proportion missed Use covariate info on animals observed (step 2) to estimate the proportion missed Sighting Probability Model 1. Model development Requires radiomarked sample of animals Use survey method on area of interest (most commonly used with aerial surveys) For each animal detected, determine whether or not it is radiomarked and measure covariates (e.g., group size, habitat) For each radiomarked animal not detected via the survey method, radio-locate and then measure covariates 1. Model development Estimate parameters of model relating detection probability to covariates, e.g. using logistic regression: Estimate parameters of model relating detection probability to covariates, e.g. using logistic regression: x ij = value of covariate i for animal j x ij = value of covariate i for animal j Data y j = {0,1} 2. Estimation Conduct survey and collect covariate information, x ij, for each detected animal, j Conduct survey and collect covariate information, x ij, for each detected animal, j Use the covariate data with sighting probability model (developed before) to estimate sighting probability for each individual j (p j ) Use the covariate data with sighting probability model (developed before) to estimate sighting probability for each individual j (p j ) Estimate abundance with Horvitz-Thompson (1952) estimator: Estimate abundance with Horvitz-Thompson (1952) estimator: Proportion missed = Sighting Probability Model Important assumptions: Important assumptions: Most important variables affecting detection probability have been identified as covariates Conditions at the times of the surveys are similar to those existing at the time of sighting probability model development Reference: Samuel et al. (1987) Reference: Samuel et al. (1987) 1. Repeated Count approach (N-mixture models) Repeated Counts In a nutshell: Use spatial and temporal replication to disentangle N from p in counts Sampling design Spatial replicates: Multiple locations i where N i are assumed equivalent (same distribution) Spatial replicates: Multiple locations i where N i are assumed equivalent (same distribution) Temporal replicates: Point counts (n ij ) at each location at multiple times (survey j) during a season Temporal replicates: Point counts (n ij ) at each location at multiple times (survey j) during a season Repeated Counts Statistics (n ij ): number of individuals detected at each occasion in each site Statistics (n ij ): number of individuals detected at each occasion in each site Hierarchical modeling approach: - Counts are conditional (on true abundance) binomials with detection parameter(s) (= Obs. component) - Abundance distribution (sites) assumed to be Poisson, neg-binomial, etc. (= Process component) Hierarchical modeling approach: - Counts are conditional (on true abundance) binomials with detection parameter(s) (= Obs. component) - Abundance distribution (sites) assumed to be Poisson, neg-binomial, etc. (= Process component) Numerical estimation based on mixture model of process and observation components Numerical estimation based on mixture model of process and observation components Repeated Counts At any site i detections are binomial: At any site i detections are binomial: N i = abundance at site i N i = abundance at site i p = detection probability p = detection probability n it = number of individuals detected at site i on occasion t n it = number of individuals detected at site i on occasion t Repeated Counts Each realization N i result from a common process (= distributional constraint on N i s) Repeated Counts Repeated Counts: Assumptions (Super)population closure during season (Super)population closure during season All birds have same detection probability (can be relaxed in various ways) All birds have same detection probability (can be relaxed in various ways) No errors in recognizing and counting individual birds that are detected No errors in recognizing and counting individual birds that are detected Birds are correctly assigned as inside or outside of sample unit (e.g., fixed radius point counts) Birds are correctly assigned as inside or outside of sample unit (e.g., fixed radius point counts) Reasonable choice of distribution for spatial process (Poisson? Neg-Binomial?) Reasonable choice of distribution for spatial process (Poisson? Neg-Binomial?) Summary / conclusion To estimate abundance you need: - Counts - To estimate p (# missed) - Replicate your counts to estimate p Options: - multiple observers - multiple surveys and a way to distinguish newly caught animals or some other assumption (N-mixture) Decomposing the Detection Process N * = number of individuals whose ranges overlap the large area of interest N * = number of individuals whose ranges overlap the large area of interest Pr (a particular member of N * is detected in a survey) Pr (a particular member of N * is detected in a survey) p s = Pr (animals home range overlaps the sample unit) p p =Pr (animal is present in sample unit at time of survey) p a = Pr (animal is available for detection at some time during the survey) p d = Pr (animal is actually detected) Interpreting Abundance Estimates: Avian Point Counts Dist. sampling, mult. observers: Dist. sampling, mult. observers: Time of detection: Time of detection: Repeated counts*: Repeated counts*: * Depends on time scale of sampling Interpreting Abundance Estimates Different estimation approaches expected to yield different estimates of abundance and Pr (detection) Different estimation approaches expected to yield different estimates of abundance and Pr (detection) Combination of approaches permits estimation of separate detection components Combination of approaches permits estimation of separate detection components Interpreting Abundance Estimates Extrapolation to larger area of inference Extrapolation to larger area of inference Single, short survey periods => snapshot => Closure (dist. sampling, multiple obs., time of detection): extrapolation straightforward Long periods between surveys (repeated counts): extrapolation requires extra information (e.g., average number of sample units overlapped by an individual bird) Key: select method that best corresponds to objectives Key: select method that best corresponds to objectives Combination of Approaches It is possible to combine the different approaches It is possible to combine the different approaches More flexible, increases precision or allow relaxing some assumptions More flexible, increases precision or allow relaxing some assumptions Examples: Examples: Distance sampling + multiple observers Time of detection + multiple observers Repeated counts + time of detection or distance sampling or multiple observers Combination Approaches Time of detection + multiple observers: Time of detection + multiple observers: Robust design (2 observers at each time interval) Permits separate estimation of p d and p a Can relax assumption of random availability process and treat as Markov process Combination Approaches: Advantages Decomposition of components of detection process Decomposition of components of detection process Increased precision Increased precision Relaxation of single-approach assumptions Relaxation of single-approach assumptions More flexible modeling More flexible modeling