1. Estimating mean abundance from repeated presence-absence surveys
Royle-Nichols Abundance Induced Heterogeneity
Repeated Count Model (ROYLE)

2. Royle-Nichols Abundance Induced Heterogeneity

3. Resources: MacKenzies’s book
Vermont Unit web site: Donovan, T. M. and J. Hines.  2007.  Exercises in occupancy modeling and estimation.
Royle, J.A. and J.D. Nichols Estimating abundance from repeated presence-absence data or point counts. Ecology 84(3):

4. Motivation Estimating abundance from presence-absence data
Repeated visits on T occasions to multiple sites (R)
Data structured as a typical encounter history

5. Main assumptions
Spatial distribution of animals across the survey sites follows some kind of prior distribution, such as Poisson distribution (alternatively negative binomial)
Probability of detecting an animal at a site is a function of how many animals are actually at that site

6. What is Poisson?
Distribution used to model the number of randomly occurring events
Number of car accidents in your home town
Number of individuals of a species within each of your survey sites
Assumptions:
Each events is independent
Number of events in any time period is random and independent of any other time period.
Animal distributions
Number of animals inhabiting one site is random and independent of the number of animals at other sites.
Each site in your survey is home to some number of animals
Number can be modeled by a specified prior distribution like the Poisson.
Number does not change over the course of your study.

7. Poisson distribution
Single parameter, λ (“lambda”), the mean (equal to the variance).
Mean abundance across the R sites.
Estimates probability of any level of abundance x from 0 to ∞ given some lambda.

8. What is lambda? Lambda is the number of animals per site
Given lambda you can find the probability that a specific number of animals will occur at a given site.

9. Detection is a function of abundance - second major assumption
Royle and Nichols call this inherent detection probability, r.
Varies by species
Constant for all individuals of a species.
Detection of a species is easier when there are many animals at the site
Even though cougars are inherently difficult to detect, you are more likely to observe cougar sign when a site is occupied by 10 cougars compared to 1.

10. site detection probability - p
is a function of the species inherent detection probability, r, and the site abundance, Ni.
Ni is the abundance at site i.)
Ntotal represent the total abundance across all sites.
Ntotal = Σ(Ni).
(1-r) is the probability of missing a single individual occupying the site.
probability of missing all Ni individuals is (1-r)Ni.
probability of detecting any animal at the site is one minus this term, or 1 –(1-r)Ni.

11. Site is home to 10 animals, but the species has an r = 0
 Site is home to 10 animals, but the species has an r = 0.10 (intrinsically hard to detect).
The probability of missing one individual is = 0.90.
The probability of missing all 10 animals is (1-0.10)^10 = 0.35.
The probability of detecting an animal at this site is 1-(1-0.10)^10, or 0.65.
If abundance was only 2 at that site, the probability of detecting an animal at this site is 1-(1-0.10)^2, or Here is what this relationship looks like graphically, given that r = 0.10.

12. Estimating abundance
assume that there is some number (it could be 0) of individuals actually inhabiting each site (Ni).
We also assume that whether or not you detect the target at that site is going to be a function of the species-specific detection probability (r) according to this formula:

13. Likelihood

14. Likelihood
Single site
Multiple sites

15. REPEATED Count Model (ROYLE)

16. Resources:
Donovan, T. M. and J. Hines.  2007.  Exercises in occupancy modeling and estimation.   <
Royle, J. A N-mixture models for estimating population size from spatially replicated counts. Biometrics 60:

17. Motivation
Suppose that you want to estimate the size of a population across a relatively large area. Abundance may vary among sites.
Can’t
Survey the entire area (need to sample)
Assume perfect detection of individuals
Can
Select representative sample of sites
Conduct multiple surveys each site
Count individuals on each survey

18. Data /*1*/ 21122 1; /*2*/ 12212 1; /*3*/ 04111 1; /*4*/ 20001 1;
Encounter histories (repeated counts)
One (row) for each site surveyed
Number of individuals counted on each survey
Number of sites with exact same encounter history
Site & sampling covariates
/*1*/ ;
/*2*/ ;
/*3*/ ;
/*4*/ ;
/*5*/ ;
/*6*/ ;
/*7*/ ;
/*8*/ ;
/*9*/ ;
/*10*/ ;
/*11*/ ;
/*12*/ ;
/*13*/ ;
/*14*/ ;
/*15*/ ;
/*16*/ ;
/*17*/ ;
/*18*/ ;
/*19*/ ;
/*20*/ ;

19. ASSUMPTIONS OF THE ROYLE COUNT MODEL
Population is demographically closed over the course of the T surveys.
Spatial distribution of the animals across the R survey sites follows some kind of prior distribution, such as the Poisson distribution,
Probability of detecting n animals at a site represents a binomial trial (Bernoulli trial) of how many animals are actually at that site.
Royle Count Model is a mixture of the Poisson and Binomial distributions

20. Likelihood
Likelihood of p (the probability of detecting an individual given present) and  (the mean of the Poisson distribution), given the observed field data {nit}, where {nit} are the encounter histories, equals….

21. Likelihood
… the product of the binomial probabilities of detecting nit animals out of N total animals at the site, given the probability of detection is p across the T occasions multiplied by …

22. Binomial distribution
Probability of n successes (detections), given N trials and model p.
Probability of n heads, given N coin flips and probability of heads, p

23. Likelihood
… the product of the binomial probabilities of detecting nit animals out of N total animals at the site, given the probability of detection is p across the T occasions multiplied by …

24. Likelihood
…the Poisson probability that Ni individuals inhabit the site given a mean density of  individuals per site.

25. Poisson probability
Single parameter, ϴ (“theta”), the mean (equal to the variance).
Mean abundance across the R sites.
Estimates probability of any level of abundance Ni from 0 to ∞ given ϴ.
Probability of count (Ni )

26. What is ϴ? ϴ is the mean number of animals per site
Given ϴ you can estimate the probability that a specific number of animals will occur at a given site.

27. Likelihood
…the Poisson probability that Ni individuals inhabit the site given a mean density of  individuals per site.

28. Likelihood Summed across all possible values of Ni
Ex. 1: Grey heron
Likelihood
Summed across all possible values of Ni
Product across all of the R sites
Maximize the likelihood by substituting values for N, p, and ϴ (or associated s).

29. Likelihood
…the Poisson probability that Ni individuals inhabit the site given a mean density of  individuals per site.

30. Modeling Site covariates – factors affecting density
Sampling covariates – factors affecting detection
Link functions
Site covariates
Log (unconstrained) – why?
Sampling covariates
Logit (or other constrained [0,1]) – why?
Model selection based on AICc
Model averaging using model weights (wi)

31. Model Selection Site covariates – factors affecting density
Sampling covariates – factors affecting detection
Link functions
Site covariates
Log (unconstrained)
Sampling covariates
Logit (or other constrained [0,1])