interpretation of detection probability and occupancy probability

[xinzhu zhang]

Hello everyone

I got two questions.

1. The interpretation of naive occupancy vs modeled occupancy. Let's say that I had 26 sites surveyed for bobcats and I detected bobcats in 9 of them. Then the naive occupancy is 9/26. Should I interpret the results here as bobcats were detected in 34.6% of our study sites? Then I had the modeled occupancy to take imperfect detection into consideration, and I run occupancy model without covariates and applied back transform function. Let's imagine the result I got was 36%. Should I interpret the result from modeled occupancy as there is 36% chance that bobcat occupies a site (assuming the occupancy probability of each site remains constant) instead of 36% of my study sites were occupied by bobcats? I'm wondering the difference between naive occupancy and modeled occupancy is not just take imperfect detection into consideration, but also the way to interpret two results?

2. The results of detection probability from the occupancy model without a covariate. should I interpret in as the probability of a species detected in each survey occasion (e.g., a week) given the site is occupied by a bobcat instead of the accumulative detection rate across the entire survey period. For example, I surveyed bobcats for four weeks in total, and the output I had was 60%. Does it mean that for each week, I had 60% chance to detect a species given the site is occupied?

[Marc Kery]

Dear Zhang,

two answers:

1. Yes, the apparent occupancy is the observed proportion of sites at which we *detect* a species. In contrast, the estimated occupancy under the model refers to the true proportion of sites at which a species really does occur (regardless of whether we detect it or not). The two are linked by detection probability. In a sense, yes, one might say that the interpretation of 'occupancy' changes. But in another sense, both the apparent and the estimated occupancy from the model refer to the same thing: the true proportion of sites at which the species occurs. Only, that apparent occupancy is a biased estimate whenever detection probability is not equal to 1. (Btw, I try to avoid the term 'naive occupancy' and prefer 'apparent occupancy' or 'observed occupancy'. I find the former term a little arrogant; plus, although it's true that interpreting the observed occupancy as if it was the true occupancy is naive, there are so many other ways in which our models may be inadequate and hence, naive.)
2. Yes, in the occupancy model detection probability p is defined for an occupied site and *for one occasion*. Under simplistic assumptions (all equal and independence), this scales up to the total detection probability, P*, over n occasions as follows:
   
   
1. P* = 1 - (1-p)^n.
3. 
   
    That is, the probability that the species is detected at least once at a site where it is present is 1 minus the probability that it is missed during all n occasions, and the latter is given by (1-p)^n

Best regards  --- Marc

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From: unma...@googlegroups.com <unma...@googlegroups.com> on behalf of xinzhu zhang <xinz...@gmail.com>
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Subject: [unmarked] interpretation of detection probability and occupancy probability

 

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[xinzhu zhang]

Thanks!

A couple follow up questions

1. Is there a way to calculate the SE for p*? I mean, the model output will give SE for p. Is there a way to calculate SE for p* based on SE for p?

2. For dynamic occupancy, when there's no covariates in the model, the percentage of colonization and extinction from the model output represent the colonization/extinction rate each year, right? I mean, assuming the rate stays constant per year.

3. I'm trying to find a rationale for how increasing the number of cameras increase detectability. It seems that the number of cameras used in the study does not play a role in the calculation of weekly or accumulative detection probabilities. But I think the number of cameras still impact detection probability somehow. For example, if you only place one camera on your study site, you may not detect the species. But if you place two cameras, you are probably more likely to detect the species. In other words, you might have more "presence" sites than "absence" sites in the model and further impact the calculation of detection and occupancy. Does this make sense?

Best,

Xinzhu

[Marc Kery]

Dear Xinzhu,

as for 1: P* is a function of the model parameter p, for which you are given a SE in the output when you fit the model. To compute the SE of that function, you could use the so-called delta rule or a sort of parametric bootstrap. For the example given in the help text of the 'occu' function in unmarked, below is some code for the latter solution, where, after fitting the model, the MLE and its SE for the detection intercept are -1.939 and 0.166.

as for 2: yes (of course !), only that the colo and ext parameters in the output are given on the logit scale. So to get to a probability you have to plogis() them (and to get to a percentage, further multiply by 100).

as for 3: you ask about the probability to detect a species that occurs somewhere in a study area. This is not equal to the detection probability in the occupancy model, which in a camtrap study is the probability to detect the species at an occupied trap location. Obviously, detection prob in the occupancy model at a site is not affected by how many other traps you have in the area, but the former probability is affected by this. I think it will be a function of both occupancy and detection probability. I can't remember whether this problem has been dealt with in the literature, but I'd be surprised if it was not.

Best regards  — Marc

# Get SE for Pstar from the MLE of p and its SE using a form of parametric bootstrap

# Draw 100000 samples from the normal distribution centered on the MLE and with SD = SE (repeat 3 times, once for each occasion)

lp1 <- rnorm(10^5, mean = -1.939, sd = 0.166)             # Logit scale

lp2 <- rnorm(10^5, mean = -1.939, sd = 0.166)

lp3 <- rnorm(10^5, mean = -1.939, sd = 0.166)

# Convert to probability scale

p1 <- plogis(lp1)    ;    p2 <- plogis(lp2)   ;   p3 <- plogis(lp3)

# Apply function for aggregate detection prob P* to each draw

Pstar <- 1 - (1 - p1)*(1 - p2)*(1 - p3)

# Plot sampling distribution of Pstar and get its mean and SD. Latter is the SE of Pstar

hist(Pstar)   ;   mean(Pstar)  ;  sd(Pstar)

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Subject: Re: [unmarked] interpretation of detection probability and occupancy probability

 

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[dsto...@gmail.com]

Hi Xinzhu

In response to your 3rd question, yes you can increase detection probability of species by incorporating more cameras at a site. I published a paper on this 10 years ago WR15083

In that study, we had 5 cameras deployed per site and I randomly sub-sampled the data to assess how detection probability changed with varying numbers of cameras per site. I'm sure there are other ways to analyse the data, but it was an effective approach for my question.

Regards

Dani