Capture probability doubts

[Puig]

Hi, I wonder if someone could give me some advice here. I estimated the capture probability (detection probability) of whales during their migration to estimate their population. I used a Huggins model because I'm considering some covariates (6). I have 2 time intervals each time interval represents the observations made by each one of the two observation teams. Each team made their observations simultaneously and independent from each other. So I did something like simple Petersen experiments.

The thing is that I’m not sure of how I handle the p parameters. if the Huggins model assumes independence in detections not only among whales counts, but also between watch stations, I’m not sure If this implies that my p parameters have to be 2 distinct (p1,p2=c)? Or just one (p1=p2=c) because the teams are simultaneously watching so they might have the same detection probability? I’m not sure.

If both p parameters should be distinct which of those I have to use as my detection probability? Do I have to do something with both just to get one “general” p value to correct my data?

And finally, one more thing how does the matrix design should look like in either case?


Thaks a lot
Puig

[jlaake]

You can fit either p1,p2 or p1=p2 or any set of models using your covariates that assume a constant intercept for watch station or observer. They could have the same detection probability or different ones by station. What you seem to be describing is dependence in capture probabilities because they are watching the same place. The covariates can remove a certain amount of dependence but not all dependence. For example, if the 2 observers only watched a very thin strip of ocean perpendicular to shore, then the estimated capture probability would be the probability that the observers saw a whale that was at the surface. Any whales that were beneath the surface as they passed by the strip would not be included. There is little you can do about that kind of positive dependence in capture probabilities without considering the design of the sampling and even then you can't entirely avoid it (eg. neither observer will be able to see gray whales in a beaufort 7). Positive dependence will create a negative bias in the abundance estimator.

Regarding your other question about which p to use. If you always had 2 observers watching then you would use p=1-(1-p1)*(1-p2)=p1+p2-p1*p2. If you only had one watch station operating then you would use the relevant p1 or p2 for that station using the appropriate covariates from your model like observer, beaufort etc.

Your question about the design matrix is too general. I suggest that you read through Cooch and White's online manual for many examples on creating design matrices.[/quote]

[jlaake]

Puig replied to me personally but I'm posting my explanation to the questions here as others may have the same questions. Also, it may be of interest to the readers to know that this 2 occasion Huggins type model can be fitted using an iterative logistic regression approach. It is described on pages 360-364 of the Advanced Distance Sampling book. The original reference is Buckland et al 1992 in Marine Mammal Science. The code to do the iterative logistic regression is contained in the mrds R package for Distance 5. So if you are an R programmer you can use it directly or you can use Distance 5.


hppuig@cicese.mx wrote:
> Hi Dr Laake:
> I decided to write you directly to your mail because I feel that it might
> be bit long my letter. You already helped me before with some doubts about
> my thesis work. I'm very sorry for all the questions I hope you don't mind
> that I write directly to you.
>
> In the forum you said that:
> _________________________________________________________________________
> You can fit either p1,p2 or p1=p2 or any set of models using your
> covariates that assume a constant intercept for watch station or observer.
> They could have the same detection probability or different ones by
> station. What you seem to be describing is dependence in capture
> probabilities because they are watching the same place. The covariates can
> remove a certain amount of dependence but not all dependence. For example,
> if the 2 observers only watched a very thin strip of ocean perpendicular
> to shore, then the estimated capture probability would be the probability
> that the observers saw a whale that was at the surface. Any whales that
> were beneath the surface as they passed by the strip would not be
> included. There is little you can do about that kind of positive
> dependence in capture probabilities without considering the design of the
> sampling and even then you can't entirely avoid it (eg. neither observer
> will be able to see gray whales in a beaufort 7). Positive dependence will
> create a negative bias in the abundance estimator.
>
> Regarding your other question about which p to use. If you always had 2
> observers watching then you would use p=1-(1-p1)*(1-p2)=p1+p2-p1*p2. If
> you only had one watch station operating then you would use the relevant
> p1 or p2 for that station using the appropriate covariates from your model
> like observer, beaufort etc.
> __________________________________________________________________________
>
> So, maybe I’m a little confuse here but when you mention that there’s some
> dependence between my watch stations, this imply that the correct model is
> the one that assumes that p1=p2 ?
No. You are confusing dependence and equality. Imagine 2 observers, one with good eyesight and one with poor eyesight. Each are watching for gray whales passing by. Clearly p1 is not equal to p2 but there will be dependence in the sighting probabilities because they can only see gray whales that surface and they are both more likely to see a gray whale breach than a mom/calf pair that slinks by. Positive dependence occurs when there are covariates that affect each observer the same way but you can't measure or include the covariate into the model. Surfacing can't be included unless you include a model of surfacing behavior. The breach/non-breach could be included (note this is just an example as they don't breach often). Thus your observers can search independently of one another but the probabilities can still be dependent because of covariates that you are unaware of or can't measure. But don't get too hung up on this point because there is probably little you can do about it. Just get it clear in your head that dependence is not the same thing as equality.
> Could you please explain me a little bit
> more which are the assumptions or the logic behind to assume that p1=p2?
> Is this because the two observers are in the same place, doing the
> observations at the same time and under the same circumstances? That makes
> that the 2 observers p1 and p2 have the same probability?
> If we add the covariates and we think that they had a different effect in
> each observer team (station), then I might think that the model now has to
> consider that p1 and p2 are different, is this true?
Even if you use the Huggins model it doesn't mean that p1 must be different from p2. Remember that 1 and 2 represent the station. Consider having 2 identical stations and each contains a twin observer so your observers are identical in their abilities and their views are identical. In that case p1=p2 but it doesn't mean that p1 and p2 are constant. They could vary depending on beaufort state, pod size etc so those would be covariates in the Huggins model. Now let's expand that idea somewhat. Consider 2 identical stations but the observers differ and observer 1 is much better than observer 2. If observer 1 is always in station 1 then p1>p2. But if they switch stations, then for some sightings p1>p2 and for others p2>p1 when the observers switch stations. But that can be modelled with an observer covariate that changes for the station.
> Is this what Huggins
> model means when it assumes independence between observers?
>
>
No see above.
> Help! I’m actually lost here now….
>
>
Hopefully the above helped.
> I’m afraid that maybe I’m asking to much! but one more thing If it turns
> out that p1 and p2 have to be different, do I have to apply the covariates
> to each one
>
You have to use the covariates for each one regardless. Hopefully that is clear from above. Let me give an example. Let's assume that you have covariates of observer (with 2 observers) and beaufort states 0-4. To make it easier, I'll assume that you have used beaufort as a numeric variable rather than as a factor variable and that you have coded observer such that observer 1 is the intercept and obs21 is a dummy variable coded as 1 when observer 2 is in station1 and 0 when observer 2 is in station 2. You also need a dummy variable obs22=1-obs21 because it is the status of observer 2 for station 2. Some example data could look as follows:

ch obs21 obs22 beaufort
10 0 1 0
01 1 0 2
11 1 0 1
...

In the first record observer 2 is in station 2 and in the last two records, observer 2 is in station 1

Your design matrix might look as follows with the first column as the intercept, the second is the observer effect and the third is the beaufort effect.

p1 1 obs21 beaufort
p2 1 obs22 beaufort


Let's say the parameter estimates are b1,b2,b3 and you used the default logit link. Then for the first observation the p's are as follows:

p1=1/(1+exp(-b1-b3*0) p2=1/(1+exp(-b1-b2-b3*0) and p=p1+p2-p1*p2

For the next observation it would be

p1=1/(1+exp(-b1-b2-b3*2) p2=1/(1+exp(-b1-b3*2) and p=p1+p2-p1*p2

For the last observation it would be

p1=1/(1+exp(-b1-b2-b3*1) p2=1/(1+exp(-b1-b3*1) and p=p1+p2-p1*p2


The estimator for abundance of pods would be 1/p_1+1/p_2+1/p_3, where p_i is the value of p for the ith observation. If there were pod sizes s1,s2,s3 then the estimated abundance of whales would be s1/p_1+s2/p_2+s3/p_3. Clearly all the calculations and sums extend for all n of your observations and in this simple example n=3.

> Thanks a lot for all your time and patience
>
>
You are welcome. I hope this helps. I'm going to also post this on the list server as others may have the same questions.

regards --jeff