Interpreting real estimate for p in MARK

questions concerning analysis/theory using program MARK

Interpreting real estimate for p in MARK

Postby heg90 » Sat Nov 09, 2019 1:45 pm

Hi all,

I'm working on a single-season occupancy model with covariates, and I wanted to pose a question about the interpretation of the real estimate for 'p' in MARK. A while back I learned that the 'real estimate' for 'psi' only reflects the first site and used CH 21 of the MARK book to generate an overall psi estimate, so I wanted to also make sure my interpretation of the real estimate for 'p' was correct.

Going off the spotted owl/barred owl example in CH 21, I see that the text reports a p-hat=0.36 when BAOW=1 and p-hat=0.71 when BAOW=0. My question is, does the p-hat (i.e., the 'real estimate' for p) reflect the daily detection probability (i.e., probability of detecting the species on one survey event), or the overall detection probability (i.e., probability of detecting the species on all n=J survey events: (1-(1-p)^J). Typically I've seen daily detection probability reported as p, and overall detection probability reported as p-hat to reflect p-hat= (1-(1-p)^J), so I was a bit confused by the annotation in CH 21. It seems like the chapter is referring to the real estimate of p-hat = 0.36 as the daily detection probability, as I see in the sidebar on conditional site occupancy that part of the equation includes (1-phat)^J, making me think that p-hat is just the daily detection multiplied by the number of survey events, J.

My detection models are similar to the example in CH 21, and my top detection model has two covariates: session (June = 0, July = 1) and duration (# days stations were active), and the number of survey events J = 5. When I hold duration constant at the average # days stations were active, the 'real estimate' for p when session = 0 is 0.24, and the 'real estimate' for p when session = 1 is 0.51, indicating detection probability is higher in the month of July. What is the best way to report daily detection probability (p), and overall detection probability (phat) for my 5 surveys based on my top model? I know I can generate p and phat from the raw data of detection histories, and from manually calculating it from the raw data I get an average daily detection probability of 0.4125 and an overall detection probability of 0.93 (i.e., 1 - (1-0.4125)^5), but I am curious if there is a way to get a daily and overall estimate from the top model, or if it's best to report that value separately as reflecting the change in month of June vs July.

Any advice would be greatly appreciated, thank you so much!

~Holly
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Re: Interpreting real estimate for p in MARK

Postby darryl » Sat Nov 09, 2019 5:51 pm

Hi Holly,
The 'hat' is just denoting that we're talking about the estimate of the parameter, rather than the true (but unknown) value of the parameter. So 'p' is the true detection probability for a single survey and 'phat' is an estimate of that detection probability. The estimated overall detection probability can be calculated in the manner you described based on the value for phat.

Hope that helps clear up your confusion.

Cheers
Darryl
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Re: Interpreting real estimate for p in MARK

Postby heg90 » Sat Nov 09, 2019 6:08 pm

Hi,

Thank you so much for your response.

So to clarify, the 'real estimate' for p that MARK reports represents the daily detection probability (or more accurately, the detection probability for a single survey), so if I am interested in generating the overall detection probability from my top model I would take my 'real estimate' for p (either p=0.24 or 0.51 depending on session) and use the formula p-overall = 1 - (1-p)^5 ?

~Holly
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Re: Interpreting real estimate for p in MARK

Postby darryl » Sat Nov 09, 2019 6:20 pm

Yep
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Re: Interpreting real estimate for p in MARK

Postby heg90 » Sat Nov 09, 2019 6:24 pm

Awesome, thank you so much for the clarification!
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Re: Interpreting real estimate for p in MARK

Postby bam59 » Sat Nov 09, 2019 7:06 pm

Hi Holly,
Just to add on a bit here -- the equation you're using (1-(1-p_hat)^J) is the probability of detecting the species of interest at least once, given J visits. The p-hat by itself is the probability of detecting the species on a single visit. As Darryl points out, the "hat" that parameters sometimes wear indicates that they are estimates, rather than true underlying values (which are essentially always unknown). Sometimes people get sloppy and drop the hats (you may have observed this somewhere) but it is good practice to keep them on any estimate you report.

Most people refer to the quantity 1-(1-p_hat)^J as p* (pronounced "p-star"). That equation is specified on page 21-14 of Chapter 21.

Brittany
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Re: Interpreting real estimate for p in MARK

Postby cooch » Sat Nov 09, 2019 7:37 pm

bam59 wrote:Most people refer to the quantity 1-(1-p_hat)^J as p* (pronounced "p-star"). That equation is specified on page 21-14 of Chapter 21.

Brittany


This also shows up in the robust design (Chapter 15) -- occupancy being a 'special case' of RD sampling.

\hat{p}^{\ast}=1-(1-\hat{p})^{J} over J visits (sampling occasions).
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Re: Interpreting real estimate for p in MARK

Postby heg90 » Sat Nov 09, 2019 7:58 pm

Hi,

Thank you all for your feedback, and for the clarification on the annotation.

~Holly
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