Meaning of pi parameter from Multistate Jolly Seber model

questions concerning analysis/theory using program MARK

Meaning of pi parameter from Multistate Jolly Seber model

Postby plelong » Mon Jul 11, 2022 7:56 am

Hi everyone,

I’m currently fitting on Mark software a Multi-state Jolly Seber (MSJS) model to live recapture data from an open population of immature green turtles. These turtles recruit on juvenile foraging ground after a short phase offshore. Then, they grow up until reach the sexual maturity, and leave for adult foraging ground. I used 6 states, 2 sites and 3 size classes on each site.
MSJS is of interest because it provides informations on the recruitment in the studied population through pent and pi.
Pent(i) meaning is clear, as this is the probability that an individual from the super-population move to given state i.
My problem is the meaning of pi. The Mark help give this definition: “pi -- probability of recruitment to each state”. But on the other way, we’ve already got this information using pent: {pent(i) x N* = number of new individuals in N(i)} where N* is the super-pop size and N(i) is the abundance of a certain state i. Thus, there is something I don’t get.
In Dupuis and Schwarz (2007), they didn’t mentioned pi in MSJS parameters. In the Mark book, MSJS is briefly mentioned in chapter 12 "Jolly Seber models in MARK" part 6 “Final words” but without information on pi.
Does anyone know how to interpret this pi ?
Thank you for your help.
Pierre

Dupuis, J. A., & Schwarz, C. J. (2007). A Bayesian Approach to the Multistate Jolly-Seber Capture-Recapture Model. Biometrics, 63(4), 1015–1022. http://www.jstor.org/stable/4541454
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Re: Meaning of pi parameter from Multistate Jolly Seber mode

Postby gwhite » Mon Jul 11, 2022 9:03 am

pent is specific to each state. So before pent can be applied to insert an animal into the process, the animal must be assigned a state with probability pi.

So pent gives the probability of the time when an animal recruits given it is in a state, whereas pi assigns the initial state.
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Re: Meaning of pi parameter from Multistate Jolly Seber mode

Postby plelong » Tue Jul 12, 2022 2:36 pm

Thank you very much for your answer.

Allow me to rephrase by my own words, to be sure I have fully understood your answer:

- Pi : gives in which state the animal is the more susceptible to recruit initially throughout the study. Sum of the pi’s must sum to 1.

- Pent : gives, for each state, when the animal have been the more susceptible to recruit. Sum per state of the pent’s must sum to 1.

So, am I wrong by saying that pi*pent gives the probability that an animal recruit in a given state, at a given moment (year for my example)?

As there is multiple states in this case, I can’t just do pent*N (where N is the super-population size) to obtain the number of new individuals per year in a given state, as POPAN model with a single state. I then need to do: pi*pent*N to take into account the initial repartition of individuals between the different states. Is this reasoning correct?

One last question, to obtain the variance of the number of new individuals, I have to use the Delta Method. Is N independent of pi and pent? As N is a derived parameter, it’s not on the same var-cov matrix than pi and pent, so I think it’s independent and cov(N, pi) and cov(N, pent) = 0 but I’m not sure.

Thanks for your help.

Pierre
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Re: Meaning of pi parameter from Multistate Jolly Seber mode

Postby gwhite » Wed Jul 13, 2022 8:37 am

So, am I wrong by saying that pi*pent gives the probability that an animal recruit in a given state, at a given moment (year for my example)?
GCW> No, I think this is corret.

As there is multiple states in this case, I can’t just do pent*N (where N is the super-population size) to obtain the number of new individuals per year in a given state, as POPAN model with a single state. I then need to do: pi*pent*N to take into account the initial repartition of individuals between the different states. Is this reasoning correct?
GCW> Yes.

One last question, to obtain the variance of the number of new individuals, I have to use the Delta Method. Is N independent of pi and pent? As N is a derived parameter, it’s not on the same var-cov matrix than pi and pent, so I think it’s independent and cov(N, pi) and cov(N, pent) = 0 but I’m not sure.
GCW> No, there is a covariance between N and pi and N and pent, but an estimate is not provided in MARK.
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Re: Meaning of pi parameter from Multistate Jolly Seber mode

Postby plelong » Wed Jul 13, 2022 10:26 am

gwhite wrote:So, am I wrong by saying that pi*pent gives the probability that an animal recruit in a given state, at a given moment (year for my example)?
GCW> No, I think this is corret.

As there is multiple states in this case, I can’t just do pent*N (where N is the super-population size) to obtain the number of new individuals per year in a given state, as POPAN model with a single state. I then need to do: pi*pent*N to take into account the initial repartition of individuals between the different states. Is this reasoning correct?
GCW> Yes.

One last question, to obtain the variance of the number of new individuals, I have to use the Delta Method. Is N independent of pi and pent? As N is a derived parameter, it’s not on the same var-cov matrix than pi and pent, so I think it’s independent and cov(N, pi) and cov(N, pent) = 0 but I’m not sure.
GCW> No, there is a covariance between N and pi and N and pent, but an estimate is not provided in MARK.



Thank you for your answer. I will look for an other way to obtain the confidence interval of the number of new individuals.
Pierre
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