process variance for monthly survival converted to annual

questions concerning analysis/theory using program MARK

process variance for monthly survival converted to annual

Postby sixtystrat » Mon Oct 25, 2021 3:50 pm

I want to estimate process SEs for annual survival rate estimates from known fate data. I have monthly data over 19 years. To convert monthly SEs to annual I did the following based on the delta method from Powell 2007:
sqrt(144 * sigma^2 * (Sm * 132)), where sigma^2 is process variance from the variance components analysis and Sm is the mean monthly survival rate (beta hat). Does that seem right? My confusion stems from the Mark book (appendix 4) which states for total variance I have to multiply SE(beta-hat) by the number of years. Do I have to do that with process variances as well? Thanks!
Joe
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Re: process variance for monthly survival converted to annua

Postby cooch » Tue Oct 26, 2021 9:13 am

sixtystrat wrote:...based on the delta method from Powell 2007: sqrt(144 * sigma^2 * (Sm * 132)),


This seems to be incorrect (there are a number of typos in this paper -- not Larkin's fault - a number of the errors I know about slipped in during typesetting). This seems to be one I missed...

The function f(S)=S^{12}, where S is the mean monthly survival rate.

So, differentiating wrt to S yields

\displaystyle\frac{\partial(f(S))}{\partial{S}}=12S^{11}

So, the first-order approximation to the variance is

\hat{\mbox{var}}(f(S))\approx\left(\displaystyle\frac{\partial f(S)}{\partial\hat{S}}\right)^2\cdot\widehat{\mbox{var}}(\hat{S})=144(S^{22})\cdot\widehat{\mbox{var}}(\hat{S})

which is not the same as what you've extracted from Larkin's paper -- is should be S^{22}, not S^{132} (again, not his fault).

More to the point, I'd simply take your estimate of mean and process variance from the method-of-moments approach (or MCMC), and plug into above.
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Re: process variance for monthly survival converted to annua

Postby sixtystrat » Tue Oct 26, 2021 10:36 am

Thanks Evan! That's too bad about the Powell paper.
Joe
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Re: process variance for monthly survival converted to annua

Postby cooch » Tue Oct 26, 2021 11:17 am

sixtystrat wrote:Thanks Evan! That's too bad about the Powell paper.
Joe


Ironically, I just cleaned up some typos in my initial reply.

As for Larkin's paper -- it is unfortunate. But, a not uncommon problem - manual typesetting used to be the rule, and the likelihood of errors in the process increases in direct proportion to the number of equations in the MS. The main concern is that Larkin's paper is so widely 'looked at' (for good reason -- it provides a ready-made 'cookbook' for some common functions, and is admittedly more accessible than Appendix B), that the few errors that crept into the paper have magnified impact. I generally tell people that its always a good idea to work out the math yourself, when practical. It isn't difficult for simple transformations.
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Re: process variance for monthly survival converted to annua

Postby sixtystrat » Wed Oct 27, 2021 10:11 am

Just to beat a dead horse even more, I assume that the process SE would be sqrt(sigma^2/n), where n is the number of years? Thanks again!
Joe
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Re: process variance for monthly survival converted to annua

Postby cooch » Thu Oct 28, 2021 3:27 pm

More or less. The only reason I can see wanting a SE is to construct some sort of CI. At which point, you might be better off doing it in an MCMC framework, since the MCMC capabilities in MARK will let you derive not only a posterior estimate for the derived parameter, but a credible bound (based on HPD - highest posterior density) which is arguably more defensible than a standard frequentist CI.

Appendix E has the details. It really isn't that difficult to implement.
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Re: process variance for monthly survival converted to annua

Postby sixtystrat » Fri Oct 29, 2021 9:26 am

Got it. Thanks!
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Re: process variance for monthly survival converted to annua

Postby cooch » Fri Oct 29, 2021 12:42 pm

Best starting point is the Addendum on (largely) 'numerically-intensive alternatives to the Delta method'. Methods (ii) and (iii) are both quite easy to implement. If you're using RMark, method (ii) is especially straightforward.
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