My aim is to estimate mortality rates of some eagles on the base of survival tables. The good toll for this is Weibull aging model (see Ricklefs, 2000, page 104 in attachment). According to Ricklefs, the survivorship curve follows the equation
lx = exp( -m0x - (alpha * xbeta + 1)/(beta + 1)),
where x is age, lx is the proportion of population surviving to the age x, m0 is the initial (accidental) mortality, alpha and beta are coefficients connected to the rates of aging.
So, I have to fit 3 parameters of the model: m0, alpha, and beta.
I ask you colleagues to help me with the correct way to do it. So far I've been using NLS (non-linear squares) method in R, and it basically works. However, the model fitting strongly depends on start values of parameters, that's why in some cases the model doesn't converge or comes to singularity.
All this prevents me to make a proper permutation test of my model (after some iterations of the nls function an exception arises and the loop breaks).
Ricklefs himself refers to 'maximum likelihood approaches' which he used to fit the model, but I do not exactly understand what specifically is it in the given context. Could anyone help me with the correct way to fit my data?
Thanks,
Michael
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Ricklefs, R.E., 2000. Intrinsic aging-related mortality in birds. Journal of Avian Biology, 31, 103–111.
Why "-m0x"? Ricklefs wanted it (initial mortality) to be independent, so it should be just -m0. Isn't it?
Well, good question. Correct, the initial mortality doesn't depend on age. It can be seen from the mortality equation:
mx = m0 + alpha * xbeta
But what I fit is the survival curve, i. e. proportion of the population, surviving to the age x. For example, if there was no ageing, it reduces to negative exponenta:
lx = exp( -m0x).
This curve begins from y = 1 and end in y = 0.
If you were right and x is not needed, lx would be constant, lx = exp(-m0), which means eternal individuals.
But suddenly I discovered that Ricklefs seems to miss this term in his paper from 2000. But in the paper from 1998, x is present (see attacnment, p.28). I still think this is just a typo in Ricklefs (2000).
This is not really a Weibull, but a combination of a constant intensity and a Weibull intensity. If you try to build a likelihood function, it is difficult to find an optimum if beta turns out to be close to 0. Then, mx becomes m0+alpha, that means any combination m0 alpha, which have the same sum will lead to the same likelihood.
I did not succeed in finding the true optimum. Producing a 3-D plot of Likelihood in terms of mx, alpha, keeping beta constant showed me, why.
Thank you, colleagues! Eventually I managed to fit the model by the algorithm described in the Appendix B to Ricklefs (1998), thanks to my mathematician friend Igor Sizov who made a working example of this tricky thing so that I understood how it is calculated. If you're curious, see the fitted curves in the attachment. Blue colour are males, red colour are females. Solid lines are the curves fitted by NLS (non-linear least squares), dashed lines are the curved fitted by ML (maximum likelihood).
It is clear for me that NLS curves better fit to the data, while ML curves overestimate the lifespan and do not pass through "older" points. Both procedures use unweighed points. If we use weights, both methods fit worse, and begin to ignore right side points since there are fewer number individuals in older age.
So, think that further I will focus on the unweighted NLS method.
Cheers!
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