I have had a headache for the last two days trying to figure out a theoretical problem related to my work. My best attempt to explain the problem, and the closest to a solution I have thus come up with, can be found in the email below, written to one of my research assistants. This is the kind of symbolic thinking that makes my forehead tie itself in a knot. Tell me if what I wrote means anything to you, 'cause it doesn't say a whole lot to me.

Hey Nik-

I have another favor to ask. I've been wracking my brains trying to figure out a problem, that we have no null hypothesis for what G should be. I try to explain the problem and what I'd like to do about it below.

Post-Reproductive Lifespan as measured by G cannot be negative, in that an individual cannot invest in reproduction after her death. G cannot even meaningfully be zero unless every individual dies at age M, the age at which fertility drops of to 5% of its former maximum. If even one individual in the study population lives past age M, G is non-zero. This has left me struggling to figure out what a meaningful null hypothesis for G could be. The answer seems to be that there isn't one. G is a parameter designed for a quantitative, rather than qualitative distinction. Human G is very different from G of non-human primates, but it isn't meaningful to ask if G of non-human primates is different than zero, because we know without knowing anything about the populations that it will be.

The relevant question is: is senescence in fertility offset in age/time from actuarial senescence? If M, is the parameter we use to demarcate the end of fertility, we can use the exactly analogous measure, Z, to demark the end of meaningful survivorship. Z is defined as the last age for which p(x)≥0.05*max(p(x)). ( By the way, in case we don't have p(x) in the data you have, p(x)=1-q(x))

So then the question becomes, how much different is M from Z? Z-M is the post reproductive period, and (Z-M)/(Z-B) is the portion of the adult lifespan that is post reproductive. If we define S=(Z-M)/(Z-B), then S gives us a decent measure of how much reproductive senescence is offset from actuarial senescence. And one that I can at least imagine being zero, in that the rates of survival and fertility can drop simultaneously, even if each individual reproduces only before she dies.

Would you be so kind as to have Access calculate Z and S for the populations we have in the database, and then send me a spreadsheet with B, Z, S and M for each population? I'd like to get a sense of how these variables behave.

Thanks,

Dan

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