divorce

Robert Chung said:

Daniel Connelly said:

The statistic has nothing to do with someone born this year.

I guess in the sense that the failure rate for light bulbs from a particular manufacturing line
has nothing to do with light bulbs manufactured today.

Quoted message said:

WRT divorce, the divorce rate in the US is approximately 1 per second (
nationmaster.comPeopleOpen ↗ for 1990 data ) This is approximately half the
marriage rate. If the marriage rate is an increasing function of time over the past 20 years or
so, this means most marriages (although perhaps not most first marriages) have ended in divorce.
So Armstrong and Kristin divorcing isn't unusual.

Dan

This would be the case for what is (mis-) named a "stable population"
(i.e., where the age structure is an eigenvector for a matrix of birth and death rates. In that
case the eigenvalue for that matrix is the equilibrium growth rate for the population) with
fixed marriage and divorce functions. On the one hand, the U.S. population is far from
stable, so that estimate you're making will be off; on the other, the divorce function (the
hazard rate) is steep enough that the non-stability of the population isn't *that* important
because the divorces tend to lag the marriages by not-so-much time. That's why the ratio of
the divorce rate to the marriage rate may not be right, but it is in the right ballpark.

The divorce rate after 10 years of marriage is around 33%:
skfriends.comdo 43 percent of marriages divorce.htmOpen ↗

So a time slice isn't sufficient.

Dan

Daniel Connelly said:

Robert Chung:

Quoted message said:

Uh, Dan, this is not what life expectancy is.

What's the formula, then?

There are several formulas, but the two that are easiest to understand use either the probability of
survivorship to a particular age or the hazard function.

Let p(a,t+a) = the probability of surviving to age a at time t. Note that p(0,t) = 1 and p(omega,
t+omega) = 0 where omega is a big number like 116 or so (unless you believe in the literal
interpretation of the bible, in which case you may think that omega is 989).

Then the cohort expectation of life (aka the lifetable expectation of life) is: int [from 0 to
omega] p(x,t+x) dx

The period expectation of life is what you've been complaining about. It's based on a "virtual"
cohort where you use p(x,today) in place of p(x,t+x).

If you use the hazard function, p(a,t) = exp ( - int [0 to a] mu(x,t) dx and then you can proceed
as before.

Your first formula was actually pretty close to the average age of death in a population (but I
confess I have a hard time reading ascii formulas so I may be slightly off), but except in a stable
population with a zero growth rate, that's not the expectation of life. Your second formula is even
closer, but I think you're subtracting off a t at the end and I can't figure out why. BTW,

Quoted message said:

Quoted message said:

Um, this isn't it, either.

However, you're right that there are two different life expectancies: one calculated for a birth
cohort (and can only be calculated after the entire cohort has died) and the period life
expectancy (which is usually what you see printed in the newspapers), which uses the
probabilities of death in the year t for people age a (i.e., born in year t-a). If you think of a
surface where one axis is age, one axis is time, and the height of the surface is the proportion
surviving of each birth cohort, the cohort expectation of life is the integral along the 45
degree diagonal while the period expectation of life is the integral along the time axis.

Right. But neither is the "expectation value of the time to death of a baby born now", which is
commonly reported.

It's no more wrong than saying that the expected value of a random variable X is E(X) without
specifying the pdf. We do this all the time when we estimate things using weighted means, or
weighted least squares, or any kind of weighting scheme. Demographic measures are concpetually
simpler to understand when they pertain to cohorts, but they are certainly more useful when they
pertain to periods. The period life expectancy uses the survivorships or the death probabilities for
*this* period.

Robert Chung said:

Daniel Connelly said:

Robert Chung:

Quoted message said:

Uh, Dan, this is not what life expectancy is.

What's the formula, then?

There are several formulas, but the two that are easiest to understand use either the probability
of survivorship to a particular age or the hazard function.

Let p(a,t+a) = the probability of surviving to age a at time t. Note that p(0,t) = 1 and p(omega,
t+omega) = 0 where omega is a big number like 116 or so (unless you believe in the literal
interpretation of the bible, in which case you may think that omega is 989).

Then the cohort expectation of life (aka the lifetable expectation of life) is: int [from 0 to
omega] p(x,t+x) dx

Right. That's essentially what I wrote. But p(x, t+x) can't be evaluated for x>0, which is in the
future for a contemporary evaluation.

Quoted message said:

The period expectation of life is what you've been complaining about. It's based on a "virtual"
cohort where you use p(x,today) in place of p(x,t+x).

Exactly. Not the same. I believe my function was equivalent.

Quoted message said:

If you use the hazard function, p(a,t) = exp ( - int [0 to a] mu(x,t) dx and then you can proceed
as before.

Right. Hazard functions aren't so simple, however. Nor are they time-invariant.

Quoted message said:

Your first formula was actually pretty close to the average age of death in a population (but I
confess I have a hard time reading ascii formulas so I may be slightly off), but except in a
stable population with a zero growth rate, that's not the expectation of life. Your second formula
is even closer, but I think you're subtracting off a t at the end and I can't figure out why. BTW,

My scalar field was the probability of living from time t to t', where times are absolute, so you
need to subtract the current time to get an age.

Daniel Connelly said:

So a time slice isn't sufficient.

Yeah, and that's why no serious researcher would do it that way. I was just explaining why when you
do the cross-sectional (aka time slice) thing you get a number in the right ballpark rather than a
number like .01 or
1..

I often do cocktail party calculations like that. I call them cocktail party calculations
because unlike Back Of The Envelope calculations you do them while leaning against a wall with a
drink in your hand (occasionally with a slight buzz). BOTE is one step up because you actually
get to use a pencil.