How do we estimate the force of mortality for lives aged x exactly?

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asked Apr 7, 2018
edited Apr 8, 2018
As stated in the title: How do we estimate the force of mortality for lives aged x exactly when the death data implies a rate interval based on the age nearest classification?

This is a question based on the UCT BUS3018F 2005 Exam Paper, as found in BUS3018F Tutorial 3, Section 4, Question 4.
1. The census data provided is based on the age next birthday definition.
2. The death data is based on the age nearest birthday definition.

We are asked to find an estimate for the force of mortality, but since the death data specifies the rate interval and due to the fact that the force of mortality applies to x+1/2 not x naturally. How do I adjust the census data to give me an estimate that would work as required?

answered Apr 8, 2018 by (1,350 points)
selected Apr 8, 2018 by Daniel

Thank you for the question. I do not seem to agree with the age to which you say the force of mortality applies to.

You are correct in saying that the death data specifies the rate interval. Now, there are TWO STEPS to find the age to which the force of mortality applies to:

1. Calculate the age at the beginning of the rate interval. This we infer from the AGE LABEL of the deaths, which is "age x nearest birthday at the point of exposure" (the time we want to age the life, in this case it is death). So, the life maintains this age label from 6 months before their xth birthday, to 6 months after. Hence, the EXACT age at the start of this rate interval is $$x - \frac{1}{2}$$.  Please will you now write out the rate interval clearly.

2. Then, find the age to which $$\mu$$ applies to - recall that we said that (in most cases) $$\mu$$ applies to the average age at death in the rate interval, and if we assume that deaths occur on average half way through the rate interval, then that age is $$x$$.

Fortuitously, we obtain the estimate for $$x$$.

Now, to adjust the census data. The first thing to note about this question is that you are given EXPOSURE data (ie. times of exposure) and NOT POPULATION DATA. I will give you a hint to get you started.

We have $$E_x^C\ast$$ , which is the years of exposure during the period of investigation, for lives aged $$x$$ next birthday. We wish to extract information on $$E_x^C$$ , which is defined to be the years of exposure during the period of investigation, for lives aged $$x$$ nearest birthday. Now, of the lives contributing exposure to  $$E_x^C$$ , 1/2 will be between the ages (x-1/2,x), and the others between ages (x, x+1/2) assuming that birthdays are uniformly distributed over the year which starts 6 months before the time point of exposure (i.e. your date of census, which is NECESSARY since we have that $$E_x^C$$ is a function of $$P_x(t)$$ and $$P_x(t+1)$$ <- here, your census dates are t and t+1), to 6 months after (a subtle concept - draw a timeline to see this!). On the basis of this information, I hope you can now extract the relevant exposure for the lives age $x$next.

commented Apr 8, 2018 by (380 points)

Hi Mario,

Thank your very much for your comprehensive reply, particularly on a weekend.

I seem to have gotten myself confused with regard to the way in which we tackled to which age the force of mortality applies for the rate intervals defined by "age nearest birthday at previous policy anniversary (PA)", where we assumed that qx applies to average age x and with the force of mortality applying to age x+1/2. Is it correct to say that the nuance of having to assume a UDB (as for the PA RI) falls away in our case in this question, since for the life rate interval here  we know that the life will hold the age label from exactly x-1/2 to x+1.2 as you mentioned? So the life rate interval is based on the individual's ageing based on his birthday rather than the PA.

// Hope that was decipherable, quite tricky to articulate some of these ideas out loud //

1. So in an attempt to answer your request to restate the rate interval correctly:
2. "We have a life year rate interval, starting for lives classified age x, 6 months before their xth birthday" [x-1/2, x+1/2]
Still grappling your closing comments regarding the exposure. I think I understand how to approach the adjustment had we been been dealing with population data - is the logic behind that conversion transferable to the exposure, or should II try look at the problem afresh?
commented Apr 8, 2018 by (1,350 points)

Hi Daniel. My comments are in boldface in your response below. I hope they help.

__________________________________________________________________________

I seem to have gotten myself confused with regard to the way in which we tackled to which age the force of mortality applies for the rate intervals (not rate interval. It is better to say  rate interval implied by an age label") defined by "age nearest birthday at previous policy anniversary (PA)", where we assumed that qx applies to average age x and with the force of mortality applying to age x+1/2 Your ages are correct. Is it correct to say (I am not quite clear which question you are referring to - the age label posted here, or the one from the previous question), since for the life rate interval here  we know that the life will hold the age label from exactly x-1/2 to x+1.2 as you mentioned?  Unfortunately, I am not sure if I understand your question. But let me make a few general comments. For the age label of "age nearest birthday", we have a life year rate interval (RI), starting for lives classified age x, 6 months before their xth birthday. So in this case, lives carry the same age label over EXACT ages x-0.5 to x+0.5. If we had the age you mention in this comment directly above, we would be dealing with a policy year rate interval, starting for lives classified age x, on the policy anniversary at which the lives were classified age x nearest birthday. That means, that on the PA at the start of the RI, lives are aged x nearest. We do not have an an exact age at the start of the RI, which is the PA, so we need to find an average age. Using UDB over the policy year (no need to worry about specifying the dates), the average age on this PA is x.   So the life rate interval is based on the individual's ageing based on his birthday rather than the PA. Yes, this is correct.

// Hope that was decipherable, quite tricky to articulate some of these ideas out loud //

1. So in an attempt to answer your request to restate the rate interval correctly:
2. "We have a life year rate interval, starting for lives classified age x, 6 months before their xth birthday" [x-1/2, x+1/2]  Excellent!
Still grappling your closing comments regarding the exposure. I think I understand how to approach the adjustment had we been been dealing with population data - is the logic behind that conversion transferable to the exposure, or should II try look at the problem afresh? In general, I think that the logic is transferrable, with one or two minor modifications. Do the following - it may help clear things up a little.
1. Suppose, you have the population, at census times 31 Dec. 2016 and 31 Dec 2017 (which we can assume delineates the period of investigation), with an age label of "age next birthday at the census times". Now, on the basis of this information, approximate the population, at census times 31 Dec. 2016 and 31 Dec. 2017, aged x nearest birthday at the census date.
2. Suppose you have the years of exposure to risk of lives aged x next birthday during the period of investigation of 31 Dec. 2016 to 31 Dec. 2017 i.e. E_x^C, for different ages x. On the basis of this information, find the years of exposure to risk of lives aged x nearest birthday during the period of investigation of 31 Dec. 2016 to 31 Dec. 2017. (Note that this is essentially what the tut question you asked about wanted you to do).
Note that in 1, you are working with population data, and in 2 with exposure, and crossing between the same age labels in each case. Hopefully, you should be able to see the differences in approach. Good luck!