# Estimating $q_x$ for rate interval of lives ages nearest birthday.

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A question asks what is an appropriate rate interval to use when death data is for age $x$ nearest birthday and exposure data is for age $x$ next birthday for a study into the mortality of pensioners.  As we've been told, to be most accurate we should keep the death data unchanged and make approximations from the exposure data.

It is then asked to approximate $q_x$ for age $x$ exact. An approximation for $P_x ^{nearest}(t) = \frac{1}{2}(P_x ^{next}(t-1) + \frac{1}{2}(P_x ^{next}(t)) could then be used assuming uniform distribution of birthdays. I used the rate interval of lives aged nearest birthday, remaining consistent with the death data.The estimate for$q$would then apply for age$x-\frac{1}{2}$. Would I be correct in using the fact that$P_x = _\frac{1}{2}P_x  _{\frac{1}{2}}P_{x+\frac{1}{2}} and then proceeding to estimate _{\frac{1}{2}}q_x with {\frac{1}{2}}q_x by the uniform distribution of deaths assumption?

The question later supposes employees retire at age $x=65$  and how that would change the rate interval, that is used in the beginning. Intuitively I would say that the rate interval used would not change as we still need match the death data although using age $x$ last birthday would perhaps be more convenient. Is this correct ?

(This question is from Tutorial 3, section 4, question 4, 2018. It is attached)

commented Apr 5 by (1,310 points)

Hi Paul

There are a couple of aspects of your answe which I unfortuntely seem not to agree with (but I may be wrong; feel free to argue your point!).

Firstly, the question tells you that you have EXPOSURE data, i.e E_x^C, and not necessarily POPULATION data, ie P_x(t). So you want to steer clear of using P_x(t)’s.

We need exposure for lives aged x nearest, so we need to extract the relevant data from the exposure for lives aged x next. The argument is exactly the same as for when you are working with population data, but you no longer consider calendar time, t, since your exposure is not dependent on time (unlike population data, which is).  So, now redo the question with this new information I have told you, and post your answer again (I would suggest).

Also, I suggest you specify your rate interval properly. It is indeed a life year rate interval, but when does it start?

I hope this helps a little.

commented Apr 7 by (100 points)

Do you treat x ages as your ts?

commented Apr 8 by (1,310 points)

Good question! No, you do not treat your x's as your t's. Note that the exposure does NOT depend upon time - it is best to just try and think of your exposure "being contributed to" by lives with a certain age label, and then make DEDUCTIONSs about the age labels of those lives on the basis of your ASSUMPTIONS.

+1 vote
answered Apr 5 by (950 points)

For part ii) what you want to do is use the relation $$q_x = 1-exp{(-\int_{0}^{1} \mu_{x+s}ds})$$ and then if we assume that the true underlying $$\mu_x$$ is linear between ages $$x$$ and $$x+1$$ nearest, $$q_x = 1- exp{(-\frac{1}{2} ( \mu_x+\mu_{x+1}))}$$

part iii) Age 65 nearest would change: you only have data on lives aged 65 to 65.5 for 65 nearest (i.e. no longer have 64.5-65). So, on average deaths will occur at age 65.25.  So $$\mu_{x+t} = \frac{d_{65}}{E_{65}^{c}}$$estimates $$\mu_{65.25}$$.

You also then need to consider $$E_{65}^{c}$$ --> this will only be half of what you previously estimated.

Now we want $$q_{65} =1- exp(-0.5(\mu_{65}+\mu_{66}))$$ using linearity assumption again,

we have $$\mu_{66}$$ but not $$\mu_{65}$$, (we only have $$\mu_{65.25}$$)

You need to linearly interpolate for these two values to find $$mu_{65}$$. (using same assumption that $$\mu_{x+s}$$ is linear.

commented Apr 5 by (440 points)

Thanks Kelly. Totally makes sense