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Finding qx when age is given in units of "months"

0 votes
asked Apr 2, 2018 in BUS 3018F - Models by anonymous

 image Tutorial 3, section 1, question 4. 

How do you calculate 

1) probability of dying in first month of life

2) q0

3) 3q0

commented Apr 5, 2018 by MarioGiuricich (1,350 points)

Hi there! Please also note that there is an error in my tutorial (I did put the update at Vula, with the correction written in red). E_x^C is in YEARS, not months. So Kelly’s aside does in fact hold - you would need to multiply this by 12.

And thank you for your insightful answer, Kelly. I like your idea of considering the case of deaths after birth occurring on average 5 days into the month. It’s a relevant (and realistic) complication to, in fact, consider.

1 Answer

+4 votes
answered Apr 5, 2018 by Kelly (970 points)

1) P(dying in the first month) = (# of deaths in first month)/ (first month initial exposure) =D1/E1 --> this E1 corresponds to your initial exposed to risk.  

D1 = 396 clearly. 

E1 is slightly more tricky. We are given the central exposed to risk: E1c = 1536.2 measured in months (as an aside, consider what you might do if this was given in years rather than months, hint: multiply by 12). So we need to estimate our initial exposed to risk using the central exposed to risk. Central exposed to risk accounts for deaths during the month and, if we assume that deaths occur on average half way through the month, then Ec will be smaller than E by 396*(0.5 months). Thus, E = Ec+396*0.5

We could also assume that deaths occur on average 5 days into the month (taking into account birth complications), in which case Ec would be smaller than E by (30-5)/30 *396. Thus you can estimate E=Ec+5/6*396. However, the former assumption of deaths occurring on average half way through the interval is more common - but its good to think about and create different problems for yourself.

2) q0 = D/E, where D is the number of deaths in the first YEAR and E is initial exposed to risk in YEARs.

Again we will assume deaths occur on average half way through each INTERVAL: 

So all you need to to calculate E = sum of central exposed to risk and add back fractions of the year that were not lived by those who died (i.e. 11.5/12*396 + 10/12*139.+...)

3) 3q0 =1-3p0   --> we can now split 3p0 into 1p0 * 1p1 * 1p2.

We have 1p0=1-1q0, which we calculated in 2

1q0 = number of deaths/ initial exposed to risk (so use same approach as 1 and 2 to calculate E from Ec)

1q2 = same as 1q0

commented Apr 5, 2018 by MarioGiuricich (1,350 points)

Thanks for your great answer, Kelly! 

I like the additional, realistic, complication to bring in - deaths occurring on average 5 days into the first month of life.

commented Apr 5, 2018 by wksjem001 (110 points)
Thanks Kelly! Can you please clarify why we multiply by 12? 
It seems more intuitive to divide by 12 when going from years to months?
commented Apr 18, 2018 by MarioGiuricich (1,350 points)

Hi Jemma

For example: if we have one year, how many months is it? This is 1*12 = 12 months. It's a similar application in the above question. We divide by 12 if we go from months to years i.e. 1 month is 1/12 of a year.

Hope this helps!