Exposed to Risk

Jake

27/04/2022

Central Exposed to Risk

  • This module focuses on the estimation of inital exposed to risk, which we can then use to approximate the inital exposed to risk (using the uniform distribution of deaths assumption):

\[ E_x \approx E_x^c + \frac{1}{2}d_x\]

  • Easy to calculate with complete data, but it is often complicated due to incomplete data:
    • Exact dates of entry and exit are not recorded
    • Definition of age does not correspond exactly to age interval \(x\) to \(x+1\) (for integer x)

Principle of Correspondence

  • A life alive at time \(t\) should be included in the exposure at age \(x\) at time \(t\) iff, were that life to die immediately, they would be counted in the deaths data \(d_x\) at age \(x\).

    • Ensure that age definition is consistent between the numerator and denominator in mortality rate estimates:

    \[ \mu_x = \frac{d_x}{E^c_x}\]

  • Issues arise when the age definition between these values differ. In these cases we take the age definition of \(d_x\) as reference and approximate the appropriate \(E_x^c\)

Age Definitions

Age Last Birthday

  • A life will be considered age \(x\) with their real age being in the range \((x, x+1)\)

Age Nearest Birthday

  • A life will be considered age \(x\) with their real age being in the range \((x-\frac{1}{2}, x+\frac{1}{2})\)

Age Next Birthday

  • A life will be considered age \(x\) with their real age being in the range \((x-1, x)\)

Rate Interval

  • A rate interval is a period of one year during which a life’s recorded age remains the same
  • Life Year Rate Interval
    • Label is dependent on a birthday, e.g nearest, last, next
  • Calender Year Rate Interval
    • Label is dependent on a yearly reference point, e.g nearest birthday at 1 Jan
  • For mortality probability estimates \(\hat{q}\), \(q\) is estimated at the actual age at the start of the rate interval.
    • Age last birthday - Actual age is \((x,x+1)\) in the interval, therefore \(\hat{q}\) is estimated at \(x\).
    • Age nearest birthday - Actual age is \((x-\frac{1}{2},x+\frac{1}{2})\) in the interval, therefore \(\hat{q}\) is estimated at \(x-\frac{1}{2}\).
    • Age next birthday - Actual age is \((x-1,x)\) in the interval, therefore \(\hat{q}\) is estimated at \(x-1\).
  • For mortality rate estimates \(\hat{\mu}\), \(\mu\) is estimated at the actual age in the middle of the rate interval.
    • Age last birthday - Actual age is \((x,x+1)\) in the interval, therefore \(\hat{q}\) is estimated at \(x + \frac{1}{2}\).
    • Age nearest birthday - Actual age is \((x-\frac{1}{2},x+\frac{1}{2})\) in the interval, therefore \(\hat{q}\) is estimated at \(x-\frac{1}{2} + \frac{1}{2} = x\).
    • Age next birthday - Actual age is \((x-1,x)\) in the interval, therefore \(\hat{q}\) is estimated at \(x-1 +\frac{1}{2} = x-\frac{1}{2}\).

Census Approximation

  • We denote the number of people at risk of age \(x\) at time \(t\) as \(P_{x,t}\), so that:

\[ E_x^c=\int_{\text{time of study}}P_{x,t}dt\] * Note that \(P_{x,t}\) is a population aggregate of age \(x\), therefore it includes people entering and exiting the population:

\[ P_{x,t+1} = P_x - \text{deaths}+\text{immigration}-\text{emmigration}\]

  • However, we often don’t have a continuous stream of this information, but rather yearly ‘census’ snapshots. Therefore, assuming \(P_{x,t}\) is linear between census dates, we have the trapezium approximation:

\[ E^c_x\approx \sum_{\text{time of study}} \frac{1}{2}[P_{x,t}+P_{x,t+1}]*\text{Period Length} \]

Adjusting Age Rate Interval

  • We need to find the census data for population that lines up with the time period that the deaths cover.

    • Define the needed census data \(P'_{x,t}\) and construct this with the census data we have \(P_{x,s}\)

  • For Example:

    • Deaths are defined as age nearest birthday
    • Census data are defined via age last birthday

  • \(P'_{x,t}\) is the number of lives under observation aged x nearest birthday at time \(t\), therefore \(x\) birthday in \((t-\frac{1}{2},t+\frac{1}{2})\)

  • \(P_{x,t}\) is the number of lives under observation aged x last birthday at time \(t\), therefore \(x\) birthday in \((t-1,t)\)

  • \(P_{x-1,t}\) is the number of lives under observation aged x last birthday at time \(t-1\), therefore \(x\) birthday in \((t,t+1)\)

  • If we assume that birthdays are uniform over a calender year, we can use the approximation:

\[ P'_{x,t}\approx \frac{1}{2}P_{x-1,t}+\frac{1}{2}P_{x,t}\]

Adjusting Calender Rate Interval

  • A calender rate interval starts on a fixed date in the calender year
    • Need to consider the average age at the start of the interval.
      • For example, assuming uniform distribution over the calender year and the age definition of birthday in same calender year, the average age at the start of the rate interval is \(x-\frac{1}{2}\) as the birthdays can range from the start to end of calender year.
      • Therefore \(\hat{q}_{x-\frac{1}{2}}\) and \(\mu_{\left(x-\frac{1}{2}\right)+\frac{1}{2}=x}\)
  • In the case of calender rate interval we use the census approximation:
    • We need to use \(P_{x+1,t+1}\) rather than \(P_{x,t+1}\) as we do not have access to that information

\[ E^c_x \approx \sum^{K+N}_{t=K}\frac{1}{2}[P_{x,t}+P_{x+1,t+1}]\]

  • We want to force the \(P_{x,t}\) to the middle of the interval for central exposed to risk calculation