Mortality Model¶

Data¶

To obtain the mortality data for each year, we used one table from Statistics Canada:

Past Data: 1996 - 2021¶

For past years, we used Table 13-10-00837-01 from StatCan.

The *.csv file can be downloaded from here: 13100837-eng.zip and is saved as: LEAP/leap/original_data/13100837.csv.

The relevant columns are:

Projected Data: 2021 - 2068¶

Statistics Canada doesn’t provide annual projections for death probabilities, but does provide a projection for specific years (which we call calibration years):

This data can be found in the Statistics Canada Population Projections Technical Report: Table 3.1, Table 3.2, Table 5.2.1, Table 5.2.2, Table 5.2.3.

Model¶

We have mortality data for past years (1996 - 2020), and life expectancy projections for specific future years; but we would like to have mortality data for all future years in our model. Statistics Canada describes how they model mortality here: Methods for Constructing Life Tables for Canada, Provinces and Territories.

In particular, the model they use is the Kannisto-Thatcher model, described in this paper: On the use of Kannisto model for mortality trajectory modelling at very old ages.

According to the Kannisto-Thatcher model, the instantaneous probability of death at age \(x\) is given by:

In mathematical terms, \(\mu(x)\) is the hazard rate. Let’s break this down further. Let \(F_X(x)\) be the cumulative distribution function for age at death, \(X\):

We want the conditional probability of death between age \(x\) and \(x + \Delta x\), given that the person has survived till age \(x\). This is given by:

Recall that for a conditional probability:

and so:

Since \(F_X(x)\) is the cumulative distribution function, by definition it must sum to 1:

Since \(X > x\) if \(x < X \leq x + \Delta x\), we can rewrite the numerator as:

Putting it all together, we have:

Now, we want to find the instantaneous rate of death; the probability of death per unit time. If we take the limit as \(\Delta x \to 0\), we will find the instantaneous probability of death at age \(x\). To get the probability of death per unit time, we need to divide by \(\Delta x\):

You will recognize the derivative of \(F_X(x)\):

and so:

The data in the Statistics Canada mortality table is the probability of death between age \(x\) and \(x + 1\), which is denoted as \(q_x\). This is the same as the probability \(P(x < X \leq x + \Delta x \mid X > x)\), with \(\Delta x = 1\). We would like to solve for \(q_x\), using the Kannisto-Thatcher Equation for \(\mu(x)\). First, we can write \(q_x\) in terms of \(F_X(x)\):

Let us define \(S_X(x)\), the survival function, for convenience:

Then we have:

and so \(\mu(x)\) can be rewritten as:

Solving this first order separable linear differential equation, we have:

Letting \(u(x) := 1 + a e^{\beta x}\), we have:

Now, we can substitute this into the equation for \(q_x\):

If we take the logit of \(q_x\), we have:

Let us now look at \(\sigma^{-1}(q_x) - \sigma^{-1}(q_{x_0})\):

Now, based on fitting the model to empirical data, typically we have [Appendix D, Table 5, [Kannisto, 1994]]:

1. \(\beta \approx \mathcal{O}(10^{-1})\)

2. \(\alpha \approx \mathcal{O}(10^{-5})\)

We can use the binomial approximation to simplify the above equation. Let us take:

In order to use the binomial approximation, we must have:

Since \(x\) represents the age in years, we have \(x \in [0, 120]\). These conditions hold for all ages. Using the binomial approximation, we have:

Going back to our equation for \(\sigma^{-1}(q_x) - \sigma^{-1}(q_{x_0})\), we have:

The last term is much smaller than the first term, and so we can ignore it. Thus, we have:

If \(x_0\) is the age of the person in the starting year of the simulation, then \((\text{year} - \text{year}_0) = (x - x_0)\):

The parameter \(\beta_{\text{sex}}\) is unknown, and so we first need to calculate it. To do so, we set \(\text{year} = \text{year}_C\), the calibration year, and use the Brent root-finding algorithm to optimize \(\beta_{\text{sex}}\) such that the life expectancy in the calibration year (which is known) matches the predicted life expectancy.

Once we have found \(\beta_{\text{sex}}\), we can use this formula to find the projected death probabilities.