Appendix: Kannisto-Thatcher Model

For the age groups 95 to 109 and the open age group of 110 years and over, Statistics Canada models mortality using the Kannisto-Thatcher model, described in: On the use of Kannisto model for mortality trajectory modelling at very old ages. See also: Methods for Constructing Life Tables for Canada, Provinces and Territories.

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

\[\begin{split}\mu(x, \Delta x, t) &= \dfrac{\alpha(t) e^{\beta(t) x}}{1 + \alpha(t) e^{\beta(t) x}} \\ &= \lim_{\Delta x \to 0} \dfrac{P(\text{death between age $x$ and $x + \Delta x$} \mid \text{survived till $x$})}{\Delta x}\end{split}\]

In mathematical terms, \(\mu(x, \Delta x, t)\) is the hazard rate. Let’s break this down further. Let \(F_X(x)\) be the cumulative distribution function for age at death, \(X\):

\[\begin{split}F_X(x) :&= P(\text{age at death} \leq \text{given age}) \\ &= P(X \leq x)\end{split}\]

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:

\[P(x < X \leq x + \Delta x \mid X > x)\]

Recall that for a conditional probability:

\[P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}\]

and so:

\[P(x < X \leq x + \Delta x \mid X > x) = \dfrac{P(x < X \leq x + \Delta x \bigcap X > x)}{P(X > x)}\]

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

\[P(X > x) = 1 - F_X(x)\]

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

\[\begin{split}P(x < X \leq x + \Delta x \bigcap X > x) &= P(x < X \leq x + \Delta x) \\ &= F_X(x + \Delta x) - F_X(x)\end{split}\]

Putting it all together, we have:

\[P(x < X \leq x + \Delta x \mid X > x) = \dfrac{F_X(x + \Delta x) - F_X(x)}{1 - F_X(x)}\]

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 \(\Delta x\):

\[\mu(x, \Delta x, t) = \lim_{\Delta x \to 0} \dfrac{F_X(x + \Delta x) - F_X(x)}{\Delta x (1 - F_X(x))}\]

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

\[\dfrac{d}{dx} F_X(x) = \lim_{\Delta x \to 0} \dfrac{F_X(x + \Delta x) - F_X(x)}{\Delta x}\]

and so:

\[\mu(x, \Delta x, t) = \dfrac{F_X'(x)}{1 - F_X(x)}\]

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 \(q(x, \Delta x, t)\), using the Kannisto-Thatcher Equation for \(\mu(x)\). First, we can write \(q(x, \Delta x, t)\) in terms of \(F_X(x)\):

\[\begin{split}q(x, \Delta x, t) &= P(x < X \leq x + 1 \mid X > x) \\ &= \dfrac{F_X(x + 1) - F_X(x)}{1 - F_X(x)}\end{split}\]

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

\[\begin{split}S_X(x) &:= 1 - F_X(x) \\ &= P(X > x)\end{split}\]

Then we have:

\[\dfrac{dS}{dx} = -F_X'(x)\]

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

\[\mu(x, \Delta x, t) = -\dfrac{dS}{dx}\dfrac{1}{S_X(x)}\]

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

\[F_X(x) = 1 - k (1 + \alpha(t) e^{\beta(t) x})^{-\frac{1}{\beta(t)}}\]
Math: \(F_X(x)\)
\[\begin{split}\int \dfrac{dS}{S_X} &= -\int \mu(x, \Delta x, t) dx \\ \ln(S_X(x)) &= -\int \mu(x, \Delta x, t) dx + C \\ &= -\int \dfrac{\alpha(t) e^{\beta(t) x}}{1 + \alpha(t) e^{\beta(t) x}} dx + C\end{split}\]

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

\[\begin{split}\ln(S_X(x)) &= - \dfrac{1}{\beta(t)} \int \dfrac{du}{u} + C \\ &= - \dfrac{1}{\beta(t)} \ln(u(x, t)) + C \\ S_X(x) &= e^C (1 + \alpha(t) e^{\beta(t) x})^{-\frac{1}{\beta(t)}} \\ &= k (1 + \alpha(t) e^{\beta(t) x})^{-\frac{1}{\beta(t)}} \\ 1 - F_X(x) &= k (1 + \alpha(t) e^{\beta(t) x})^{-\frac{1}{\beta(t)}} \\ F_X(x) &= 1 - k (1 + \alpha(t) e^{\beta(t) x})^{-\frac{1}{\beta(t)}}\end{split}\]

Now, we can substitute this into the equation for \(q(x, \Delta x, t)\):

\[\begin{split}q(x, \Delta x, t) &= \dfrac{F_X(x + \Delta x) - F_X(x)}{1 - F_X(x)} \\ &= 1 - \left( \dfrac{1 + \alpha(t) e^{\beta(t) x}}{1 + \alpha(t) e^{\beta(t) (x + \Delta x)}} \right)^{\frac{1}{\beta(t)}}\end{split}\]
Math: \(q(x, \Delta x, t)\)
\[\begin{split}q(x, \Delta x, t) &= \dfrac{F_X(x + \Delta x) - F_X(x)}{1 - F_X(x)} \\ &= \dfrac{ 1 - k (1 + \alpha(t) e^{\beta(t) (x + \Delta x)})^{-\frac{1}{\beta(t)}} - 1 + k (1 + \alpha(t) e^{\beta(t) x})^{-\frac{1}{\beta(t)}} }{k (1 + \alpha(t) e^{\beta(t) x})^{-\frac{1}{\beta(t)}}} \\ &= \dfrac{ - k (1 + \alpha(t) e^{\beta(t) (x + \Delta x)})^{-\frac{1}{\beta(t)}} + k (1 + \alpha(t) e^{\beta x})^{-\frac{1}{\beta(t)}} }{k (1 + \alpha(t) e^{\beta(t) x})^{-\frac{1}{\beta(t)}}} \\ &= 1 - \left(\dfrac{1 + a e^{\beta(t) (x + \Delta x)}}{1 + \alpha(t) e^{\beta(t) x}}\right)^{-\frac{1}{\beta}} \\ &= 1 - \left( \dfrac{1 + \alpha(t) e^{\beta(t) x}}{1 + \alpha(t) e^{\beta(t) (x + \Delta x)}} \right)^{\frac{1}{\beta(t)}}\end{split}\]

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