Multiple Decrement Build Methods

Introduction

The property MultiDecBuildMethod specifies the method that is used to build complete cause-specific mortality curves from a finite set of death ratios.It is assumed that a standard single-decrement mortality curve has been built, that is the survival fraction \(\sfrac(x)\) and person-years lived \(\ssfrac(x)\) functions are known for all ages \(x\), and the objective is to build cause-specific survival fractions \(\sfrac_\cause(x)\) \begin{equation} \sfrac_\cause(x) := \int_x^\infty \sfrac(a) \force_\cause(a) da \end{equation} from cause-specific death ratios \(\dratio_\cause^i\) for age interval \([x_i,x_{i+1})\) , \(i=0,\ldots,n-2\) and \([x_{n-1},\infty)\) for \(i=n-1\). The cause-specific survival fraction \(\sfrac_\cause(x)\) is the fraction of the cohort that will die from cause \(\cause\) on or after age \(x\).

Backwards Recurrence

Survival fractions at \(x_i\) can be found using the following backwards recurrence \begin{eqnarray} \sfrac_\cause(x_{n-1}) &=& \dratio_\cause^{n-1} \sfrac(x_{n-1})\\ \sfrac_\cause(x_i) &=&\sfrac_\cause( x_{i+1}) + \dratio_\cause^i \left(\sfrac(x_i)-\sfrac(x_{i+1})\right), \quad i=n-2,\ldots,0 \end{eqnarray} The problem is then to find \(\sfrac_\cause(x)\) for any \(x\) given \(\sfrac_\cause^i\deq\sfrac_\cause(x_i)\).

Interpolating Ratios

The difference in survival fractions is related to the cause-specific force \(\force_\cause(x)\) by \begin{equation} \sfrac_\cause(x)-\sfrac_\cause(x_{i+1}) = \int_x^{x_{i+1}} \sfrac(a) \force_\cause(a) da,\quad x\in [x_i,x_{i+1}) \end{equation} Define the force ratio \(\fratio_\cause\) by \begin{equation} \force_\cause (x) = \fratio_\cause(x) \force(x) \end{equation} and thinking of \(\fratio_\cause\) as a function of \(\sfrac\) the above equation can be rewritten as \begin{equation} \sfrac_\cause(x)-\sfrac_\cause^{i+1} = \int_{\sfrac_{i+1}}^{\sfrac(x)} \fratio_\cause(\sfrac) d\sfrac,\quad x\in [x_i,x_{i+1}) \end{equation} which leads to interpolations based on \(\fratio_\cause(\sfrac)\).

Single Decrement Survival Fraction

The single decrement survival function \begin{equation} \sdsfrac_\cause(x) := \exp\left(-\int_0^x \force_\cause(a) da\right) \end{equation} is given by the forward recurrence \begin{eqnarray} \sdsfrac_\cause(0)&=&1\\ \sdsfrac_\cause(x)&=&\sdsfrac_\cause(x_i)\cdot % \exp\left(\int_{\sfrac_i}^{\sfrac_x} \frac{\fratio_\cause(\sfrac)}{\sfrac} d\sfrac\right),% \quad x\in [x_i,x_{i+1}) \end{eqnarray}

Constant Ratio Build Method

\( \newcommand{\confratio}{\dratio} \)

In the CONSTANT_RATIO build method the force ratio is constant between nodes and equal to the death ratio \begin{equation} \fratio(\sfrac)=\dratio_i, \quad \sfrac\in (\sfrac_{i+1},\sfrac_i]. \end{equation}

Hybrid Ratio Build Method

\( \newcommand{\curvature}{\lambda} \)

In the HYBRID_RATIO build method the force between nodes is a quadratice function of \(\sfrac\) \begin{equation} \fratio(\sfrac)=\frac{\sfrac_i-\sfrac}{\sfrac_i-\sfrac_{i+1}}\fratio_{i+1}+ \frac{\sfrac-\sfrac_{i+1}}{\sfrac_i-\sfrac_{i+1}}\fratio_i+ \frac{\curvature_i}{2} (\sfrac_i-\sfrac)(\sfrac-\sfrac_{i+1}), \quad \sfrac\in (\sfrac_{i+1},\sfrac_i] \end{equation} where \(\sfrac_n=0\). For internal nodes \(i=1,\ldots,n-1\) \begin{equation} \fratio_i=\left\{ \begin{array}{cc} \frac{\sfrac_{i+1}-\sfrac_i}{\sfrac_{i+1}-\sfrac_{i-1}} \confratio_{i-1} +\frac{\sfrac_i-\sfrac_{i-1}}{\sfrac_{i+1}-\sfrac_{i-1}} \confratio_i & 1\le i < n-1\\ \frac{\sfrac_i}{\sfrac_{i-1}} \confratio_{i-1} +\frac{\sfrac_{i-1}-\sfrac_i}{\sfrac_{i-1}}\confratio_i & i=n-1 \end{array}\right. \end{equation} and end nodes \(i=0,n\) \begin{eqnarray} \fratio_0&=&2\confratio_0-\fratio_1\\ \fratio_n&=&2\confratio_{n-1}-\fratio_{n-1} \end{eqnarray} The curvature is given by \begin{equation} \frac{\curvature_i}{2}=\frac{6}{(\sfrac_i-\sfrac_{i+1})^2}\left(-\confratio_i + \frac{\fratio_i+\fratio_{i+1}}{2}\right),\quad 1\le i < n-1 \end{equation} for internal intervals and \begin{equation} \curvature_0=\curvature_{n-1}=0 \end{equation} for end intervals