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\section*{Estimation of the force of mortality -independently of the
  initial health state- from cross-longitudinal surveys using IMaCh
  version 0.97}

\newcommand{\thetah}{{\hat{\theta}}}
\newcommand{\thetat}{{\underset{\tilde{~}}{\theta}}}
\newcommand{\thetaht}{{\hat{\underset{\tilde{~}}{\theta}}}}





The starting point (origin of time) of the duration of survival of
each individual is the date of entry in the study, i.e. its age at the
date of the first wave. The time of survival is measured until the
date of the death if the subject dies before the last interview
or until the age at the last interview if the subject is still alive.
The models classically used in analysis of the biographies consider
only the duration of survival and suppose that all the individuals are
interviewed at the same time.  Because of the great disparities of the
ages at the first wave, it is mandatory to take into account the age
in the model of analysis of survival.  The estimated parameters are
calculated with the method of the maximum of probability.


Let be $x_i$ the age at the first interview of individual $i$, $x_i^d$ is the
age at death, $x_i^c$ is the age at the last interview and
$\delta_i$ a dummy variable indicating the status ($\delta_i$=0 if
the individual is dead and 1 otherwise).

If the subject is dead, its contribution to the likelihood is the
product of the survival probability between age $x_i$ and $x_i^d$ by
the probability of dying between age $x_i^d$ and $x_i^d+1$. This
contribution is
\begin{eqnarray}
\mu (x_i^d)\exp\left(-\int_{x_i}^{x_i^d}\mu(u)du\right).
\end{eqnarray} 

The contribution of a surviving suject to the date of the last wave is the
survival probability between age $x_i$ and $x_i^c$, i.e.
\begin{eqnarray}
\exp\left(-\int_{x_i}^{x_i^c}\mu(u)du\right).
\end{eqnarray}


\bigskip The total likelihood $L$ of $n$ independant sujects,
indexed by $i$, is the product of the contributions of each individuals:
\begin{eqnarray}
L = \Pi_{i=1}^{n} \left[\mu
  (x_i^d)\exp\left(-\int_{x_i}^{x_i^d}\mu(u)du\right)\right]^{(1-\delta_i)}\left[\exp\left(-\int_{x_i}^{x_i^c}\mu(u)du\right)\right]^{(\delta_i)} 
\end{eqnarray}
where $\mu(x)$ is the force of mortality at age $x$. By definition,
$\mu(x)dx$ is the probability for an individual aged $x$ to die
between ages $x$ and $x+dx$.

\bigskip The log-likelihood is then
\begin{eqnarray}
\label{e:loglik}
l =
\sum_{i=1}^{n}(1-\delta_i)\left[\left(-\int_{x_i}^{x_i^d}\mu(u)du\right)
+\log(\mu(x_i^d))\right]+\delta_i\left[-\int_{x_i}^{x_i^c}\mu(u)du\right]
\end{eqnarray}

\bigskip

Suppose that the force of mortality is modelled by a Gompertz law
where the two parameters are $\mu_{100}$ and $\theta_1$. The force of
mortality is $\mu(x) = \mu_{100} \exp(\theta_1 (x-100))$. The
parameter $\mu_{100}$ is the force at age 100 ans and $\theta_1$ is
the slope.

\bigskip Then the log-likelihood is
\begin{eqnarray}
\label{e:llgompertz}
  l(\mu_{100},\theta_1) &=& \sum (1-\delta_i) \left[ - \frac{\mu_{100}}{\theta1} 
    \left( \exp(\theta_1x_i^d)-\exp(\theta_1x_i)\right)  
    + \log(\mu_{100}) + \theta_1(x_i^d) \right] \nonumber\\
  &&
  + \delta_i \left[ - \frac{\mu_{100}}{\theta1} \left( \exp(\theta_1x_i^c)
      -\exp(\theta_1x_i)\right)\right]
\end{eqnarray}



\bigskip The usual software of statistics cannot be employed to
implement this parametric model because their procedures making it
possible to carry out biographical analyses do not take into account
the age.  All the estimates and the construction of the confidence
intervals were carried out with a program written in language C. We
used a function of maximization based on the algorithm of Powell
describes in the book { \em Numerical Recipes in C }
(1992).  The matrix of covariance is calculated
like the reverse of the matrix hessienne to the optimum. 






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