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   27: \makeatletter
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   31: 
   32: \section*{Estimation of the force of mortality -independently of the
   33:   initial health state- from cross-longitudinal surveys using IMaCh
   34:   version 0.97}
   35: 
   36: \newcommand{\thetah}{{\hat{\theta}}}
   37: \newcommand{\thetat}{{\underset{\tilde{~}}{\theta}}}
   38: \newcommand{\thetaht}{{\hat{\underset{\tilde{~}}{\theta}}}}
   39: 
   40: 
   41: 
   42: 
   43: 
   44: The starting point (origin of time) of the duration of survival of
   45: each individual is the date of entry in the study, i.e. its age at the
   46: date of the first wave. The time of survival is measured until the
   47: date of the death if the subject dies before the last interview
   48: or until the age at the last interview if the subject is still alive.
   49: The models classically used in analysis of the biographies consider
   50: only the duration of survival and suppose that all the individuals are
   51: interviewed at the same time.  Because of the great disparities of the
   52: ages at the first wave, it is mandatory to take into account the age
   53: in the model of analysis of survival.  The estimated parameters are
   54: calculated with the method of the maximum of probability.
   55: 
   56: 
   57: Let be $x_i$ the age at the first interview of individual $i$, $x_i^d$ is the
   58: age at death, $x_i^c$ is the age at the last interview and
   59: $\delta_i$ a dummy variable indicating the status ($\delta_i$=0 if
   60: the individual is dead and 1 otherwise).
   61: 
   62: If the subject is dead, its contribution to the likelihood is the
   63: product of the survival probability between age $x_i$ and $x_i^d$ by
   64: the probability of dying between age $x_i^d$ and $x_i^d+1$. This
   65: contribution is
   66: \begin{eqnarray}
   67: \mu (x_i^d)\exp\left(-\int_{x_i}^{x_i^d}\mu(u)du\right).
   68: \end{eqnarray} 
   69: 
   70: The contribution of a surviving suject to the date of the last wave is the
   71: survival probability between age $x_i$ and $x_i^c$, i.e.
   72: \begin{eqnarray}
   73: \exp\left(-\int_{x_i}^{x_i^c}\mu(u)du\right).
   74: \end{eqnarray}
   75: 
   76: 
   77: \bigskip The total likelihood $L$ of $n$ independant sujects,
   78: indexed by $i$, is the product of the contributions of each individuals:
   79: \begin{eqnarray}
   80: L = \Pi_{i=1}^{n} \left[\mu
   81:   (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)} 
   82: \end{eqnarray}
   83: where $\mu(x)$ is the force of mortality at age $x$. By definition,
   84: $\mu(x)dx$ is the probability for an individual aged $x$ to die
   85: between ages $x$ and $x+dx$.
   86: 
   87: \bigskip The log-likelihood is then
   88: \begin{eqnarray}
   89: \label{e:loglik}
   90: l =
   91: \sum_{i=1}^{n}(1-\delta_i)\left[\left(-\int_{x_i}^{x_i^d}\mu(u)du\right)
   92: +\log(\mu(x_i^d))\right]+\delta_i\left[-\int_{x_i}^{x_i^c}\mu(u)du\right]
   93: \end{eqnarray}
   94: 
   95: \bigskip
   96: 
   97: Suppose that the force of mortality is modelled by a Gompertz law
   98: where the two parameters are $\mu_{100}$ and $\theta_1$. The force of
   99: mortality is $\mu(x) = \mu_{100} \exp(\theta_1 (x-100))$. The
  100: parameter $\mu_{100}$ is the force at age 100 ans and $\theta_1$ is
  101: the slope.
  102: 
  103: \bigskip Then the log-likelihood is
  104: \begin{eqnarray}
  105: \label{e:llgompertz}
  106:   l(\mu_{100},\theta_1) &=& \sum (1-\delta_i) \left[ - \frac{\mu_{100}}{\theta1} 
  107:     \left( \exp(\theta_1x_i^d)-\exp(\theta_1x_i)\right)  
  108:     + \log(\mu_{100}) + \theta_1(x_i^d) \right] \nonumber\\
  109:   &&
  110:   + \delta_i \left[ - \frac{\mu_{100}}{\theta1} \left( \exp(\theta_1x_i^c)
  111:       -\exp(\theta_1x_i)\right)\right]
  112: \end{eqnarray}
  113: 
  114: 
  115: 
  116: \bigskip The usual software of statistics cannot be employed to
  117: implement this parametric model because their procedures making it
  118: possible to carry out biographical analyses do not take into account
  119: the age.  All the estimates and the construction of the confidence
  120: intervals were carried out with a program written in language C. We
  121: used a function of maximization based on the algorithm of Powell
  122: describes in the book { \em Numerical Recipes in C }
  123: (1992).  The matrix of covariance is calculated
  124: like the reverse of the matrix hessienne to the optimum. 
  125: 
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  130: 
  131: \end{document}
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