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1.2     ! brouard     2: % $Id: docmortweb.tex,v 1.1 2004/08/02 10:46:13 brouard Exp $
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                     24: \begin{document}
                     25: %\maketitle
                     26: 
                     27: \makeatletter
                     28: \renewcommand{\@biblabel}[1]{}
                     29: \makeatother
                     30: \bibliographystyle{apalike}
                     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
1.2     ! brouard    47: date of the death if the subject dies before the last interview
1.1       brouard    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: 
                    126: 
                    127: 
                    128: 
                    129: 
                    130: 
                    131: \end{document}
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