Return to docmortweb.tex CVS log

Up to [Local Repository] / imach / html / doc

Annotation of imach/html/doc/docmortweb.tex, revision 1.1

1.1     ! brouard     1: % -*-Latex-*- 
        !             2: % $Id: $
        !             3: %\documentstyle[11pt,epsf,a4]{article}
        !             4: \documentclass[12pt,oneside]{article} 
        !             5: \usepackage[T1]{fontenc}
        !             6: \usepackage{mathmacr,amsmath} %\usepackage{english}
        !             7: \usepackage[francais]{babel} %\selectlanguage{francais}
        !             8: \usepackage{graphicx,a4,indentfirst,latexsym,color}
        !             9: \usepackage[cyr]{aeguill}
        !            10: \usepackage{vmargin}
        !            11: \usepackage{amsmath}
        !            12: %\usepackage{times}
        !            13: %\usepackage{shorttoc}
        !            14: %\pagestyle{empty}
        !            15: %\pagestyle{headings}
        !            16: %\setmarginsrb{3cm}{1.7cm}{2.5cm}{3cm}{0cm}{2cm}{1cm}{1cm}
        !            17: %\renewcommand{\baselinestretch}{1.5}
        !            18: %\usepackage{fancyheadings}
        !            19: %\pagestyle{headings}
        !            20: \interfootnotelinepenalty=10000
        !            21: 
        !            22: %\rfoot{\leftmark\\\rightmark}
        !            23: %\cfoot{}
        !            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
        !            47: date of the death if the subject died 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: 
        !           126: 
        !           127: 
        !           128: 
        !           129: 
        !           130: 
        !           131: \end{document}
        !           132: 
        !           133: 
        !           134: 
        !           135: 
        !           136: 
        !           137: 
        !           138: 
        !           139: 
        !           140: