From: N. Brouard <brouard@ined.fr>
Date: Mon, 2 Aug 2004 10:46:13 +0000 (+0000)
Subject: Documentation on pure mortality analysus (i.e mle=-3)
X-Git-Tag: imach-099s7~577
X-Git-Url: https://henry.ined.fr/git/?a=commitdiff_plain;h=7473b8170276a29b598dd7544d4c274f406e774b;p=.git

Documentation on pure mortality analysus (i.e mle=-3)
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+% -*-Latex-*- 
+% $Id$
+%\documentstyle[11pt,epsf,a4]{article}
+\documentclass[12pt,oneside]{article} 
+\usepackage[T1]{fontenc}
+\usepackage{mathmacr,amsmath} %\usepackage{english}
+\usepackage[francais]{babel} %\selectlanguage{francais}
+\usepackage{graphicx,a4,indentfirst,latexsym,color}
+\usepackage[cyr]{aeguill}
+\usepackage{vmargin}
+\usepackage{amsmath}
+%\usepackage{times}
+%\usepackage{shorttoc}
+%\pagestyle{empty}
+%\pagestyle{headings}
+%\setmarginsrb{3cm}{1.7cm}{2.5cm}{3cm}{0cm}{2cm}{1cm}{1cm}
+%\renewcommand{\baselinestretch}{1.5}
+%\usepackage{fancyheadings}
+%\pagestyle{headings}
+\interfootnotelinepenalty=10000
+
+%\rfoot{\leftmark\\\rightmark}
+%\cfoot{}
+\begin{document}
+%\maketitle
+
+\makeatletter
+\renewcommand{\@biblabel}[1]{}
+\makeatother
+\bibliographystyle{apalike}
+
+\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 died 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. 
+
+
+
+
+
+
+\end{document}
+
+
+
+
+
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+
+
+