Documentation on pure mortality analysus (i.e mle=-3)

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+% -*-Latex-*- \r
+% $Id$\r
+%\documentstyle[11pt,epsf,a4]{article}\r
+\documentclass[12pt,oneside]{article} \r
+\usepackage[T1]{fontenc}\r
+\usepackage{mathmacr,amsmath} %\usepackage{english}\r
+\usepackage[francais]{babel} %\selectlanguage{francais}\r
+\usepackage{graphicx,a4,indentfirst,latexsym,color}\r
+\usepackage[cyr]{aeguill}\r
+\usepackage{vmargin}\r
+\usepackage{amsmath}\r
+%\usepackage{times}\r
+%\usepackage{shorttoc}\r
+%\pagestyle{empty}\r
+%\pagestyle{headings}\r
+%\setmarginsrb{3cm}{1.7cm}{2.5cm}{3cm}{0cm}{2cm}{1cm}{1cm}\r
+%\renewcommand{\baselinestretch}{1.5}\r
+%\usepackage{fancyheadings}\r
+%\pagestyle{headings}\r
+\interfootnotelinepenalty=10000\r
+\r
+%\rfoot{\leftmark\\\rightmark}\r
+%\cfoot{}\r
+\begin{document}\r
+%\maketitle\r
+\r
+\makeatletter\r
+\renewcommand{\@biblabel}[1]{}\r
+\makeatother\r
+\bibliographystyle{apalike}\r
+\r
+\section*{Estimation of the force of mortality -independently of the\r
+  initial health state- from cross-longitudinal surveys using IMaCh\r
+  version 0.97}\r
+\r
+\newcommand{\thetah}{{\hat{\theta}}}\r
+\newcommand{\thetat}{{\underset{\tilde{~}}{\theta}}}\r
+\newcommand{\thetaht}{{\hat{\underset{\tilde{~}}{\theta}}}}\r
+\r
+\r
+\r
+\r
+\r
+The starting point (origin of time) of the duration of survival of\r
+each individual is the date of entry in the study, i.e. its age at the\r
+date of the first wave. The time of survival is measured until the\r
+date of the death if the subject died before the last interview\r
+or until the age at the last interview if the subject is still alive.\r
+The models classically used in analysis of the biographies consider\r
+only the duration of survival and suppose that all the individuals are\r
+interviewed at the same time.  Because of the great disparities of the\r
+ages at the first wave, it is mandatory to take into account the age\r
+in the model of analysis of survival.  The estimated parameters are\r
+calculated with the method of the maximum of probability.\r
+\r
+\r
+Let be $x_i$ the age at the first interview of individual $i$, $x_i^d$ is the\r
+age at death, $x_i^c$ is the age at the last interview and\r
+$\delta_i$ a dummy variable indicating the status ($\delta_i$=0 if\r
+the individual is dead and 1 otherwise).\r
+\r
+If the subject is dead, its contribution to the likelihood is the\r
+product of the survival probability between age $x_i$ and $x_i^d$ by\r
+the probability of dying between age $x_i^d$ and $x_i^d+1$. This\r
+contribution is\r
+\begin{eqnarray}\r
+\mu (x_i^d)\exp\left(-\int_{x_i}^{x_i^d}\mu(u)du\right).\r
+\end{eqnarray} \r
+\r
+The contribution of a surviving suject to the date of the last wave is the\r
+survival probability between age $x_i$ and $x_i^c$, i.e.\r
+\begin{eqnarray}\r
+\exp\left(-\int_{x_i}^{x_i^c}\mu(u)du\right).\r
+\end{eqnarray}\r
+\r
+\r
+\bigskip The total likelihood $L$ of $n$ independant sujects,\r
+indexed by $i$, is the product of the contributions of each individuals:\r
+\begin{eqnarray}\r
+L = \Pi_{i=1}^{n} \left[\mu\r
+  (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)} \r
+\end{eqnarray}\r
+where $\mu(x)$ is the force of mortality at age $x$. By definition,\r
+$\mu(x)dx$ is the probability for an individual aged $x$ to die\r
+between ages $x$ and $x+dx$.\r
+\r
+\bigskip The log-likelihood is then\r
+\begin{eqnarray}\r
+\label{e:loglik}\r
+l =\r
+\sum_{i=1}^{n}(1-\delta_i)\left[\left(-\int_{x_i}^{x_i^d}\mu(u)du\right)\r
++\log(\mu(x_i^d))\right]+\delta_i\left[-\int_{x_i}^{x_i^c}\mu(u)du\right]\r
+\end{eqnarray}\r
+\r
+\bigskip\r
+\r
+Suppose that the force of mortality is modelled by a Gompertz law\r
+where the two parameters are $\mu_{100}$ and $\theta_1$. The force of\r
+mortality is $\mu(x) = \mu_{100} \exp(\theta_1 (x-100))$. The\r
+parameter $\mu_{100}$ is the force at age 100 ans and $\theta_1$ is\r
+the slope.\r
+\r
+\bigskip Then the log-likelihood is\r
+\begin{eqnarray}\r
+\label{e:llgompertz}\r
+  l(\mu_{100},\theta_1) &=& \sum (1-\delta_i) \left[ - \frac{\mu_{100}}{\theta1} \r
+    \left( \exp(\theta_1x_i^d)-\exp(\theta_1x_i)\right)  \r
+    + \log(\mu_{100}) + \theta_1(x_i^d) \right] \nonumber\\\r
+  &&\r
+  + \delta_i \left[ - \frac{\mu_{100}}{\theta1} \left( \exp(\theta_1x_i^c)\r
+      -\exp(\theta_1x_i)\right)\right]\r
+\end{eqnarray}\r
+\r
+\r
+\r
+\bigskip The usual software of statistics cannot be employed to\r
+implement this parametric model because their procedures making it\r
+possible to carry out biographical analyses do not take into account\r
+the age.  All the estimates and the construction of the confidence\r
+intervals were carried out with a program written in language C. We\r
+used a function of maximization based on the algorithm of Powell\r
+describes in the book { \em Numerical Recipes in C }\r
+(1992).  The matrix of covariance is calculated\r
+like the reverse of the matrix hessienne to the optimum. \r
+\r
+\r
+\r
+\r
+\r
+\r
+\end{document}\r
+\r
+\r
+\r
+\r
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+\r
+\r