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 <h1 align="center"><font color="#00006A">Computing Health  content="text/html; charset=iso-8859-1">
 Expectancies using IMaCh</font></h1>  <meta name="GENERATOR" content="Microsoft FrontPage Express 2.0">
   <title></title>
 <h1 align="center"><font color="#00006A" size="5">(a Maximum  </head>
 Likelihood Computer Program using Interpolation of Markov Chains)</font></h1>  
   <body bgcolor="#FFFFFF">
 <p align="center">&nbsp;</p>  
   <hr size="3" color="#EC5E5E">
 <p align="center"><a href="http://www.ined.fr/"><img  
 src="logo-ined.gif" border="0" width="151" height="76"></a><img  <h1 align="center"><font color="#00006A">Computing Health
 src="euroreves2.gif" width="151" height="75"></p>  Expectancies using IMaCh</font></h1>
   
 <h3 align="center"><a href="http://www.ined.fr/"><font  <h1 align="center"><font color="#00006A" size="5">(a Maximum
 color="#00006A">INED</font></a><font color="#00006A"> and </font><a  Likelihood Computer Program using Interpolation of Markov Chains)</font></h1>
 href="http://euroreves.ined.fr"><font color="#00006A">EUROREVES</font></a></h3>  
   <p align="center">&nbsp;</p>
 <p align="center"><font color="#00006A" size="4"><strong>March  
 2000</strong></font></p>  <p align="center"><a href="http://www.ined.fr/"><img
   src="logo-ined.gif" border="0" width="151" height="76"></a><img
 <hr size="3" color="#EC5E5E">  src="euroreves2.gif" width="151" height="75"></p>
   
 <p align="center"><font color="#00006A"><strong>Authors of the  <h3 align="center"><a href="http://www.ined.fr/"><font
 program: </strong></font><a href="http://sauvy.ined.fr/brouard"><font  color="#00006A">INED</font></a><font color="#00006A"> and </font><a
 color="#00006A"><strong>Nicolas Brouard</strong></font></a><font  href="http://euroreves.ined.fr"><font color="#00006A">EUROREVES</font></a></h3>
 color="#00006A"><strong>, senior researcher at the </strong></font><a  
 href="http://www.ined.fr"><font color="#00006A"><strong>Institut  <p align="center"><font color="#00006A" size="4"><strong>Version
 National d'Etudes Démographiques</strong></font></a><font  0.71a, March 2002</strong></font></p>
 color="#00006A"><strong> (INED, Paris) in the &quot;Mortality,  
 Health and Epidemiology&quot; Research Unit </strong></font></p>  <hr size="3" color="#EC5E5E">
   
 <p align="center"><font color="#00006A"><strong>and Agnès  <p align="center"><font color="#00006A"><strong>Authors of the
 Lièvre<br clear="left">  program: </strong></font><a href="http://sauvy.ined.fr/brouard"><font
 </strong></font></p>  color="#00006A"><strong>Nicolas Brouard</strong></font></a><font
   color="#00006A"><strong>, senior researcher at the </strong></font><a
 <h4><font color="#00006A">Contribution to the mathematics: C. R.  href="http://www.ined.fr"><font color="#00006A"><strong>Institut
 Heathcote </font><font color="#00006A" size="2">(Australian  National d'Etudes Démographiques</strong></font></a><font
 National University, Canberra).</font></h4>  color="#00006A"><strong> (INED, Paris) in the &quot;Mortality,
   Health and Epidemiology&quot; Research Unit </strong></font></p>
 <h4><font color="#00006A">Contact: Agnès Lièvre (</font><a  
 href="mailto:lievre@ined.fr"><font color="#00006A"><i>lievre@ined.fr</i></font></a><font  <p align="center"><font color="#00006A"><strong>and Agnès
 color="#00006A">) </font></h4>  Lièvre<br clear="left">
   </strong></font></p>
 <hr>  
   <h4><font color="#00006A">Contribution to the mathematics: C. R.
 <ul>  Heathcote </font><font color="#00006A" size="2">(Australian
     <li><a href="#intro">Introduction</a> </li>  National University, Canberra).</font></h4>
     <li>The detailed statistical model (<a href="docmath.pdf">PDF  
         version</a>),(<a href="docmath.ps">ps version</a>) </li>  <h4><font color="#00006A">Contact: Agnès Lièvre (</font><a
     <li><a href="#data">On what kind of data can it be used?</a></li>  href="mailto:lievre@ined.fr"><font color="#00006A"><i>lievre@ined.fr</i></font></a><font
     <li><a href="#datafile">The data file</a> </li>  color="#00006A">) </font></h4>
     <li><a href="#biaspar">The parameter file</a> </li>  
     <li><a href="#running">Running Imach</a> </li>  <hr>
     <li><a href="#output">Output files and graphs</a> </li>  
     <li><a href="#example">Exemple</a> </li>  <ul>
 </ul>      <li><a href="#intro">Introduction</a> </li>
       <li><a href="#data">On what kind of data can it be used?</a></li>
 <hr>      <li><a href="#datafile">The data file</a> </li>
       <li><a href="#biaspar">The parameter file</a> </li>
 <h2><a name="intro"><font color="#00006A">Introduction</font></a></h2>      <li><a href="#running">Running Imach</a> </li>
       <li><a href="#output">Output files and graphs</a> </li>
 <p>This program computes <b>Healthy Life Expectancies</b> from <b>cross-longitudinal      <li><a href="#example">Exemple</a> </li>
 data</b>. Within the family of Health Expectancies (HE),  </ul>
 Disability-free life expectancy (DFLE) is probably the most  
 important index to monitor. In low mortality countries, there is  <hr>
 a fear that when mortality declines, the increase in DFLE is not  
 proportionate to the increase in total Life expectancy. This case  <h2><a name="intro"><font color="#00006A">Introduction</font></a></h2>
 is called the <em>Expansion of morbidity</em>. Most of the data  
 collected today, in particular by the international <a  <p>This program computes <b>Healthy Life Expectancies</b> from <b>cross-longitudinal
 href="http://euroreves/reves">REVES</a> network on Health  data</b> using the methodology pioneered by Laditka and Wolf (1).
 expectancy, and most HE indices based on these data, are <em>cross-sectional</em>.  Within the family of Health Expectancies (HE), Disability-free
 It means that the information collected comes from a single  life expectancy (DFLE) is probably the most important index to
 cross-sectional survey: people from various ages (but mostly old  monitor. In low mortality countries, there is a fear that when
 people) are surveyed on their health status at a single date.  mortality declines, the increase in DFLE is not proportionate to
 Proportion of people disabled at each age, can then be measured  the increase in total Life expectancy. This case is called the <em>Expansion
 at that date. This age-specific prevalence curve is then used to  of morbidity</em>. Most of the data collected today, in
 distinguish, within the stationary population (which, by  particular by the international <a href="http://www.reves.org">REVES</a>
 definition, is the life table estimated from the vital statistics  network on Health expectancy, and most HE indices based on these
 on mortality at the same date), the disable population from the  data, are <em>cross-sectional</em>. It means that the information
 disability-free population. Life expectancy (LE) (or total  collected comes from a single cross-sectional survey: people from
 population divided by the yearly number of births or deaths of  various ages (but mostly old people) are surveyed on their health
 this stationary population) is then decomposed into DFLE and DLE.  status at a single date. Proportion of people disabled at each
 This method of computing HE is usually called the Sullivan method  age, can then be measured at that date. This age-specific
 (from the name of the author who first described it).</p>  prevalence curve is then used to distinguish, within the
   stationary population (which, by definition, is the life table
 <p>Age-specific proportions of people disable are very difficult  estimated from the vital statistics on mortality at the same
 to forecast because each proportion corresponds to historical  date), the disable population from the disability-free
 conditions of the cohort and it is the result of the historical  population. Life expectancy (LE) (or total population divided by
 flows from entering disability and recovering in the past until  the yearly number of births or deaths of this stationary
 today. The age-specific intensities (or incidence rates) of  population) is then decomposed into DFLE and DLE. This method of
 entering disability or recovering a good health, are reflecting  computing HE is usually called the Sullivan method (from the name
 actual conditions and therefore can be used at each age to  of the author who first described it).</p>
 forecast the future of this cohort. For example if a country is  
 improving its technology of prosthesis, the incidence of  <p>Age-specific proportions of people disable are very difficult
 recovering the ability to walk will be higher at each (old) age,  to forecast because each proportion corresponds to historical
 but the prevalence of disability will only slightly reflect an  conditions of the cohort and it is the result of the historical
 improve because the prevalence is mostly affected by the history  flows from entering disability and recovering in the past until
 of the cohort and not by recent period effects. To measure the  today. The age-specific intensities (or incidence rates) of
 period improvement we have to simulate the future of a cohort of  entering disability or recovering a good health, are reflecting
 new-borns entering or leaving at each age the disability state or  actual conditions and therefore can be used at each age to
 dying according to the incidence rates measured today on  forecast the future of this cohort. For example if a country is
 different cohorts. The proportion of people disabled at each age  improving its technology of prosthesis, the incidence of
 in this simulated cohort will be much lower (using the exemple of  recovering the ability to walk will be higher at each (old) age,
 an improvement) that the proportions observed at each age in a  but the prevalence of disability will only slightly reflect an
 cross-sectional survey. This new prevalence curve introduced in a  improve because the prevalence is mostly affected by the history
 life table will give a much more actual and realistic HE level  of the cohort and not by recent period effects. To measure the
 than the Sullivan method which mostly measured the History of  period improvement we have to simulate the future of a cohort of
 health conditions in this country.</p>  new-borns entering or leaving at each age the disability state or
   dying according to the incidence rates measured today on
 <p>Therefore, the main question is how to measure incidence rates  different cohorts. The proportion of people disabled at each age
 from cross-longitudinal surveys? This is the goal of the IMaCH  in this simulated cohort will be much lower (using the exemple of
 program. From your data and using IMaCH you can estimate period  an improvement) that the proportions observed at each age in a
 HE and not only Sullivan's HE. Also the standard errors of the HE  cross-sectional survey. This new prevalence curve introduced in a
 are computed.</p>  life table will give a much more actual and realistic HE level
   than the Sullivan method which mostly measured the History of
 <p>A cross-longitudinal survey consists in a first survey  health conditions in this country.</p>
 (&quot;cross&quot;) where individuals from different ages are  
 interviewed on their health status or degree of disability. At  <p>Therefore, the main question is how to measure incidence rates
 least a second wave of interviews (&quot;longitudinal&quot;)  from cross-longitudinal surveys? This is the goal of the IMaCH
 should measure each new individual health status. Health  program. From your data and using IMaCH you can estimate period
 expectancies are computed from the transitions observed between  HE and not only Sullivan's HE. Also the standard errors of the HE
 waves and are computed for each degree of severity of disability  are computed.</p>
 (number of life states). More degrees you consider, more time is  
 necessary to reach the Maximum Likelihood of the parameters  <p>A cross-longitudinal survey consists in a first survey
 involved in the model. Considering only two states of disability  (&quot;cross&quot;) where individuals from different ages are
 (disable and healthy) is generally enough but the computer  interviewed on their health status or degree of disability. At
 program works also with more health statuses.<br>  least a second wave of interviews (&quot;longitudinal&quot;)
 <br>  should measure each new individual health status. Health
 The simplest model is the multinomial logistic model where <i>pij</i>  expectancies are computed from the transitions observed between
 is the probability to be observed in state <i>j</i> at the second  waves and are computed for each degree of severity of disability
 wave conditional to be observed in state <em>i</em> at the first  (number of life states). More degrees you consider, more time is
 wave. Therefore a simple model is: log<em>(pij/pii)= aij +  necessary to reach the Maximum Likelihood of the parameters
 bij*age+ cij*sex,</em> where '<i>age</i>' is age and '<i>sex</i>'  involved in the model. Considering only two states of disability
 is a covariate. The advantage that this computer program claims,  (disable and healthy) is generally enough but the computer
 comes from that if the delay between waves is not identical for  program works also with more health statuses.<br>
 each individual, or if some individual missed an interview, the  <br>
 information is not rounded or lost, but taken into account using  The simplest model is the multinomial logistic model where <i>pij</i>
 an interpolation or extrapolation. <i>hPijx</i> is the  is the probability to be observed in state <i>j</i> at the second
 probability to be observed in state <i>i</i> at age <i>x+h</i>  wave conditional to be observed in state <em>i</em> at the first
 conditional to the observed state <i>i</i> at age <i>x</i>. The  wave. Therefore a simple model is: log<em>(pij/pii)= aij +
 delay '<i>h</i>' can be split into an exact number (<i>nh*stepm</i>)  bij*age+ cij*sex,</em> where '<i>age</i>' is age and '<i>sex</i>'
 of unobserved intermediate states. This elementary transition (by  is a covariate. The advantage that this computer program claims,
 month or quarter trimester, semester or year) is modeled as a  comes from that if the delay between waves is not identical for
 multinomial logistic. The <i>hPx</i> matrix is simply the matrix  each individual, or if some individual missed an interview, the
 product of <i>nh*stepm</i> elementary matrices and the  information is not rounded or lost, but taken into account using
 contribution of each individual to the likelihood is simply <i>hPijx</i>.  an interpolation or extrapolation. <i>hPijx</i> is the
 <br>  probability to be observed in state <i>i</i> at age <i>x+h</i>
 </p>  conditional to the observed state <i>i</i> at age <i>x</i>. The
   delay '<i>h</i>' can be split into an exact number (<i>nh*stepm</i>)
 <p>The program presented in this manual is a quite general  of unobserved intermediate states. This elementary transition (by
 program named <strong>IMaCh</strong> (for <strong>I</strong>nterpolated  month or quarter trimester, semester or year) is modeled as a
 <strong>MA</strong>rkov <strong>CH</strong>ain), designed to  multinomial logistic. The <i>hPx</i> matrix is simply the matrix
 analyse transition data from longitudinal surveys. The first step  product of <i>nh*stepm</i> elementary matrices and the
 is the parameters estimation of a transition probabilities model  contribution of each individual to the likelihood is simply <i>hPijx</i>.
 between an initial status and a final status. From there, the  <br>
 computer program produces some indicators such as observed and  </p>
 stationary prevalence, life expectancies and their variances and  
 graphs. Our transition model consists in absorbing and  <p>The program presented in this manual is a quite general
 non-absorbing states with the possibility of return across the  program named <strong>IMaCh</strong> (for <strong>I</strong>nterpolated
 non-absorbing states. The main advantage of this package,  <strong>MA</strong>rkov <strong>CH</strong>ain), designed to
 compared to other programs for the analysis of transition data  analyse transition data from longitudinal surveys. The first step
 (For example: Proc Catmod of SAS<sup>®</sup>) is that the whole  is the parameters estimation of a transition probabilities model
 individual information is used even if an interview is missing, a  between an initial status and a final status. From there, the
 status or a date is unknown or when the delay between waves is  computer program produces some indicators such as observed and
 not identical for each individual. The program can be executed  stationary prevalence, life expectancies and their variances and
 according to parameters: selection of a sub-sample, number of  graphs. Our transition model consists in absorbing and
 absorbing and non-absorbing states, number of waves taken in  non-absorbing states with the possibility of return across the
 account (the user inputs the first and the last interview), a  non-absorbing states. The main advantage of this package,
 tolerance level for the maximization function, the periodicity of  compared to other programs for the analysis of transition data
 the transitions (we can compute annual, quaterly or monthly  (For example: Proc Catmod of SAS<sup>®</sup>) is that the whole
 transitions), covariates in the model. It works on Windows or on  individual information is used even if an interview is missing, a
 Unix.<br>  status or a date is unknown or when the delay between waves is
 </p>  not identical for each individual. The program can be executed
   according to parameters: selection of a sub-sample, number of
 <hr>  absorbing and non-absorbing states, number of waves taken in
   account (the user inputs the first and the last interview), a
 <h2><a name="data"><font color="#00006A">On what kind of data can  tolerance level for the maximization function, the periodicity of
 it be used?</font></a></h2>  the transitions (we can compute annual, quarterly or monthly
   transitions), covariates in the model. It works on Windows or on
 <p>The minimum data required for a transition model is the  Unix.<br>
 recording of a set of individuals interviewed at a first date and  </p>
 interviewed again at least one another time. From the  
 observations of an individual, we obtain a follow-up over time of  <hr>
 the occurrence of a specific event. In this documentation, the  
 event is related to health status at older ages, but the program  <p>(1) Laditka, Sarah B. and Wolf, Douglas A. (1998), &quot;New
 can be applied on a lot of longitudinal studies in different  Methods for Analyzing Active Life Expectancy&quot;. <i>Journal of
 contexts. To build the data file explained into the next section,  Aging and Health</i>. Vol 10, No. 2. </p>
 you must have the month and year of each interview and the  
 corresponding health status. But in order to get age, date of  <hr>
 birth (month and year) is required (missing values is allowed for  
 month). Date of death (month and year) is an important  <h2><a name="data"><font color="#00006A">On what kind of data can
 information also required if the individual is dead. Shorter  it be used?</font></a></h2>
 steps (i.e. a month) will more closely take into account the  
 survival time after the last interview.</p>  <p>The minimum data required for a transition model is the
   recording of a set of individuals interviewed at a first date and
 <hr>  interviewed again at least one another time. From the
   observations of an individual, we obtain a follow-up over time of
 <h2><a name="datafile"><font color="#00006A">The data file</font></a></h2>  the occurrence of a specific event. In this documentation, the
   event is related to health status at older ages, but the program
 <p>In this example, 8,000 people have been interviewed in a  can be applied on a lot of longitudinal studies in different
 cross-longitudinal survey of 4 waves (1984, 1986, 1988, 1990).  contexts. To build the data file explained into the next section,
 Some people missed 1, 2 or 3 interviews. Health statuses are  you must have the month and year of each interview and the
 healthy (1) and disable (2). The survey is not a real one. It is  corresponding health status. But in order to get age, date of
 a simulation of the American Longitudinal Survey on Aging. The  birth (month and year) is required (missing values is allowed for
 disability state is defined if the individual missed one of four  month). Date of death (month and year) is an important
 ADL (Activity of daily living, like bathing, eating, walking).  information also required if the individual is dead. Shorter
 Therefore, even is the individuals interviewed in the sample are  steps (i.e. a month) will more closely take into account the
 virtual, the information brought with this sample is close to the  survival time after the last interview.</p>
 situation of the United States. Sex is not recorded is this  
 sample.</p>  <hr>
   
 <p>Each line of the data set (named <a href="data1.txt">data1.txt</a>  <h2><a name="datafile"><font color="#00006A">The data file</font></a></h2>
 in this first example) is an individual record which fields are: </p>  
   <p>In this example, 8,000 people have been interviewed in a
 <ul>  cross-longitudinal survey of 4 waves (1984, 1986, 1988, 1990).
     <li><b>Index number</b>: positive number (field 1) </li>  Some people missed 1, 2 or 3 interviews. Health statuses are
     <li><b>First covariate</b> positive number (field 2) </li>  healthy (1) and disable (2). The survey is not a real one. It is
     <li><b>Second covariate</b> positive number (field 3) </li>  a simulation of the American Longitudinal Survey on Aging. The
     <li><a name="Weight"><b>Weight</b></a>: positive number  disability state is defined if the individual missed one of four
         (field 4) . In most surveys individuals are weighted  ADL (Activity of daily living, like bathing, eating, walking).
         according to the stratification of the sample.</li>  Therefore, even is the individuals interviewed in the sample are
     <li><b>Date of birth</b>: coded as mm/yyyy. Missing dates are  virtual, the information brought with this sample is close to the
         coded as 99/9999 (field 5) </li>  situation of the United States. Sex is not recorded is this
     <li><b>Date of death</b>: coded as mm/yyyy. Missing dates are  sample.</p>
         coded as 99/9999 (field 6) </li>  
     <li><b>Date of first interview</b>: coded as mm/yyyy. Missing  <p>Each line of the data set (named <a href="data1.txt">data1.txt</a>
         dates are coded as 99/9999 (field 7) </li>  in this first example) is an individual record which fields are: </p>
     <li><b>Status at first interview</b>: positive number.  
         Missing values ar coded -1. (field 8) </li>  <ul>
     <li><b>Date of second interview</b>: coded as mm/yyyy.      <li><b>Index number</b>: positive number (field 1) </li>
         Missing dates are coded as 99/9999 (field 9) </li>      <li><b>First covariate</b> positive number (field 2) </li>
     <li><strong>Status at second interview</strong> positive      <li><b>Second covariate</b> positive number (field 3) </li>
         number. Missing values ar coded -1. (field 10) </li>      <li><a name="Weight"><b>Weight</b></a>: positive number
     <li><b>Date of third interview</b>: coded as mm/yyyy. Missing          (field 4) . In most surveys individuals are weighted
         dates are coded as 99/9999 (field 11) </li>          according to the stratification of the sample.</li>
     <li><strong>Status at third interview</strong> positive      <li><b>Date of birth</b>: coded as mm/yyyy. Missing dates are
         number. Missing values ar coded -1. (field 12) </li>          coded as 99/9999 (field 5) </li>
     <li><b>Date of fourth interview</b>: coded as mm/yyyy.      <li><b>Date of death</b>: coded as mm/yyyy. Missing dates are
         Missing dates are coded as 99/9999 (field 13) </li>          coded as 99/9999 (field 6) </li>
     <li><strong>Status at fourth interview</strong> positive      <li><b>Date of first interview</b>: coded as mm/yyyy. Missing
         number. Missing values are coded -1. (field 14) </li>          dates are coded as 99/9999 (field 7) </li>
     <li>etc</li>      <li><b>Status at first interview</b>: positive number.
 </ul>          Missing values ar coded -1. (field 8) </li>
       <li><b>Date of second interview</b>: coded as mm/yyyy.
 <p>&nbsp;</p>          Missing dates are coded as 99/9999 (field 9) </li>
       <li><strong>Status at second interview</strong> positive
 <p>If your longitudinal survey do not include information about          number. Missing values ar coded -1. (field 10) </li>
 weights or covariates, you must fill the column with a number      <li><b>Date of third interview</b>: coded as mm/yyyy. Missing
 (e.g. 1) because a missing field is not allowed.</p>          dates are coded as 99/9999 (field 11) </li>
       <li><strong>Status at third interview</strong> positive
 <hr>          number. Missing values ar coded -1. (field 12) </li>
       <li><b>Date of fourth interview</b>: coded as mm/yyyy.
 <h2><font color="#00006A">Your first example parameter file</font><a          Missing dates are coded as 99/9999 (field 13) </li>
 href="http://euroreves.ined.fr/imach"></a><a name="uio"></a></h2>      <li><strong>Status at fourth interview</strong> positive
           number. Missing values are coded -1. (field 14) </li>
 <h2><a name="biaspar"></a>#Imach version 0.63, February 2000,      <li>etc</li>
 INED-EUROREVES </h2>  </ul>
   
 <p>This is a comment. Comments start with a '#'.</p>  <p>&nbsp;</p>
   
 <h4><font color="#FF0000">First uncommented line</font></h4>  <p>If your longitudinal survey do not include information about
   weights or covariates, you must fill the column with a number
 <pre>title=1st_example datafile=data1.txt lastobs=8600 firstpass=1 lastpass=4</pre>  (e.g. 1) because a missing field is not allowed.</p>
   
 <ul>  <hr>
     <li><b>title=</b> 1st_example is title of the run. </li>  
     <li><b>datafile=</b>data1.txt is the name of the data set.  <h2><font color="#00006A">Your first example parameter file</font><a
         Our example is a six years follow-up survey. It consists  href="http://euroreves.ined.fr/imach"></a><a name="uio"></a></h2>
         in a baseline followed by 3 reinterviews. </li>  
     <li><b>lastobs=</b> 8600 the program is able to run on a  <h2><a name="biaspar"></a>#Imach version 0.71a, March 2002,
         subsample where the last observation number is lastobs.  INED-EUROREVES </h2>
         It can be set a bigger number than the real number of  
         observations (e.g. 100000). In this example, maximisation  <p>This is a comment. Comments start with a '#'.</p>
         will be done on the 8600 first records. </li>  
     <li><b>firstpass=1</b> , <b>lastpass=4 </b>In case of more  <h4><font color="#FF0000">First uncommented line</font></h4>
         than two interviews in the survey, the program can be run  
         on selected transitions periods. firstpass=1 means the  <pre>title=1st_example datafile=data1.txt lastobs=8600 firstpass=1 lastpass=4</pre>
         first interview included in the calculation is the  
         baseline survey. lastpass=4 means that the information  <ul>
         brought by the 4th interview is taken into account.</li>      <li><b>title=</b> 1st_example is title of the run. </li>
 </ul>      <li><b>datafile=</b>data1.txt is the name of the data set.
           Our example is a six years follow-up survey. It consists
 <p>&nbsp;</p>          in a baseline followed by 3 reinterviews. </li>
       <li><b>lastobs=</b> 8600 the program is able to run on a
 <h4><a name="biaspar-2"><font color="#FF0000">Second uncommented          subsample where the last observation number is lastobs.
 line</font></a></h4>          It can be set a bigger number than the real number of
           observations (e.g. 100000). In this example, maximisation
 <pre>ftol=1.e-08 stepm=1 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre>          will be done on the 8600 first records. </li>
       <li><b>firstpass=1</b> , <b>lastpass=4 </b>In case of more
 <ul>          than two interviews in the survey, the program can be run
     <li><b>ftol=1e-8</b> Convergence tolerance on the function          on selected transitions periods. firstpass=1 means the
         value in the maximisation of the likelihood. Choosing a          first interview included in the calculation is the
         correct value for ftol is difficult. 1e-8 is a correct          baseline survey. lastpass=4 means that the information
         value for a 32 bits computer.</li>          brought by the 4th interview is taken into account.</li>
     <li><b>stepm=1</b> Time unit in months for interpolation.  </ul>
         Examples:<ul>  
             <li>If stepm=1, the unit is a month </li>  <p>&nbsp;</p>
             <li>If stepm=4, the unit is a trimester</li>  
             <li>If stepm=12, the unit is a year </li>  <h4><a name="biaspar-2"><font color="#FF0000">Second uncommented
             <li>If stepm=24, the unit is two years</li>  line</font></a></h4>
             <li>... </li>  
         </ul>  <pre>ftol=1.e-08 stepm=1 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre>
     </li>  
     <li><b>ncov=2</b> Number of covariates to be add to the  <ul>
         model. The intercept and the age parameter are counting      <li><b>ftol=1e-8</b> Convergence tolerance on the function
         for 2 covariates. For example, if you want to add gender          value in the maximisation of the likelihood. Choosing a
         in the covariate vector you must write ncov=3 else          correct value for ftol is difficult. 1e-8 is a correct
         ncov=2. </li>          value for a 32 bits computer.</li>
     <li><b>nlstate=2</b> Number of non-absorbing (live) states.      <li><b>stepm=1</b> Time unit in months for interpolation.
         Here we have two alive states: disability-free is coded 1          Examples:<ul>
         and disability is coded 2. </li>              <li>If stepm=1, the unit is a month </li>
     <li><b>ndeath=1</b> Number of absorbing states. The absorbing              <li>If stepm=4, the unit is a trimester</li>
         state death is coded 3. </li>              <li>If stepm=12, the unit is a year </li>
     <li><b>maxwav=4</b> Maximum number of waves. The program can              <li>If stepm=24, the unit is two years</li>
         not include more than 4 interviews. </li>              <li>... </li>
     <li><a name="mle"><b>mle</b></a><b>=1</b> Option for the          </ul>
         Maximisation Likelihood Estimation. <ul>      </li>
             <li>If mle=1 the program does the maximisation and      <li><b>ncov=2</b> Number of covariates in the datafile. </li>
                 the calculation of heath expectancies </li>      <li><b>nlstate=2</b> Number of non-absorbing (alive) states.
             <li>If mle=0 the program only does the calculation of          Here we have two alive states: disability-free is coded 1
                 the health expectancies. </li>          and disability is coded 2. </li>
         </ul>      <li><b>ndeath=1</b> Number of absorbing states. The absorbing
     </li>          state death is coded 3. </li>
     <li><b>weight=0</b> Possibility to add weights. <ul>      <li><b>maxwav=4</b> Number of waves in the datafile.</li>
             <li>If weight=0 no weights are included </li>      <li><a name="mle"><b>mle</b></a><b>=1</b> Option for the
             <li>If weight=1 the maximisation integrates the          Maximisation Likelihood Estimation. <ul>
                 weights which are in field <a href="#Weight">4</a></li>              <li>If mle=1 the program does the maximisation and
         </ul>                  the calculation of health expectancies </li>
     </li>              <li>If mle=0 the program only does the calculation of
 </ul>                  the health expectancies. </li>
           </ul>
 <h4><font color="#FF0000">Guess values for optimization</font><font      </li>
 color="#00006A"> </font></h4>      <li><b>weight=0</b> Possibility to add weights. <ul>
               <li>If weight=0 no weights are included </li>
 <p>You must write the initial guess values of the parameters for              <li>If weight=1 the maximisation integrates the
 optimization. The number of parameters, <em>N</em> depends on the                  weights which are in field <a href="#Weight">4</a></li>
 number of absorbing states and non-absorbing states and on the          </ul>
 number of covariates. <br>      </li>
 <em>N</em> is given by the formula <em>N</em>=(<em>nlstate</em> +  </ul>
 <em>ndeath</em>-1)*<em>nlstate</em>*<em>ncov</em>&nbsp;. <br>  
 <br>  <h4><font color="#FF0000">Covariates</font></h4>
 Thus in the simple case with 2 covariates (the model is log  
 (pij/pii) = aij + bij * age where intercept and age are the two  <p>Intercept and age are systematically included in the model.
 covariates), and 2 health degrees (1 for disability-free and 2  Additional covariates can be included with the command: </p>
 for disability) and 1 absorbing state (3), you must enter 8  
 initials values, a12, b12, a13, b13, a21, b21, a23, b23. You can  <pre>model=<em>list of covariates</em></pre>
 start with zeros as in this example, but if you have a more  
 precise set (for example from an earlier run) you can enter it  <ul>
 and it will speed up them<br>      <li>if<strong> model=. </strong>then no covariates are
 Each of the four lines starts with indices &quot;ij&quot;: <br>          included</li>
 <br>      <li>if <strong>model=V1</strong> the model includes the first
 <b>ij aij bij</b> </p>          covariate (field 2)</li>
       <li>if <strong>model=V2 </strong>the model includes the
 <blockquote>          second covariate (field 3)</li>
     <pre># Guess values of aij and bij in log (pij/pii) = aij + bij * age      <li>if <strong>model=V1+V2 </strong>the model includes the
 12 -14.155633  0.110794           first and the second covariate (fields 2 and 3)</li>
 13  -7.925360  0.032091       <li>if <strong>model=V1*V2 </strong>the model includes the
 21  -1.890135 -0.029473           product of the first and the second covariate (fields 2
 23  -6.234642  0.022315 </pre>          and 3)</li>
 </blockquote>      <li>if <strong>model=V1+V1*age</strong> the model includes
           the product covariate*age</li>
 <p>or, to simplify: </p>  </ul>
   
 <blockquote>  <p>In this example, we have two covariates in the data file
     <pre>12 0.0 0.0  (fields 2 and 3). The number of covariates is defined with
 13 0.0 0.0  statement ncov=2. If now you have 3 covariates in the datafile
 21 0.0 0.0  (fields 2, 3 and 4), you have to set ncov=3. Then you can run the
 23 0.0 0.0</pre>  programme with a new parametrisation taking into account the
 </blockquote>  third covariate. For example, <strong>model=V1+V3 </strong>estimates
   a model with the first and third covariates. More complicated
 <h4><font color="#FF0000">Guess values for computing variances</font></h4>  models can be used, but it will takes more time to converge. With
   a simple model (no covariates), the programme estimates 8
 <p>This is an output if <a href="#mle">mle</a>=1. But it can be  parameters. Adding covariates increases the number of parameters
 used as an input to get the vairous output data files (Health  : 12 for <strong>model=V1, </strong>16 for <strong>model=V1+V1*age
 expectancies, stationary prevalence etc.) and figures without  </strong>and 20 for <strong>model=V1+V2+V3.</strong></p>
 rerunning the rather long maximisation phase (mle=0). </p>  
   <h4><font color="#FF0000">Guess values for optimization</font><font
 <p>The scales are small values for the evaluation of numerical  color="#00006A"> </font></h4>
 derivatives. These derivatives are used to compute the hessian  
 matrix of the parameters, that is the inverse of the covariance  <p>You must write the initial guess values of the parameters for
 matrix, and the variances of health expectancies. Each line  optimization. The number of parameters, <em>N</em> depends on the
 consists in indices &quot;ij&quot; followed by the initial scales  number of absorbing states and non-absorbing states and on the
 (zero to simplify) associated with aij and bij. </p>  number of covariates. <br>
   <em>N</em> is given by the formula <em>N</em>=(<em>nlstate</em> +
 <ul>  <em>ndeath</em>-1)*<em>nlstate</em>*<em>ncov</em>&nbsp;. <br>
     <li>If mle=1 you can enter zeros:</li>  <br>
 </ul>  Thus in the simple case with 2 covariates (the model is log
   (pij/pii) = aij + bij * age where intercept and age are the two
 <blockquote>  covariates), and 2 health degrees (1 for disability-free and 2
     <pre># Scales (for hessian or gradient estimation)  for disability) and 1 absorbing state (3), you must enter 8
 12 0. 0.   initials values, a12, b12, a13, b13, a21, b21, a23, b23. You can
 13 0. 0.   start with zeros as in this example, but if you have a more
 21 0. 0.   precise set (for example from an earlier run) you can enter it
 23 0. 0. </pre>  and it will speed up them<br>
 </blockquote>  Each of the four lines starts with indices &quot;ij&quot;: <b>ij
   aij bij</b> </p>
 <ul>  
     <li>If mle=0 you must enter a covariance matrix (usually  <blockquote>
         obtained from an earlier run).</li>      <pre># Guess values of aij and bij in log (pij/pii) = aij + bij * age
 </ul>  12 -14.155633  0.110794
   13  -7.925360  0.032091
 <h4><font color="#FF0000">Covariance matrix of parameters</font></h4>  21  -1.890135 -0.029473
   23  -6.234642  0.022315 </pre>
 <p>This is an output if <a href="#mle">mle</a>=1. But it can be  </blockquote>
 used as an input to get the vairous output data files (Health  
 expectancies, stationary prevalence etc.) and figures without  <p>or, to simplify (in most of cases it converges but there is no
 rerunning the rather long maximisation phase (mle=0). </p>  warranty!): </p>
   
 <p>Each line starts with indices &quot;ijk&quot; followed by the  <blockquote>
 covariances between aij and bij: </p>      <pre>12 0.0 0.0
   13 0.0 0.0
 <pre>  21 0.0 0.0
    121 Var(a12)   23 0.0 0.0</pre>
    122 Cov(b12,a12)  Var(b12)   </blockquote>
           ...  
    232 Cov(b23,a12)  Cov(b23,b12) ... Var (b23) </pre>  <p> In order to speed up the convergence you can make a first run with
   a large stepm i.e stepm=12 or 24 and then decrease the stepm until
 <ul>  stepm=1 month. If newstepm is the new shorter stepm and stepm can be
     <li>If mle=1 you can enter zeros. </li>  expressed as a multiple of newstepm, like newstepm=n stepm, then the
 </ul>  following approximation holds:
   <pre>aij(stepm) = aij(n . stepm) - ln(n)
 <blockquote>  </pre> and
     <pre># Covariance matrix  <pre>bij(stepm) = bij(n . stepm) .</pre>
 121 0.  
 122 0. 0.  <p> For example if you already ran for a 6 months interval and
 131 0. 0. 0.   got:<br>
 132 0. 0. 0. 0.    <pre># Parameters
 211 0. 0. 0. 0. 0.   12 -13.390179  0.126133
 212 0. 0. 0. 0. 0. 0.   13  -7.493460  0.048069
 231 0. 0. 0. 0. 0. 0. 0.   21   0.575975 -0.041322
 232 0. 0. 0. 0. 0. 0. 0. 0.</pre>  23  -4.748678  0.030626
 </blockquote>  </pre>
   If you now want to get the monthly estimates, you can guess the aij by
 <ul>  substracting ln(6)= 1,7917<br> and running<br>
     <li>If mle=0 you must enter a covariance matrix (usually  <pre>12 -15.18193847  0.126133
         obtained from an earlier run).<br>  13 -9.285219469  0.048069
         </li>  21 -1.215784469 -0.041322
 </ul>  23 -6.540437469  0.030626
   </pre>
 <h4><a name="biaspar-l"></a><font color="#FF0000">last  and get<br>
 uncommented line</font></h4>  <pre>12 -15.029768 0.124347
   13 -8.472981 0.036599
 <pre>agemin=70 agemax=100 bage=50 fage=100</pre>  21 -1.472527 -0.038394
   23 -6.553602 0.029856
 <p>Once we obtained the estimated parameters, the program is able  </br>
 to calculated stationary prevalence, transitions probabilities  which is closer to the results. The approximation is probably useful
 and life expectancies at any age. Choice of age ranges is useful  only for very small intervals and we don't have enough experience to
 for extrapolation. In our data file, ages varies from age 70 to  know if you will speed up the convergence or not.
 102. Setting bage=50 and fage=100, makes the program computing  <pre>         -ln(12)= -2.484
 life expectancy from age bage to age fage. As we use a model, we   -ln(6/1)=-ln(6)= -1.791
 can compute life expectancy on a wider age range than the age   -ln(3/1)=-ln(3)= -1.0986
 range from the data. But the model can be rather wrong on big  -ln(12/6)=-ln(2)= -0.693
 intervals.</p>  </pre>
   
 <p>Similarly, it is possible to get extrapolated stationary  <h4><font color="#FF0000">Guess values for computing variances</font></h4>
 prevalence by age raning from agemin to agemax. </p>  
   <p>This is an output if <a href="#mle">mle</a>=1. But it can be
 <ul>  used as an input to get the various output data files (Health
     <li><b>agemin=</b> Minimum age for calculation of the  expectancies, stationary prevalence etc.) and figures without
         stationary prevalence </li>  rerunning the rather long maximisation phase (mle=0). </p>
     <li><b>agemax=</b> Maximum age for calculation of the  
         stationary prevalence </li>  <p>The scales are small values for the evaluation of numerical
     <li><b>bage=</b> Minimum age for calculation of the health  derivatives. These derivatives are used to compute the hessian
         expectancies </li>  matrix of the parameters, that is the inverse of the covariance
     <li><b>fage=</b> Maximum ages for calculation of the health  matrix, and the variances of health expectancies. Each line
         expectancies </li>  consists in indices &quot;ij&quot; followed by the initial scales
 </ul>  (zero to simplify) associated with aij and bij. </p>
   <ul> <li>If mle=1 you can enter zeros:</li>
 <hr>  <blockquote><pre># Scales (for hessian or gradient estimation)
   12 0. 0.
 <h2><a name="running"></a><font color="#00006A">Running Imach  13 0. 0.
 with this example</font></h2>  21 0. 0.
   23 0. 0. </pre>
 <p>We assume that you entered your <a href="biaspar.txt">1st_example  </blockquote>
 parameter file</a> as explained <a href="#biaspar">above</a>. To      <li>If mle=0 you must enter a covariance matrix (usually
 run the program you should click on the imach.exe icon and enter          obtained from an earlier run).</li>
 the name of the parameter file which is for example <a  </ul>
 href="C:\usr\imach\mle\biaspar.txt">C:\usr\imach\mle\biaspar.txt</a>  
 (you also can click on the biaspar.txt icon located in <br>  <h4><font color="#FF0000">Covariance matrix of parameters</font></h4>
 <a href="C:\usr\imach\mle">C:\usr\imach\mle</a> and put it with  
 the mouse on the imach window).<br>  <p>This is an output if <a href="#mle">mle</a>=1. But it can be
 </p>  used as an input to get the various output data files (Health
   expectancies, stationary prevalence etc.) and figures without
 <p>The time to converge depends on the step unit that you used (1  rerunning the rather long maximisation phase (mle=0). <br>
 month is cpu consuming), on the number of cases, and on the  Each line starts with indices &quot;ijk&quot; followed by the
 number of variables.</p>  covariances between aij and bij:<br>
   <pre>
 <p>The program outputs many files. Most of them are files which     121 Var(a12)
 will be plotted for better understanding.</p>     122 Cov(b12,a12)  Var(b12)
             ...
 <hr>     232 Cov(b23,a12)  Cov(b23,b12) ... Var (b23) </pre>
   <ul>
 <h2><a name="output"><font color="#00006A">Output of the program      <li>If mle=1 you can enter zeros. </li>
 and graphs</font> </a></h2>      <pre># Covariance matrix
   121 0.
 <p>Once the optimization is finished, some graphics can be made  122 0. 0.
 with a grapher. We use Gnuplot which is an interactive plotting  131 0. 0. 0.
 program copyrighted but freely distributed. Imach outputs the  132 0. 0. 0. 0.
 source of a gnuplot file, named 'graph.gp', which can be directly  211 0. 0. 0. 0. 0.
 input into gnuplot.<br>  212 0. 0. 0. 0. 0. 0.
 When the running is finished, the user should enter a caracter  231 0. 0. 0. 0. 0. 0. 0.
 for plotting and output editing. </p>  232 0. 0. 0. 0. 0. 0. 0. 0.</pre>
       <li>If mle=0 you must enter a covariance matrix (usually
 <p>These caracters are:</p>          obtained from an earlier run). </li>
   </ul>
 <ul>  
     <li>'c' to start again the program from the beginning.</li>  <h4><font color="#FF0000">Age range for calculation of stationary
     <li>'g' to made graphics. The output graphs are in GIF format  prevalences and health expectancies</font></h4>
         and you have no control over which is produced. If you  
         want to modify the graphics or make another one, you  <pre>agemin=70 agemax=100 bage=50 fage=100</pre>
         should modify the parameters in the file <b>graph.gp</b>  
         located in imach\bin. A gnuplot reference manual is  <br>Once we obtained the estimated parameters, the program is able
         available <a  to calculated stationary prevalence, transitions probabilities
         href="http://www.cs.dartmouth.edu/gnuplot/gnuplot.html">here</a>.  and life expectancies at any age. Choice of age range is useful
     </li>  for extrapolation. In our data file, ages varies from age 70 to
     <li>'e' opens the <strong>index.htm</strong> file to edit the  102. It is possible to get extrapolated stationary prevalence by
         output files and graphs. </li>  age ranging from agemin to agemax.
     <li>'q' for exiting.</li>  
 </ul>  <br>Setting bage=50 (begin age) and fage=100 (final age), makes
   the program computing life expectancy from age 'bage' to age
 <h5><font size="4"><strong>Results files </strong></font><br>  'fage'. As we use a model, we can interessingly compute life
 <br>  expectancy on a wider age range than the age range from the data.
 <font color="#EC5E5E" size="3"><strong>- </strong></font><a  But the model can be rather wrong on much larger intervals.
 name="Observed prevalence in each state"><font color="#EC5E5E"  Program is limited to around 120 for upper age!
 size="3"><strong>Observed prevalence in each state</strong></font></a><font  <ul>
 color="#EC5E5E" size="3"><strong> (and at first pass)</strong></font><b>:      <li><b>agemin=</b> Minimum age for calculation of the
 </b><a href="prbiaspar.txt"><b>prbiaspar.txt</b></a><br>          stationary prevalence </li>
 </h5>      <li><b>agemax=</b> Maximum age for calculation of the
           stationary prevalence </li>
 <p>The first line is the title and displays each field of the      <li><b>bage=</b> Minimum age for calculation of the health
 file. The first column is age. The fields 2 and 6 are the          expectancies </li>
 proportion of individuals in states 1 and 2 respectively as      <li><b>fage=</b> Maximum age for calculation of the health
 observed during the first exam. Others fields are the numbers of          expectancies </li>
 people in states 1, 2 or more. The number of columns increases if  </ul>
 the number of states is higher than 2.<br>  
 The header of the file is </p>  <h4><a name="Computing"><font color="#FF0000">Computing</font></a><font
   color="#FF0000"> the observed prevalence</font></h4>
 <pre># Age Prev(1) N(1) N Age Prev(2) N(2) N  
 70 1.00000 631 631 70 0.00000 0 631  <pre>begin-prev-date=1/1/1984 end-prev-date=1/6/1988 </pre>
 71 0.99681 625 627 71 0.00319 2 627   
 72 0.97125 1115 1148 72 0.02875 33 1148 </pre>  <br>Statements 'begin-prev-date' and 'end-prev-date' allow to
   select the period in which we calculate the observed prevalences
 <pre># Age Prev(1) N(1) N Age Prev(2) N(2) N  in each state. In this example, the prevalences are calculated on
     70 0.95721 604 631 70 0.04279 27 631</pre>  data survey collected between 1 january 1984 and 1 june 1988.
   <ul>
 <p>It means that at age 70, the prevalence in state 1 is 1.000      <li><strong>begin-prev-date= </strong>Starting date
 and in state 2 is 0.00 . At age 71 the number of individuals in          (day/month/year)</li>
 state 1 is 625 and in state 2 is 2, hence the total number of      <li><strong>end-prev-date= </strong>Final date
 people aged 71 is 625+2=627. <br>          (day/month/year)</li>
 </p>  </ul>
   
 <h5><font color="#EC5E5E" size="3"><b>- Estimated parameters and  <h4><font color="#FF0000">Population- or status-based health
 covariance matrix</b></font><b>: </b><a href="rbiaspar.txt"><b>rbiaspar.txt</b></a></h5>  expectancies</font></h4>
   
 <p>This file contains all the maximisation results: </p>  <pre>pop_based=0</pre>
   
 <pre> Number of iterations=47  <p>The program computes status-based health expectancies, i.e
  -2 log likelihood=46553.005854373667    health expectancies which depends on your initial health state.
  Estimated parameters: a12 = -12.691743 b12 = 0.095819   If you are healthy your healthy life expectancy (e11) is higher
                        a13 = -7.815392   b13 = 0.031851   than if you were disabled (e21, with e11 &gt; e21).<br>
                        a21 = -1.809895 b21 = -0.030470   To compute a healthy life expectancy independant of the initial
                        a23 = -7.838248  b23 = 0.039490    status we have to weight e11 and e21 according to the probability
  Covariance matrix: Var(a12) = 1.03611e-001  to be in each state at initial age or, with other word, according
                     Var(b12) = 1.51173e-005  to the proportion of people in each state.<br>
                     Var(a13) = 1.08952e-001  We prefer computing a 'pure' period healthy life expectancy based
                     Var(b13) = 1.68520e-005    only on the transtion forces. Then the weights are simply the
                     Var(a21) = 4.82801e-001  stationnary prevalences or 'implied' prevalences at the initial
                     Var(b21) = 6.86392e-005  age.<br>
                     Var(a23) = 2.27587e-001  Some other people would like to use the cross-sectional
                     Var(b23) = 3.04465e-005   prevalences (the &quot;Sullivan prevalences&quot;) observed at
  </pre>  the initial age during a period of time <a href="#Computing">defined
   just above</a>. <br>
 <h5><font color="#EC5E5E" size="3"><b>- Transition probabilities</b></font><b>:  
 </b><a href="pijrbiaspar.txt"><b>pijrbiaspar.txt</b></a></h5>  <ul>
       <li><strong>popbased= 0 </strong>Health expectancies are
 <p>Here are the transitions probabilities Pij(x, x+nh) where nh          computed at each age from stationary prevalences
 is a multiple of 2 years. The first column is the starting age x          'expected' at this initial age.</li>
 (from age 50 to 100), the second is age (x+nh) and the others are      <li><strong>popbased= 1 </strong>Health expectancies are
 the transition probabilities p11, p12, p13, p21, p22, p23. For          computed at each age from cross-sectional 'observed'
 example, line 5 of the file is: </p>          prevalence at this initial age. As all the population is
           not observed at the same exact date we define a short
 <pre> 100 106 0.03286 0.23512 0.73202 0.02330 0.19210 0.78460 </pre>          period were the observed prevalence is computed.</li>
   </ul>
 <p>and this means: </p>  
   <h4><font color="#FF0000">Prevalence forecasting ( Experimental)</font></h4>
 <pre>p11(100,106)=0.03286  
 p12(100,106)=0.23512  <pre>starting-proj-date=1/1/1989 final-proj-date=1/1/1992 mov_average=0 </pre>
 p13(100,106)=0.73202  
 p21(100,106)=0.02330  <p>Prevalence and population projections are only available if
 p22(100,106)=0.19210   the interpolation unit is a month, i.e. stepm=1 and if there are
 p22(100,106)=0.78460 </pre>  no covariate. The programme estimates the prevalence in each
   state at a precise date expressed in day/month/year. The
 <h5><font color="#EC5E5E" size="3"><b>- </b></font><a  programme computes one forecasted prevalence a year from a
 name="Stationary prevalence in each state"><font color="#EC5E5E"  starting date (1 january of 1989 in this example) to a final date
 size="3"><b>Stationary prevalence in each state</b></font></a><b>:  (1 january 1992). The statement mov_average allows to compute
 </b><a href="plrbiaspar.txt"><b>plrbiaspar.txt</b></a></h5>  smoothed forecasted prevalences with a five-age moving average
   centered at the mid-age of the five-age period. <br>
 <pre>#Age 1-1 2-2   
 70 0.92274 0.07726   <ul>
 71 0.91420 0.08580       <li><strong>starting-proj-date</strong>= starting date
 72 0.90481 0.09519           (day/month/year) of forecasting</li>
 73 0.89453 0.10547</pre>      <li><strong>final-proj-date= </strong>final date
           (day/month/year) of forecasting</li>
 <p>At age 70 the stationary prevalence is 0.92274 in state 1 and      <li><strong>mov_average</strong>= smoothing with a five-age
 0.07726 in state 2. This stationary prevalence differs from          moving average centered at the mid-age of the five-age
 observed prevalence. Here is the point. The observed prevalence          period. The command<strong> mov_average</strong> takes
 at age 70 results from the incidence of disability, incidence of          value 1 if the prevalences are smoothed and 0 otherwise.</li>
 recovery and mortality which occurred in the past of the cohort.  </ul>
 Stationary prevalence results from a simulation with actual  
 incidences and mortality (estimated from this cross-longitudinal  <h4><font color="#FF0000">Last uncommented line : Population
 survey). It is the best predictive value of the prevalence in the  forecasting </font></h4>
 future if &quot;nothing changes in the future&quot;. This is  
 exactly what demographers do with a Life table. Life expectancy  <pre>popforecast=0 popfile=pyram.txt popfiledate=1/1/1989 last-popfiledate=1/1/1992</pre>
 is the expected mean time to survive if observed mortality rates  
 (incidence of mortality) &quot;remains constant&quot; in the  <p>This command is available if the interpolation unit is a
 future. </p>  month, i.e. stepm=1 and if popforecast=1. From a data file
   including age and number of persons alive at the precise date
 <h5><font color="#EC5E5E" size="3"><b>- Standard deviation of  &#145;popfiledate&#146;, you can forecast the number of persons
 stationary prevalence</b></font><b>: </b><a  in each state until date &#145;last-popfiledate&#146;. In this
 href="vplrbiaspar.txt"><b>vplrbiaspar.txt</b></a></h5>  example, the popfile <a href="pyram.txt"><b>pyram.txt</b></a>
   includes real data which are the Japanese population in 1989.<br>
 <p>The stationary prevalence has to be compared with the observed  
 prevalence by age. But both are statistical estimates and  <ul type="disc">
 subjected to stochastic errors due to the size of the sample, the      <li class="MsoNormal"
 design of the survey, and, for the stationary prevalence to the      style="TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-list: l10 level1 lfo36; tab-stops: list 36.0pt"><b>popforecast=
 model used and fitted. It is possible to compute the standard          0 </b>Option for population forecasting. If
 deviation of the stationary prevalence at each age.</p>          popforecast=1, the programme does the forecasting<b>.</b></li>
       <li class="MsoNormal"
 <h6><font color="#EC5E5E" size="3">Observed and stationary      style="TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-list: l10 level1 lfo36; tab-stops: list 36.0pt"><b>popfile=
 prevalence in state (2=disable) with the confident interval</font>:<b>          </b>name of the population file</li>
 vbiaspar2.gif</b></h6>      <li class="MsoNormal"
       style="TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-list: l10 level1 lfo36; tab-stops: list 36.0pt"><b>popfiledate=</b>
 <p><br>          date of the population population</li>
 This graph exhibits the stationary prevalence in state (2) with      <li class="MsoNormal"
 the confidence interval in red. The green curve is the observed      style="TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-list: l10 level1 lfo36; tab-stops: list 36.0pt"><b>last-popfiledate</b>=
 prevalence (or proportion of individuals in state (2)). Without          date of the last population projection&nbsp;</li>
 discussing the results (it is not the purpose here), we observe  </ul>
 that the green curve is rather below the stationary prevalence.  
 It suggests an increase of the disability prevalence in the  <hr>
 future.</p>  
   <h2><a name="running"></a><font color="#00006A">Running Imach
 <p><img src="vbiaspar2.gif" width="400" height="300"></p>  with this example</font></h2>
   
 <h6><font color="#EC5E5E" size="3"><b>Convergence to the  We assume that you typed in your <a href="biaspar.imach">1st_example
 stationary prevalence of disability</b></font><b>: pbiaspar1.gif</b><br>  parameter file</a> as explained <a href="#biaspar">above</a>.
 <img src="pbiaspar1.gif" width="400" height="300"> </h6>  <br>To run the program you should either:
   <ul> <li> click on the imach.exe icon and enter
 <p>This graph plots the conditional transition probabilities from  the name of the parameter file which is for example <a
 an initial state (1=healthy in red at the bottom, or 2=disable in  href="C:\usr\imach\mle\biaspar.imach">C:\usr\imach\mle\biaspar.imach</a>
 green on top) at age <em>x </em>to the final state 2=disable<em> </em>at  <li> You also can locate the biaspar.imach icon in
 age <em>x+h. </em>Conditional means at the condition to be alive  <a href="C:\usr\imach\mle">C:\usr\imach\mle</a> with your mouse and drag it with
 at age <em>x+h </em>which is <i>hP12x</i> + <em>hP22x</em>. The  the mouse on the imach window).
 curves <i>hP12x/(hP12x</i> + <em>hP22x) </em>and <i>hP22x/(hP12x</i>  <li> With latest version (0.7 and higher) if you setup windows in order to
 + <em>hP22x) </em>converge with <em>h, </em>to the <em>stationary  understand ".imach" extension you can right click the
 prevalence of disability</em>. In order to get the stationary  biaspar.imach icon and either edit with notepad the parameter file or
 prevalence at age 70 we should start the process at an earlier  execute it with imach or whatever.
 age, i.e.50. If the disability state is defined by severe  </ul>  
 disability criteria with only a few chance to recover, then the  
 incidence of recovery is low and the time to convergence is  The time to converge depends on the step unit that you used (1
 probably longer. But we don't have experience yet.</p>  month is cpu consuming), on the number of cases, and on the
   number of variables.
 <h5><font color="#EC5E5E" size="3"><b>- Life expectancies by age  
 and initial health status</b></font><b>: </b><a  <br>The program outputs many files. Most of them are files which
 href="erbiaspar.txt"><b>erbiaspar.txt</b></a></h5>  will be plotted for better understanding.
   
 <pre># Health expectancies   <hr>
 # Age 1-1 1-2 2-1 2-2   
 70 10.7297 2.7809 6.3440 5.9813   <h2><a name="output"><font color="#00006A">Output of the program
 71 10.3078 2.8233 5.9295 5.9959   and graphs</font> </a></h2>
 72 9.8927 2.8643 5.5305 6.0033   
 73 9.4848 2.9036 5.1474 6.0035 </pre>  <p>Once the optimization is finished, some graphics can be made
   with a grapher. We use Gnuplot which is an interactive plotting
 <pre>For example 70 10.7297 2.7809 6.3440 5.9813 means:  program copyrighted but freely distributed. A gnuplot reference
 e11=10.7297 e12=2.7809 e21=6.3440 e22=5.9813</pre>  manual is available <a href="http://www.gnuplot.info/">here</a>. <br>
   When the running is finished, the user should enter a caracter
 <pre><img src="exbiaspar1.gif" width="400" height="300"><img  for plotting and output editing.
 src="exbiaspar2.gif" width="400" height="300"></pre>  
   <br>These caracters are:<br>
 <p>For example, life expectancy of a healthy individual at age 70  
 is 10.73 in the healthy state and 2.78 in the disability state  <ul>
 (=13.51 years). If he was disable at age 70, his life expectancy      <li>'c' to start again the program from the beginning.</li>
 will be shorter, 6.34 in the healthy state and 5.98 in the      <li>'e' opens the <a href="biaspar.htm"><strong>biaspar.htm</strong></a>
 disability state (=12.32 years). The total life expectancy is a          file to edit the output files and graphs. </li>
 weighted mean of both, 13.51 and 12.32; weight is the proportion      <li>'q' for exiting.</li>
 of people disabled at age 70. In order to get a pure period index  </ul>
 (i.e. based only on incidences) we use the <a  
 href="#Stationary prevalence in each state">computed or  <h5><font size="4"><strong>Results files </strong></font><br>
 stationary prevalence</a> at age 70 (i.e. computed from  <br>
 incidences at earlier ages) instead of the <a  <font color="#EC5E5E" size="3"><strong>- </strong></font><a
 href="#Observed prevalence in each state">observed prevalence</a>  name="Observed prevalence in each state"><font color="#EC5E5E"
 (for example at first exam) (<a href="#Health expectancies">see  size="3"><strong>Observed prevalence in each state</strong></font></a><font
 below</a>).</p>  color="#EC5E5E" size="3"><strong> (and at first pass)</strong></font><b>:
   </b><a href="prbiaspar.txt"><b>prbiaspar.txt</b></a><br>
 <h5><font color="#EC5E5E" size="3"><b>- Variances of life  </h5>
 expectancies by age and initial health status</b></font><b>: </b><a  
 href="vrbiaspar.txt"><b>vrbiaspar.txt</b></a></h5>  <p>The first line is the title and displays each field of the
   file. The first column is age. The fields 2 and 6 are the
 <p>For example, the covariances of life expectancies Cov(ei,ej)  proportion of individuals in states 1 and 2 respectively as
 at age 50 are (line 3) </p>  observed during the first exam. Others fields are the numbers of
   people in states 1, 2 or more. The number of columns increases if
 <pre>   Cov(e1,e1)=0.4667  Cov(e1,e2)=0.0605=Cov(e2,e1)  Cov(e2,e2)=0.0183</pre>  the number of states is higher than 2.<br>
   The header of the file is </p>
 <h5><font color="#EC5E5E" size="3"><b>- </b></font><a  
 name="Health expectancies"><font color="#EC5E5E" size="3"><b>Health  <pre># Age Prev(1) N(1) N Age Prev(2) N(2) N
 expectancies</b></font></a><font color="#EC5E5E" size="3"><b>  70 1.00000 631 631 70 0.00000 0 631
 with standard errors in parentheses</b></font><b>: </b><a  71 0.99681 625 627 71 0.00319 2 627
 href="trbiaspar.txt"><font face="Courier New"><b>trbiaspar.txt</b></font></a></h5>  72 0.97125 1115 1148 72 0.02875 33 1148 </pre>
   
 <pre>#Total LEs with variances: e.. (std) e.1 (std) e.2 (std) </pre>  <p>It means that at age 70, the prevalence in state 1 is 1.000
   and in state 2 is 0.00 . At age 71 the number of individuals in
 <pre>70 13.42 (0.18) 10.39 (0.15) 3.03 (0.10)70 13.81 (0.18) 11.28 (0.14) 2.53 (0.09) </pre>  state 1 is 625 and in state 2 is 2, hence the total number of
   people aged 71 is 625+2=627. <br>
 <p>Thus, at age 70 the total life expectancy, e..=13.42 years is  </p>
 the weighted mean of e1.=13.51 and e2.=12.32 by the stationary  
 prevalence at age 70 which are 0.92274 in state 1 and 0.07726 in  <h5><font color="#EC5E5E" size="3"><b>- Estimated parameters and
 state 2, respectively (the sum is equal to one). e.1=10.39 is the  covariance matrix</b></font><b>: </b><a href="rbiaspar.txt"><b>rbiaspar.imach</b></a></h5>
 Disability-free life expectancy at age 70 (it is again a weighted  
 mean of e11 and e21). e.2=3.03 is also the life expectancy at age  <p>This file contains all the maximisation results: </p>
 70 to be spent in the disability state.</p>  
   <pre> -2 log likelihood= 21660.918613445392
 <h6><font color="#EC5E5E" size="3"><b>Total life expectancy by   Estimated parameters: a12 = -12.290174 b12 = 0.092161
 age and health expectancies in states (1=healthy) and (2=disable)</b></font><b>:                         a13 = -9.155590  b13 = 0.046627
 ebiaspar.gif</b></h6>                         a21 = -2.629849  b21 = -0.022030
                          a23 = -7.958519  b23 = 0.042614  
 <p>This figure represents the health expectancies and the total   Covariance matrix: Var(a12) = 1.47453e-001
 life expectancy with the confident interval in dashed curve. </p>                      Var(b12) = 2.18676e-005
                       Var(a13) = 2.09715e-001
 <pre>        <img src="ebiaspar.gif" width="400" height="300"></pre>                      Var(b13) = 3.28937e-005  
                       Var(a21) = 9.19832e-001
 <p>Standard deviations (obtained from the information matrix of                      Var(b21) = 1.29229e-004
 the model) of these quantities are very useful.                      Var(a23) = 4.48405e-001
 Cross-longitudinal surveys are costly and do not involve huge                      Var(b23) = 5.85631e-005
 samples, generally a few thousands; therefore it is very   </pre>
 important to have an idea of the standard deviation of our  
 estimates. It has been a big challenge to compute the Health  <p>By substitution of these parameters in the regression model,
 Expectancy standard deviations. Don't be confuse: life expectancy  we obtain the elementary transition probabilities:</p>
 is, as any expected value, the mean of a distribution; but here  
 we are not computing the standard deviation of the distribution,  <p><img src="pebiaspar1.gif" width="400" height="300"></p>
 but the standard deviation of the estimate of the mean.</p>  
   <h5><font color="#EC5E5E" size="3"><b>- Transition probabilities</b></font><b>:
 <p>Our health expectancies estimates vary according to the sample  </b><a href="pijrbiaspar.txt"><b>pijrbiaspar.txt</b></a></h5>
 size (and the standard deviations give confidence intervals of  
 the estimate) but also according to the model fitted. Let us  <p>Here are the transitions probabilities Pij(x, x+nh) where nh
 explain it in more details.</p>  is a multiple of 2 years. The first column is the starting age x
   (from age 50 to 100), the second is age (x+nh) and the others are
 <p>Choosing a model means ar least two kind of choices. First we  the transition probabilities p11, p12, p13, p21, p22, p23. For
 have to decide the number of disability states. Second we have to  example, line 5 of the file is: </p>
 design, within the logit model family, the model: variables,  
 covariables, confonding factors etc. to be included.</p>  <pre> 100 106 0.02655 0.17622 0.79722 0.01809 0.13678 0.84513 </pre>
   
 <p>More disability states we have, better is our demographical  <p>and this means: </p>
 approach of the disability process, but smaller are the number of  
 transitions between each state and higher is the noise in the  <pre>p11(100,106)=0.02655
 measurement. We do not have enough experiments of the various  p12(100,106)=0.17622
 models to summarize the advantages and disadvantages, but it is  p13(100,106)=0.79722
 important to say that even if we had huge and unbiased samples,  p21(100,106)=0.01809
 the total life expectancy computed from a cross-longitudinal  p22(100,106)=0.13678
 survey, varies with the number of states. If we define only two  p22(100,106)=0.84513 </pre>
 states, alive or dead, we find the usual life expectancy where it  
 is assumed that at each age, people are at the same risk to die.  <h5><font color="#EC5E5E" size="3"><b>- </b></font><a
 If we are differentiating the alive state into healthy and  name="Stationary prevalence in each state"><font color="#EC5E5E"
 disable, and as the mortality from the disability state is higher  size="3"><b>Stationary prevalence in each state</b></font></a><b>:
 than the mortality from the healthy state, we are introducing  </b><a href="plrbiaspar.txt"><b>plrbiaspar.txt</b></a></h5>
 heterogeneity in the risk of dying. The total mortality at each  
 age is the weighted mean of the mortality in each state by the  <pre>#Prevalence
 prevalence in each state. Therefore if the proportion of people  #Age 1-1 2-2
 at each age and in each state is different from the stationary  
 equilibrium, there is no reason to find the same total mortality  #************
 at a particular age. Life expectancy, even if it is a very useful  70 0.90134 0.09866
 tool, has a very strong hypothesis of homogeneity of the  71 0.89177 0.10823
 population. Our main purpose is not to measure differential  72 0.88139 0.11861
 mortality but to measure the expected time in a healthy or  73 0.87015 0.12985 </pre>
 disability state in order to maximise the former and minimize the  
 latter. But the differential in mortality complexifies the  <p>At age 70 the stationary prevalence is 0.90134 in state 1 and
 measurement.</p>  0.09866 in state 2. This stationary prevalence differs from
   observed prevalence. Here is the point. The observed prevalence
 <p>Incidences of disability or recovery are not affected by the  at age 70 results from the incidence of disability, incidence of
 number of states if these states are independant. But incidences  recovery and mortality which occurred in the past of the cohort.
 estimates are dependant on the specification of the model. More  Stationary prevalence results from a simulation with actual
 covariates we added in the logit model better is the model, but  incidences and mortality (estimated from this cross-longitudinal
 some covariates are not well measured, some are confounding  survey). It is the best predictive value of the prevalence in the
 factors like in any statistical model. The procedure to &quot;fit  future if &quot;nothing changes in the future&quot;. This is
 the best model' is similar to logistic regression which itself is  exactly what demographers do with a Life table. Life expectancy
 similar to regression analysis. We haven't yet been sofar because  is the expected mean time to survive if observed mortality rates
 we also have a severe limitation which is the speed of the  (incidence of mortality) &quot;remains constant&quot; in the
 convergence. On a Pentium III, 500 MHz, even the simplest model,  future. </p>
 estimated by month on 8,000 people may take 4 hours to converge.  
 Also, the program is not yet a statistical package, which permits  <h5><font color="#EC5E5E" size="3"><b>- Standard deviation of
 a simple writing of the variables and the model to take into  stationary prevalence</b></font><b>: </b><a
 account in the maximisation. The actual program allows only to  href="vplrbiaspar.txt"><b>vplrbiaspar.txt</b></a></h5>
 add simple variables without covariations, like age+sex but  
 without age+sex+ age*sex . This can be done from the source code  <p>The stationary prevalence has to be compared with the observed
 (you have to change three lines in the source code) but will  prevalence by age. But both are statistical estimates and
 never be general enough. But what is to remember, is that  subjected to stochastic errors due to the size of the sample, the
 incidences or probability of change from one state to another is  design of the survey, and, for the stationary prevalence to the
 affected by the variables specified into the model.</p>  model used and fitted. It is possible to compute the standard
   deviation of the stationary prevalence at each age.</p>
 <p>Also, the age range of the people interviewed has a link with  
 the age range of the life expectancy which can be estimated by  <h5><font color="#EC5E5E" size="3">-Observed and stationary
 extrapolation. If your sample ranges from age 70 to 95, you can  prevalence in state (2=disable) with the confident interval</font>:<b>
 clearly estimate a life expectancy at age 70 and trust your  </b><a href="vbiaspar21.htm"><b>vbiaspar21.gif</b></a></h5>
 confidence interval which is mostly based on your sample size,  
 but if you want to estimate the life expectancy at age 50, you  <p>This graph exhibits the stationary prevalence in state (2)
 should rely in your model, but fitting a logistic model on a age  with the confidence interval in red. The green curve is the
 range of 70-95 and estimating probabilties of transition out of  observed prevalence (or proportion of individuals in state (2)).
 this age range, say at age 50 is very dangerous. At least you  Without discussing the results (it is not the purpose here), we
 should remember that the confidence interval given by the  observe that the green curve is rather below the stationary
 standard deviation of the health expectancies, are under the  prevalence. It suggests an increase of the disability prevalence
 strong assumption that your model is the 'true model', which is  in the future.</p>
 probably not the case.</p>  
   <p><img src="vbiaspar21.gif" width="400" height="300"></p>
 <h5><font color="#EC5E5E" size="3"><b>- Copy of the parameter  
 file</b></font><b>: </b><a href="orbiaspar.txt"><b>orbiaspar.txt</b></a></h5>  <h5><font color="#EC5E5E" size="3"><b>-Convergence to the
   stationary prevalence of disability</b></font><b>: </b><a
 <p>This copy of the parameter file can be useful to re-run the  href="pbiaspar11.gif"><b>pbiaspar11.gif</b></a><br>
 program while saving the old output files. </p>  <img src="pbiaspar11.gif" width="400" height="300"> </h5>
   
 <hr>  <p>This graph plots the conditional transition probabilities from
   an initial state (1=healthy in red at the bottom, or 2=disable in
 <h2><a name="example" </a><font color="#00006A">Trying an example</font></a></h2>  green on top) at age <em>x </em>to the final state 2=disable<em> </em>at
   age <em>x+h. </em>Conditional means at the condition to be alive
 <p>Since you know how to run the program, it is time to test it  at age <em>x+h </em>which is <i>hP12x</i> + <em>hP22x</em>. The
 on your own computer. Try for example on a parameter file named <a  curves <i>hP12x/(hP12x</i> + <em>hP22x) </em>and <i>hP22x/(hP12x</i>
 href="file://../mytry/imachpar.txt">imachpar.txt</a> which is a  + <em>hP22x) </em>converge with <em>h, </em>to the <em>stationary
 copy of <font size="2" face="Courier New">mypar.txt</font>  prevalence of disability</em>. In order to get the stationary
 included in the subdirectory of imach, <font size="2"  prevalence at age 70 we should start the process at an earlier
 face="Courier New">mytry</font>. Edit it to change the name of  age, i.e.50. If the disability state is defined by severe
 the data file to <font size="2" face="Courier New">..\data\mydata.txt</font>  disability criteria with only a few chance to recover, then the
 if you don't want to copy it on the same directory. The file <font  incidence of recovery is low and the time to convergence is
 face="Courier New">mydata.txt</font> is a smaller file of 3,000  probably longer. But we don't have experience yet.</p>
 people but still with 4 waves. </p>  
   <h5><font color="#EC5E5E" size="3"><b>- Life expectancies by age
 <p>Click on the imach.exe icon to open a window. Answer to the  and initial health status</b></font><b>: </b><a
 question:'<strong>Enter the parameter file name:'</strong></p>  href="erbiaspar.txt"><b>erbiaspar.txt</b></a></h5>
   
 <table border="1">  <pre># Health expectancies
     <tr>  # Age 1-1 1-2 2-1 2-2
         <td width="100%"><strong>IMACH, Version 0.63</strong><p><strong>Enter  70 10.9226 3.0401 5.6488 6.2122
         the parameter file name: ..\mytry\imachpar.txt</strong></p>  71 10.4384 3.0461 5.2477 6.1599
         </td>  72 9.9667 3.0502 4.8663 6.1025
     </tr>  73 9.5077 3.0524 4.5044 6.0401 </pre>
 </table>  
   <pre>For example 70 10.4227 3.0402 5.6488 5.7123 means:
 <p>Most of the data files or image files generated, will use the  e11=10.4227 e12=3.0402 e21=5.6488 e22=5.7123</pre>
 'imachpar' string into their name. The running time is about 2-3  
 minutes on a Pentium III. If the execution worked correctly, the  <pre><img src="expbiaspar21.gif" width="400" height="300"><img
 outputs files are created in the current directory, and should be  src="expbiaspar11.gif" width="400" height="300"></pre>
 the same as the mypar files initially included in the directory <font  
 size="2" face="Courier New">mytry</font>.</p>  <p>For example, life expectancy of a healthy individual at age 70
   is 10.42 in the healthy state and 3.04 in the disability state
 <ul>  (=13.46 years). If he was disable at age 70, his life expectancy
     <li><pre><u>Output on the screen</u> The output screen looks like <a  will be shorter, 5.64 in the healthy state and 5.71 in the
 href="imachrun.LOG">this Log file</a>  disability state (=11.35 years). The total life expectancy is a
 #  weighted mean of both, 13.46 and 11.35; weight is the proportion
   of people disabled at age 70. In order to get a pure period index
 title=MLE datafile=..\data\mydata.txt lastobs=3000 firstpass=1 lastpass=3  (i.e. based only on incidences) we use the <a
 ftol=1.000000e-008 stepm=24 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre>  href="#Stationary prevalence in each state">computed or
     </li>  stationary prevalence</a> at age 70 (i.e. computed from
     <li><pre>Total number of individuals= 2965, Agemin = 70.00, Agemax= 100.92  incidences at earlier ages) instead of the <a
   href="#Observed prevalence in each state">observed prevalence</a>
 Warning, no any valid information for:126 line=126  (for example at first exam) (<a href="#Health expectancies">see
 Warning, no any valid information for:2307 line=2307  below</a>).</p>
 Delay (in months) between two waves Min=21 Max=51 Mean=24.495826  
 <font face="Times New Roman">These lines give some warnings on the data file and also some raw statistics on frequencies of transitions.</font>  <h5><font color="#EC5E5E" size="3"><b>- Variances of life
 Age 70 1.=230 loss[1]=3.5% 2.=16 loss[2]=12.5% 1.=222 prev[1]=94.1% 2.=14  expectancies by age and initial health status</b></font><b>: </b><a
  prev[2]=5.9% 1-1=8 11=200 12=7 13=15 2-1=2 21=6 22=7 23=1  href="vrbiaspar.txt"><b>vrbiaspar.txt</b></a></h5>
 Age 102 1.=0 loss[1]=NaNQ% 2.=0 loss[2]=NaNQ% 1.=0 prev[1]=NaNQ% 2.=0 </pre>  
     </li>  <p>For example, the covariances of life expectancies Cov(ei,ej)
 </ul>  at age 50 are (line 3) </p>
   
 <p>&nbsp;</p>  <pre>   Cov(e1,e1)=0.4776  Cov(e1,e2)=0.0488=Cov(e2,e1)  Cov(e2,e2)=0.0424</pre>
   
 <ul>  <h5><font color="#EC5E5E" size="3"><b>- </b></font><a
     <li>Maximisation with the Powell algorithm. 8 directions are  name="Health expectancies"><font color="#EC5E5E" size="3"><b>Health
         given corresponding to the 8 parameters. this can be  expectancies</b></font></a><font color="#EC5E5E" size="3"><b>
         rather long to get convergence.<br>  with standard errors in parentheses</b></font><b>: </b><a
         <font size="1" face="Courier New"><br>  href="trbiaspar.txt"><font face="Courier New"><b>trbiaspar.txt</b></font></a></h5>
         Powell iter=1 -2*LL=11531.405658264877 1 0.000000000000 2  
         0.000000000000 3<br>  <pre>#Total LEs with variances: e.. (std) e.1 (std) e.2 (std) </pre>
         0.000000000000 4 0.000000000000 5 0.000000000000 6  
         0.000000000000 7 <br>  <pre>70 13.26 (0.22) 9.95 (0.20) 3.30 (0.14) </pre>
         0.000000000000 8 0.000000000000<br>  
         1..........2.................3..........4.................5.........<br>  <p>Thus, at age 70 the total life expectancy, e..=13.26 years is
         6................7........8...............<br>  the weighted mean of e1.=13.46 and e2.=11.35 by the stationary
         Powell iter=23 -2*LL=6744.954108371555 1 -12.967632334283  prevalence at age 70 which are 0.90134 in state 1 and 0.09866 in
         <br>  state 2, respectively (the sum is equal to one). e.1=9.95 is the
         2 0.135136681033 3 -7.402109728262 4 0.067844593326 <br>  Disability-free life expectancy at age 70 (it is again a weighted
         5 -0.673601538129 6 -0.006615504377 7 -5.051341616718 <br>  mean of e11 and e21). e.2=3.30 is also the life expectancy at age
         8 0.051272038506<br>  70 to be spent in the disability state.</p>
         1..............2...........3..............4...........<br>  
         5..........6................7...........8.........<br>  <h5><font color="#EC5E5E" size="3"><b>-Total life expectancy by
         #Number of iterations = 23, -2 Log likelihood =  age and health expectancies in states (1=healthy) and (2=disable)</b></font><b>:
         6744.954042573691<br>  </b><a href="ebiaspar1.gif"><b>ebiaspar1.gif</b></a></h5>
         # Parameters<br>  
         12 -12.966061 0.135117 <br>  <p>This figure represents the health expectancies and the total
         13 -7.401109 0.067831 <br>  life expectancy with the confident interval in dashed curve. </p>
         21 -0.672648 -0.006627 <br>  
         23 -5.051297 0.051271 </font><br>  <pre>        <img src="ebiaspar1.gif" width="400" height="300"></pre>
         </li>  
     <li><pre><font size="2">Calculation of the hessian matrix. Wait...  <p>Standard deviations (obtained from the information matrix of
 12345678.12.13.14.15.16.17.18.23.24.25.26.27.28.34.35.36.37.38.45.46.47.48.56.57.58.67.68.78  the model) of these quantities are very useful.
   Cross-longitudinal surveys are costly and do not involve huge
 Inverting the hessian to get the covariance matrix. Wait...  samples, generally a few thousands; therefore it is very
   important to have an idea of the standard deviation of our
 #Hessian matrix#  estimates. It has been a big challenge to compute the Health
 3.344e+002 2.708e+004 -4.586e+001 -3.806e+003 -1.577e+000 -1.313e+002 3.914e-001 3.166e+001   Expectancy standard deviations. Don't be confuse: life expectancy
 2.708e+004 2.204e+006 -3.805e+003 -3.174e+005 -1.303e+002 -1.091e+004 2.967e+001 2.399e+003   is, as any expected value, the mean of a distribution; but here
 -4.586e+001 -3.805e+003 4.044e+002 3.197e+004 2.431e-002 1.995e+000 1.783e-001 1.486e+001   we are not computing the standard deviation of the distribution,
 -3.806e+003 -3.174e+005 3.197e+004 2.541e+006 2.436e+000 2.051e+002 1.483e+001 1.244e+003   but the standard deviation of the estimate of the mean.</p>
 -1.577e+000 -1.303e+002 2.431e-002 2.436e+000 1.093e+002 8.979e+003 -3.402e+001 -2.843e+003   
 -1.313e+002 -1.091e+004 1.995e+000 2.051e+002 8.979e+003 7.420e+005 -2.842e+003 -2.388e+005   <p>Our health expectancies estimates vary according to the sample
 3.914e-001 2.967e+001 1.783e-001 1.483e+001 -3.402e+001 -2.842e+003 1.494e+002 1.251e+004   size (and the standard deviations give confidence intervals of
 3.166e+001 2.399e+003 1.486e+001 1.244e+003 -2.843e+003 -2.388e+005 1.251e+004 1.053e+006   the estimate) but also according to the model fitted. Let us
 # Scales  explain it in more details.</p>
 12 1.00000e-004 1.00000e-006  
 13 1.00000e-004 1.00000e-006  <p>Choosing a model means ar least two kind of choices. First we
 21 1.00000e-003 1.00000e-005  have to decide the number of disability states. Second we have to
 23 1.00000e-004 1.00000e-005  design, within the logit model family, the model: variables,
 # Covariance  covariables, confonding factors etc. to be included.</p>
   1 5.90661e-001  
   2 -7.26732e-003 8.98810e-005  <p>More disability states we have, better is our demographical
   3 8.80177e-002 -1.12706e-003 5.15824e-001  approach of the disability process, but smaller are the number of
   4 -1.13082e-003 1.45267e-005 -6.50070e-003 8.23270e-005  transitions between each state and higher is the noise in the
   5 9.31265e-003 -1.16106e-004 6.00210e-004 -8.04151e-006 1.75753e+000  measurement. We do not have enough experiments of the various
   6 -1.15664e-004 1.44850e-006 -7.79995e-006 1.04770e-007 -2.12929e-002 2.59422e-004  models to summarize the advantages and disadvantages, but it is
   7 1.35103e-003 -1.75392e-005 -6.38237e-004 7.85424e-006 4.02601e-001 -4.86776e-003 1.32682e+000  important to say that even if we had huge and unbiased samples,
   8 -1.82421e-005 2.35811e-007 7.75503e-006 -9.58687e-008 -4.86589e-003 5.91641e-005 -1.57767e-002 1.88622e-004  the total life expectancy computed from a cross-longitudinal
 # agemin agemax for lifexpectancy, bage fage (if mle==0 ie no data nor Max likelihood).  survey, varies with the number of states. If we define only two
   states, alive or dead, we find the usual life expectancy where it
   is assumed that at each age, people are at the same risk to die.
 agemin=70 agemax=100 bage=50 fage=100  If we are differentiating the alive state into healthy and
 Computing prevalence limit: result on file 'plrmypar.txt'   disable, and as the mortality from the disability state is higher
 Computing pij: result on file 'pijrmypar.txt'   than the mortality from the healthy state, we are introducing
 Computing Health Expectancies: result on file 'ermypar.txt'   heterogeneity in the risk of dying. The total mortality at each
 Computing Variance-covariance of DFLEs: file 'vrmypar.txt'   age is the weighted mean of the mortality in each state by the
 Computing Total LEs with variances: file 'trmypar.txt'   prevalence in each state. Therefore if the proportion of people
 Computing Variance-covariance of Prevalence limit: file 'vplrmypar.txt'   at each age and in each state is different from the stationary
 End of Imach  equilibrium, there is no reason to find the same total mortality
 </font></pre>  at a particular age. Life expectancy, even if it is a very useful
     </li>  tool, has a very strong hypothesis of homogeneity of the
 </ul>  population. Our main purpose is not to measure differential
   mortality but to measure the expected time in a healthy or
 <p><font size="3">Once the running is finished, the program  disability state in order to maximise the former and minimize the
 requires a caracter:</font></p>  latter. But the differential in mortality complexifies the
   measurement.</p>
 <table border="1">  
     <tr>  <p>Incidences of disability or recovery are not affected by the
         <td width="100%"><strong>Type g for plotting (available  number of states if these states are independant. But incidences
         if mle=1), e to edit output files, c to start again,</strong><p><strong>and  estimates are dependant on the specification of the model. More
         q for exiting:</strong></p>  covariates we added in the logit model better is the model, but
         </td>  some covariates are not well measured, some are confounding
     </tr>  factors like in any statistical model. The procedure to &quot;fit
 </table>  the best model' is similar to logistic regression which itself is
   similar to regression analysis. We haven't yet been sofar because
 <p><font size="3">First you should enter <strong>g</strong> to  we also have a severe limitation which is the speed of the
 make the figures and then you can edit all the results by typing <strong>e</strong>.  convergence. On a Pentium III, 500 MHz, even the simplest model,
 </font></p>  estimated by month on 8,000 people may take 4 hours to converge.
   Also, the program is not yet a statistical package, which permits
 <ul>  a simple writing of the variables and the model to take into
     <li><u>Outputs files</u> <br>  account in the maximisation. The actual program allows only to
         - index.htm, this file is the master file on which you  add simple variables like age+sex or age+sex+ age*sex but will
         should click first.<br>  never be general enough. But what is to remember, is that
         - Observed prevalence in each state: <a  incidences or probability of change from one state to another is
         href="..\mytry\prmypar.txt">mypar.txt</a> <br>  affected by the variables specified into the model.</p>
         - Estimated parameters and the covariance matrix: <a  
         href="..\mytry\rmypar.txt">rmypar.txt</a> <br>  <p>Also, the age range of the people interviewed has a link with
         - Stationary prevalence in each state: <a  the age range of the life expectancy which can be estimated by
         href="..\mytry\plrmypar.txt">plrmypar.txt</a> <br>  extrapolation. If your sample ranges from age 70 to 95, you can
         - Transition probabilities: <a  clearly estimate a life expectancy at age 70 and trust your
         href="..\mytry\pijrmypar.txt">pijrmypar.txt</a> <br>  confidence interval which is mostly based on your sample size,
         - Copy of the parameter file: <a  but if you want to estimate the life expectancy at age 50, you
         href="..\mytry\ormypar.txt">ormypar.txt</a> <br>  should rely in your model, but fitting a logistic model on a age
         - Life expectancies by age and initial health status: <a  range of 70-95 and estimating probabilties of transition out of
         href="..\mytry\ermypar.txt">ermypar.txt</a> <br>  this age range, say at age 50 is very dangerous. At least you
         - Variances of life expectancies by age and initial  should remember that the confidence interval given by the
         health status: <a href="..\mytry\vrmypar.txt">vrmypar.txt</a>  standard deviation of the health expectancies, are under the
         <br>  strong assumption that your model is the 'true model', which is
         - Health expectancies with their variances: <a  probably not the case.</p>
         href="..\mytry\trmypar.txt">trmypar.txt</a> <br>  
         - Standard deviation of stationary prevalence: <a  <h5><font color="#EC5E5E" size="3"><b>- Copy of the parameter
         href="..\mytry\vplrmypar.txt">vplrmypar.txt</a> <br>  file</b></font><b>: </b><a href="orbiaspar.txt"><b>orbiaspar.txt</b></a></h5>
         <br>  
         </li>  <p>This copy of the parameter file can be useful to re-run the
     <li><u>Graphs</u> <br>  program while saving the old output files. </p>
         <br>  
         -<a href="..\mytry\vmypar1.gif">Observed and stationary  <h5><font color="#EC5E5E" size="3"><b>- Prevalence forecasting</b></font><b>:
         prevalence in state (1) with the confident interval</a> <br>  </b><a href="frbiaspar.txt"><b>frbiaspar.txt</b></a></h5>
         -<a href="..\mytry\vmypar2.gif">Observed and stationary  
         prevalence in state (2) with the confident interval</a> <br>  <p
         -<a href="..\mytry\exmypar1.gif">Health life expectancies  style="TEXT-ALIGN: justify; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt">First,
         by age and initial health state (1)</a> <br>  we have estimated the observed prevalence between 1/1/1984 and
         -<a href="..\mytry\exmypar2.gif">Health life expectancies  1/6/1988. The mean date of interview (weighed average of the
         by age and initial health state (2)</a> <br>  interviews performed between1/1/1984 and 1/6/1988) is estimated
         -<a href="..\mytry\emypar.gif">Total life expectancy by  to be 13/9/1985, as written on the top on the file. Then we
         age and health expectancies in states (1) and (2).</a> </li>  forecast the probability to be in each state. </p>
 </ul>  
   <p
 <p>This software have been partly granted by <a  style="TEXT-ALIGN: justify; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt">Example,
 href="http://euroreves.ined.fr">Euro-REVES</a>, a concerted  at date 1/1/1989 : </p>
 action from the European Union. It will be copyrighted  
 identically to a GNU software product, i.e. program and software  <pre class="MsoNormal"># StartingAge FinalAge P.1 P.2 P.3
 can be distributed freely for non commercial use. Sources are not  # Forecasting at date 1/1/1989
 widely distributed today. You can get them by asking us with a    73 0.807 0.078 0.115</pre>
 simple justification (name, email, institute) <a  
 href="mailto:brouard@ined.fr">mailto:brouard@ined.fr</a> and <a  <p
 href="mailto:lievre@ined.fr">mailto:lievre@ined.fr</a> .</p>  style="TEXT-ALIGN: justify; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt">Since
   the minimum age is 70 on the 13/9/1985, the youngest forecasted
 <p>Latest version (0.63 of 16 march 2000) can be accessed at <a  age is 73. This means that at age a person aged 70 at 13/9/1989
 href="http://euroeves.ined.fr/imach">http://euroreves.ined.fr/imach</a><br>  has a probability to enter state1 of 0.807 at age 73 on 1/1/1989.
 </p>  Similarly, the probability to be in state 2 is 0.078 and the
 </body>  probability to die is 0.115. Then, on the 1/1/1989, the
 </html>  prevalence of disability at age 73 is estimated to be 0.088.</p>
   
   <h5><font color="#EC5E5E" size="3"><b>- Population forecasting</b></font><b>:
   </b><a href="poprbiaspar.txt"><b>poprbiaspar.txt</b></a></h5>
   
   <pre># Age P.1 P.2 P.3 [Population]
   # Forecasting at date 1/1/1989
   75 572685.22 83798.08
   74 621296.51 79767.99
   73 645857.70 69320.60 </pre>
   
   <pre># Forecasting at date 1/1/19909
   76 442986.68 92721.14 120775.48
   75 487781.02 91367.97 121915.51
   74 512892.07 85003.47 117282.76 </pre>
   
   <p>From the population file, we estimate the number of people in
   each state. At age 73, 645857 persons are in state 1 and 69320
   are in state 2. One year latter, 512892 are still in state 1,
   85003 are in state 2 and 117282 died before 1/1/1990.</p>
   
   <hr>
   
   <h2><a name="example"></a><font color="#00006A">Trying an example</font></h2>
   
   <p>Since you know how to run the program, it is time to test it
   on your own computer. Try for example on a parameter file named <a
   href="..\mytry\imachpar.imach">imachpar.imach</a> which is a copy of <font
   size="2" face="Courier New">mypar.imach</font> included in the
   subdirectory of imach, <font size="2" face="Courier New">mytry</font>.
   Edit it to change the name of the data file to <font size="2"
   face="Courier New">..\data\mydata.txt</font> if you don't want to
   copy it on the same directory. The file <font face="Courier New">mydata.txt</font>
   is a smaller file of 3,000 people but still with 4 waves. </p>
   
   <p>Click on the imach.exe icon to open a window. Answer to the
   question:'<strong>Enter the parameter file name:'</strong></p>
   
   <table border="1">
       <tr>
           <td width="100%"><strong>IMACH, Version 0.71</strong><p><strong>Enter
           the parameter file name: ..\mytry\imachpar.imach</strong></p>
           </td>
       </tr>
   </table>
   
   <p>Most of the data files or image files generated, will use the
   'imachpar' string into their name. The running time is about 2-3
   minutes on a Pentium III. If the execution worked correctly, the
   outputs files are created in the current directory, and should be
   the same as the mypar files initially included in the directory <font
   size="2" face="Courier New">mytry</font>.</p>
   
   <ul>
       <li><pre><u>Output on the screen</u> The output screen looks like <a
   href="imachrun.LOG">this Log file</a>
   #
   
   title=MLE datafile=..\data\mydata.txt lastobs=3000 firstpass=1 lastpass=3
   ftol=1.000000e-008 stepm=24 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre>
       </li>
       <li><pre>Total number of individuals= 2965, Agemin = 70.00, Agemax= 100.92
   
   Warning, no any valid information for:126 line=126
   Warning, no any valid information for:2307 line=2307
   Delay (in months) between two waves Min=21 Max=51 Mean=24.495826
   <font face="Times New Roman">These lines give some warnings on the data file and also some raw statistics on frequencies of transitions.</font>
   Age 70 1.=230 loss[1]=3.5% 2.=16 loss[2]=12.5% 1.=222 prev[1]=94.1% 2.=14
    prev[2]=5.9% 1-1=8 11=200 12=7 13=15 2-1=2 21=6 22=7 23=1
   Age 102 1.=0 loss[1]=NaNQ% 2.=0 loss[2]=NaNQ% 1.=0 prev[1]=NaNQ% 2.=0 </pre>
       </li>
   </ul>
   
   <p>&nbsp;</p>
   
   <ul>
       <li>Maximisation with the Powell algorithm. 8 directions are
           given corresponding to the 8 parameters. this can be
           rather long to get convergence.<br>
           <font size="1" face="Courier New"><br>
           Powell iter=1 -2*LL=11531.405658264877 1 0.000000000000 2
           0.000000000000 3<br>
           0.000000000000 4 0.000000000000 5 0.000000000000 6
           0.000000000000 7 <br>
           0.000000000000 8 0.000000000000<br>
           1..........2.................3..........4.................5.........<br>
           6................7........8...............<br>
           Powell iter=23 -2*LL=6744.954108371555 1 -12.967632334283
           <br>
           2 0.135136681033 3 -7.402109728262 4 0.067844593326 <br>
           5 -0.673601538129 6 -0.006615504377 7 -5.051341616718 <br>
           8 0.051272038506<br>
           1..............2...........3..............4...........<br>
           5..........6................7...........8.........<br>
           #Number of iterations = 23, -2 Log likelihood =
           6744.954042573691<br>
           # Parameters<br>
           12 -12.966061 0.135117 <br>
           13 -7.401109 0.067831 <br>
           21 -0.672648 -0.006627 <br>
           23 -5.051297 0.051271 </font><br>
           </li>
       <li><pre><font size="2">Calculation of the hessian matrix. Wait...
   12345678.12.13.14.15.16.17.18.23.24.25.26.27.28.34.35.36.37.38.45.46.47.48.56.57.58.67.68.78
   
   Inverting the hessian to get the covariance matrix. Wait...
   
   #Hessian matrix#
   3.344e+002 2.708e+004 -4.586e+001 -3.806e+003 -1.577e+000 -1.313e+002 3.914e-001 3.166e+001
   2.708e+004 2.204e+006 -3.805e+003 -3.174e+005 -1.303e+002 -1.091e+004 2.967e+001 2.399e+003
   -4.586e+001 -3.805e+003 4.044e+002 3.197e+004 2.431e-002 1.995e+000 1.783e-001 1.486e+001
   -3.806e+003 -3.174e+005 3.197e+004 2.541e+006 2.436e+000 2.051e+002 1.483e+001 1.244e+003
   -1.577e+000 -1.303e+002 2.431e-002 2.436e+000 1.093e+002 8.979e+003 -3.402e+001 -2.843e+003
   -1.313e+002 -1.091e+004 1.995e+000 2.051e+002 8.979e+003 7.420e+005 -2.842e+003 -2.388e+005
   3.914e-001 2.967e+001 1.783e-001 1.483e+001 -3.402e+001 -2.842e+003 1.494e+002 1.251e+004
   3.166e+001 2.399e+003 1.486e+001 1.244e+003 -2.843e+003 -2.388e+005 1.251e+004 1.053e+006
   # Scales
   12 1.00000e-004 1.00000e-006
   13 1.00000e-004 1.00000e-006
   21 1.00000e-003 1.00000e-005
   23 1.00000e-004 1.00000e-005
   # Covariance
     1 5.90661e-001
     2 -7.26732e-003 8.98810e-005
     3 8.80177e-002 -1.12706e-003 5.15824e-001
     4 -1.13082e-003 1.45267e-005 -6.50070e-003 8.23270e-005
     5 9.31265e-003 -1.16106e-004 6.00210e-004 -8.04151e-006 1.75753e+000
     6 -1.15664e-004 1.44850e-006 -7.79995e-006 1.04770e-007 -2.12929e-002 2.59422e-004
     7 1.35103e-003 -1.75392e-005 -6.38237e-004 7.85424e-006 4.02601e-001 -4.86776e-003 1.32682e+000
     8 -1.82421e-005 2.35811e-007 7.75503e-006 -9.58687e-008 -4.86589e-003 5.91641e-005 -1.57767e-002 1.88622e-004
   # agemin agemax for lifexpectancy, bage fage (if mle==0 ie no data nor Max likelihood).
   
   
   agemin=70 agemax=100 bage=50 fage=100
   Computing prevalence limit: result on file 'plrmypar.txt'
   Computing pij: result on file 'pijrmypar.txt'
   Computing Health Expectancies: result on file 'ermypar.txt'
   Computing Variance-covariance of DFLEs: file 'vrmypar.txt'
   Computing Total LEs with variances: file 'trmypar.txt'
   Computing Variance-covariance of Prevalence limit: file 'vplrmypar.txt'
   End of Imach
   </font></pre>
       </li>
   </ul>
   
   <p><font size="3">Once the running is finished, the program
   requires a caracter:</font></p>
   
   <table border="1">
       <tr>
           <td width="100%"><strong>Type e to edit output files, c
           to start again, and q for exiting:</strong></td>
       </tr>
   </table>
   
   <p><font size="3">First you should enter <strong>e </strong>to
   edit the master file mypar.htm. </font></p>
   
   <ul>
       <li><u>Outputs files</u> <br>
           <br>
           - Observed prevalence in each state: <a
           href="..\mytry\prmypar.txt">pmypar.txt</a> <br>
           - Estimated parameters and the covariance matrix: <a
           href="..\mytry\rmypar.txt">rmypar.imach</a> <br>
           - Stationary prevalence in each state: <a
           href="..\mytry\plrmypar.txt">plrmypar.txt</a> <br>
           - Transition probabilities: <a
           href="..\mytry\pijrmypar.txt">pijrmypar.txt</a> <br>
           - Copy of the parameter file: <a
           href="..\mytry\ormypar.txt">ormypar.txt</a> <br>
           - Life expectancies by age and initial health status: <a
           href="..\mytry\ermypar.txt">ermypar.txt</a> <br>
           - Variances of life expectancies by age and initial
           health status: <a href="..\mytry\vrmypar.txt">vrmypar.txt</a>
           <br>
           - Health expectancies with their variances: <a
           href="..\mytry\trmypar.txt">trmypar.txt</a> <br>
           - Standard deviation of stationary prevalence: <a
           href="..\mytry\vplrmypar.txt">vplrmypar.txt</a><br>
           - Prevalences forecasting: <a href="frmypar.txt">frmypar.txt</a>
           <br>
           - Population forecasting (if popforecast=1): <a
           href="poprmypar.txt">poprmypar.txt</a> <br>
           </li>
       <li><u>Graphs</u> <br>
           <br>
           -<a href="../mytry/pemypar1.gif">One-step transition probabilities</a><br>
           -<a href="../mytry/pmypar11.gif">Convergence to the stationary prevalence</a><br>
           -<a href="..\mytry\vmypar11.gif">Observed and stationary prevalence in state (1) with the confident interval</a> <br>
           -<a href="..\mytry\vmypar21.gif">Observed and stationary prevalence in state (2) with the confident interval</a> <br>
           -<a href="..\mytry\expmypar11.gif">Health life expectancies by age and initial health state (1)</a> <br>
           -<a href="..\mytry\expmypar21.gif">Health life expectancies by age and initial health state (2)</a> <br>
           -<a href="..\mytry\emypar1.gif">Total life expectancy by age and health expectancies in states (1) and (2).</a> </li>
   </ul>
   
   <p>This software have been partly granted by <a
   href="http://euroreves.ined.fr">Euro-REVES</a>, a concerted
   action from the European Union. It will be copyrighted
   identically to a GNU software product, i.e. program and software
   can be distributed freely for non commercial use. Sources are not
   widely distributed today. You can get them by asking us with a
   simple justification (name, email, institute) <a
   href="mailto:brouard@ined.fr">mailto:brouard@ined.fr</a> and <a
   href="mailto:lievre@ined.fr">mailto:lievre@ined.fr</a> .</p>
   
   <p>Latest version (0.71d of March 2002) can be accessed at <a
   href="http://euroreves.ined.fr/imach">http://euroreves.ined.fr/imach</a><br>
   </p>
   </body>
   </html>

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