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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>  <h4><font color="#FF0000">Guess values for computing variances</font></h4>
   
 <ul>  <p>This is an output if <a href="#mle">mle</a>=1. But it can be
     <li>If mle=1 you can enter zeros. </li>  used as an input to get the various output data files (Health
 </ul>  expectancies, stationary prevalence etc.) and figures without
   rerunning the rather long maximisation phase (mle=0). </p>
 <blockquote>  
     <pre># Covariance matrix  <p>The scales are small values for the evaluation of numerical
 121 0.  derivatives. These derivatives are used to compute the hessian
 122 0. 0.  matrix of the parameters, that is the inverse of the covariance
 131 0. 0. 0.   matrix, and the variances of health expectancies. Each line
 132 0. 0. 0. 0.   consists in indices &quot;ij&quot; followed by the initial scales
 211 0. 0. 0. 0. 0.   (zero to simplify) associated with aij and bij. </p>
 212 0. 0. 0. 0. 0. 0.   
 231 0. 0. 0. 0. 0. 0. 0.   <ul>
 232 0. 0. 0. 0. 0. 0. 0. 0.</pre>      <li>If mle=1 you can enter zeros:</li>
 </blockquote>  </ul>
   
 <ul>  <blockquote>
     <li>If mle=0 you must enter a covariance matrix (usually      <pre># Scales (for hessian or gradient estimation)
         obtained from an earlier run).<br>  12 0. 0.
         </li>  13 0. 0.
 </ul>  21 0. 0.
   23 0. 0. </pre>
 <h4><a name="biaspar-l"></a><font color="#FF0000">last  </blockquote>
 uncommented line</font></h4>  
   <ul>
 <pre>agemin=70 agemax=100 bage=50 fage=100</pre>      <li>If mle=0 you must enter a covariance matrix (usually
           obtained from an earlier run).</li>
 <p>Once we obtained the estimated parameters, the program is able  </ul>
 to calculated stationary prevalence, transitions probabilities  
 and life expectancies at any age. Choice of age ranges is useful  <h4><font color="#FF0000">Covariance matrix of parameters</font></h4>
 for extrapolation. In our data file, ages varies from age 70 to  
 102. Setting bage=50 and fage=100, makes the program computing  <p>This is an output if <a href="#mle">mle</a>=1. But it can be
 life expectancy from age bage to age fage. As we use a model, we  used as an input to get the various output data files (Health
 can compute life expectancy on a wider age range than the age  expectancies, stationary prevalence etc.) and figures without
 range from the data. But the model can be rather wrong on big  rerunning the rather long maximisation phase (mle=0). </p>
 intervals.</p>  
   <p>Each line starts with indices &quot;ijk&quot; followed by the
 <p>Similarly, it is possible to get extrapolated stationary  covariances between aij and bij: </p>
 prevalence by age raning from agemin to agemax. </p>  
   <pre>
 <ul>     121 Var(a12)
     <li><b>agemin=</b> Minimum age for calculation of the     122 Cov(b12,a12)  Var(b12)
         stationary prevalence </li>            ...
     <li><b>agemax=</b> Maximum age for calculation of the     232 Cov(b23,a12)  Cov(b23,b12) ... Var (b23) </pre>
         stationary prevalence </li>  
     <li><b>bage=</b> Minimum age for calculation of the health  <ul>
         expectancies </li>      <li>If mle=1 you can enter zeros. </li>
     <li><b>fage=</b> Maximum ages for calculation of the health  </ul>
         expectancies </li>  
 </ul>  <blockquote>
       <pre># Covariance matrix
 <hr>  121 0.
   122 0. 0.
 <h2><a name="running"></a><font color="#00006A">Running Imach  131 0. 0. 0.
 with this example</font></h2>  132 0. 0. 0. 0.
   211 0. 0. 0. 0. 0.
 <p>We assume that you entered your <a href="biaspar.txt">1st_example  212 0. 0. 0. 0. 0. 0.
 parameter file</a> as explained <a href="#biaspar">above</a>. To  231 0. 0. 0. 0. 0. 0. 0.
 run the program you should click on the imach.exe icon and enter  232 0. 0. 0. 0. 0. 0. 0. 0.</pre>
 the name of the parameter file which is for example <a  </blockquote>
 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>  <ul>
 <a href="C:\usr\imach\mle">C:\usr\imach\mle</a> and put it with      <li>If mle=0 you must enter a covariance matrix (usually
 the mouse on the imach window).<br>          obtained from an earlier run).<br>
 </p>          </li>
   </ul>
 <p>The time to converge depends on the step unit that you used (1  
 month is cpu consuming), on the number of cases, and on the  <h4><font color="#FF0000">Age range for calculation of stationary
 number of variables.</p>  prevalences and health expectancies</font></h4>
   
 <p>The program outputs many files. Most of them are files which  <pre>agemin=70 agemax=100 bage=50 fage=100</pre>
 will be plotted for better understanding.</p>  
   <p>Once we obtained the estimated parameters, the program is able
 <hr>  to calculated stationary prevalence, transitions probabilities
   and life expectancies at any age. Choice of age range is useful
 <h2><a name="output"><font color="#00006A">Output of the program  for extrapolation. In our data file, ages varies from age 70 to
 and graphs</font> </a></h2>  102. It is possible to get extrapolated stationary prevalence by
   age ranging from agemin to agemax. </p>
 <p>Once the optimization is finished, some graphics can be made  
 with a grapher. We use Gnuplot which is an interactive plotting  <p>Setting bage=50 (begin age) and fage=100 (final age), makes
 program copyrighted but freely distributed. Imach outputs the  the program computing life expectancy from age 'bage' to age
 source of a gnuplot file, named 'graph.gp', which can be directly  'fage'. As we use a model, we can interessingly compute life
 input into gnuplot.<br>  expectancy on a wider age range than the age range from the data.
 When the running is finished, the user should enter a caracter  But the model can be rather wrong on much larger intervals.
 for plotting and output editing. </p>  Program is limited to around 120 for upper age!</p>
   
 <p>These caracters are:</p>  <ul>
       <li><b>agemin=</b> Minimum age for calculation of the
 <ul>          stationary prevalence </li>
     <li>'c' to start again the program from the beginning.</li>      <li><b>agemax=</b> Maximum age for calculation of the
     <li>'g' to made graphics. The output graphs are in GIF format          stationary prevalence </li>
         and you have no control over which is produced. If you      <li><b>bage=</b> Minimum age for calculation of the health
         want to modify the graphics or make another one, you          expectancies </li>
         should modify the parameters in the file <b>graph.gp</b>      <li><b>fage=</b> Maximum age for calculation of the health
         located in imach\bin. A gnuplot reference manual is          expectancies </li>
         available <a  </ul>
         href="http://www.cs.dartmouth.edu/gnuplot/gnuplot.html">here</a>.  
     </li>  <h4><a name="Computing"><font color="#FF0000">Computing</font></a><font
     <li>'e' opens the <strong>index.htm</strong> file to edit the  color="#FF0000"> the observed prevalence</font></h4>
         output files and graphs. </li>  
     <li>'q' for exiting.</li>  <pre>begin-prev-date=1/1/1984 end-prev-date=1/6/1988 </pre>
 </ul>  
   <p>Statements 'begin-prev-date' and 'end-prev-date' allow to
 <h5><font size="4"><strong>Results files </strong></font><br>  select the period in which we calculate the observed prevalences
 <br>  in each state. In this example, the prevalences are calculated on
 <font color="#EC5E5E" size="3"><strong>- </strong></font><a  data survey collected between 1 january 1984 and 1 june 1988. </p>
 name="Observed prevalence in each state"><font color="#EC5E5E"  
 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><strong>begin-prev-date= </strong>Starting date
 </b><a href="prbiaspar.txt"><b>prbiaspar.txt</b></a><br>          (day/month/year)</li>
 </h5>      <li><strong>end-prev-date= </strong>Final date
           (day/month/year)</li>
 <p>The first line is the title and displays each field of the  </ul>
 file. The first column is age. The fields 2 and 6 are the  
 proportion of individuals in states 1 and 2 respectively as  <h4><font color="#FF0000">Population- or status-based health
 observed during the first exam. Others fields are the numbers of  expectancies</font></h4>
 people in states 1, 2 or more. The number of columns increases if  
 the number of states is higher than 2.<br>  <pre>pop_based=0</pre>
 The header of the file is </p>  
   <p>The program computes status-based health expectancies, i.e
 <pre># Age Prev(1) N(1) N Age Prev(2) N(2) N  health expectancies which depends on your initial health state.
 70 1.00000 631 631 70 0.00000 0 631  If you are healthy your healthy life expectancy (e11) is higher
 71 0.99681 625 627 71 0.00319 2 627   than if you were disabled (e21, with e11 &gt; e21).<br>
 72 0.97125 1115 1148 72 0.02875 33 1148 </pre>  To compute a healthy life expectancy independant of the initial
   status we have to weight e11 and e21 according to the probability
 <pre># Age Prev(1) N(1) N Age Prev(2) N(2) N  to be in each state at initial age or, with other word, according
     70 0.95721 604 631 70 0.04279 27 631</pre>  to the proportion of people in each state.<br>
   We prefer computing a 'pure' period healthy life expectancy based
 <p>It means that at age 70, the prevalence in state 1 is 1.000  only on the transtion forces. Then the weights are simply the
 and in state 2 is 0.00 . At age 71 the number of individuals in  stationnary prevalences or 'implied' prevalences at the initial
 state 1 is 625 and in state 2 is 2, hence the total number of  age.<br>
 people aged 71 is 625+2=627. <br>  Some other people would like to use the cross-sectional
 </p>  prevalences (the &quot;Sullivan prevalences&quot;) observed at
   the initial age during a period of time <a href="#Computing">defined
 <h5><font color="#EC5E5E" size="3"><b>- Estimated parameters and  just above</a>. </p>
 covariance matrix</b></font><b>: </b><a href="rbiaspar.txt"><b>rbiaspar.txt</b></a></h5>  
   <ul>
 <p>This file contains all the maximisation results: </p>      <li><strong>popbased= 0 </strong>Health expectancies are
           computed at each age from stationary prevalences
 <pre> Number of iterations=47          'expected' at this initial age.</li>
  -2 log likelihood=46553.005854373667        <li><strong>popbased= 1 </strong>Health expectancies are
  Estimated parameters: a12 = -12.691743 b12 = 0.095819           computed at each age from cross-sectional 'observed'
                        a13 = -7.815392   b13 = 0.031851           prevalence at this initial age. As all the population is
                        a21 = -1.809895 b21 = -0.030470           not observed at the same exact date we define a short
                        a23 = -7.838248  b23 = 0.039490            period were the observed prevalence is computed.</li>
  Covariance matrix: Var(a12) = 1.03611e-001  </ul>
                     Var(b12) = 1.51173e-005  
                     Var(a13) = 1.08952e-001  <h4><font color="#FF0000">Prevalence forecasting ( Experimental)</font></h4>
                     Var(b13) = 1.68520e-005    
                     Var(a21) = 4.82801e-001  <pre>starting-proj-date=1/1/1989 final-proj-date=1/1/1992 mov_average=0 </pre>
                     Var(b21) = 6.86392e-005  
                     Var(a23) = 2.27587e-001  <p>Prevalence and population projections are only available if
                     Var(b23) = 3.04465e-005   the interpolation unit is a month, i.e. stepm=1 and if there are
  </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>- Transition probabilities</b></font><b>:  programme computes one forecasted prevalence a year from a
 </b><a href="pijrbiaspar.txt"><b>pijrbiaspar.txt</b></a></h5>  starting date (1 january of 1989 in this example) to a final date
   (1 january 1992). The statement mov_average allows to compute
 <p>Here are the transitions probabilities Pij(x, x+nh) where nh  smoothed forecasted prevalences with a five-age moving average
 is a multiple of 2 years. The first column is the starting age x  centered at the mid-age of the five-age period. </p>
 (from age 50 to 100), the second is age (x+nh) and the others are  
 the transition probabilities p11, p12, p13, p21, p22, p23. For  <ul>
 example, line 5 of the file is: </p>      <li><strong>starting-proj-date</strong>= starting date
           (day/month/year) of forecasting</li>
 <pre> 100 106 0.03286 0.23512 0.73202 0.02330 0.19210 0.78460 </pre>      <li><strong>final-proj-date= </strong>final date
           (day/month/year) of forecasting</li>
 <p>and this means: </p>      <li><strong>mov_average</strong>= smoothing with a five-age
           moving average centered at the mid-age of the five-age
 <pre>p11(100,106)=0.03286          period. The command<strong> mov_average</strong> takes
 p12(100,106)=0.23512          value 1 if the prevalences are smoothed and 0 otherwise.</li>
 p13(100,106)=0.73202  </ul>
 p21(100,106)=0.02330  
 p22(100,106)=0.19210   <h4><font color="#FF0000">Last uncommented line : Population
 p22(100,106)=0.78460 </pre>  forecasting </font></h4>
   
 <h5><font color="#EC5E5E" size="3"><b>- </b></font><a  <pre>popforecast=0 popfile=pyram.txt popfiledate=1/1/1989 last-popfiledate=1/1/1992</pre>
 name="Stationary prevalence in each state"><font color="#EC5E5E"  
 size="3"><b>Stationary prevalence in each state</b></font></a><b>:  <p>This command is available if the interpolation unit is a
 </b><a href="plrbiaspar.txt"><b>plrbiaspar.txt</b></a></h5>  month, i.e. stepm=1 and if popforecast=1. From a data file
   including age and number of persons alive at the precise date
 <pre>#Age 1-1 2-2   &#145;popfiledate&#146;, you can forecast the number of persons
 70 0.92274 0.07726   in each state until date &#145;last-popfiledate&#146;. In this
 71 0.91420 0.08580   example, the popfile <a href="pyram.txt"><b>pyram.txt</b></a>
 72 0.90481 0.09519   includes real data which are the Japanese population in 1989.</p>
 73 0.89453 0.10547</pre>  
   <ul type="disc">
 <p>At age 70 the stationary prevalence is 0.92274 in state 1 and      <li class="MsoNormal"
 0.07726 in state 2. This stationary prevalence differs from      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=
 observed prevalence. Here is the point. The observed prevalence          0 </b>Option for population forecasting. If
 at age 70 results from the incidence of disability, incidence of          popforecast=1, the programme does the forecasting<b>.</b></li>
 recovery and mortality which occurred in the past of the cohort.      <li class="MsoNormal"
 Stationary prevalence results from a simulation with actual      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=
 incidences and mortality (estimated from this cross-longitudinal          </b>name of the population file</li>
 survey). It is the best predictive value of the prevalence in the      <li class="MsoNormal"
 future if &quot;nothing changes in the future&quot;. This is      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>
 exactly what demographers do with a Life table. Life expectancy          date of the population population</li>
 is the expected mean time to survive if observed mortality rates      <li class="MsoNormal"
 (incidence of mortality) &quot;remains constant&quot; in 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>last-popfiledate</b>=
 future. </p>          date of the last population projection&nbsp;</li>
   </ul>
 <h5><font color="#EC5E5E" size="3"><b>- Standard deviation of  
 stationary prevalence</b></font><b>: </b><a  <hr>
 href="vplrbiaspar.txt"><b>vplrbiaspar.txt</b></a></h5>  
   <h2><a name="running"></a><font color="#00006A">Running Imach
 <p>The stationary prevalence has to be compared with the observed  with this example</font></h2>
 prevalence by age. But both are statistical estimates and  
 subjected to stochastic errors due to the size of the sample, the  <p>We assume that you entered your <a href="biaspar.imach">1st_example
 design of the survey, and, for the stationary prevalence to the  parameter file</a> as explained <a href="#biaspar">above</a>. To
 model used and fitted. It is possible to compute the standard  run the program you should click on the imach.exe icon and enter
 deviation of the stationary prevalence at each age.</p>  the name of the parameter file which is for example <a
   href="C:\usr\imach\mle\biaspar.txt">C:\usr\imach\mle\biaspar.txt</a>
 <h6><font color="#EC5E5E" size="3">Observed and stationary  (you also can click on the biaspar.txt icon located in <br>
 prevalence in state (2=disable) with the confident interval</font>:<b>  <a href="C:\usr\imach\mle">C:\usr\imach\mle</a> and put it with
 vbiaspar2.gif</b></h6>  the mouse on the imach window).<br>
   </p>
 <p><br>  
 This graph exhibits the stationary prevalence in state (2) with  <p>The time to converge depends on the step unit that you used (1
 the confidence interval in red. The green curve is the observed  month is cpu consuming), on the number of cases, and on the
 prevalence (or proportion of individuals in state (2)). Without  number of variables.</p>
 discussing the results (it is not the purpose here), we observe  
 that the green curve is rather below the stationary prevalence.  <p>The program outputs many files. Most of them are files which
 It suggests an increase of the disability prevalence in the  will be plotted for better understanding.</p>
 future.</p>  
   <hr>
 <p><img src="vbiaspar2.gif" width="400" height="300"></p>  
   <h2><a name="output"><font color="#00006A">Output of the program
 <h6><font color="#EC5E5E" size="3"><b>Convergence to the  and graphs</font> </a></h2>
 stationary prevalence of disability</b></font><b>: pbiaspar1.gif</b><br>  
 <img src="pbiaspar1.gif" width="400" height="300"> </h6>  <p>Once the optimization is finished, some graphics can be made
   with a grapher. We use Gnuplot which is an interactive plotting
 <p>This graph plots the conditional transition probabilities from  program copyrighted but freely distributed. A gnuplot reference
 an initial state (1=healthy in red at the bottom, or 2=disable in  manual is available <a href="http://www.gnuplot.info/">here</a>. <br>
 green on top) at age <em>x </em>to the final state 2=disable<em> </em>at  When the running is finished, the user should enter a caracter
 age <em>x+h. </em>Conditional means at the condition to be alive  for plotting and output editing. </p>
 at age <em>x+h </em>which is <i>hP12x</i> + <em>hP22x</em>. The  
 curves <i>hP12x/(hP12x</i> + <em>hP22x) </em>and <i>hP22x/(hP12x</i>  <p>These caracters are:</p>
 + <em>hP22x) </em>converge with <em>h, </em>to the <em>stationary  
 prevalence of disability</em>. In order to get the stationary  <ul>
 prevalence at age 70 we should start the process at an earlier      <li>'c' to start again the program from the beginning.</li>
 age, i.e.50. If the disability state is defined by severe      <li>'e' opens the <a href="biaspar.htm"><strong>biaspar.htm</strong></a>
 disability criteria with only a few chance to recover, then the          file to edit the output files and graphs. </li>
 incidence of recovery is low and the time to convergence is      <li>'q' for exiting.</li>
 probably longer. But we don't have experience yet.</p>  </ul>
   
 <h5><font color="#EC5E5E" size="3"><b>- Life expectancies by age  <h5><font size="4"><strong>Results files </strong></font><br>
 and initial health status</b></font><b>: </b><a  <br>
 href="erbiaspar.txt"><b>erbiaspar.txt</b></a></h5>  <font color="#EC5E5E" size="3"><strong>- </strong></font><a
   name="Observed prevalence in each state"><font color="#EC5E5E"
 <pre># Health expectancies   size="3"><strong>Observed prevalence in each state</strong></font></a><font
 # Age 1-1 1-2 2-1 2-2   color="#EC5E5E" size="3"><strong> (and at first pass)</strong></font><b>:
 70 10.7297 2.7809 6.3440 5.9813   </b><a href="prbiaspar.txt"><b>prbiaspar.txt</b></a><br>
 71 10.3078 2.8233 5.9295 5.9959   </h5>
 72 9.8927 2.8643 5.5305 6.0033   
 73 9.4848 2.9036 5.1474 6.0035 </pre>  <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
 <pre>For example 70 10.7297 2.7809 6.3440 5.9813 means:  proportion of individuals in states 1 and 2 respectively as
 e11=10.7297 e12=2.7809 e21=6.3440 e22=5.9813</pre>  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><img src="exbiaspar1.gif" width="400" height="300"><img  the number of states is higher than 2.<br>
 src="exbiaspar2.gif" width="400" height="300"></pre>  The header of the file is </p>
   
 <p>For example, life expectancy of a healthy individual at age 70  <pre># Age Prev(1) N(1) N Age Prev(2) N(2) N
 is 10.73 in the healthy state and 2.78 in the disability state  70 1.00000 631 631 70 0.00000 0 631
 (=13.51 years). If he was disable at age 70, his life expectancy  71 0.99681 625 627 71 0.00319 2 627
 will be shorter, 6.34 in the healthy state and 5.98 in the  72 0.97125 1115 1148 72 0.02875 33 1148 </pre>
 disability state (=12.32 years). The total life expectancy is a  
 weighted mean of both, 13.51 and 12.32; weight is the proportion  <p>It means that at age 70, the prevalence in state 1 is 1.000
 of people disabled at age 70. In order to get a pure period index  and in state 2 is 0.00 . At age 71 the number of individuals in
 (i.e. based only on incidences) we use the <a  state 1 is 625 and in state 2 is 2, hence the total number of
 href="#Stationary prevalence in each state">computed or  people aged 71 is 625+2=627. <br>
 stationary prevalence</a> at age 70 (i.e. computed from  </p>
 incidences at earlier ages) instead of the <a  
 href="#Observed prevalence in each state">observed prevalence</a>  <h5><font color="#EC5E5E" size="3"><b>- Estimated parameters and
 (for example at first exam) (<a href="#Health expectancies">see  covariance matrix</b></font><b>: </b><a href="rbiaspar.txt"><b>rbiaspar.txt</b></a></h5>
 below</a>).</p>  
   <p>This file contains all the maximisation results: </p>
 <h5><font color="#EC5E5E" size="3"><b>- Variances of life  
 expectancies by age and initial health status</b></font><b>: </b><a  <pre> -2 log likelihood= 21660.918613445392
 href="vrbiaspar.txt"><b>vrbiaspar.txt</b></a></h5>   Estimated parameters: a12 = -12.290174 b12 = 0.092161
                          a13 = -9.155590  b13 = 0.046627
 <p>For example, the covariances of life expectancies Cov(ei,ej)                         a21 = -2.629849  b21 = -0.022030
 at age 50 are (line 3) </p>                         a23 = -7.958519  b23 = 0.042614  
    Covariance matrix: Var(a12) = 1.47453e-001
 <pre>   Cov(e1,e1)=0.4667  Cov(e1,e2)=0.0605=Cov(e2,e1)  Cov(e2,e2)=0.0183</pre>                      Var(b12) = 2.18676e-005
                       Var(a13) = 2.09715e-001
 <h5><font color="#EC5E5E" size="3"><b>- </b></font><a                      Var(b13) = 3.28937e-005  
 name="Health expectancies"><font color="#EC5E5E" size="3"><b>Health                      Var(a21) = 9.19832e-001
 expectancies</b></font></a><font color="#EC5E5E" size="3"><b>                      Var(b21) = 1.29229e-004
 with standard errors in parentheses</b></font><b>: </b><a                      Var(a23) = 4.48405e-001
 href="trbiaspar.txt"><font face="Courier New"><b>trbiaspar.txt</b></font></a></h5>                      Var(b23) = 5.85631e-005
    </pre>
 <pre>#Total LEs with variances: e.. (std) e.1 (std) e.2 (std) </pre>  
   <p>By substitution of these parameters in the regression model,
 <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>  we obtain the elementary transition probabilities:</p>
   
 <p>Thus, at age 70 the total life expectancy, e..=13.42 years is  <p><img src="pebiaspar1.gif" width="400" height="300"></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>- Transition probabilities</b></font><b>:
 state 2, respectively (the sum is equal to one). e.1=10.39 is the  </b><a href="pijrbiaspar.txt"><b>pijrbiaspar.txt</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>Here are the transitions probabilities Pij(x, x+nh) where nh
 70 to be spent in the disability state.</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
 <h6><font color="#EC5E5E" size="3"><b>Total life expectancy by  the transition probabilities p11, p12, p13, p21, p22, p23. For
 age and health expectancies in states (1=healthy) and (2=disable)</b></font><b>:  example, line 5 of the file is: </p>
 ebiaspar.gif</b></h6>  
   <pre> 100 106 0.02655 0.17622 0.79722 0.01809 0.13678 0.84513 </pre>
 <p>This figure represents the health expectancies and the total  
 life expectancy with the confident interval in dashed curve. </p>  <p>and this means: </p>
   
 <pre>        <img src="ebiaspar.gif" width="400" height="300"></pre>  <pre>p11(100,106)=0.02655
   p12(100,106)=0.17622
 <p>Standard deviations (obtained from the information matrix of  p13(100,106)=0.79722
 the model) of these quantities are very useful.  p21(100,106)=0.01809
 Cross-longitudinal surveys are costly and do not involve huge  p22(100,106)=0.13678
 samples, generally a few thousands; therefore it is very  p22(100,106)=0.84513 </pre>
 important to have an idea of the standard deviation of our  
 estimates. It has been a big challenge to compute the Health  <h5><font color="#EC5E5E" size="3"><b>- </b></font><a
 Expectancy standard deviations. Don't be confuse: life expectancy  name="Stationary prevalence in each state"><font color="#EC5E5E"
 is, as any expected value, the mean of a distribution; but here  size="3"><b>Stationary prevalence in each state</b></font></a><b>:
 we are not computing the standard deviation of the distribution,  </b><a href="plrbiaspar.txt"><b>plrbiaspar.txt</b></a></h5>
 but the standard deviation of the estimate of the mean.</p>  
   <pre>#Prevalence
 <p>Our health expectancies estimates vary according to the sample  #Age 1-1 2-2
 size (and the standard deviations give confidence intervals of  
 the estimate) but also according to the model fitted. Let us  #************
 explain it in more details.</p>  70 0.90134 0.09866
   71 0.89177 0.10823
 <p>Choosing a model means ar least two kind of choices. First we  72 0.88139 0.11861
 have to decide the number of disability states. Second we have to  73 0.87015 0.12985 </pre>
 design, within the logit model family, the model: variables,  
 covariables, confonding factors etc. to be included.</p>  <p>At age 70 the stationary prevalence is 0.90134 in state 1 and
   0.09866 in state 2. This stationary prevalence differs from
 <p>More disability states we have, better is our demographical  observed prevalence. Here is the point. The observed prevalence
 approach of the disability process, but smaller are the number of  at age 70 results from the incidence of disability, incidence of
 transitions between each state and higher is the noise in the  recovery and mortality which occurred in the past of the cohort.
 measurement. We do not have enough experiments of the various  Stationary prevalence results from a simulation with actual
 models to summarize the advantages and disadvantages, but it is  incidences and mortality (estimated from this cross-longitudinal
 important to say that even if we had huge and unbiased samples,  survey). It is the best predictive value of the prevalence in the
 the total life expectancy computed from a cross-longitudinal  future if &quot;nothing changes in the future&quot;. This is
 survey, varies with the number of states. If we define only two  exactly what demographers do with a Life table. Life expectancy
 states, alive or dead, we find the usual life expectancy where it  is the expected mean time to survive if observed mortality rates
 is assumed that at each age, people are at the same risk to die.  (incidence of mortality) &quot;remains constant&quot; in the
 If we are differentiating the alive state into healthy and  future. </p>
 disable, and as the mortality from the disability state is higher  
 than the mortality from the healthy state, we are introducing  <h5><font color="#EC5E5E" size="3"><b>- Standard deviation of
 heterogeneity in the risk of dying. The total mortality at each  stationary prevalence</b></font><b>: </b><a
 age is the weighted mean of the mortality in each state by the  href="vplrbiaspar.txt"><b>vplrbiaspar.txt</b></a></h5>
 prevalence in each state. Therefore if the proportion of people  
 at each age and in each state is different from the stationary  <p>The stationary prevalence has to be compared with the observed
 equilibrium, there is no reason to find the same total mortality  prevalence by age. But both are statistical estimates and
 at a particular age. Life expectancy, even if it is a very useful  subjected to stochastic errors due to the size of the sample, the
 tool, has a very strong hypothesis of homogeneity of the  design of the survey, and, for the stationary prevalence to the
 population. Our main purpose is not to measure differential  model used and fitted. It is possible to compute the standard
 mortality but to measure the expected time in a healthy or  deviation of the stationary prevalence at each age.</p>
 disability state in order to maximise the former and minimize the  
 latter. But the differential in mortality complexifies the  <h5><font color="#EC5E5E" size="3">-Observed and stationary
 measurement.</p>  prevalence in state (2=disable) with the confident interval</font>:<b>
   </b><a href="vbiaspar21.htm"><b>vbiaspar21.gif</b></a></h5>
 <p>Incidences of disability or recovery are not affected by the  
 number of states if these states are independant. But incidences  <p>This graph exhibits the stationary prevalence in state (2)
 estimates are dependant on the specification of the model. More  with the confidence interval in red. The green curve is the
 covariates we added in the logit model better is the model, but  observed prevalence (or proportion of individuals in state (2)).
 some covariates are not well measured, some are confounding  Without discussing the results (it is not the purpose here), we
 factors like in any statistical model. The procedure to &quot;fit  observe that the green curve is rather below the stationary
 the best model' is similar to logistic regression which itself is  prevalence. It suggests an increase of the disability prevalence
 similar to regression analysis. We haven't yet been sofar because  in the future.</p>
 we also have a severe limitation which is the speed of the  
 convergence. On a Pentium III, 500 MHz, even the simplest model,  <p><img src="vbiaspar21.gif" width="400" height="300"></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>-Convergence to the
 a simple writing of the variables and the model to take into  stationary prevalence of disability</b></font><b>: </b><a
 account in the maximisation. The actual program allows only to  href="pbiaspar11.gif"><b>pbiaspar11.gif</b></a><br>
 add simple variables without covariations, like age+sex but  <img src="pbiaspar11.gif" width="400" height="300"> </h5>
 without age+sex+ age*sex . This can be done from the source code  
 (you have to change three lines in the source code) but will  <p>This graph plots the conditional transition probabilities from
 never be general enough. But what is to remember, is that  an initial state (1=healthy in red at the bottom, or 2=disable in
 incidences or probability of change from one state to another is  green on top) at age <em>x </em>to the final state 2=disable<em> </em>at
 affected by the variables specified into the model.</p>  age <em>x+h. </em>Conditional means at the condition to be alive
   at age <em>x+h </em>which is <i>hP12x</i> + <em>hP22x</em>. The
 <p>Also, the age range of the people interviewed has a link with  curves <i>hP12x/(hP12x</i> + <em>hP22x) </em>and <i>hP22x/(hP12x</i>
 the age range of the life expectancy which can be estimated by  + <em>hP22x) </em>converge with <em>h, </em>to the <em>stationary
 extrapolation. If your sample ranges from age 70 to 95, you can  prevalence of disability</em>. In order to get the stationary
 clearly estimate a life expectancy at age 70 and trust your  prevalence at age 70 we should start the process at an earlier
 confidence interval which is mostly based on your sample size,  age, i.e.50. If the disability state is defined by severe
 but if you want to estimate the life expectancy at age 50, you  disability criteria with only a few chance to recover, then the
 should rely in your model, but fitting a logistic model on a age  incidence of recovery is low and the time to convergence is
 range of 70-95 and estimating probabilties of transition out of  probably longer. But we don't have experience yet.</p>
 this age range, say at age 50 is very dangerous. At least you  
 should remember that the confidence interval given by the  <h5><font color="#EC5E5E" size="3"><b>- Life expectancies by age
 standard deviation of the health expectancies, are under the  and initial health status</b></font><b>: </b><a
 strong assumption that your model is the 'true model', which is  href="erbiaspar.txt"><b>erbiaspar.txt</b></a></h5>
 probably not the case.</p>  
   <pre># Health expectancies
 <h5><font color="#EC5E5E" size="3"><b>- Copy of the parameter  # Age 1-1 1-2 2-1 2-2
 file</b></font><b>: </b><a href="orbiaspar.txt"><b>orbiaspar.txt</b></a></h5>  70 10.9226 3.0401 5.6488 6.2122
   71 10.4384 3.0461 5.2477 6.1599
 <p>This copy of the parameter file can be useful to re-run the  72 9.9667 3.0502 4.8663 6.1025
 program while saving the old output files. </p>  73 9.5077 3.0524 4.5044 6.0401 </pre>
   
 <hr>  <pre>For example 70 10.4227 3.0402 5.6488 5.7123 means:
   e11=10.4227 e12=3.0402 e21=5.6488 e22=5.7123</pre>
 <h2><a name="example" </a><font color="#00006A">Trying an example</font></a></h2>  
   <pre><img src="expbiaspar21.gif" width="400" height="300"><img
 <p>Since you know how to run the program, it is time to test it  src="expbiaspar11.gif" width="400" height="300"></pre>
 on your own computer. Try for example on a parameter file named <a  
 href="file://../mytry/imachpar.txt">imachpar.txt</a> which is a  <p>For example, life expectancy of a healthy individual at age 70
 copy of <font size="2" face="Courier New">mypar.txt</font>  is 10.42 in the healthy state and 3.04 in the disability state
 included in the subdirectory of imach, <font size="2"  (=13.46 years). If he was disable at age 70, his life expectancy
 face="Courier New">mytry</font>. Edit it to change the name of  will be shorter, 5.64 in the healthy state and 5.71 in the
 the data file to <font size="2" face="Courier New">..\data\mydata.txt</font>  disability state (=11.35 years). The total life expectancy is a
 if you don't want to copy it on the same directory. The file <font  weighted mean of both, 13.46 and 11.35; weight is the proportion
 face="Courier New">mydata.txt</font> is a smaller file of 3,000  of people disabled at age 70. In order to get a pure period index
 people but still with 4 waves. </p>  (i.e. based only on incidences) we use the <a
   href="#Stationary prevalence in each state">computed or
 <p>Click on the imach.exe icon to open a window. Answer to the  stationary prevalence</a> at age 70 (i.e. computed from
 question:'<strong>Enter the parameter file name:'</strong></p>  incidences at earlier ages) instead of the <a
   href="#Observed prevalence in each state">observed prevalence</a>
 <table border="1">  (for example at first exam) (<a href="#Health expectancies">see
     <tr>  below</a>).</p>
         <td width="100%"><strong>IMACH, Version 0.63</strong><p><strong>Enter  
         the parameter file name: ..\mytry\imachpar.txt</strong></p>  <h5><font color="#EC5E5E" size="3"><b>- Variances of life
         </td>  expectancies by age and initial health status</b></font><b>: </b><a
     </tr>  href="vrbiaspar.txt"><b>vrbiaspar.txt</b></a></h5>
 </table>  
   <p>For example, the covariances of life expectancies Cov(ei,ej)
 <p>Most of the data files or image files generated, will use the  at age 50 are (line 3) </p>
 'imachpar' string into their name. The running time is about 2-3  
 minutes on a Pentium III. If the execution worked correctly, the  <pre>   Cov(e1,e1)=0.4776  Cov(e1,e2)=0.0488=Cov(e2,e1)  Cov(e2,e2)=0.0424</pre>
 outputs files are created in the current directory, and should be  
 the same as the mypar files initially included in the directory <font  <h5><font color="#EC5E5E" size="3"><b>- </b></font><a
 size="2" face="Courier New">mytry</font>.</p>  name="Health expectancies"><font color="#EC5E5E" size="3"><b>Health
   expectancies</b></font></a><font color="#EC5E5E" size="3"><b>
 <ul>  with standard errors in parentheses</b></font><b>: </b><a
     <li><pre><u>Output on the screen</u> The output screen looks like <a  href="trbiaspar.txt"><font face="Courier New"><b>trbiaspar.txt</b></font></a></h5>
 href="imachrun.LOG">this Log file</a>  
 #  <pre>#Total LEs with variances: e.. (std) e.1 (std) e.2 (std) </pre>
   
 title=MLE datafile=..\data\mydata.txt lastobs=3000 firstpass=1 lastpass=3  <pre>70 13.26 (0.22) 9.95 (0.20) 3.30 (0.14) </pre>
 ftol=1.000000e-008 stepm=24 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre>  
     </li>  <p>Thus, at age 70 the total life expectancy, e..=13.26 years is
     <li><pre>Total number of individuals= 2965, Agemin = 70.00, Agemax= 100.92  the weighted mean of e1.=13.46 and e2.=11.35 by the stationary
   prevalence at age 70 which are 0.90134 in state 1 and 0.09866 in
 Warning, no any valid information for:126 line=126  state 2, respectively (the sum is equal to one). e.1=9.95 is the
 Warning, no any valid information for:2307 line=2307  Disability-free life expectancy at age 70 (it is again a weighted
 Delay (in months) between two waves Min=21 Max=51 Mean=24.495826  mean of e11 and e21). e.2=3.30 is also the life expectancy at age
 <font face="Times New Roman">These lines give some warnings on the data file and also some raw statistics on frequencies of transitions.</font>  70 to be spent in the disability state.</p>
 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  <h5><font color="#EC5E5E" size="3"><b>-Total life expectancy by
 Age 102 1.=0 loss[1]=NaNQ% 2.=0 loss[2]=NaNQ% 1.=0 prev[1]=NaNQ% 2.=0 </pre>  age and health expectancies in states (1=healthy) and (2=disable)</b></font><b>:
     </li>  </b><a href="ebiaspar1.gif"><b>ebiaspar1.gif</b></a></h5>
 </ul>  
   <p>This figure represents the health expectancies and the total
 <p>&nbsp;</p>  life expectancy with the confident interval in dashed curve. </p>
   
 <ul>  <pre>        <img src="ebiaspar1.gif" width="400" height="300"></pre>
     <li>Maximisation with the Powell algorithm. 8 directions are  
         given corresponding to the 8 parameters. this can be  <p>Standard deviations (obtained from the information matrix of
         rather long to get convergence.<br>  the model) of these quantities are very useful.
         <font size="1" face="Courier New"><br>  Cross-longitudinal surveys are costly and do not involve huge
         Powell iter=1 -2*LL=11531.405658264877 1 0.000000000000 2  samples, generally a few thousands; therefore it is very
         0.000000000000 3<br>  important to have an idea of the standard deviation of our
         0.000000000000 4 0.000000000000 5 0.000000000000 6  estimates. It has been a big challenge to compute the Health
         0.000000000000 7 <br>  Expectancy standard deviations. Don't be confuse: life expectancy
         0.000000000000 8 0.000000000000<br>  is, as any expected value, the mean of a distribution; but here
         1..........2.................3..........4.................5.........<br>  we are not computing the standard deviation of the distribution,
         6................7........8...............<br>  but the standard deviation of the estimate of the mean.</p>
         Powell iter=23 -2*LL=6744.954108371555 1 -12.967632334283  
         <br>  <p>Our health expectancies estimates vary according to the sample
         2 0.135136681033 3 -7.402109728262 4 0.067844593326 <br>  size (and the standard deviations give confidence intervals of
         5 -0.673601538129 6 -0.006615504377 7 -5.051341616718 <br>  the estimate) but also according to the model fitted. Let us
         8 0.051272038506<br>  explain it in more details.</p>
         1..............2...........3..............4...........<br>  
         5..........6................7...........8.........<br>  <p>Choosing a model means ar least two kind of choices. First we
         #Number of iterations = 23, -2 Log likelihood =  have to decide the number of disability states. Second we have to
         6744.954042573691<br>  design, within the logit model family, the model: variables,
         # Parameters<br>  covariables, confonding factors etc. to be included.</p>
         12 -12.966061 0.135117 <br>  
         13 -7.401109 0.067831 <br>  <p>More disability states we have, better is our demographical
         21 -0.672648 -0.006627 <br>  approach of the disability process, but smaller are the number of
         23 -5.051297 0.051271 </font><br>  transitions between each state and higher is the noise in the
         </li>  measurement. We do not have enough experiments of the various
     <li><pre><font size="2">Calculation of the hessian matrix. Wait...  models to summarize the advantages and disadvantages, but it is
 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  important to say that even if we had huge and unbiased samples,
   the total life expectancy computed from a cross-longitudinal
 Inverting the hessian to get the covariance matrix. Wait...  survey, varies with the number of states. If we define only two
   states, alive or dead, we find the usual life expectancy where it
 #Hessian matrix#  is assumed that at each age, people are at the same risk to die.
 3.344e+002 2.708e+004 -4.586e+001 -3.806e+003 -1.577e+000 -1.313e+002 3.914e-001 3.166e+001   If we are differentiating the alive state into healthy and
 2.708e+004 2.204e+006 -3.805e+003 -3.174e+005 -1.303e+002 -1.091e+004 2.967e+001 2.399e+003   disable, and as the mortality from the disability state is higher
 -4.586e+001 -3.805e+003 4.044e+002 3.197e+004 2.431e-002 1.995e+000 1.783e-001 1.486e+001   than the mortality from the healthy state, we are introducing
 -3.806e+003 -3.174e+005 3.197e+004 2.541e+006 2.436e+000 2.051e+002 1.483e+001 1.244e+003   heterogeneity in the risk of dying. The total mortality at each
 -1.577e+000 -1.303e+002 2.431e-002 2.436e+000 1.093e+002 8.979e+003 -3.402e+001 -2.843e+003   age is the weighted mean of the mortality in each state by the
 -1.313e+002 -1.091e+004 1.995e+000 2.051e+002 8.979e+003 7.420e+005 -2.842e+003 -2.388e+005   prevalence in each state. Therefore if the proportion of people
 3.914e-001 2.967e+001 1.783e-001 1.483e+001 -3.402e+001 -2.842e+003 1.494e+002 1.251e+004   at each age and in each state is different from the stationary
 3.166e+001 2.399e+003 1.486e+001 1.244e+003 -2.843e+003 -2.388e+005 1.251e+004 1.053e+006   equilibrium, there is no reason to find the same total mortality
 # Scales  at a particular age. Life expectancy, even if it is a very useful
 12 1.00000e-004 1.00000e-006  tool, has a very strong hypothesis of homogeneity of the
 13 1.00000e-004 1.00000e-006  population. Our main purpose is not to measure differential
 21 1.00000e-003 1.00000e-005  mortality but to measure the expected time in a healthy or
 23 1.00000e-004 1.00000e-005  disability state in order to maximise the former and minimize the
 # Covariance  latter. But the differential in mortality complexifies the
   1 5.90661e-001  measurement.</p>
   2 -7.26732e-003 8.98810e-005  
   3 8.80177e-002 -1.12706e-003 5.15824e-001  <p>Incidences of disability or recovery are not affected by the
   4 -1.13082e-003 1.45267e-005 -6.50070e-003 8.23270e-005  number of states if these states are independant. But incidences
   5 9.31265e-003 -1.16106e-004 6.00210e-004 -8.04151e-006 1.75753e+000  estimates are dependant on the specification of the model. More
   6 -1.15664e-004 1.44850e-006 -7.79995e-006 1.04770e-007 -2.12929e-002 2.59422e-004  covariates we added in the logit model better is the model, but
   7 1.35103e-003 -1.75392e-005 -6.38237e-004 7.85424e-006 4.02601e-001 -4.86776e-003 1.32682e+000  some covariates are not well measured, some are confounding
   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  factors like in any statistical model. The procedure to &quot;fit
 # agemin agemax for lifexpectancy, bage fage (if mle==0 ie no data nor Max likelihood).  the best model' is similar to logistic regression which itself is
   similar to regression analysis. We haven't yet been sofar because
   we also have a severe limitation which is the speed of the
 agemin=70 agemax=100 bage=50 fage=100  convergence. On a Pentium III, 500 MHz, even the simplest model,
 Computing prevalence limit: result on file 'plrmypar.txt'   estimated by month on 8,000 people may take 4 hours to converge.
 Computing pij: result on file 'pijrmypar.txt'   Also, the program is not yet a statistical package, which permits
 Computing Health Expectancies: result on file 'ermypar.txt'   a simple writing of the variables and the model to take into
 Computing Variance-covariance of DFLEs: file 'vrmypar.txt'   account in the maximisation. The actual program allows only to
 Computing Total LEs with variances: file 'trmypar.txt'   add simple variables like age+sex or age+sex+ age*sex but will
 Computing Variance-covariance of Prevalence limit: file 'vplrmypar.txt'   never be general enough. But what is to remember, is that
 End of Imach  incidences or probability of change from one state to another is
 </font></pre>  affected by the variables specified into the model.</p>
     </li>  
 </ul>  <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
 <p><font size="3">Once the running is finished, the program  extrapolation. If your sample ranges from age 70 to 95, you can
 requires a caracter:</font></p>  clearly estimate a life expectancy at age 70 and trust your
   confidence interval which is mostly based on your sample size,
 <table border="1">  but if you want to estimate the life expectancy at age 50, you
     <tr>  should rely in your model, but fitting a logistic model on a age
         <td width="100%"><strong>Type g for plotting (available  range of 70-95 and estimating probabilties of transition out of
         if mle=1), e to edit output files, c to start again,</strong><p><strong>and  this age range, say at age 50 is very dangerous. At least you
         q for exiting:</strong></p>  should remember that the confidence interval given by the
         </td>  standard deviation of the health expectancies, are under the
     </tr>  strong assumption that your model is the 'true model', which is
 </table>  probably not the case.</p>
   
 <p><font size="3">First you should enter <strong>g</strong> to  <h5><font color="#EC5E5E" size="3"><b>- Copy of the parameter
 make the figures and then you can edit all the results by typing <strong>e</strong>.  file</b></font><b>: </b><a href="orbiaspar.txt"><b>orbiaspar.txt</b></a></h5>
 </font></p>  
   <p>This copy of the parameter file can be useful to re-run the
 <ul>  program while saving the old output files. </p>
     <li><u>Outputs files</u> <br>  
         - index.htm, this file is the master file on which you  <h5><font color="#EC5E5E" size="3"><b>- Prevalence forecasting</b></font><b>:
         should click first.<br>  </b><a href="frbiaspar.txt"><b>frbiaspar.txt</b></a></h5>
         - Observed prevalence in each state: <a  
         href="..\mytry\prmypar.txt">mypar.txt</a> <br>  <p
         - Estimated parameters and the covariance matrix: <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">First,
         href="..\mytry\rmypar.txt">rmypar.txt</a> <br>  we have estimated the observed prevalence between 1/1/1984 and
         - Stationary prevalence in each state: <a  1/6/1988. The mean date of interview (weighed average of the
         href="..\mytry\plrmypar.txt">plrmypar.txt</a> <br>  interviews performed between1/1/1984 and 1/6/1988) is estimated
         - Transition probabilities: <a  to be 13/9/1985, as written on the top on the file. Then we
         href="..\mytry\pijrmypar.txt">pijrmypar.txt</a> <br>  forecast the probability to be in each state. </p>
         - Copy of the parameter file: <a  
         href="..\mytry\ormypar.txt">ormypar.txt</a> <br>  <p
         - Life expectancies by age and initial health status: <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="..\mytry\ermypar.txt">ermypar.txt</a> <br>  at date 1/1/1989 : </p>
         - Variances of life expectancies by age and initial  
         health status: <a href="..\mytry\vrmypar.txt">vrmypar.txt</a>  <pre class="MsoNormal"># StartingAge FinalAge P.1 P.2 P.3
         <br>  # Forecasting at date 1/1/1989
         - Health expectancies with their variances: <a    73 0.807 0.078 0.115</pre>
         href="..\mytry\trmypar.txt">trmypar.txt</a> <br>  
         - Standard deviation of stationary prevalence: <a  <p
         href="..\mytry\vplrmypar.txt">vplrmypar.txt</a> <br>  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
         <br>  the minimum age is 70 on the 13/9/1985, the youngest forecasted
         </li>  age is 73. This means that at age a person aged 70 at 13/9/1989
     <li><u>Graphs</u> <br>  has a probability to enter state1 of 0.807 at age 73 on 1/1/1989.
         <br>  Similarly, the probability to be in state 2 is 0.078 and the
         -<a href="..\mytry\vmypar1.gif">Observed and stationary  probability to die is 0.115. Then, on the 1/1/1989, the
         prevalence in state (1) with the confident interval</a> <br>  prevalence of disability at age 73 is estimated to be 0.088.</p>
         -<a href="..\mytry\vmypar2.gif">Observed and stationary  
         prevalence in state (2) with the confident interval</a> <br>  <h5><font color="#EC5E5E" size="3"><b>- Population forecasting</b></font><b>:
         -<a href="..\mytry\exmypar1.gif">Health life expectancies  </b><a href="poprbiaspar.txt"><b>poprbiaspar.txt</b></a></h5>
         by age and initial health state (1)</a> <br>  
         -<a href="..\mytry\exmypar2.gif">Health life expectancies  <pre># Age P.1 P.2 P.3 [Population]
         by age and initial health state (2)</a> <br>  # Forecasting at date 1/1/1989
         -<a href="..\mytry\emypar.gif">Total life expectancy by  75 572685.22 83798.08
         age and health expectancies in states (1) and (2).</a> </li>  74 621296.51 79767.99
 </ul>  73 645857.70 69320.60 </pre>
   
 <p>This software have been partly granted by <a  <pre># Forecasting at date 1/1/19909
 href="http://euroreves.ined.fr">Euro-REVES</a>, a concerted  76 442986.68 92721.14 120775.48
 action from the European Union. It will be copyrighted  75 487781.02 91367.97 121915.51
 identically to a GNU software product, i.e. program and software  74 512892.07 85003.47 117282.76 </pre>
 can be distributed freely for non commercial use. Sources are not  
 widely distributed today. You can get them by asking us with a  <p>From the population file, we estimate the number of people in
 simple justification (name, email, institute) <a  each state. At age 73, 645857 persons are in state 1 and 69320
 href="mailto:brouard@ined.fr">mailto:brouard@ined.fr</a> and <a  are in state 2. One year latter, 512892 are still in state 1,
 href="mailto:lievre@ined.fr">mailto:lievre@ined.fr</a> .</p>  85003 are in state 2 and 117282 died before 1/1/1990.</p>
   
 <p>Latest version (0.63 of 16 march 2000) can be accessed at <a  <hr>
 href="http://euroeves.ined.fr/imach">http://euroreves.ined.fr/imach</a><br>  
 </p>  <h2><a name="example"></a><font color="#00006A">Trying an example</font></h2>
 </body>  
 </html>  <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.txt">imachpar.txt</a> which is a copy of <font
   size="2" face="Courier New">mypar.txt</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.txt</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.txt</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 (0.71a 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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