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

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