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-
-<h1 align="center"><font color="#00006A">Computing Health
-Expectancies using IMaCh</font></h1>
-
-<h1 align="center"><font color="#00006A" size="5">(a Maximum
-Likelihood Computer Program using Interpolation of Markov Chains)</font></h1>
-
-<p align="center"> </p>
-
-<p align="center"><a href="http://www.ined.fr/"><img
-src="logo-ined.gif" border="0" width="151" height="76"></a><img
-src="euroreves2.gif" width="151" height="75"></p>
-
-<h3 align="center"><a href="http://www.ined.fr/"><font
-color="#00006A">INED</font></a><font color="#00006A"> and </font><a
-href="http://euroreves.ined.fr"><font color="#00006A">EUROREVES</font></a></h3>
-
-<p align="center"><font color="#00006A" size="4"><strong>March
-2000</strong></font></p>
-
-<hr size="3" color="#EC5E5E">
-
-<p align="center"><font color="#00006A"><strong>Authors of the
-program: </strong></font><a href="http://sauvy.ined.fr/brouard"><font
-color="#00006A"><strong>Nicolas Brouard</strong></font></a><font
-color="#00006A"><strong>, senior researcher at the </strong></font><a
-href="http://www.ined.fr"><font color="#00006A"><strong>Institut
-National d'Etudes Démographiques</strong></font></a><font
-color="#00006A"><strong> (INED, Paris) in the "Mortality,
-Health and Epidemiology" Research Unit </strong></font></p>
-
-<p align="center"><font color="#00006A"><strong>and Agnès
-Lièvre<br clear="left">
-</strong></font></p>
-
-<h4><font color="#00006A">Contribution to the mathematics: C. R.
-Heathcote </font><font color="#00006A" size="2">(Australian
-National University, Canberra).</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
-color="#00006A">) </font></h4>
-
-<hr>
-
-<ul>
- <li><a href="#intro">Introduction</a> </li>
- <li>The detailed statistical model (<a href="docmath.pdf">PDF
- version</a>),(<a href="docmath.ps">ps version</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="#biaspar">The parameter file</a> </li>
- <li><a href="#running">Running Imach</a> </li>
- <li><a href="#output">Output files and graphs</a> </li>
- <li><a href="#example">Exemple</a> </li>
-</ul>
-
-<hr>
-
-<h2><a name="intro"><font color="#00006A">Introduction</font></a></h2>
-
-<p>This program computes <b>Healthy Life Expectancies</b> from <b>cross-longitudinal
-data</b>. Within the family of Health Expectancies (HE),
-Disability-free life expectancy (DFLE) is probably the most
-important index to monitor. In low mortality countries, there is
-a fear that when mortality declines, the increase in DFLE is not
-proportionate to the increase in total Life expectancy. This case
-is called the <em>Expansion of morbidity</em>. Most of the data
-collected today, in particular by the international <a
-href="http://euroreves/reves">REVES</a> network on Health
-expectancy, and most HE indices based on these data, are <em>cross-sectional</em>.
-It means that the information collected comes from a single
-cross-sectional survey: people from various ages (but mostly old
-people) are surveyed on their health status at a single date.
-Proportion of people disabled at each age, can then be measured
-at that date. This age-specific prevalence curve is then used to
-distinguish, within the stationary population (which, by
-definition, is the life table estimated from the vital statistics
-on mortality at the same date), the disable population from the
-disability-free population. Life expectancy (LE) (or total
-population divided by the yearly number of births or deaths of
-this stationary population) is then decomposed into DFLE and DLE.
-This method of 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
-conditions of the cohort and it is the result of the historical
-flows from entering disability and recovering in the past until
-today. The age-specific intensities (or incidence rates) of
-entering disability or recovering a good health, are reflecting
-actual conditions and therefore can be used at each age to
-forecast the future of this cohort. For example if a country is
-improving its technology of prosthesis, the incidence of
-recovering the ability to walk will be higher at each (old) age,
-but the prevalence of disability will only slightly reflect an
-improve because the prevalence is mostly affected by the history
-of the cohort and not by recent period effects. To measure the
-period improvement we have to simulate the future of a cohort of
-new-borns entering or leaving at each age the disability state or
-dying according to the incidence rates measured today on
-different cohorts. The proportion of people disabled at each age
-in this simulated cohort will be much lower (using the exemple of
-an improvement) that the proportions observed at each age in a
-cross-sectional survey. This new prevalence curve introduced in a
-life table will give a much more actual and realistic HE level
-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
-program. From your data and using IMaCH you can estimate period
-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
-("cross") where individuals from different ages are
-interviewed on their health status or degree of disability. At
-least a second wave of interviews ("longitudinal")
-should measure each new individual health status. Health
-expectancies are computed from the transitions observed between
-waves and are computed for each degree of severity of disability
-(number of life states). More degrees you consider, more time is
-necessary to reach the Maximum Likelihood of the parameters
-involved in the model. Considering only two states of disability
-(disable and healthy) is generally enough but the computer
-program works also with more health statuses.<br>
-<br>
-The simplest model is the multinomial logistic model where <i>pij</i>
-is the probability to be observed in state <i>j</i> at the second
-wave conditional to be observed in state <em>i</em> at the first
-wave. Therefore a simple model is: log<em>(pij/pii)= aij +
-bij*age+ cij*sex,</em> where '<i>age</i>' is age and '<i>sex</i>'
-is a covariate. The advantage that this computer program claims,
-comes from that if the delay between waves is not identical for
-each individual, or if some individual missed an interview, the
-information is not rounded or lost, but taken into account using
-an interpolation or extrapolation. <i>hPijx</i> is the
-probability to be observed in state <i>i</i> at age <i>x+h</i>
-conditional to the observed state <i>i</i> at age <i>x</i>. The
-delay '<i>h</i>' can be split into an exact number (<i>nh*stepm</i>)
-of unobserved intermediate states. This elementary transition (by
-month or quarter trimester, semester or year) is modeled as a
-multinomial logistic. The <i>hPx</i> matrix is simply the matrix
-product of <i>nh*stepm</i> elementary matrices and the
-contribution of each individual to the likelihood is simply <i>hPijx</i>.
-<br>
-</p>
-
-<p>The program presented in this manual is a quite general
-program named <strong>IMaCh</strong> (for <strong>I</strong>nterpolated
-<strong>MA</strong>rkov <strong>CH</strong>ain), designed to
-analyse transition data from longitudinal surveys. The first step
-is the parameters estimation of a transition probabilities model
-between an initial status and a final status. From there, the
-computer program produces some indicators such as observed and
-stationary prevalence, life expectancies and their variances and
-graphs. Our transition model consists in absorbing and
-non-absorbing states with the possibility of return across the
-non-absorbing states. The main advantage of this package,
-compared to other programs for the analysis of transition data
-(For example: Proc Catmod of SAS<sup>®</sup>) is that the whole
-individual information is used even if an interview is missing, a
-status or a date is unknown or when the delay between waves is
-not identical for each individual. The program can be executed
-according to parameters: selection of a sub-sample, number of
-absorbing and non-absorbing states, number of waves taken in
-account (the user inputs the first and the last interview), a
-tolerance level for the maximization function, the periodicity of
-the transitions (we can compute annual, quaterly or monthly
-transitions), covariates in the model. It works on Windows or on
-Unix.<br>
-</p>
-
-<hr>
-
-<h2><a name="data"><font color="#00006A">On what kind of data can
-it be used?</font></a></h2>
-
-<p>The minimum data required for a transition model is the
-recording of a set of individuals interviewed at a first date and
-interviewed again at least one another time. From the
-observations of an individual, we obtain a follow-up over time of
-the occurrence of a specific event. In this documentation, the
-event is related to health status at older ages, but the program
-can be applied on a lot of longitudinal studies in different
-contexts. To build the data file explained into the next section,
-you must have the month and year of each interview and the
-corresponding health status. But in order to get age, date of
-birth (month and year) is required (missing values is allowed for
-month). Date of death (month and year) is an important
-information also required if the individual is dead. Shorter
-steps (i.e. a month) will more closely take into account the
-survival time after the last interview.</p>
-
-<hr>
-
-<h2><a name="datafile"><font color="#00006A">The data file</font></a></h2>
-
-<p>In this example, 8,000 people have been interviewed in a
-cross-longitudinal survey of 4 waves (1984, 1986, 1988, 1990).
-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
-a simulation of the American Longitudinal Survey on Aging. The
-disability state is defined if the individual missed one of four
-ADL (Activity of daily living, like bathing, eating, walking).
-Therefore, even is the individuals interviewed in the sample are
-virtual, the information brought with this sample is close to the
-situation of the United States. Sex is not recorded is this
-sample.</p>
-
-<p>Each line of the data set (named <a href="data1.txt">data1.txt</a>
-in this first example) is an individual record which fields are: </p>
-
-<ul>
- <li><b>Index number</b>: positive number (field 1) </li>
- <li><b>First covariate</b> positive number (field 2) </li>
- <li><b>Second covariate</b> positive number (field 3) </li>
- <li><a name="Weight"><b>Weight</b></a>: positive number
- (field 4) . In most surveys individuals are weighted
- according to the stratification of the sample.</li>
- <li><b>Date of birth</b>: coded as mm/yyyy. Missing dates are
- coded as 99/9999 (field 5) </li>
- <li><b>Date of death</b>: coded as mm/yyyy. Missing dates are
- coded as 99/9999 (field 6) </li>
- <li><b>Date of first interview</b>: coded as mm/yyyy. Missing
- dates are coded as 99/9999 (field 7) </li>
- <li><b>Status at first interview</b>: positive number.
- Missing values ar coded -1. (field 8) </li>
- <li><b>Date of second interview</b>: coded as mm/yyyy.
- Missing dates are coded as 99/9999 (field 9) </li>
- <li><strong>Status at second interview</strong> positive
- number. Missing values ar coded -1. (field 10) </li>
- <li><b>Date of third interview</b>: coded as mm/yyyy. Missing
- dates are coded as 99/9999 (field 11) </li>
- <li><strong>Status at third interview</strong> positive
- number. Missing values ar coded -1. (field 12) </li>
- <li><b>Date of fourth interview</b>: coded as mm/yyyy.
- Missing dates are coded as 99/9999 (field 13) </li>
- <li><strong>Status at fourth interview</strong> positive
- number. Missing values are coded -1. (field 14) </li>
- <li>etc</li>
-</ul>
-
-<p> </p>
-
-<p>If your longitudinal survey do not include information about
-weights or covariates, you must fill the column with a number
-(e.g. 1) because a missing field is not allowed.</p>
-
-<hr>
-
-<h2><font color="#00006A">Your first example parameter file</font><a
-href="http://euroreves.ined.fr/imach"></a><a name="uio"></a></h2>
-
-<h2><a name="biaspar"></a>#Imach version 0.63, February 2000,
-INED-EUROREVES </h2>
-
-<p>This is a comment. Comments start with a '#'.</p>
-
-<h4><font color="#FF0000">First uncommented line</font></h4>
-
-<pre>title=1st_example datafile=data1.txt lastobs=8600 firstpass=1 lastpass=4</pre>
-
-<ul>
- <li><b>title=</b> 1st_example is title of the run. </li>
- <li><b>datafile=</b>data1.txt is the name of the data set.
- Our example is a six years follow-up survey. It consists
- in a baseline followed by 3 reinterviews. </li>
- <li><b>lastobs=</b> 8600 the program is able to run on a
- subsample where the last observation number is lastobs.
- It can be set a bigger number than the real number of
- observations (e.g. 100000). In this example, maximisation
- will be done on the 8600 first records. </li>
- <li><b>firstpass=1</b> , <b>lastpass=4 </b>In case of more
- than two interviews in the survey, the program can be run
- on selected transitions periods. firstpass=1 means the
- first interview included in the calculation is the
- baseline survey. lastpass=4 means that the information
- brought by the 4th interview is taken into account.</li>
-</ul>
-
-<p> </p>
-
-<h4><a name="biaspar-2"><font color="#FF0000">Second uncommented
-line</font></a></h4>
-
-<pre>ftol=1.e-08 stepm=1 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre>
-
-<ul>
- <li><b>ftol=1e-8</b> Convergence tolerance on the function
- value in the maximisation of the likelihood. Choosing a
- correct value for ftol is difficult. 1e-8 is a correct
- value for a 32 bits computer.</li>
- <li><b>stepm=1</b> Time unit in months for interpolation.
- Examples:<ul>
- <li>If stepm=1, the unit is a month </li>
- <li>If stepm=4, the unit is a trimester</li>
- <li>If stepm=12, the unit is a year </li>
- <li>If stepm=24, the unit is two years</li>
- <li>... </li>
- </ul>
- </li>
- <li><b>ncov=2</b> Number of covariates to be add to the
- model. The intercept and the age parameter are counting
- for 2 covariates. For example, if you want to add gender
- in the covariate vector you must write ncov=3 else
- ncov=2. </li>
- <li><b>nlstate=2</b> Number of non-absorbing (live) states.
- Here we have two alive states: disability-free is coded 1
- and disability is coded 2. </li>
- <li><b>ndeath=1</b> Number of absorbing states. The absorbing
- state death is coded 3. </li>
- <li><b>maxwav=4</b> Maximum number of waves. The program can
- not include more than 4 interviews. </li>
- <li><a name="mle"><b>mle</b></a><b>=1</b> Option for the
- Maximisation Likelihood Estimation. <ul>
- <li>If mle=1 the program does the maximisation and
- the calculation of heath expectancies </li>
- <li>If mle=0 the program only does the calculation of
- the health expectancies. </li>
- </ul>
- </li>
- <li><b>weight=0</b> Possibility to add weights. <ul>
- <li>If weight=0 no weights are included </li>
- <li>If weight=1 the maximisation integrates the
- weights which are in field <a href="#Weight">4</a></li>
- </ul>
- </li>
-</ul>
-
-<h4><font color="#FF0000">Guess values for optimization</font><font
-color="#00006A"> </font></h4>
-
-<p>You must write the initial guess values of the parameters for
-optimization. The number of parameters, <em>N</em> depends on the
-number of absorbing states and non-absorbing states and on the
-number of covariates. <br>
-<em>N</em> is given by the formula <em>N</em>=(<em>nlstate</em> +
-<em>ndeath</em>-1)*<em>nlstate</em>*<em>ncov</em> . <br>
-<br>
-Thus in the simple case with 2 covariates (the model is log
-(pij/pii) = aij + bij * age where intercept and age are the two
-covariates), and 2 health degrees (1 for disability-free and 2
-for disability) and 1 absorbing state (3), you must enter 8
-initials values, a12, b12, a13, b13, a21, b21, a23, b23. You can
-start with zeros as in this example, but if you have a more
-precise set (for example from an earlier run) you can enter it
-and it will speed up them<br>
-Each of the four lines starts with indices "ij": <br>
-<br>
-<b>ij aij bij</b> </p>
-
-<blockquote>
- <pre># Guess values of aij and bij in log (pij/pii) = aij + bij * age
-12 -14.155633 0.110794
-13 -7.925360 0.032091
-21 -1.890135 -0.029473
-23 -6.234642 0.022315 </pre>
-</blockquote>
-
-<p>or, to simplify: </p>
-
-<blockquote>
- <pre>12 0.0 0.0
-13 0.0 0.0
-21 0.0 0.0
-23 0.0 0.0</pre>
-</blockquote>
-
-<h4><font color="#FF0000">Guess values for computing variances</font></h4>
-
-<p>This is an output if <a href="#mle">mle</a>=1. But it can be
-used as an input to get the vairous output data files (Health
-expectancies, stationary prevalence etc.) and figures without
-rerunning the rather long maximisation phase (mle=0). </p>
-
-<p>The scales are small values for the evaluation of numerical
-derivatives. These derivatives are used to compute the hessian
-matrix of the parameters, that is the inverse of the covariance
-matrix, and the variances of health expectancies. Each line
-consists in indices "ij" followed by the initial scales
-(zero to simplify) associated with aij and bij. </p>
-
-<ul>
- <li>If mle=1 you can enter zeros:</li>
-</ul>
-
-<blockquote>
- <pre># Scales (for hessian or gradient estimation)
-12 0. 0.
-13 0. 0.
-21 0. 0.
-23 0. 0. </pre>
-</blockquote>
-
-<ul>
- <li>If mle=0 you must enter a covariance matrix (usually
- obtained from an earlier run).</li>
-</ul>
-
-<h4><font color="#FF0000">Covariance matrix of parameters</font></h4>
-
-<p>This is an output if <a href="#mle">mle</a>=1. But it can be
-used as an input to get the vairous output data files (Health
-expectancies, stationary prevalence etc.) and figures without
-rerunning the rather long maximisation phase (mle=0). </p>
-
-<p>Each line starts with indices "ijk" followed by the
-covariances between aij and bij: </p>
-
-<pre>
- 121 Var(a12)
- 122 Cov(b12,a12) Var(b12)
- ...
- 232 Cov(b23,a12) Cov(b23,b12) ... Var (b23) </pre>
-
-<ul>
- <li>If mle=1 you can enter zeros. </li>
-</ul>
-
-<blockquote>
- <pre># Covariance matrix
-121 0.
-122 0. 0.
-131 0. 0. 0.
-132 0. 0. 0. 0.
-211 0. 0. 0. 0. 0.
-212 0. 0. 0. 0. 0. 0.
-231 0. 0. 0. 0. 0. 0. 0.
-232 0. 0. 0. 0. 0. 0. 0. 0.</pre>
-</blockquote>
-
-<ul>
- <li>If mle=0 you must enter a covariance matrix (usually
- obtained from an earlier run).<br>
- </li>
-</ul>
-
-<h4><a name="biaspar-l"></a><font color="#FF0000">last
-uncommented line</font></h4>
-
-<pre>agemin=70 agemax=100 bage=50 fage=100</pre>
-
-<p>Once we obtained the estimated parameters, the program is able
-to calculated stationary prevalence, transitions probabilities
-and life expectancies at any age. Choice of age ranges is useful
-for extrapolation. In our data file, ages varies from age 70 to
-102. Setting bage=50 and fage=100, makes the program computing
-life expectancy from age bage to age fage. As we use a model, we
-can compute life expectancy on a wider age range than the age
-range from the data. But the model can be rather wrong on big
-intervals.</p>
-
-<p>Similarly, it is possible to get extrapolated stationary
-prevalence by age raning from agemin to agemax. </p>
-
-<ul>
- <li><b>agemin=</b> Minimum age for calculation of the
- stationary prevalence </li>
- <li><b>agemax=</b> Maximum age for calculation of the
- stationary prevalence </li>
- <li><b>bage=</b> Minimum age for calculation of the health
- expectancies </li>
- <li><b>fage=</b> Maximum ages for calculation of the health
- expectancies </li>
-</ul>
-
-<hr>
-
-<h2><a name="running"></a><font color="#00006A">Running Imach
-with this example</font></h2>
-
-<p>We assume that you entered your <a href="biaspar.txt">1st_example
-parameter file</a> as explained <a href="#biaspar">above</a>. To
-run the program you should click on the imach.exe icon and enter
-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>
-<a href="C:\usr\imach\mle">C:\usr\imach\mle</a> and put it with
-the mouse on the imach window).<br>
-</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
-number of variables.</p>
-
-<p>The program outputs many files. Most of them are files which
-will be plotted for better understanding.</p>
-
-<hr>
-
-<h2><a name="output"><font color="#00006A">Output of the program
-and graphs</font> </a></h2>
-
-<p>Once the optimization is finished, some graphics can be made
-with a grapher. We use Gnuplot which is an interactive plotting
-program copyrighted but freely distributed. Imach outputs the
-source of a gnuplot file, named 'graph.gp', which can be directly
-input into gnuplot.<br>
-When the running is finished, the user should enter a caracter
-for plotting and output editing. </p>
-
-<p>These caracters are:</p>
-
-<ul>
- <li>'c' to start again the program from the beginning.</li>
- <li>'g' to made graphics. The output graphs are in GIF format
- and you have no control over which is produced. If you
- want to modify the graphics or make another one, you
- should modify the parameters in the file <b>graph.gp</b>
- located in imach\bin. A gnuplot reference manual is
- available <a
- href="http://www.cs.dartmouth.edu/gnuplot/gnuplot.html">here</a>.
- </li>
- <li>'e' opens the <strong>index.htm</strong> file to edit the
- output files and graphs. </li>
- <li>'q' for exiting.</li>
-</ul>
-
-<h5><font size="4"><strong>Results files </strong></font><br>
-<br>
-<font color="#EC5E5E" size="3"><strong>- </strong></font><a
-name="Observed prevalence in each state"><font color="#EC5E5E"
-size="3"><strong>Observed prevalence in each state</strong></font></a><font
-color="#EC5E5E" size="3"><strong> (and at first pass)</strong></font><b>:
-</b><a href="prbiaspar.txt"><b>prbiaspar.txt</b></a><br>
-</h5>
-
-<p>The first line is the title and displays each field of the
-file. The first column is age. The fields 2 and 6 are the
-proportion of individuals in states 1 and 2 respectively as
-observed during the first exam. Others fields are the numbers of
-people in states 1, 2 or more. The number of columns increases if
-the number of states is higher than 2.<br>
-The header of the file is </p>
-
-<pre># Age Prev(1) N(1) N Age Prev(2) N(2) N
-70 1.00000 631 631 70 0.00000 0 631
-71 0.99681 625 627 71 0.00319 2 627
-72 0.97125 1115 1148 72 0.02875 33 1148 </pre>
-
-<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>It means that at age 70, the prevalence in state 1 is 1.000
-and in state 2 is 0.00 . At age 71 the number of individuals in
-state 1 is 625 and in state 2 is 2, hence the total number of
-people aged 71 is 625+2=627. <br>
-</p>
-
-<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>
-
-<p>This file contains all the maximisation results: </p>
-
-<pre> Number of iterations=47
- -2 log likelihood=46553.005854373667
- Estimated parameters: a12 = -12.691743 b12 = 0.095819
- a13 = -7.815392 b13 = 0.031851
- a21 = -1.809895 b21 = -0.030470
- a23 = -7.838248 b23 = 0.039490
- Covariance matrix: Var(a12) = 1.03611e-001
- Var(b12) = 1.51173e-005
- Var(a13) = 1.08952e-001
- Var(b13) = 1.68520e-005
- Var(a21) = 4.82801e-001
- Var(b21) = 6.86392e-005
- Var(a23) = 2.27587e-001
- Var(b23) = 3.04465e-005
- </pre>
-
-<h5><font color="#EC5E5E" size="3"><b>- Transition probabilities</b></font><b>:
-</b><a href="pijrbiaspar.txt"><b>pijrbiaspar.txt</b></a></h5>
-
-<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
-(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
-example, line 5 of the file is: </p>
-
-<pre> 100 106 0.03286 0.23512 0.73202 0.02330 0.19210 0.78460 </pre>
-
-<p>and this means: </p>
-
-<pre>p11(100,106)=0.03286
-p12(100,106)=0.23512
-p13(100,106)=0.73202
-p21(100,106)=0.02330
-p22(100,106)=0.19210
-p22(100,106)=0.78460 </pre>
-
-<h5><font color="#EC5E5E" size="3"><b>- </b></font><a
-name="Stationary prevalence in each state"><font color="#EC5E5E"
-size="3"><b>Stationary prevalence in each state</b></font></a><b>:
-</b><a href="plrbiaspar.txt"><b>plrbiaspar.txt</b></a></h5>
-
-<pre>#Age 1-1 2-2
-70 0.92274 0.07726
-71 0.91420 0.08580
-72 0.90481 0.09519
-73 0.89453 0.10547</pre>
-
-<p>At age 70 the stationary prevalence is 0.92274 in state 1 and
-0.07726 in state 2. This stationary prevalence differs from
-observed prevalence. Here is the point. The observed prevalence
-at age 70 results from the incidence of disability, incidence of
-recovery and mortality which occurred in the past of the cohort.
-Stationary prevalence results from a simulation with actual
-incidences and mortality (estimated from this cross-longitudinal
-survey). It is the best predictive value of the prevalence in the
-future if "nothing changes in the future". This is
-exactly what demographers do with a Life table. Life expectancy
-is the expected mean time to survive if observed mortality rates
-(incidence of mortality) "remains constant" in the
-future. </p>
-
-<h5><font color="#EC5E5E" size="3"><b>- Standard deviation of
-stationary prevalence</b></font><b>: </b><a
-href="vplrbiaspar.txt"><b>vplrbiaspar.txt</b></a></h5>
-
-<p>The stationary prevalence has to be compared with the observed
-prevalence by age. But both are statistical estimates and
-subjected to stochastic errors due to the size of the sample, the
-design of the survey, and, for the stationary prevalence to the
-model used and fitted. It is possible to compute the standard
-deviation of the stationary prevalence at each age.</p>
-
-<h6><font color="#EC5E5E" size="3">Observed and stationary
-prevalence in state (2=disable) with the confident interval</font>:<b>
-vbiaspar2.gif</b></h6>
-
-<p><br>
-This graph exhibits the stationary prevalence in state (2) with
-the confidence interval in red. The green curve is the observed
-prevalence (or proportion of individuals in state (2)). Without
-discussing the results (it is not the purpose here), we observe
-that the green curve is rather below the stationary prevalence.
-It suggests an increase of the disability prevalence in the
-future.</p>
-
-<p><img src="vbiaspar2.gif" width="400" height="300"></p>
-
-<h6><font color="#EC5E5E" size="3"><b>Convergence to the
-stationary prevalence of disability</b></font><b>: pbiaspar1.gif</b><br>
-<img src="pbiaspar1.gif" width="400" height="300"> </h6>
-
-<p>This graph plots the conditional transition probabilities from
-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
-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
-curves <i>hP12x/(hP12x</i> + <em>hP22x) </em>and <i>hP22x/(hP12x</i>
-+ <em>hP22x) </em>converge with <em>h, </em>to the <em>stationary
-prevalence of disability</em>. In order to get the stationary
-prevalence at age 70 we should start the process at an earlier
-age, i.e.50. If the disability state is defined by severe
-disability criteria with only a few chance to recover, then the
-incidence of recovery is low and the time to convergence is
-probably longer. But we don't have experience yet.</p>
-
-<h5><font color="#EC5E5E" size="3"><b>- Life expectancies by age
-and initial health status</b></font><b>: </b><a
-href="erbiaspar.txt"><b>erbiaspar.txt</b></a></h5>
-
-<pre># Health expectancies
-# Age 1-1 1-2 2-1 2-2
-70 10.7297 2.7809 6.3440 5.9813
-71 10.3078 2.8233 5.9295 5.9959
-72 9.8927 2.8643 5.5305 6.0033
-73 9.4848 2.9036 5.1474 6.0035 </pre>
-
-<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>
-
-<pre><img src="exbiaspar1.gif" width="400" height="300"><img
-src="exbiaspar2.gif" width="400" height="300"></pre>
-
-<p>For example, life expectancy of a healthy individual at age 70
-is 10.73 in the healthy state and 2.78 in the disability state
-(=13.51 years). If he was disable at age 70, his life expectancy
-will be shorter, 6.34 in the healthy state and 5.98 in the
-disability state (=12.32 years). The total life expectancy is a
-weighted mean of both, 13.51 and 12.32; weight is the proportion
-of people disabled at age 70. In order to get a pure period index
-(i.e. based only on incidences) we use the <a
-href="#Stationary prevalence in each state">computed or
-stationary prevalence</a> at age 70 (i.e. computed from
-incidences at earlier ages) instead of the <a
-href="#Observed prevalence in each state">observed prevalence</a>
-(for example at first exam) (<a href="#Health expectancies">see
-below</a>).</p>
-
-<h5><font color="#EC5E5E" size="3"><b>- Variances of life
-expectancies by age and initial health status</b></font><b>: </b><a
-href="vrbiaspar.txt"><b>vrbiaspar.txt</b></a></h5>
-
-<p>For example, the covariances of life expectancies Cov(ei,ej)
-at age 50 are (line 3) </p>
-
-<pre> Cov(e1,e1)=0.4667 Cov(e1,e2)=0.0605=Cov(e2,e1) Cov(e2,e2)=0.0183</pre>
-
-<h5><font color="#EC5E5E" size="3"><b>- </b></font><a
-name="Health expectancies"><font color="#EC5E5E" size="3"><b>Health
-expectancies</b></font></a><font color="#EC5E5E" size="3"><b>
-with standard errors in parentheses</b></font><b>: </b><a
-href="trbiaspar.txt"><font face="Courier New"><b>trbiaspar.txt</b></font></a></h5>
-
-<pre>#Total LEs with variances: e.. (std) e.1 (std) e.2 (std) </pre>
-
-<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>
-
-<p>Thus, at age 70 the total life expectancy, e..=13.42 years is
-the weighted mean of e1.=13.51 and e2.=12.32 by the stationary
-prevalence at age 70 which are 0.92274 in state 1 and 0.07726 in
-state 2, respectively (the sum is equal to one). e.1=10.39 is the
-Disability-free life expectancy at age 70 (it is again a weighted
-mean of e11 and e21). e.2=3.03 is also the life expectancy at age
-70 to be spent in the disability state.</p>
-
-<h6><font color="#EC5E5E" size="3"><b>Total life expectancy by
-age and health expectancies in states (1=healthy) and (2=disable)</b></font><b>:
-ebiaspar.gif</b></h6>
-
-<p>This figure represents the health expectancies and the total
-life expectancy with the confident interval in dashed curve. </p>
-
-<pre> <img src="ebiaspar.gif" width="400" height="300"></pre>
-
-<p>Standard deviations (obtained from the information matrix of
-the model) of these quantities are very useful.
-Cross-longitudinal surveys are costly and do not involve huge
-samples, generally a few thousands; therefore it is very
-important to have an idea of the standard deviation of our
-estimates. It has been a big challenge to compute the Health
-Expectancy standard deviations. Don't be confuse: life expectancy
-is, as any expected value, the mean of a distribution; but here
-we are not computing the standard deviation of the distribution,
-but the standard deviation of the estimate of the mean.</p>
-
-<p>Our health expectancies estimates vary according to the sample
-size (and the standard deviations give confidence intervals of
-the estimate) but also according to the model fitted. Let us
-explain it in more details.</p>
-
-<p>Choosing a model means ar least two kind of choices. First we
-have to decide the number of disability states. Second we have to
-design, within the logit model family, the model: variables,
-covariables, confonding factors etc. to be included.</p>
-
-<p>More disability states we have, better is our demographical
-approach of the disability process, but smaller are the number of
-transitions between each state and higher is the noise in the
-measurement. We do not have enough experiments of the various
-models to summarize the advantages and disadvantages, but it is
-important to say that even if we had huge and unbiased samples,
-the total life expectancy computed from a cross-longitudinal
-survey, varies with the number of states. If we define only two
-states, alive or dead, we find the usual life expectancy where it
-is assumed that at each age, people are at the same risk to die.
-If we are differentiating the alive state into healthy and
-disable, and as the mortality from the disability state is higher
-than the mortality from the healthy state, we are introducing
-heterogeneity in the risk of dying. The total mortality at each
-age is the weighted mean of the mortality in each state by the
-prevalence in each state. Therefore if the proportion of people
-at each age and in each state is different from the stationary
-equilibrium, there is no reason to find the same total mortality
-at a particular age. Life expectancy, even if it is a very useful
-tool, has a very strong hypothesis of homogeneity of the
-population. Our main purpose is not to measure differential
-mortality but to measure the expected time in a healthy or
-disability state in order to maximise the former and minimize the
-latter. But the differential in mortality complexifies the
-measurement.</p>
-
-<p>Incidences of disability or recovery are not affected by the
-number of states if these states are independant. But incidences
-estimates are dependant on the specification of the model. More
-covariates we added in the logit model better is the model, but
-some covariates are not well measured, some are confounding
-factors like in any statistical model. The procedure to "fit
-the best model' is similar to logistic regression which itself is
-similar to regression analysis. We haven't yet been sofar because
-we also have a severe limitation which is the speed of the
-convergence. On a Pentium III, 500 MHz, even the simplest model,
-estimated by month on 8,000 people may take 4 hours to converge.
-Also, the program is not yet a statistical package, which permits
-a simple writing of the variables and the model to take into
-account in the maximisation. The actual program allows only to
-add simple variables without covariations, like age+sex but
-without age+sex+ age*sex . This can be done from the source code
-(you have to change three lines in the source code) but will
-never be general enough. But what is to remember, is that
-incidences or probability of change from one state to another is
-affected by the variables specified into the model.</p>
-
-<p>Also, the age range of the people interviewed has a link with
-the age range of the life expectancy which can be estimated by
-extrapolation. If your sample ranges from age 70 to 95, you can
-clearly estimate a life expectancy at age 70 and trust your
-confidence interval which is mostly based on your sample size,
-but if you want to estimate the life expectancy at age 50, you
-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
-should remember that the confidence interval given by the
-standard deviation of the health expectancies, are under the
-strong assumption that your model is the 'true model', which is
-probably not the case.</p>
-
-<h5><font color="#EC5E5E" size="3"><b>- Copy of the parameter
-file</b></font><b>: </b><a href="orbiaspar.txt"><b>orbiaspar.txt</b></a></h5>
-
-<p>This copy of the parameter file can be useful to re-run the
-program while saving the old output files. </p>
-
-<hr>
-
-<h2><a name="example" </a><font color="#00006A">Trying an example</font></a></h2>
-
-<p>Since you know how to run the program, it is time to test it
-on your own computer. Try for example on a parameter file named <a
-href="file://../mytry/imachpar.txt">imachpar.txt</a> which is a
-copy of <font size="2" face="Courier New">mypar.txt</font>
-included in the subdirectory of imach, <font size="2"
-face="Courier New">mytry</font>. Edit it to change the name of
-the data file to <font size="2" face="Courier New">..\data\mydata.txt</font>
-if you don't want to copy it on the same directory. The file <font
-face="Courier New">mydata.txt</font> is a smaller file of 3,000
-people but still with 4 waves. </p>
-
-<p>Click on the imach.exe icon to open a window. Answer to the
-question:'<strong>Enter the parameter file name:'</strong></p>
-
-<table border="1">
- <tr>
- <td width="100%"><strong>IMACH, Version 0.63</strong><p><strong>Enter
- the parameter file name: ..\mytry\imachpar.txt</strong></p>
- </td>
- </tr>
-</table>
-
-<p>Most of the data files or image files generated, will use the
-'imachpar' string into their name. The running time is about 2-3
-minutes on a Pentium III. If the execution worked correctly, the
-outputs files are created in the current directory, and should be
-the same as the mypar files initially included in the directory <font
-size="2" face="Courier New">mytry</font>.</p>
-
-<ul>
- <li><pre><u>Output on the screen</u> The output screen looks like <a
-href="imachrun.LOG">this Log file</a>
-#
-
-title=MLE datafile=..\data\mydata.txt lastobs=3000 firstpass=1 lastpass=3
-ftol=1.000000e-008 stepm=24 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre>
- </li>
- <li><pre>Total number of individuals= 2965, Agemin = 70.00, Agemax= 100.92
-
-Warning, no any valid information for:126 line=126
-Warning, no any valid information for:2307 line=2307
-Delay (in months) between two waves Min=21 Max=51 Mean=24.495826
-<font face="Times New Roman">These lines give some warnings on the data file and also some raw statistics on frequencies of transitions.</font>
-Age 70 1.=230 loss[1]=3.5% 2.=16 loss[2]=12.5% 1.=222 prev[1]=94.1% 2.=14
- prev[2]=5.9% 1-1=8 11=200 12=7 13=15 2-1=2 21=6 22=7 23=1
-Age 102 1.=0 loss[1]=NaNQ% 2.=0 loss[2]=NaNQ% 1.=0 prev[1]=NaNQ% 2.=0 </pre>
- </li>
-</ul>
-
-<p> </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 g for plotting (available
- if mle=1), e to edit output files, c to start again,</strong><p><strong>and
- q for exiting:</strong></p>
- </td>
- </tr>
-</table>
-
-<p><font size="3">First you should enter <strong>g</strong> to
-make the figures and then you can edit all the results by typing <strong>e</strong>.
-</font></p>
-
-<ul>
- <li><u>Outputs files</u> <br>
- - index.htm, this file is the master file on which you
- should click first.<br>
- - Observed prevalence in each state: <a
- href="..\mytry\prmypar.txt">mypar.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>
- <br>
- </li>
- <li><u>Graphs</u> <br>
- <br>
- -<a href="..\mytry\vmypar1.gif">Observed and stationary
- prevalence in state (1) with the confident interval</a> <br>
- -<a href="..\mytry\vmypar2.gif">Observed and stationary
- prevalence in state (2) with the confident interval</a> <br>
- -<a href="..\mytry\exmypar1.gif">Health life expectancies
- by age and initial health state (1)</a> <br>
- -<a href="..\mytry\exmypar2.gif">Health life expectancies
- by age and initial health state (2)</a> <br>
- -<a href="..\mytry\emypar.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 (0.63 of 16 march 2000) can be accessed at <a
-href="http://euroeves.ined.fr/imach">http://euroreves.ined.fr/imach</a><br>
-</p>
-</body>
-</html>
git://henry.ined.fr / .git / commitdiff