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                      7: <title>Computing Health Expectancies using IMaCh</title>
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                     14: 
                     15: <h1 align="center"><font color="#00006A">Computing Health
                     16: Expectancies using IMaCh</font></h1>
                     17: 
                     18: <h1 align="center"><font color="#00006A" size="5">(a Maximum
                     19: Likelihood Computer Program using Interpolation of Markov Chains)</font></h1>
                     20: 
                     21: <p align="center">&nbsp;</p>
                     22: 
                     23: <p align="center"><a href="http://www.ined.fr/"><img
                     24: src="logo-ined.gif" border="0" width="151" height="76"></a><img
                     25: src="euroreves2.gif" width="151" height="75"></p>
                     26: 
                     27: <h3 align="center"><a href="http://www.ined.fr/"><font
                     28: color="#00006A">INED</font></a><font color="#00006A"> and </font><a
                     29: href="http://euroreves.ined.fr"><font color="#00006A">EUROREVES</font></a></h3>
                     30: 
1.3       lievre     31: <p align="center"><font color="#00006A" size="4"><strong>Version
1.4     ! lievre     32: 0.7, February 2002</strong></font></p>
1.2       lievre     33: 
                     34: <hr size="3" color="#EC5E5E">
                     35: 
                     36: <p align="center"><font color="#00006A"><strong>Authors of the
                     37: program: </strong></font><a href="http://sauvy.ined.fr/brouard"><font
                     38: color="#00006A"><strong>Nicolas Brouard</strong></font></a><font
                     39: color="#00006A"><strong>, senior researcher at the </strong></font><a
                     40: href="http://www.ined.fr"><font color="#00006A"><strong>Institut
                     41: National d'Etudes Démographiques</strong></font></a><font
                     42: color="#00006A"><strong> (INED, Paris) in the &quot;Mortality,
                     43: Health and Epidemiology&quot; Research Unit </strong></font></p>
                     44: 
                     45: <p align="center"><font color="#00006A"><strong>and Agnès
                     46: Lièvre<br clear="left">
                     47: </strong></font></p>
                     48: 
                     49: <h4><font color="#00006A">Contribution to the mathematics: C. R.
                     50: Heathcote </font><font color="#00006A" size="2">(Australian
                     51: National University, Canberra).</font></h4>
                     52: 
                     53: <h4><font color="#00006A">Contact: Agnès Lièvre (</font><a
                     54: href="mailto:lievre@ined.fr"><font color="#00006A"><i>lievre@ined.fr</i></font></a><font
                     55: color="#00006A">) </font></h4>
                     56: 
                     57: <hr>
                     58: 
                     59: <ul>
                     60:     <li><a href="#intro">Introduction</a> </li>
                     61:     <li>The detailed statistical model (<a href="docmath.pdf">PDF
                     62:         version</a>),(<a href="docmath.ps">ps version</a>) </li>
                     63:     <li><a href="#data">On what kind of data can it be used?</a></li>
                     64:     <li><a href="#datafile">The data file</a> </li>
                     65:     <li><a href="#biaspar">The parameter file</a> </li>
                     66:     <li><a href="#running">Running Imach</a> </li>
                     67:     <li><a href="#output">Output files and graphs</a> </li>
                     68:     <li><a href="#example">Exemple</a> </li>
                     69: </ul>
                     70: 
                     71: <hr>
                     72: 
                     73: <h2><a name="intro"><font color="#00006A">Introduction</font></a></h2>
                     74: 
                     75: <p>This program computes <b>Healthy Life Expectancies</b> from <b>cross-longitudinal
                     76: data</b> using the methodology pioneered by Laditka and Wolf (1).
                     77: Within the family of Health Expectancies (HE), Disability-free
                     78: life expectancy (DFLE) is probably the most important index to
                     79: monitor. In low mortality countries, there is a fear that when
                     80: mortality declines, the increase in DFLE is not proportionate to
                     81: the increase in total Life expectancy. This case is called the <em>Expansion
                     82: of morbidity</em>. Most of the data collected today, in
                     83: particular by the international <a href="http://euroreves/reves">REVES</a>
                     84: network on Health expectancy, and most HE indices based on these
                     85: data, are <em>cross-sectional</em>. It means that the information
                     86: collected comes from a single cross-sectional survey: people from
                     87: various ages (but mostly old people) are surveyed on their health
                     88: status at a single date. Proportion of people disabled at each
                     89: age, can then be measured at that date. This age-specific
                     90: prevalence curve is then used to distinguish, within the
                     91: stationary population (which, by definition, is the life table
                     92: estimated from the vital statistics on mortality at the same
                     93: date), the disable population from the disability-free
                     94: population. Life expectancy (LE) (or total population divided by
                     95: the yearly number of births or deaths of this stationary
                     96: population) is then decomposed into DFLE and DLE. This method of
                     97: computing HE is usually called the Sullivan method (from the name
                     98: of the author who first described it).</p>
                     99: 
                    100: <p>Age-specific proportions of people disable are very difficult
                    101: to forecast because each proportion corresponds to historical
                    102: conditions of the cohort and it is the result of the historical
                    103: flows from entering disability and recovering in the past until
                    104: today. The age-specific intensities (or incidence rates) of
                    105: entering disability or recovering a good health, are reflecting
                    106: actual conditions and therefore can be used at each age to
                    107: forecast the future of this cohort. For example if a country is
                    108: improving its technology of prosthesis, the incidence of
                    109: recovering the ability to walk will be higher at each (old) age,
                    110: but the prevalence of disability will only slightly reflect an
                    111: improve because the prevalence is mostly affected by the history
                    112: of the cohort and not by recent period effects. To measure the
                    113: period improvement we have to simulate the future of a cohort of
                    114: new-borns entering or leaving at each age the disability state or
                    115: dying according to the incidence rates measured today on
                    116: different cohorts. The proportion of people disabled at each age
                    117: in this simulated cohort will be much lower (using the exemple of
                    118: an improvement) that the proportions observed at each age in a
                    119: cross-sectional survey. This new prevalence curve introduced in a
                    120: life table will give a much more actual and realistic HE level
                    121: than the Sullivan method which mostly measured the History of
                    122: health conditions in this country.</p>
                    123: 
                    124: <p>Therefore, the main question is how to measure incidence rates
                    125: from cross-longitudinal surveys? This is the goal of the IMaCH
                    126: program. From your data and using IMaCH you can estimate period
                    127: HE and not only Sullivan's HE. Also the standard errors of the HE
                    128: are computed.</p>
                    129: 
                    130: <p>A cross-longitudinal survey consists in a first survey
                    131: (&quot;cross&quot;) where individuals from different ages are
                    132: interviewed on their health status or degree of disability. At
                    133: least a second wave of interviews (&quot;longitudinal&quot;)
                    134: should measure each new individual health status. Health
                    135: expectancies are computed from the transitions observed between
                    136: waves and are computed for each degree of severity of disability
                    137: (number of life states). More degrees you consider, more time is
                    138: necessary to reach the Maximum Likelihood of the parameters
                    139: involved in the model. Considering only two states of disability
                    140: (disable and healthy) is generally enough but the computer
                    141: program works also with more health statuses.<br>
                    142: <br>
                    143: The simplest model is the multinomial logistic model where <i>pij</i>
                    144: is the probability to be observed in state <i>j</i> at the second
                    145: wave conditional to be observed in state <em>i</em> at the first
                    146: wave. Therefore a simple model is: log<em>(pij/pii)= aij +
                    147: bij*age+ cij*sex,</em> where '<i>age</i>' is age and '<i>sex</i>'
                    148: is a covariate. The advantage that this computer program claims,
                    149: comes from that if the delay between waves is not identical for
                    150: each individual, or if some individual missed an interview, the
                    151: information is not rounded or lost, but taken into account using
                    152: an interpolation or extrapolation. <i>hPijx</i> is the
                    153: probability to be observed in state <i>i</i> at age <i>x+h</i>
                    154: conditional to the observed state <i>i</i> at age <i>x</i>. The
                    155: delay '<i>h</i>' can be split into an exact number (<i>nh*stepm</i>)
                    156: of unobserved intermediate states. This elementary transition (by
                    157: month or quarter trimester, semester or year) is modeled as a
                    158: multinomial logistic. The <i>hPx</i> matrix is simply the matrix
                    159: product of <i>nh*stepm</i> elementary matrices and the
                    160: contribution of each individual to the likelihood is simply <i>hPijx</i>.
                    161: <br>
                    162: </p>
                    163: 
                    164: <p>The program presented in this manual is a quite general
                    165: program named <strong>IMaCh</strong> (for <strong>I</strong>nterpolated
                    166: <strong>MA</strong>rkov <strong>CH</strong>ain), designed to
                    167: analyse transition data from longitudinal surveys. The first step
                    168: is the parameters estimation of a transition probabilities model
                    169: between an initial status and a final status. From there, the
                    170: computer program produces some indicators such as observed and
                    171: stationary prevalence, life expectancies and their variances and
                    172: graphs. Our transition model consists in absorbing and
                    173: non-absorbing states with the possibility of return across the
                    174: non-absorbing states. The main advantage of this package,
                    175: compared to other programs for the analysis of transition data
                    176: (For example: Proc Catmod of SAS<sup>®</sup>) is that the whole
                    177: individual information is used even if an interview is missing, a
                    178: status or a date is unknown or when the delay between waves is
                    179: not identical for each individual. The program can be executed
                    180: according to parameters: selection of a sub-sample, number of
                    181: absorbing and non-absorbing states, number of waves taken in
                    182: account (the user inputs the first and the last interview), a
                    183: tolerance level for the maximization function, the periodicity of
                    184: the transitions (we can compute annual, quaterly or monthly
                    185: transitions), covariates in the model. It works on Windows or on
                    186: Unix.<br>
                    187: </p>
                    188: 
                    189: <hr>
                    190: 
                    191: <p>(1) Laditka, Sarah B. and Wolf, Douglas A. (1998), &quot;New
                    192: Methods for Analyzing Active Life Expectancy&quot;. <i>Journal of
                    193: Aging and Health</i>. Vol 10, No. 2. </p>
                    194: 
                    195: <hr>
                    196: 
                    197: <h2><a name="data"><font color="#00006A">On what kind of data can
                    198: it be used?</font></a></h2>
                    199: 
                    200: <p>The minimum data required for a transition model is the
                    201: recording of a set of individuals interviewed at a first date and
                    202: interviewed again at least one another time. From the
                    203: observations of an individual, we obtain a follow-up over time of
                    204: the occurrence of a specific event. In this documentation, the
                    205: event is related to health status at older ages, but the program
                    206: can be applied on a lot of longitudinal studies in different
                    207: contexts. To build the data file explained into the next section,
                    208: you must have the month and year of each interview and the
                    209: corresponding health status. But in order to get age, date of
                    210: birth (month and year) is required (missing values is allowed for
                    211: month). Date of death (month and year) is an important
                    212: information also required if the individual is dead. Shorter
                    213: steps (i.e. a month) will more closely take into account the
                    214: survival time after the last interview.</p>
                    215: 
                    216: <hr>
                    217: 
                    218: <h2><a name="datafile"><font color="#00006A">The data file</font></a></h2>
                    219: 
                    220: <p>In this example, 8,000 people have been interviewed in a
                    221: cross-longitudinal survey of 4 waves (1984, 1986, 1988, 1990).
                    222: Some people missed 1, 2 or 3 interviews. Health statuses are
                    223: healthy (1) and disable (2). The survey is not a real one. It is
                    224: a simulation of the American Longitudinal Survey on Aging. The
                    225: disability state is defined if the individual missed one of four
                    226: ADL (Activity of daily living, like bathing, eating, walking).
                    227: Therefore, even is the individuals interviewed in the sample are
                    228: virtual, the information brought with this sample is close to the
                    229: situation of the United States. Sex is not recorded is this
                    230: sample.</p>
                    231: 
                    232: <p>Each line of the data set (named <a href="data1.txt">data1.txt</a>
                    233: in this first example) is an individual record which fields are: </p>
                    234: 
                    235: <ul>
                    236:     <li><b>Index number</b>: positive number (field 1) </li>
                    237:     <li><b>First covariate</b> positive number (field 2) </li>
                    238:     <li><b>Second covariate</b> positive number (field 3) </li>
                    239:     <li><a name="Weight"><b>Weight</b></a>: positive number
                    240:         (field 4) . In most surveys individuals are weighted
                    241:         according to the stratification of the sample.</li>
                    242:     <li><b>Date of birth</b>: coded as mm/yyyy. Missing dates are
                    243:         coded as 99/9999 (field 5) </li>
                    244:     <li><b>Date of death</b>: coded as mm/yyyy. Missing dates are
                    245:         coded as 99/9999 (field 6) </li>
                    246:     <li><b>Date of first interview</b>: coded as mm/yyyy. Missing
                    247:         dates are coded as 99/9999 (field 7) </li>
                    248:     <li><b>Status at first interview</b>: positive number.
                    249:         Missing values ar coded -1. (field 8) </li>
                    250:     <li><b>Date of second interview</b>: coded as mm/yyyy.
                    251:         Missing dates are coded as 99/9999 (field 9) </li>
                    252:     <li><strong>Status at second interview</strong> positive
                    253:         number. Missing values ar coded -1. (field 10) </li>
                    254:     <li><b>Date of third interview</b>: coded as mm/yyyy. Missing
                    255:         dates are coded as 99/9999 (field 11) </li>
                    256:     <li><strong>Status at third interview</strong> positive
                    257:         number. Missing values ar coded -1. (field 12) </li>
                    258:     <li><b>Date of fourth interview</b>: coded as mm/yyyy.
                    259:         Missing dates are coded as 99/9999 (field 13) </li>
                    260:     <li><strong>Status at fourth interview</strong> positive
                    261:         number. Missing values are coded -1. (field 14) </li>
                    262:     <li>etc</li>
                    263: </ul>
                    264: 
                    265: <p>&nbsp;</p>
                    266: 
                    267: <p>If your longitudinal survey do not include information about
                    268: weights or covariates, you must fill the column with a number
                    269: (e.g. 1) because a missing field is not allowed.</p>
                    270: 
                    271: <hr>
                    272: 
                    273: <h2><font color="#00006A">Your first example parameter file</font><a
                    274: href="http://euroreves.ined.fr/imach"></a><a name="uio"></a></h2>
                    275: 
1.4     ! lievre    276: <h2><a name="biaspar"></a>#Imach version 0.7, February 2002,
1.2       lievre    277: INED-EUROREVES </h2>
                    278: 
                    279: <p>This is a comment. Comments start with a '#'.</p>
                    280: 
                    281: <h4><font color="#FF0000">First uncommented line</font></h4>
                    282: 
                    283: <pre>title=1st_example datafile=data1.txt lastobs=8600 firstpass=1 lastpass=4</pre>
                    284: 
                    285: <ul>
                    286:     <li><b>title=</b> 1st_example is title of the run. </li>
                    287:     <li><b>datafile=</b>data1.txt is the name of the data set.
                    288:         Our example is a six years follow-up survey. It consists
                    289:         in a baseline followed by 3 reinterviews. </li>
                    290:     <li><b>lastobs=</b> 8600 the program is able to run on a
                    291:         subsample where the last observation number is lastobs.
                    292:         It can be set a bigger number than the real number of
                    293:         observations (e.g. 100000). In this example, maximisation
                    294:         will be done on the 8600 first records. </li>
                    295:     <li><b>firstpass=1</b> , <b>lastpass=4 </b>In case of more
                    296:         than two interviews in the survey, the program can be run
                    297:         on selected transitions periods. firstpass=1 means the
                    298:         first interview included in the calculation is the
                    299:         baseline survey. lastpass=4 means that the information
                    300:         brought by the 4th interview is taken into account.</li>
                    301: </ul>
                    302: 
                    303: <p>&nbsp;</p>
                    304: 
                    305: <h4><a name="biaspar-2"><font color="#FF0000">Second uncommented
                    306: line</font></a></h4>
                    307: 
                    308: <pre>ftol=1.e-08 stepm=1 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre>
                    309: 
                    310: <ul>
                    311:     <li><b>ftol=1e-8</b> Convergence tolerance on the function
                    312:         value in the maximisation of the likelihood. Choosing a
                    313:         correct value for ftol is difficult. 1e-8 is a correct
                    314:         value for a 32 bits computer.</li>
                    315:     <li><b>stepm=1</b> Time unit in months for interpolation.
                    316:         Examples:<ul>
                    317:             <li>If stepm=1, the unit is a month </li>
                    318:             <li>If stepm=4, the unit is a trimester</li>
                    319:             <li>If stepm=12, the unit is a year </li>
                    320:             <li>If stepm=24, the unit is two years</li>
                    321:             <li>... </li>
                    322:         </ul>
                    323:     </li>
                    324:     <li><b>ncov=2</b> Number of covariates in the datafile. The
                    325:         intercept and the age parameter are counting for 2
                    326:         covariates.</li>
                    327:     <li><b>nlstate=2</b> Number of non-absorbing (alive) states.
                    328:         Here we have two alive states: disability-free is coded 1
                    329:         and disability is coded 2. </li>
                    330:     <li><b>ndeath=1</b> Number of absorbing states. The absorbing
                    331:         state death is coded 3. </li>
                    332:     <li><b>maxwav=4</b> Number of waves in the datafile.</li>
                    333:     <li><a name="mle"><b>mle</b></a><b>=1</b> Option for the
                    334:         Maximisation Likelihood Estimation. <ul>
                    335:             <li>If mle=1 the program does the maximisation and
                    336:                 the calculation of health expectancies </li>
                    337:             <li>If mle=0 the program only does the calculation of
                    338:                 the health expectancies. </li>
                    339:         </ul>
                    340:     </li>
                    341:     <li><b>weight=0</b> Possibility to add weights. <ul>
                    342:             <li>If weight=0 no weights are included </li>
                    343:             <li>If weight=1 the maximisation integrates the
                    344:                 weights which are in field <a href="#Weight">4</a></li>
                    345:         </ul>
                    346:     </li>
                    347: </ul>
                    348: 
                    349: <h4><font color="#FF0000">Covariates</font></h4>
                    350: 
                    351: <p>Intercept and age are systematically included in the model.
                    352: Additional covariates can be included with the command </p>
                    353: 
                    354: <pre>model=<em>list of covariates</em></pre>
                    355: 
                    356: <ul>
                    357:     <li>if<strong> model=. </strong>then no covariates are
                    358:         included</li>
                    359:     <li>if <strong>model=V1</strong> the model includes the first
                    360:         covariate (field 2)</li>
                    361:     <li>if <strong>model=V2 </strong>the model includes the
                    362:         second covariate (field 3)</li>
                    363:     <li>if <strong>model=V1+V2 </strong>the model includes the
                    364:         first and the second covariate (fields 2 and 3)</li>
                    365:     <li>if <strong>model=V1*V2 </strong>the model includes the
                    366:         product of the first and the second covariate (fields 2
                    367:         and 3)</li>
1.3       lievre    368:     <li>if <strong>model=V1+V1*age</strong> the model includes
                    369:         the product covariate*age</li>
1.2       lievre    370: </ul>
                    371: 
                    372: <h4><font color="#FF0000">Guess values for optimization</font><font
                    373: color="#00006A"> </font></h4>
                    374: 
                    375: <p>You must write the initial guess values of the parameters for
                    376: optimization. The number of parameters, <em>N</em> depends on the
                    377: number of absorbing states and non-absorbing states and on the
                    378: number of covariates. <br>
                    379: <em>N</em> is given by the formula <em>N</em>=(<em>nlstate</em> +
                    380: <em>ndeath</em>-1)*<em>nlstate</em>*<em>ncov</em>&nbsp;. <br>
                    381: <br>
                    382: Thus in the simple case with 2 covariates (the model is log
                    383: (pij/pii) = aij + bij * age where intercept and age are the two
                    384: covariates), and 2 health degrees (1 for disability-free and 2
                    385: for disability) and 1 absorbing state (3), you must enter 8
                    386: initials values, a12, b12, a13, b13, a21, b21, a23, b23. You can
                    387: start with zeros as in this example, but if you have a more
                    388: precise set (for example from an earlier run) you can enter it
                    389: and it will speed up them<br>
1.4     ! lievre    390: Each of the four lines starts with indices &quot;ij&quot;: <b>ij
        !           391: aij bij</b> </p>
1.2       lievre    392: 
                    393: <blockquote>
                    394:     <pre># Guess values of aij and bij in log (pij/pii) = aij + bij * age
                    395: 12 -14.155633  0.110794 
                    396: 13  -7.925360  0.032091 
                    397: 21  -1.890135 -0.029473 
                    398: 23  -6.234642  0.022315 </pre>
                    399: </blockquote>
                    400: 
                    401: <p>or, to simplify: </p>
                    402: 
                    403: <blockquote>
                    404:     <pre>12 0.0 0.0
                    405: 13 0.0 0.0
                    406: 21 0.0 0.0
                    407: 23 0.0 0.0</pre>
                    408: </blockquote>
                    409: 
                    410: <h4><font color="#FF0000">Guess values for computing variances</font></h4>
                    411: 
                    412: <p>This is an output if <a href="#mle">mle</a>=1. But it can be
1.4     ! lievre    413: used as an input to get the various output data files (Health
1.2       lievre    414: expectancies, stationary prevalence etc.) and figures without
                    415: rerunning the rather long maximisation phase (mle=0). </p>
                    416: 
                    417: <p>The scales are small values for the evaluation of numerical
                    418: derivatives. These derivatives are used to compute the hessian
                    419: matrix of the parameters, that is the inverse of the covariance
                    420: matrix, and the variances of health expectancies. Each line
                    421: consists in indices &quot;ij&quot; followed by the initial scales
                    422: (zero to simplify) associated with aij and bij. </p>
                    423: 
                    424: <ul>
                    425:     <li>If mle=1 you can enter zeros:</li>
                    426: </ul>
                    427: 
                    428: <blockquote>
                    429:     <pre># Scales (for hessian or gradient estimation)
                    430: 12 0. 0. 
                    431: 13 0. 0. 
                    432: 21 0. 0. 
                    433: 23 0. 0. </pre>
                    434: </blockquote>
                    435: 
                    436: <ul>
                    437:     <li>If mle=0 you must enter a covariance matrix (usually
                    438:         obtained from an earlier run).</li>
                    439: </ul>
                    440: 
                    441: <h4><font color="#FF0000">Covariance matrix of parameters</font></h4>
                    442: 
                    443: <p>This is an output if <a href="#mle">mle</a>=1. But it can be
1.4     ! lievre    444: used as an input to get the various output data files (Health
1.2       lievre    445: expectancies, stationary prevalence etc.) and figures without
                    446: rerunning the rather long maximisation phase (mle=0). </p>
                    447: 
                    448: <p>Each line starts with indices &quot;ijk&quot; followed by the
                    449: covariances between aij and bij: </p>
                    450: 
                    451: <pre>
                    452:    121 Var(a12) 
                    453:    122 Cov(b12,a12)  Var(b12) 
                    454:           ...
                    455:    232 Cov(b23,a12)  Cov(b23,b12) ... Var (b23) </pre>
                    456: 
                    457: <ul>
                    458:     <li>If mle=1 you can enter zeros. </li>
                    459: </ul>
                    460: 
                    461: <blockquote>
                    462:     <pre># Covariance matrix
                    463: 121 0.
                    464: 122 0. 0.
                    465: 131 0. 0. 0. 
                    466: 132 0. 0. 0. 0. 
                    467: 211 0. 0. 0. 0. 0. 
                    468: 212 0. 0. 0. 0. 0. 0. 
                    469: 231 0. 0. 0. 0. 0. 0. 0. 
                    470: 232 0. 0. 0. 0. 0. 0. 0. 0.</pre>
                    471: </blockquote>
                    472: 
                    473: <ul>
                    474:     <li>If mle=0 you must enter a covariance matrix (usually
                    475:         obtained from an earlier run).<br>
                    476:         </li>
                    477: </ul>
                    478: 
1.4     ! lievre    479: <h4><font color="#FF0000">Age range for calculation of stationary
        !           480: prevalences and health expectancies</font></h4>
1.2       lievre    481: 
                    482: <pre>agemin=70 agemax=100 bage=50 fage=100</pre>
                    483: 
                    484: <p>Once we obtained the estimated parameters, the program is able
                    485: to calculated stationary prevalence, transitions probabilities
1.4     ! lievre    486: and life expectancies at any age. Choice of age range is useful
1.2       lievre    487: for extrapolation. In our data file, ages varies from age 70 to
                    488: 102. Setting bage=50 and fage=100, makes the program computing
                    489: life expectancy from age bage to age fage. As we use a model, we
                    490: can compute life expectancy on a wider age range than the age
                    491: range from the data. But the model can be rather wrong on big
                    492: intervals.</p>
                    493: 
                    494: <p>Similarly, it is possible to get extrapolated stationary
1.4     ! lievre    495: prevalence by age ranging from agemin to agemax. </p>
1.2       lievre    496: 
                    497: <ul>
                    498:     <li><b>agemin=</b> Minimum age for calculation of the
                    499:         stationary prevalence </li>
                    500:     <li><b>agemax=</b> Maximum age for calculation of the
                    501:         stationary prevalence </li>
                    502:     <li><b>bage=</b> Minimum age for calculation of the health
                    503:         expectancies </li>
1.4     ! lievre    504:     <li><b>fage=</b> Maximum age for calculation of the health
1.2       lievre    505:         expectancies </li>
                    506: </ul>
                    507: 
1.4     ! lievre    508: <h4><a name="Computing"><font color="#FF0000">Computing</font></a><font
        !           509: color="#FF0000"> the observed prevalence</font></h4>
        !           510: 
        !           511: <pre>begin-prev-date=1/1/1984 end-prev-date=1/6/1988 </pre>
        !           512: 
        !           513: <p>Statements 'begin-prev-date' and 'end-prev-date' allow to
        !           514: select the period in which we calculate the observed prevalences
        !           515: in each state. In this example, the prevalences are calculated on
        !           516: data survey collected between 1 january 1984 and 1 june 1988. </p>
        !           517: 
        !           518: <ul>
        !           519:     <li><strong>begin-prev-date= </strong>Starting date
        !           520:         (day/month/year)</li>
        !           521:     <li><strong>end-prev-date= </strong>Final date
        !           522:         (day/month/year)</li>
        !           523: </ul>
        !           524: 
        !           525: <h4><font color="#FF0000">Population- or status-based health
        !           526: expectancies</font></h4>
        !           527: 
        !           528: <pre>pop_based=0</pre>
        !           529: 
        !           530: <p>The user has the possibility to choose between
        !           531: population-based or status-based health expectancies. If
        !           532: pop_based=0 then status-based health expectancies are computed
        !           533: and if pop_based=1, the programme computes population-based
        !           534: health expectancies. Health expectancies are weighted averages of
        !           535: health expectancies respective of the initial state. For a
        !           536: status-based index, the weights are the cross-sectional
        !           537: prevalences observed between two dates, as <a href="#Computing">previously
        !           538: explained</a>, whereas for a population-based index, the weights
        !           539: are the stationary prevalences.</p>
        !           540: 
        !           541: <h4><font color="#FF0000">Prevalence forecasting </font></h4>
        !           542: 
        !           543: <pre>starting-proj-date=1/1/1989 final-proj-date=1/1/1992 mov_average=0 </pre>
        !           544: 
        !           545: <p>Prevalence and population projections are only available if
        !           546: the interpolation unit is a month, i.e. stepm=1. The programme
        !           547: estimates the prevalence in each state at a precise date
        !           548: expressed in day/month/year. The programme computes one
        !           549: forecasted prevalence a year from a starting date (1 january of
        !           550: 1989 in this example) to a final date (1 january 1992). The
        !           551: statement mov_average allows to compute smoothed forecasted
        !           552: prevalences with a five-age moving average centered at the
        !           553: mid-age of the five-age period. </p>
        !           554: 
        !           555: <ul>
        !           556:     <li><strong>starting-proj-date</strong>= starting date
        !           557:         (day/month/year) of forecasting</li>
        !           558:     <li><strong>final-proj-date= </strong>final date
        !           559:         (day/month/year) of forecasting</li>
        !           560:     <li><strong>mov_average</strong>= smoothing with a five-age
        !           561:         moving average centered at the mid-age of the five-age
        !           562:         period. The command<strong> mov_average</strong> takes
        !           563:         value 1 if the prevalences are smoothed and 0 otherwise.</li>
        !           564: </ul>
        !           565: 
        !           566: <h4><font color="#FF0000">Last uncommented line : Population
        !           567: forecasting </font></h4>
        !           568: 
        !           569: <pre>popforecast=0 popfile=pyram.txt popfiledate=1/1/1989 last-popfiledate=1/1/1992</pre>
        !           570: 
        !           571: <p>This command is available if the interpolation unit is a
        !           572: month, i.e. stepm=1 and if popforecast=1. From a data file </p>
        !           573: 
        !           574: <p>Structure of the data file <a href="pyram.txt"><b>pyram.txt</b></a><b>
        !           575: : </b>age numbers</p>
        !           576: 
        !           577: <p>&nbsp;</p>
        !           578: 
1.2       lievre    579: <hr>
                    580: 
                    581: <h2><a name="running"></a><font color="#00006A">Running Imach
                    582: with this example</font></h2>
                    583: 
1.4     ! lievre    584: <p>We assume that you entered your <a href="biaspar.imach">1st_example
1.2       lievre    585: parameter file</a> as explained <a href="#biaspar">above</a>. To
                    586: run the program you should click on the imach.exe icon and enter
                    587: the name of the parameter file which is for example <a
                    588: href="C:\usr\imach\mle\biaspar.txt">C:\usr\imach\mle\biaspar.txt</a>
                    589: (you also can click on the biaspar.txt icon located in <br>
                    590: <a href="C:\usr\imach\mle">C:\usr\imach\mle</a> and put it with
                    591: the mouse on the imach window).<br>
                    592: </p>
                    593: 
                    594: <p>The time to converge depends on the step unit that you used (1
                    595: month is cpu consuming), on the number of cases, and on the
                    596: number of variables.</p>
                    597: 
                    598: <p>The program outputs many files. Most of them are files which
                    599: will be plotted for better understanding.</p>
                    600: 
                    601: <hr>
                    602: 
                    603: <h2><a name="output"><font color="#00006A">Output of the program
                    604: and graphs</font> </a></h2>
                    605: 
                    606: <p>Once the optimization is finished, some graphics can be made
                    607: with a grapher. We use Gnuplot which is an interactive plotting
1.3       lievre    608: program copyrighted but freely distributed. A gnuplot reference
1.4     ! lievre    609: manual is available <a href="http://www.gnuplot.info/">here</a>. <br>
1.2       lievre    610: When the running is finished, the user should enter a caracter
                    611: for plotting and output editing. </p>
                    612: 
                    613: <p>These caracters are:</p>
                    614: 
                    615: <ul>
                    616:     <li>'c' to start again the program from the beginning.</li>
1.3       lievre    617:     <li>'e' opens the <a href="biaspar.htm"><strong>biaspar.htm</strong></a>
                    618:         file to edit the output files and graphs. </li>
1.2       lievre    619:     <li>'q' for exiting.</li>
                    620: </ul>
                    621: 
                    622: <h5><font size="4"><strong>Results files </strong></font><br>
                    623: <br>
                    624: <font color="#EC5E5E" size="3"><strong>- </strong></font><a
                    625: name="Observed prevalence in each state"><font color="#EC5E5E"
                    626: size="3"><strong>Observed prevalence in each state</strong></font></a><font
                    627: color="#EC5E5E" size="3"><strong> (and at first pass)</strong></font><b>:
                    628: </b><a href="prbiaspar.txt"><b>prbiaspar.txt</b></a><br>
                    629: </h5>
                    630: 
                    631: <p>The first line is the title and displays each field of the
                    632: file. The first column is age. The fields 2 and 6 are the
                    633: proportion of individuals in states 1 and 2 respectively as
                    634: observed during the first exam. Others fields are the numbers of
                    635: people in states 1, 2 or more. The number of columns increases if
                    636: the number of states is higher than 2.<br>
                    637: The header of the file is </p>
                    638: 
                    639: <pre># Age Prev(1) N(1) N Age Prev(2) N(2) N
                    640: 70 1.00000 631 631 70 0.00000 0 631
                    641: 71 0.99681 625 627 71 0.00319 2 627 
                    642: 72 0.97125 1115 1148 72 0.02875 33 1148 </pre>
                    643: 
                    644: <p>It means that at age 70, the prevalence in state 1 is 1.000
                    645: and in state 2 is 0.00 . At age 71 the number of individuals in
                    646: state 1 is 625 and in state 2 is 2, hence the total number of
                    647: people aged 71 is 625+2=627. <br>
                    648: </p>
                    649: 
                    650: <h5><font color="#EC5E5E" size="3"><b>- Estimated parameters and
                    651: covariance matrix</b></font><b>: </b><a href="rbiaspar.txt"><b>rbiaspar.txt</b></a></h5>
                    652: 
                    653: <p>This file contains all the maximisation results: </p>
                    654: 
1.3       lievre    655: <pre> -2 log likelihood= 21660.918613445392
                    656:  Estimated parameters: a12 = -12.290174 b12 = 0.092161 
                    657:                        a13 = -9.155590  b13 = 0.046627 
                    658:                        a21 = -2.629849  b21 = -0.022030 
                    659:                        a23 = -7.958519  b23 = 0.042614  
                    660:  Covariance matrix: Var(a12) = 1.47453e-001
                    661:                     Var(b12) = 2.18676e-005
                    662:                     Var(a13) = 2.09715e-001
                    663:                     Var(b13) = 3.28937e-005  
                    664:                     Var(a21) = 9.19832e-001
                    665:                     Var(b21) = 1.29229e-004
                    666:                     Var(a23) = 4.48405e-001
                    667:                     Var(b23) = 5.85631e-005 
1.2       lievre    668:  </pre>
                    669: 
1.3       lievre    670: <p>By substitution of these parameters in the regression model,
                    671: we obtain the elementary transition probabilities:</p>
                    672: 
                    673: <p><img src="pebiaspar1.gif" width="400" height="300"></p>
                    674: 
1.2       lievre    675: <h5><font color="#EC5E5E" size="3"><b>- Transition probabilities</b></font><b>:
                    676: </b><a href="pijrbiaspar.txt"><b>pijrbiaspar.txt</b></a></h5>
                    677: 
                    678: <p>Here are the transitions probabilities Pij(x, x+nh) where nh
                    679: is a multiple of 2 years. The first column is the starting age x
                    680: (from age 50 to 100), the second is age (x+nh) and the others are
                    681: the transition probabilities p11, p12, p13, p21, p22, p23. For
                    682: example, line 5 of the file is: </p>
                    683: 
1.3       lievre    684: <pre> 100 106 0.02655 0.17622 0.79722 0.01809 0.13678 0.84513 </pre>
1.2       lievre    685: 
                    686: <p>and this means: </p>
                    687: 
1.3       lievre    688: <pre>p11(100,106)=0.02655
                    689: p12(100,106)=0.17622
                    690: p13(100,106)=0.79722
                    691: p21(100,106)=0.01809
                    692: p22(100,106)=0.13678
                    693: p22(100,106)=0.84513 </pre>
1.2       lievre    694: 
                    695: <h5><font color="#EC5E5E" size="3"><b>- </b></font><a
                    696: name="Stationary prevalence in each state"><font color="#EC5E5E"
                    697: size="3"><b>Stationary prevalence in each state</b></font></a><b>:
                    698: </b><a href="plrbiaspar.txt"><b>plrbiaspar.txt</b></a></h5>
                    699: 
1.3       lievre    700: <pre>#Prevalence
                    701: #Age 1-1 2-2
                    702: 
                    703: #************ 
                    704: 70 0.90134 0.09866
                    705: 71 0.89177 0.10823 
                    706: 72 0.88139 0.11861 
                    707: 73 0.87015 0.12985 </pre>
1.2       lievre    708: 
1.3       lievre    709: <p>At age 70 the stationary prevalence is 0.90134 in state 1 and
                    710: 0.09866 in state 2. This stationary prevalence differs from
1.2       lievre    711: observed prevalence. Here is the point. The observed prevalence
                    712: at age 70 results from the incidence of disability, incidence of
                    713: recovery and mortality which occurred in the past of the cohort.
                    714: Stationary prevalence results from a simulation with actual
                    715: incidences and mortality (estimated from this cross-longitudinal
                    716: survey). It is the best predictive value of the prevalence in the
                    717: future if &quot;nothing changes in the future&quot;. This is
                    718: exactly what demographers do with a Life table. Life expectancy
                    719: is the expected mean time to survive if observed mortality rates
                    720: (incidence of mortality) &quot;remains constant&quot; in the
                    721: future. </p>
                    722: 
                    723: <h5><font color="#EC5E5E" size="3"><b>- Standard deviation of
                    724: stationary prevalence</b></font><b>: </b><a
                    725: href="vplrbiaspar.txt"><b>vplrbiaspar.txt</b></a></h5>
                    726: 
                    727: <p>The stationary prevalence has to be compared with the observed
                    728: prevalence by age. But both are statistical estimates and
                    729: subjected to stochastic errors due to the size of the sample, the
                    730: design of the survey, and, for the stationary prevalence to the
                    731: model used and fitted. It is possible to compute the standard
                    732: deviation of the stationary prevalence at each age.</p>
                    733: 
1.3       lievre    734: <h5><font color="#EC5E5E" size="3">-Observed and stationary
1.2       lievre    735: prevalence in state (2=disable) with the confident interval</font>:<b>
1.3       lievre    736: </b><a href="vbiaspar21.htm"><b>vbiaspar21.gif</b></a></h5>
1.2       lievre    737: 
1.3       lievre    738: <p>This graph exhibits the stationary prevalence in state (2)
                    739: with the confidence interval in red. The green curve is the
                    740: observed prevalence (or proportion of individuals in state (2)).
                    741: Without discussing the results (it is not the purpose here), we
                    742: observe that the green curve is rather below the stationary
                    743: prevalence. It suggests an increase of the disability prevalence
                    744: in the future.</p>
                    745: 
                    746: <p><img src="vbiaspar21.gif" width="400" height="300"></p>
                    747: 
                    748: <h5><font color="#EC5E5E" size="3"><b>-Convergence to the
                    749: stationary prevalence of disability</b></font><b>: </b><a
                    750: href="pbiaspar11.gif"><b>pbiaspar11.gif</b></a><br>
                    751: <img src="pbiaspar11.gif" width="400" height="300"> </h5>
1.2       lievre    752: 
                    753: <p>This graph plots the conditional transition probabilities from
                    754: an initial state (1=healthy in red at the bottom, or 2=disable in
                    755: green on top) at age <em>x </em>to the final state 2=disable<em> </em>at
                    756: age <em>x+h. </em>Conditional means at the condition to be alive
                    757: at age <em>x+h </em>which is <i>hP12x</i> + <em>hP22x</em>. The
                    758: curves <i>hP12x/(hP12x</i> + <em>hP22x) </em>and <i>hP22x/(hP12x</i>
                    759: + <em>hP22x) </em>converge with <em>h, </em>to the <em>stationary
                    760: prevalence of disability</em>. In order to get the stationary
                    761: prevalence at age 70 we should start the process at an earlier
                    762: age, i.e.50. If the disability state is defined by severe
                    763: disability criteria with only a few chance to recover, then the
                    764: incidence of recovery is low and the time to convergence is
                    765: probably longer. But we don't have experience yet.</p>
                    766: 
                    767: <h5><font color="#EC5E5E" size="3"><b>- Life expectancies by age
                    768: and initial health status</b></font><b>: </b><a
                    769: href="erbiaspar.txt"><b>erbiaspar.txt</b></a></h5>
                    770: 
                    771: <pre># Health expectancies 
                    772: # Age 1-1 1-2 2-1 2-2 
1.3       lievre    773: 70 10.9226 3.0401 5.6488 6.2122 
                    774: 71 10.4384 3.0461 5.2477 6.1599 
                    775: 72 9.9667 3.0502 4.8663 6.1025 
                    776: 73 9.5077 3.0524 4.5044 6.0401 </pre>
1.2       lievre    777: 
1.3       lievre    778: <pre>For example 70 10.9226 3.0401 5.6488 6.2122 means:
                    779: e11=10.9226 e12=3.0401 e21=5.6488 e22=6.2122</pre>
1.2       lievre    780: 
1.3       lievre    781: <pre><img src="expbiaspar21.gif" width="400" height="300"><img
                    782: src="expbiaspar11.gif" width="400" height="300"></pre>
1.2       lievre    783: 
                    784: <p>For example, life expectancy of a healthy individual at age 70
1.3       lievre    785: is 10.92 in the healthy state and 3.04 in the disability state
                    786: (=13.96 years). If he was disable at age 70, his life expectancy
                    787: will be shorter, 5.64 in the healthy state and 6.21 in the
                    788: disability state (=11.85 years). The total life expectancy is a
                    789: weighted mean of both, 13.96 and 11.85; weight is the proportion
1.2       lievre    790: of people disabled at age 70. In order to get a pure period index
                    791: (i.e. based only on incidences) we use the <a
                    792: href="#Stationary prevalence in each state">computed or
                    793: stationary prevalence</a> at age 70 (i.e. computed from
                    794: incidences at earlier ages) instead of the <a
                    795: href="#Observed prevalence in each state">observed prevalence</a>
                    796: (for example at first exam) (<a href="#Health expectancies">see
                    797: below</a>).</p>
                    798: 
                    799: <h5><font color="#EC5E5E" size="3"><b>- Variances of life
                    800: expectancies by age and initial health status</b></font><b>: </b><a
                    801: href="vrbiaspar.txt"><b>vrbiaspar.txt</b></a></h5>
                    802: 
                    803: <p>For example, the covariances of life expectancies Cov(ei,ej)
                    804: at age 50 are (line 3) </p>
                    805: 
1.3       lievre    806: <pre>   Cov(e1,e1)=0.4776  Cov(e1,e2)=0.0488=Cov(e2,e1)  Cov(e2,e2)=0.0424</pre>
1.2       lievre    807: 
                    808: <h5><font color="#EC5E5E" size="3"><b>- </b></font><a
                    809: name="Health expectancies"><font color="#EC5E5E" size="3"><b>Health
                    810: expectancies</b></font></a><font color="#EC5E5E" size="3"><b>
                    811: with standard errors in parentheses</b></font><b>: </b><a
                    812: href="trbiaspar.txt"><font face="Courier New"><b>trbiaspar.txt</b></font></a></h5>
                    813: 
                    814: <pre>#Total LEs with variances: e.. (std) e.1 (std) e.2 (std) </pre>
                    815: 
1.3       lievre    816: <pre>70 13.76 (0.22) 10.40 (0.20) 3.35 (0.14) </pre>
1.2       lievre    817: 
1.3       lievre    818: <p>Thus, at age 70 the total life expectancy, e..=13.76years is
                    819: the weighted mean of e1.=13.96 and e2.=11.85 by the stationary
                    820: prevalence at age 70 which are 0.90134 in state 1 and 0.09866 in
                    821: state 2, respectively (the sum is equal to one). e.1=10.40 is the
1.2       lievre    822: Disability-free life expectancy at age 70 (it is again a weighted
1.3       lievre    823: mean of e11 and e21). e.2=3.35 is also the life expectancy at age
1.2       lievre    824: 70 to be spent in the disability state.</p>
                    825: 
1.3       lievre    826: <h5><font color="#EC5E5E" size="3"><b>-Total life expectancy by
1.2       lievre    827: age and health expectancies in states (1=healthy) and (2=disable)</b></font><b>:
1.3       lievre    828: </b><a href="ebiaspar1.gif"><b>ebiaspar1.gif</b></a></h5>
1.2       lievre    829: 
                    830: <p>This figure represents the health expectancies and the total
                    831: life expectancy with the confident interval in dashed curve. </p>
                    832: 
1.3       lievre    833: <pre>        <img src="ebiaspar1.gif" width="400" height="300"></pre>
1.2       lievre    834: 
                    835: <p>Standard deviations (obtained from the information matrix of
                    836: the model) of these quantities are very useful.
                    837: Cross-longitudinal surveys are costly and do not involve huge
                    838: samples, generally a few thousands; therefore it is very
                    839: important to have an idea of the standard deviation of our
                    840: estimates. It has been a big challenge to compute the Health
                    841: Expectancy standard deviations. Don't be confuse: life expectancy
                    842: is, as any expected value, the mean of a distribution; but here
                    843: we are not computing the standard deviation of the distribution,
                    844: but the standard deviation of the estimate of the mean.</p>
                    845: 
                    846: <p>Our health expectancies estimates vary according to the sample
                    847: size (and the standard deviations give confidence intervals of
                    848: the estimate) but also according to the model fitted. Let us
                    849: explain it in more details.</p>
                    850: 
                    851: <p>Choosing a model means ar least two kind of choices. First we
                    852: have to decide the number of disability states. Second we have to
                    853: design, within the logit model family, the model: variables,
                    854: covariables, confonding factors etc. to be included.</p>
                    855: 
                    856: <p>More disability states we have, better is our demographical
                    857: approach of the disability process, but smaller are the number of
                    858: transitions between each state and higher is the noise in the
                    859: measurement. We do not have enough experiments of the various
                    860: models to summarize the advantages and disadvantages, but it is
                    861: important to say that even if we had huge and unbiased samples,
                    862: the total life expectancy computed from a cross-longitudinal
                    863: survey, varies with the number of states. If we define only two
                    864: states, alive or dead, we find the usual life expectancy where it
                    865: is assumed that at each age, people are at the same risk to die.
                    866: If we are differentiating the alive state into healthy and
                    867: disable, and as the mortality from the disability state is higher
                    868: than the mortality from the healthy state, we are introducing
                    869: heterogeneity in the risk of dying. The total mortality at each
                    870: age is the weighted mean of the mortality in each state by the
                    871: prevalence in each state. Therefore if the proportion of people
                    872: at each age and in each state is different from the stationary
                    873: equilibrium, there is no reason to find the same total mortality
                    874: at a particular age. Life expectancy, even if it is a very useful
                    875: tool, has a very strong hypothesis of homogeneity of the
                    876: population. Our main purpose is not to measure differential
                    877: mortality but to measure the expected time in a healthy or
                    878: disability state in order to maximise the former and minimize the
                    879: latter. But the differential in mortality complexifies the
                    880: measurement.</p>
                    881: 
                    882: <p>Incidences of disability or recovery are not affected by the
                    883: number of states if these states are independant. But incidences
                    884: estimates are dependant on the specification of the model. More
                    885: covariates we added in the logit model better is the model, but
                    886: some covariates are not well measured, some are confounding
                    887: factors like in any statistical model. The procedure to &quot;fit
                    888: the best model' is similar to logistic regression which itself is
                    889: similar to regression analysis. We haven't yet been sofar because
                    890: we also have a severe limitation which is the speed of the
                    891: convergence. On a Pentium III, 500 MHz, even the simplest model,
                    892: estimated by month on 8,000 people may take 4 hours to converge.
                    893: Also, the program is not yet a statistical package, which permits
                    894: a simple writing of the variables and the model to take into
                    895: account in the maximisation. The actual program allows only to
1.3       lievre    896: add simple variables like age+sex or age+sex+ age*sex but will
1.2       lievre    897: never be general enough. But what is to remember, is that
                    898: incidences or probability of change from one state to another is
                    899: affected by the variables specified into the model.</p>
                    900: 
                    901: <p>Also, the age range of the people interviewed has a link with
                    902: the age range of the life expectancy which can be estimated by
                    903: extrapolation. If your sample ranges from age 70 to 95, you can
                    904: clearly estimate a life expectancy at age 70 and trust your
                    905: confidence interval which is mostly based on your sample size,
                    906: but if you want to estimate the life expectancy at age 50, you
                    907: should rely in your model, but fitting a logistic model on a age
                    908: range of 70-95 and estimating probabilties of transition out of
                    909: this age range, say at age 50 is very dangerous. At least you
                    910: should remember that the confidence interval given by the
                    911: standard deviation of the health expectancies, are under the
                    912: strong assumption that your model is the 'true model', which is
                    913: probably not the case.</p>
                    914: 
                    915: <h5><font color="#EC5E5E" size="3"><b>- Copy of the parameter
                    916: file</b></font><b>: </b><a href="orbiaspar.txt"><b>orbiaspar.txt</b></a></h5>
                    917: 
                    918: <p>This copy of the parameter file can be useful to re-run the
                    919: program while saving the old output files. </p>
                    920: 
1.4     ! lievre    921: <h5><font color="#EC5E5E" size="3"><b>- Prevalence forecasting</b></font><b>:
        !           922: </b><a href="frbiaspar.txt"><b>frbiaspar.txt</b></a></h5>
        !           923: 
        !           924: <p>On a d'abord estimé la date moyenne des interviaew. ie
        !           925: 13/9/1995. This file contains </p>
        !           926: 
        !           927: <p>Example, at date 1/1/1989 : </p>
        !           928: 
        !           929: <p>73 0.807 0.078 0.115 </p>
        !           930: 
        !           931: <p>This means that at age 73, the probability for a person age 70
        !           932: at 13/9/1989 to be in state 1 is 0.807, in state 2 is 0.078 and
        !           933: to die is 0.115 at 1/1/1989.</p>
        !           934: 
        !           935: <h5><font color="#EC5E5E" size="3"><b>- Population forecasting</b></font><b>:
        !           936: </b><a href="poprbiaspar.txt"><b>poprbiaspar.txt</b></a></h5>
        !           937: 
        !           938: <pre># Age P.1 P.2 P.3 [Population]
        !           939: # Forecasting at date 1/1/1989 
        !           940: 75 572685.22 83798.08 
        !           941: 74 621296.51 79767.99 
        !           942: 73 645857.70 69320.60 </pre>
        !           943: 
        !           944: <pre># Forecasting at date 1/1/19909 
        !           945: 76 442986.68 92721.14 120775.48
        !           946: 75 487781.02 91367.97 121915.51
        !           947: 74 512892.07 85003.47 117282.76 </pre>
        !           948: 
1.2       lievre    949: <hr>
                    950: 
                    951: <h2><a name="example" </a><font color="#00006A">Trying an example</font></a></h2>
                    952: 
                    953: <p>Since you know how to run the program, it is time to test it
                    954: on your own computer. Try for example on a parameter file named <a
1.3       lievre    955: href="..\mytry\imachpar.txt">imachpar.txt</a> which is a copy of <font
                    956: size="2" face="Courier New">mypar.txt</font> included in the
                    957: subdirectory of imach, <font size="2" face="Courier New">mytry</font>.
                    958: Edit it to change the name of the data file to <font size="2"
                    959: face="Courier New">..\data\mydata.txt</font> if you don't want to
                    960: copy it on the same directory. The file <font face="Courier New">mydata.txt</font>
                    961: is a smaller file of 3,000 people but still with 4 waves. </p>
1.2       lievre    962: 
                    963: <p>Click on the imach.exe icon to open a window. Answer to the
                    964: question:'<strong>Enter the parameter file name:'</strong></p>
                    965: 
                    966: <table border="1">
                    967:     <tr>
1.4     ! lievre    968:         <td width="100%"><strong>IMACH, Version 0.7</strong><p><strong>Enter
1.2       lievre    969:         the parameter file name: ..\mytry\imachpar.txt</strong></p>
                    970:         </td>
                    971:     </tr>
                    972: </table>
                    973: 
                    974: <p>Most of the data files or image files generated, will use the
                    975: 'imachpar' string into their name. The running time is about 2-3
                    976: minutes on a Pentium III. If the execution worked correctly, the
                    977: outputs files are created in the current directory, and should be
                    978: the same as the mypar files initially included in the directory <font
                    979: size="2" face="Courier New">mytry</font>.</p>
                    980: 
                    981: <ul>
                    982:     <li><pre><u>Output on the screen</u> The output screen looks like <a
                    983: href="imachrun.LOG">this Log file</a>
                    984: #
                    985: 
                    986: title=MLE datafile=..\data\mydata.txt lastobs=3000 firstpass=1 lastpass=3
                    987: ftol=1.000000e-008 stepm=24 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre>
                    988:     </li>
                    989:     <li><pre>Total number of individuals= 2965, Agemin = 70.00, Agemax= 100.92
                    990: 
                    991: Warning, no any valid information for:126 line=126
                    992: Warning, no any valid information for:2307 line=2307
                    993: Delay (in months) between two waves Min=21 Max=51 Mean=24.495826
                    994: <font face="Times New Roman">These lines give some warnings on the data file and also some raw statistics on frequencies of transitions.</font>
                    995: Age 70 1.=230 loss[1]=3.5% 2.=16 loss[2]=12.5% 1.=222 prev[1]=94.1% 2.=14
                    996:  prev[2]=5.9% 1-1=8 11=200 12=7 13=15 2-1=2 21=6 22=7 23=1
                    997: Age 102 1.=0 loss[1]=NaNQ% 2.=0 loss[2]=NaNQ% 1.=0 prev[1]=NaNQ% 2.=0 </pre>
                    998:     </li>
                    999: </ul>
                   1000: 
                   1001: <p>&nbsp;</p>
                   1002: 
                   1003: <ul>
                   1004:     <li>Maximisation with the Powell algorithm. 8 directions are
                   1005:         given corresponding to the 8 parameters. this can be
                   1006:         rather long to get convergence.<br>
                   1007:         <font size="1" face="Courier New"><br>
                   1008:         Powell iter=1 -2*LL=11531.405658264877 1 0.000000000000 2
                   1009:         0.000000000000 3<br>
                   1010:         0.000000000000 4 0.000000000000 5 0.000000000000 6
                   1011:         0.000000000000 7 <br>
                   1012:         0.000000000000 8 0.000000000000<br>
                   1013:         1..........2.................3..........4.................5.........<br>
                   1014:         6................7........8...............<br>
                   1015:         Powell iter=23 -2*LL=6744.954108371555 1 -12.967632334283
                   1016:         <br>
                   1017:         2 0.135136681033 3 -7.402109728262 4 0.067844593326 <br>
                   1018:         5 -0.673601538129 6 -0.006615504377 7 -5.051341616718 <br>
                   1019:         8 0.051272038506<br>
                   1020:         1..............2...........3..............4...........<br>
                   1021:         5..........6................7...........8.........<br>
                   1022:         #Number of iterations = 23, -2 Log likelihood =
                   1023:         6744.954042573691<br>
                   1024:         # Parameters<br>
                   1025:         12 -12.966061 0.135117 <br>
                   1026:         13 -7.401109 0.067831 <br>
                   1027:         21 -0.672648 -0.006627 <br>
                   1028:         23 -5.051297 0.051271 </font><br>
                   1029:         </li>
                   1030:     <li><pre><font size="2">Calculation of the hessian matrix. Wait...
                   1031: 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
                   1032: 
                   1033: Inverting the hessian to get the covariance matrix. Wait...
                   1034: 
                   1035: #Hessian matrix#
                   1036: 3.344e+002 2.708e+004 -4.586e+001 -3.806e+003 -1.577e+000 -1.313e+002 3.914e-001 3.166e+001 
                   1037: 2.708e+004 2.204e+006 -3.805e+003 -3.174e+005 -1.303e+002 -1.091e+004 2.967e+001 2.399e+003 
                   1038: -4.586e+001 -3.805e+003 4.044e+002 3.197e+004 2.431e-002 1.995e+000 1.783e-001 1.486e+001 
                   1039: -3.806e+003 -3.174e+005 3.197e+004 2.541e+006 2.436e+000 2.051e+002 1.483e+001 1.244e+003 
                   1040: -1.577e+000 -1.303e+002 2.431e-002 2.436e+000 1.093e+002 8.979e+003 -3.402e+001 -2.843e+003 
                   1041: -1.313e+002 -1.091e+004 1.995e+000 2.051e+002 8.979e+003 7.420e+005 -2.842e+003 -2.388e+005 
                   1042: 3.914e-001 2.967e+001 1.783e-001 1.483e+001 -3.402e+001 -2.842e+003 1.494e+002 1.251e+004 
                   1043: 3.166e+001 2.399e+003 1.486e+001 1.244e+003 -2.843e+003 -2.388e+005 1.251e+004 1.053e+006 
                   1044: # Scales
                   1045: 12 1.00000e-004 1.00000e-006
                   1046: 13 1.00000e-004 1.00000e-006
                   1047: 21 1.00000e-003 1.00000e-005
                   1048: 23 1.00000e-004 1.00000e-005
                   1049: # Covariance
                   1050:   1 5.90661e-001
                   1051:   2 -7.26732e-003 8.98810e-005
                   1052:   3 8.80177e-002 -1.12706e-003 5.15824e-001
                   1053:   4 -1.13082e-003 1.45267e-005 -6.50070e-003 8.23270e-005
                   1054:   5 9.31265e-003 -1.16106e-004 6.00210e-004 -8.04151e-006 1.75753e+000
                   1055:   6 -1.15664e-004 1.44850e-006 -7.79995e-006 1.04770e-007 -2.12929e-002 2.59422e-004
                   1056:   7 1.35103e-003 -1.75392e-005 -6.38237e-004 7.85424e-006 4.02601e-001 -4.86776e-003 1.32682e+000
                   1057:   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
                   1058: # agemin agemax for lifexpectancy, bage fage (if mle==0 ie no data nor Max likelihood).
                   1059: 
                   1060: 
                   1061: agemin=70 agemax=100 bage=50 fage=100
                   1062: Computing prevalence limit: result on file 'plrmypar.txt' 
                   1063: Computing pij: result on file 'pijrmypar.txt' 
                   1064: Computing Health Expectancies: result on file 'ermypar.txt' 
                   1065: Computing Variance-covariance of DFLEs: file 'vrmypar.txt' 
                   1066: Computing Total LEs with variances: file 'trmypar.txt' 
                   1067: Computing Variance-covariance of Prevalence limit: file 'vplrmypar.txt' 
                   1068: End of Imach
                   1069: </font></pre>
                   1070:     </li>
                   1071: </ul>
                   1072: 
                   1073: <p><font size="3">Once the running is finished, the program
                   1074: requires a caracter:</font></p>
                   1075: 
                   1076: <table border="1">
                   1077:     <tr>
1.3       lievre   1078:         <td width="100%"><strong>Type e to edit output files, c
                   1079:         to start again, and q for exiting:</strong></td>
1.2       lievre   1080:     </tr>
                   1081: </table>
                   1082: 
1.3       lievre   1083: <p><font size="3">First you should enter <strong>e </strong>to
                   1084: edit the master file mypar.htm. </font></p>
1.2       lievre   1085: 
                   1086: <ul>
                   1087:     <li><u>Outputs files</u> <br>
1.3       lievre   1088:         <br>
1.2       lievre   1089:         - Observed prevalence in each state: <a
1.3       lievre   1090:         href="..\mytry\prmypar.txt">pmypar.txt</a> <br>
1.2       lievre   1091:         - Estimated parameters and the covariance matrix: <a
                   1092:         href="..\mytry\rmypar.txt">rmypar.txt</a> <br>
                   1093:         - Stationary prevalence in each state: <a
                   1094:         href="..\mytry\plrmypar.txt">plrmypar.txt</a> <br>
                   1095:         - Transition probabilities: <a
                   1096:         href="..\mytry\pijrmypar.txt">pijrmypar.txt</a> <br>
                   1097:         - Copy of the parameter file: <a
                   1098:         href="..\mytry\ormypar.txt">ormypar.txt</a> <br>
                   1099:         - Life expectancies by age and initial health status: <a
                   1100:         href="..\mytry\ermypar.txt">ermypar.txt</a> <br>
                   1101:         - Variances of life expectancies by age and initial
                   1102:         health status: <a href="..\mytry\vrmypar.txt">vrmypar.txt</a>
                   1103:         <br>
                   1104:         - Health expectancies with their variances: <a
                   1105:         href="..\mytry\trmypar.txt">trmypar.txt</a> <br>
                   1106:         - Standard deviation of stationary prevalence: <a
1.4     ! lievre   1107:         href="..\mytry\vplrmypar.txt">vplrmypar.txt</a><br>
        !          1108:         - Prevalences forecasting: <a href="frmypar.txt">frmypar.txt</a>
1.2       lievre   1109:         <br>
1.4     ! lievre   1110:         - Population forecasting (if popforecast=1): <a
        !          1111:         href="poprmypar.txt">poprmypar.txt</a> <br>
1.2       lievre   1112:         </li>
                   1113:     <li><u>Graphs</u> <br>
                   1114:         <br>
1.3       lievre   1115:         -<a href="../mytry/pemypar1.gif">One-step transition
                   1116:         probabilities</a><br>
                   1117:         -<a href="../mytry/pmypar11.gif">Convergence to the
                   1118:         stationary prevalence</a><br>
                   1119:         -<a href="..\mytry\vmypar11.gif">Observed and stationary
1.2       lievre   1120:         prevalence in state (1) with the confident interval</a> <br>
1.3       lievre   1121:         -<a href="..\mytry\vmypar21.gif">Observed and stationary
1.2       lievre   1122:         prevalence in state (2) with the confident interval</a> <br>
1.3       lievre   1123:         -<a href="..\mytry\expmypar11.gif">Health life
                   1124:         expectancies by age and initial health state (1)</a> <br>
                   1125:         -<a href="..\mytry\expmypar21.gif">Health life
                   1126:         expectancies by age and initial health state (2)</a> <br>
                   1127:         -<a href="..\mytry\emypar1.gif">Total life expectancy by
1.2       lievre   1128:         age and health expectancies in states (1) and (2).</a> </li>
                   1129: </ul>
                   1130: 
                   1131: <p>This software have been partly granted by <a
                   1132: href="http://euroreves.ined.fr">Euro-REVES</a>, a concerted
                   1133: action from the European Union. It will be copyrighted
                   1134: identically to a GNU software product, i.e. program and software
                   1135: can be distributed freely for non commercial use. Sources are not
                   1136: widely distributed today. You can get them by asking us with a
                   1137: simple justification (name, email, institute) <a
                   1138: href="mailto:brouard@ined.fr">mailto:brouard@ined.fr</a> and <a
                   1139: href="mailto:lievre@ined.fr">mailto:lievre@ined.fr</a> .</p>
                   1140: 
1.4     ! lievre   1141: <p>Latest version (0.7 of February 2002) can be accessed at <a
1.2       lievre   1142: href="http://euroeves.ined.fr/imach">http://euroreves.ined.fr/imach</a><br>
                   1143: </p>
                   1144: </body>
                   1145: </html>

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