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! 8: <title>Computing Health Expectancies using IMaCh</title>
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! 21:
! 22: <h1 align="center"><font color="#00006A">Computing Health
! 23: Expectancies using IMaCh</font></h1>
! 24:
! 25: <h1 align="center"><font color="#00006A" size="5">(a Maximum
! 26: Likelihood Computer Program using Interpolation of Markov Chains)</font></h1>
! 27:
! 28: <p align="center"> </p>
! 29:
! 30: <p align="center"><a href="http://www.ined.fr/"><img
! 31: src="logo-ined.gif" border="0" width="151" height="76"></a><img
! 32: src="euroreves2.gif" width="151" height="75"></p>
! 33:
! 34: <h3 align="center"><a href="http://www.ined.fr/"><font
! 35: color="#00006A">INED</font></a><font color="#00006A"> and </font><a
! 36: href="http://euroreves.ined.fr"><font color="#00006A">EUROREVES</font></a></h3>
! 37:
! 38: <p align="center"><font color="#00006A" size="4"><strong>Version
! 39: 0.8a, May 2002</strong></font></p>
! 40:
! 41: <hr size="3" color="#EC5E5E">
! 42:
! 43: <p align="center"><font color="#00006A"><strong>Authors of the
! 44: program: </strong></font><a href="http://sauvy.ined.fr/brouard"><font
! 45: color="#00006A"><strong>Nicolas Brouard</strong></font></a><font
! 46: color="#00006A"><strong>, senior researcher at the </strong></font><a
! 47: href="http://www.ined.fr"><font color="#00006A"><strong>Institut
! 48: National d'Etudes Démographiques</strong></font></a><font
! 49: color="#00006A"><strong> (INED, Paris) in the "Mortality,
! 50: Health and Epidemiology" Research Unit </strong></font></p>
! 51:
! 52: <p align="center"><font color="#00006A"><strong>and Agnès
! 53: Lièvre<br clear="left">
! 54: </strong></font></p>
! 55:
! 56: <h4><font color="#00006A">Contribution to the mathematics: C. R.
! 57: Heathcote </font><font color="#00006A" size="2">(Australian
! 58: National University, Canberra).</font></h4>
! 59:
! 60: <h4><font color="#00006A">Contact: Agnès Lièvre (</font><a
! 61: href="mailto:lievre@ined.fr"><font color="#00006A"><i>lievre@ined.fr</i></font></a><font
! 62: color="#00006A">) </font></h4>
! 63:
! 64: <hr>
! 65:
! 66: <ul>
! 67: <li><a href="#intro">Introduction</a> </li>
! 68: <li><a href="#data">On what kind of data can it be used?</a></li>
! 69: <li><a href="#datafile">The data file</a> </li>
! 70: <li><a href="#biaspar">The parameter file</a> </li>
! 71: <li><a href="#running">Running Imach</a> </li>
! 72: <li><a href="#output">Output files and graphs</a> </li>
! 73: <li><a href="#example">Exemple</a> </li>
! 74: </ul>
! 75:
! 76: <hr>
! 77:
! 78: <h2><a name="intro"><font color="#00006A">Introduction</font></a></h2>
! 79:
! 80: <p>This program computes <b>Healthy Life Expectancies</b> from <b>cross-longitudinal
! 81: data</b> using the methodology pioneered by Laditka and Wolf (1).
! 82: Within the family of Health Expectancies (HE), Disability-free
! 83: life expectancy (DFLE) is probably the most important index to
! 84: monitor. In low mortality countries, there is a fear that when
! 85: mortality declines, the increase in DFLE is not proportionate to
! 86: the increase in total Life expectancy. This case is called the <em>Expansion
! 87: of morbidity</em>. Most of the data collected today, in
! 88: particular by the international <a href="http://www.reves.org">REVES</a>
! 89: network on Health expectancy, and most HE indices based on these
! 90: data, are <em>cross-sectional</em>. It means that the information
! 91: collected comes from a single cross-sectional survey: people from
! 92: various ages (but mostly old people) are surveyed on their health
! 93: status at a single date. Proportion of people disabled at each
! 94: age, can then be measured at that date. This age-specific
! 95: prevalence curve is then used to distinguish, within the
! 96: stationary population (which, by definition, is the life table
! 97: estimated from the vital statistics on mortality at the same
! 98: date), the disable population from the disability-free
! 99: population. Life expectancy (LE) (or total population divided by
! 100: the yearly number of births or deaths of this stationary
! 101: population) is then decomposed into DFLE and DLE. This method of
! 102: computing HE is usually called the Sullivan method (from the name
! 103: of the author who first described it).</p>
! 104:
! 105: <p>Age-specific proportions of people disable are very difficult
! 106: to forecast because each proportion corresponds to historical
! 107: conditions of the cohort and it is the result of the historical
! 108: flows from entering disability and recovering in the past until
! 109: today. The age-specific intensities (or incidence rates) of
! 110: entering disability or recovering a good health, are reflecting
! 111: actual conditions and therefore can be used at each age to
! 112: forecast the future of this cohort. For example if a country is
! 113: improving its technology of prosthesis, the incidence of
! 114: recovering the ability to walk will be higher at each (old) age,
! 115: but the prevalence of disability will only slightly reflect an
! 116: improve because the prevalence is mostly affected by the history
! 117: of the cohort and not by recent period effects. To measure the
! 118: period improvement we have to simulate the future of a cohort of
! 119: new-borns entering or leaving at each age the disability state or
! 120: dying according to the incidence rates measured today on
! 121: different cohorts. The proportion of people disabled at each age
! 122: in this simulated cohort will be much lower (using the exemple of
! 123: an improvement) that the proportions observed at each age in a
! 124: cross-sectional survey. This new prevalence curve introduced in a
! 125: life table will give a much more actual and realistic HE level
! 126: than the Sullivan method which mostly measured the History of
! 127: health conditions in this country.</p>
! 128:
! 129: <p>Therefore, the main question is how to measure incidence rates
! 130: from cross-longitudinal surveys? This is the goal of the IMaCH
! 131: program. From your data and using IMaCH you can estimate period
! 132: HE and not only Sullivan's HE. Also the standard errors of the HE
! 133: are computed.</p>
! 134:
! 135: <p>A cross-longitudinal survey consists in a first survey
! 136: ("cross") where individuals from different ages are
! 137: interviewed on their health status or degree of disability. At
! 138: least a second wave of interviews ("longitudinal")
! 139: should measure each new individual health status. Health
! 140: expectancies are computed from the transitions observed between
! 141: waves and are computed for each degree of severity of disability
! 142: (number of life states). More degrees you consider, more time is
! 143: necessary to reach the Maximum Likelihood of the parameters
! 144: involved in the model. Considering only two states of disability
! 145: (disable and healthy) is generally enough but the computer
! 146: program works also with more health statuses.<br>
! 147: <br>
! 148: The simplest model is the multinomial logistic model where <i>pij</i>
! 149: is the probability to be observed in state <i>j</i> at the second
! 150: wave conditional to be observed in state <em>i</em> at the first
! 151: wave. Therefore a simple model is: log<em>(pij/pii)= aij +
! 152: bij*age+ cij*sex,</em> where '<i>age</i>' is age and '<i>sex</i>'
! 153: is a covariate. The advantage that this computer program claims,
! 154: comes from that if the delay between waves is not identical for
! 155: each individual, or if some individual missed an interview, the
! 156: information is not rounded or lost, but taken into account using
! 157: an interpolation or extrapolation. <i>hPijx</i> is the
! 158: probability to be observed in state <i>i</i> at age <i>x+h</i>
! 159: conditional to the observed state <i>i</i> at age <i>x</i>. The
! 160: delay '<i>h</i>' can be split into an exact number (<i>nh*stepm</i>)
! 161: of unobserved intermediate states. This elementary transition (by
! 162: month or quarter trimester, semester or year) is modeled as a
! 163: multinomial logistic. The <i>hPx</i> matrix is simply the matrix
! 164: product of <i>nh*stepm</i> elementary matrices and the
! 165: contribution of each individual to the likelihood is simply <i>hPijx</i>.
! 166: <br>
! 167: </p>
! 168:
! 169: <p>The program presented in this manual is a quite general
! 170: program named <strong>IMaCh</strong> (for <strong>I</strong>nterpolated
! 171: <strong>MA</strong>rkov <strong>CH</strong>ain), designed to
! 172: analyse transition data from longitudinal surveys. The first step
! 173: is the parameters estimation of a transition probabilities model
! 174: between an initial status and a final status. From there, the
! 175: computer program produces some indicators such as observed and
! 176: stationary prevalence, life expectancies and their variances and
! 177: graphs. Our transition model consists in absorbing and
! 178: non-absorbing states with the possibility of return across the
! 179: non-absorbing states. The main advantage of this package,
! 180: compared to other programs for the analysis of transition data
! 181: (For example: Proc Catmod of SAS<sup>®</sup>) is that the whole
! 182: individual information is used even if an interview is missing, a
! 183: status or a date is unknown or when the delay between waves is
! 184: not identical for each individual. The program can be executed
! 185: according to parameters: selection of a sub-sample, number of
! 186: absorbing and non-absorbing states, number of waves taken in
! 187: account (the user inputs the first and the last interview), a
! 188: tolerance level for the maximization function, the periodicity of
! 189: the transitions (we can compute annual, quarterly or monthly
! 190: transitions), covariates in the model. It works on Windows or on
! 191: Unix.<br>
! 192: </p>
! 193:
! 194: <hr>
! 195:
! 196: <p>(1) Laditka, Sarah B. and Wolf, Douglas A. (1998), "New
! 197: Methods for Analyzing Active Life Expectancy". <i>Journal of
! 198: Aging and Health</i>. Vol 10, No. 2. </p>
! 199:
! 200: <hr>
! 201:
! 202: <h2><a name="data"><font color="#00006A">On what kind of data can
! 203: it be used?</font></a></h2>
! 204:
! 205: <p>The minimum data required for a transition model is the
! 206: recording of a set of individuals interviewed at a first date and
! 207: interviewed again at least one another time. From the
! 208: observations of an individual, we obtain a follow-up over time of
! 209: the occurrence of a specific event. In this documentation, the
! 210: event is related to health status at older ages, but the program
! 211: can be applied on a lot of longitudinal studies in different
! 212: contexts. To build the data file explained into the next section,
! 213: you must have the month and year of each interview and the
! 214: corresponding health status. But in order to get age, date of
! 215: birth (month and year) is required (missing values is allowed for
! 216: month). Date of death (month and year) is an important
! 217: information also required if the individual is dead. Shorter
! 218: steps (i.e. a month) will more closely take into account the
! 219: survival time after the last interview.</p>
! 220:
! 221: <hr>
! 222:
! 223: <h2><a name="datafile"><font color="#00006A">The data file</font></a></h2>
! 224:
! 225: <p>In this example, 8,000 people have been interviewed in a
! 226: cross-longitudinal survey of 4 waves (1984, 1986, 1988, 1990).
! 227: Some people missed 1, 2 or 3 interviews. Health statuses are
! 228: healthy (1) and disable (2). The survey is not a real one. It is
! 229: a simulation of the American Longitudinal Survey on Aging. The
! 230: disability state is defined if the individual missed one of four
! 231: ADL (Activity of daily living, like bathing, eating, walking).
! 232: Therefore, even is the individuals interviewed in the sample are
! 233: virtual, the information brought with this sample is close to the
! 234: situation of the United States. Sex is not recorded is this
! 235: sample.</p>
! 236:
! 237: <p>Each line of the data set (named <a href="data1.txt">data1.txt</a>
! 238: in this first example) is an individual record which fields are: </p>
! 239:
! 240: <ul>
! 241: <li><b>Index number</b>: positive number (field 1) </li>
! 242: <li><b>First covariate</b> positive number (field 2) </li>
! 243: <li><b>Second covariate</b> positive number (field 3) </li>
! 244: <li><a name="Weight"><b>Weight</b></a>: positive number
! 245: (field 4) . In most surveys individuals are weighted
! 246: according to the stratification of the sample.</li>
! 247: <li><b>Date of birth</b>: coded as mm/yyyy. Missing dates are
! 248: coded as 99/9999 (field 5) </li>
! 249: <li><b>Date of death</b>: coded as mm/yyyy. Missing dates are
! 250: coded as 99/9999 (field 6) </li>
! 251: <li><b>Date of first interview</b>: coded as mm/yyyy. Missing
! 252: dates are coded as 99/9999 (field 7) </li>
! 253: <li><b>Status at first interview</b>: positive number.
! 254: Missing values ar coded -1. (field 8) </li>
! 255: <li><b>Date of second interview</b>: coded as mm/yyyy.
! 256: Missing dates are coded as 99/9999 (field 9) </li>
! 257: <li><strong>Status at second interview</strong> positive
! 258: number. Missing values ar coded -1. (field 10) </li>
! 259: <li><b>Date of third interview</b>: coded as mm/yyyy. Missing
! 260: dates are coded as 99/9999 (field 11) </li>
! 261: <li><strong>Status at third interview</strong> positive
! 262: number. Missing values ar coded -1. (field 12) </li>
! 263: <li><b>Date of fourth interview</b>: coded as mm/yyyy.
! 264: Missing dates are coded as 99/9999 (field 13) </li>
! 265: <li><strong>Status at fourth interview</strong> positive
! 266: number. Missing values are coded -1. (field 14) </li>
! 267: <li>etc</li>
! 268: </ul>
! 269:
! 270: <p> </p>
! 271:
! 272: <p>If your longitudinal survey do not include information about
! 273: weights or covariates, you must fill the column with a number
! 274: (e.g. 1) because a missing field is not allowed.</p>
! 275:
! 276: <hr>
! 277:
! 278: <h2><font color="#00006A">Your first example parameter file</font><a
! 279: href="http://euroreves.ined.fr/imach"></a><a name="uio"></a></h2>
! 280:
! 281: <h2><a name="biaspar"></a>#Imach version 0.8a, May 2002,
! 282: INED-EUROREVES </h2>
! 283:
! 284: <p>This is a comment. Comments start with a '#'.</p>
! 285:
! 286: <h4><font color="#FF0000">First uncommented line</font></h4>
! 287:
! 288: <pre>title=1st_example datafile=data1.txt lastobs=8600 firstpass=1 lastpass=4</pre>
! 289:
! 290: <ul>
! 291: <li><b>title=</b> 1st_example is title of the run. </li>
! 292: <li><b>datafile=</b> data1.txt is the name of the data set.
! 293: Our example is a six years follow-up survey. It consists
! 294: in a baseline followed by 3 reinterviews. </li>
! 295: <li><b>lastobs=</b> 8600 the program is able to run on a
! 296: subsample where the last observation number is lastobs.
! 297: It can be set a bigger number than the real number of
! 298: observations (e.g. 100000). In this example, maximisation
! 299: will be done on the 8600 first records. </li>
! 300: <li><b>firstpass=1</b> , <b>lastpass=4 </b>In case of more
! 301: than two interviews in the survey, the program can be run
! 302: on selected transitions periods. firstpass=1 means the
! 303: first interview included in the calculation is the
! 304: baseline survey. lastpass=4 means that the information
! 305: brought by the 4th interview is taken into account.</li>
! 306: </ul>
! 307:
! 308: <p> </p>
! 309:
! 310: <h4><a name="biaspar-2"><font color="#FF0000">Second uncommented
! 311: line</font></a></h4>
! 312:
! 313: <pre>ftol=1.e-08 stepm=1 ncovcol=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre>
! 314:
! 315: <ul>
! 316: <li><b>ftol=1e-8</b> Convergence tolerance on the function
! 317: value in the maximisation of the likelihood. Choosing a
! 318: correct value for ftol is difficult. 1e-8 is a correct
! 319: value for a 32 bits computer.</li>
! 320: <li><b>stepm=1</b> Time unit in months for interpolation.
! 321: Examples:<ul>
! 322: <li>If stepm=1, the unit is a month </li>
! 323: <li>If stepm=4, the unit is a trimester</li>
! 324: <li>If stepm=12, the unit is a year </li>
! 325: <li>If stepm=24, the unit is two years</li>
! 326: <li>... </li>
! 327: </ul>
! 328: </li>
! 329: <li><b>ncovcol=2</b> Number of covariate columns in the
! 330: datafile which precede the date of birth. Here you can
! 331: put variables that won't necessary be used during the
! 332: run. It is not the number of covariates that will be
! 333: specified by the model. The 'model' syntax describe the
! 334: covariates to take into account. </li>
! 335: <li><b>nlstate=2</b> Number of non-absorbing (alive) states.
! 336: Here we have two alive states: disability-free is coded 1
! 337: and disability is coded 2. </li>
! 338: <li><b>ndeath=1</b> Number of absorbing states. The absorbing
! 339: state death is coded 3. </li>
! 340: <li><b>maxwav=4</b> Number of waves in the datafile.</li>
! 341: <li><a name="mle"><b>mle</b></a><b>=1</b> Option for the
! 342: Maximisation Likelihood Estimation. <ul>
! 343: <li>If mle=1 the program does the maximisation and
! 344: the calculation of health expectancies </li>
! 345: <li>If mle=0 the program only does the calculation of
! 346: the health expectancies. </li>
! 347: </ul>
! 348: </li>
! 349: <li><b>weight=0</b> Possibility to add weights. <ul>
! 350: <li>If weight=0 no weights are included </li>
! 351: <li>If weight=1 the maximisation integrates the
! 352: weights which are in field <a href="#Weight">4</a></li>
! 353: </ul>
! 354: </li>
! 355: </ul>
! 356:
! 357: <h4><font color="#FF0000">Covariates</font></h4>
! 358:
! 359: <p>Intercept and age are systematically included in the model.
! 360: Additional covariates can be included with the command: </p>
! 361:
! 362: <pre>model=<em>list of covariates</em></pre>
! 363:
! 364: <ul>
! 365: <li>if<strong> model=. </strong>then no covariates are
! 366: included</li>
! 367: <li>if <strong>model=V1</strong> the model includes the first
! 368: covariate (field 2)</li>
! 369: <li>if <strong>model=V2 </strong>the model includes the
! 370: second covariate (field 3)</li>
! 371: <li>if <strong>model=V1+V2 </strong>the model includes the
! 372: first and the second covariate (fields 2 and 3)</li>
! 373: <li>if <strong>model=V1*V2 </strong>the model includes the
! 374: product of the first and the second covariate (fields 2
! 375: and 3)</li>
! 376: <li>if <strong>model=V1+V1*age</strong> the model includes
! 377: the product covariate*age</li>
! 378: </ul>
! 379:
! 380: <p>In this example, we have two covariates in the data file
! 381: (fields 2 and 3). The number of covariates included in the data
! 382: file between the id and the date of birth is ncovcol=2 (it was
! 383: named ncov in version prior to 0.8). If you have 3 covariates in
! 384: the datafile (fields 2, 3 and 4), you will set ncovcol=3. Then
! 385: you can run the programme with a new parametrisation taking into
! 386: account the third covariate. For example, <strong>model=V1+V3 </strong>estimates
! 387: a model with the first and third covariates. More complicated
! 388: models can be used, but it will takes more time to converge. With
! 389: a simple model (no covariates), the programme estimates 8
! 390: parameters. Adding covariates increases the number of parameters
! 391: : 12 for <strong>model=V1, </strong>16 for <strong>model=V1+V1*age
! 392: </strong>and 20 for <strong>model=V1+V2+V3.</strong></p>
! 393:
! 394: <h4><font color="#FF0000">Guess values for optimization</font><font
! 395: color="#00006A"> </font></h4>
! 396:
! 397: <p>You must write the initial guess values of the parameters for
! 398: optimization. The number of parameters, <em>N</em> depends on the
! 399: number of absorbing states and non-absorbing states and on the
! 400: number of covariates. <br>
! 401: <em>N</em> is given by the formula <em>N</em>=(<em>nlstate</em> +
! 402: <em>ndeath</em>-1)*<em>nlstate</em>*<em>ncovmodel</em> . <br>
! 403: <br>
! 404: Thus in the simple case with 2 covariates (the model is log
! 405: (pij/pii) = aij + bij * age where intercept and age are the two
! 406: covariates), and 2 health degrees (1 for disability-free and 2
! 407: for disability) and 1 absorbing state (3), you must enter 8
! 408: initials values, a12, b12, a13, b13, a21, b21, a23, b23. You can
! 409: start with zeros as in this example, but if you have a more
! 410: precise set (for example from an earlier run) you can enter it
! 411: and it will speed up them<br>
! 412: Each of the four lines starts with indices "ij": <b>ij
! 413: aij bij</b> </p>
! 414:
! 415: <blockquote>
! 416: <pre># Guess values of aij and bij in log (pij/pii) = aij + bij * age
! 417: 12 -14.155633 0.110794
! 418: 13 -7.925360 0.032091
! 419: 21 -1.890135 -0.029473
! 420: 23 -6.234642 0.022315 </pre>
! 421: </blockquote>
! 422:
! 423: <p>or, to simplify (in most of cases it converges but there is no
! 424: warranty!): </p>
! 425:
! 426: <blockquote>
! 427: <pre>12 0.0 0.0
! 428: 13 0.0 0.0
! 429: 21 0.0 0.0
! 430: 23 0.0 0.0</pre>
! 431: </blockquote>
! 432:
! 433: <p>In order to speed up the convergence you can make a first run
! 434: with a large stepm i.e stepm=12 or 24 and then decrease the stepm
! 435: until stepm=1 month. If newstepm is the new shorter stepm and
! 436: stepm can be expressed as a multiple of newstepm, like newstepm=n
! 437: stepm, then the following approximation holds: </p>
! 438:
! 439: <pre>aij(stepm) = aij(n . stepm) - ln(n)
! 440: </pre>
! 441:
! 442: <p>and </p>
! 443:
! 444: <pre>bij(stepm) = bij(n . stepm) .</pre>
! 445:
! 446: <p>For example if you already ran for a 6 months interval and
! 447: got:<br>
! 448: </p>
! 449:
! 450: <pre># Parameters
! 451: 12 -13.390179 0.126133
! 452: 13 -7.493460 0.048069
! 453: 21 0.575975 -0.041322
! 454: 23 -4.748678 0.030626
! 455: </pre>
! 456:
! 457: <p>If you now want to get the monthly estimates, you can guess
! 458: the aij by substracting ln(6)= 1,7917<br>
! 459: and running<br>
! 460: </p>
! 461:
! 462: <pre>12 -15.18193847 0.126133
! 463: 13 -9.285219469 0.048069
! 464: 21 -1.215784469 -0.041322
! 465: 23 -6.540437469 0.030626
! 466: </pre>
! 467:
! 468: <p>and get<br>
! 469: </p>
! 470:
! 471: <pre>12 -15.029768 0.124347
! 472: 13 -8.472981 0.036599
! 473: 21 -1.472527 -0.038394
! 474: 23 -6.553602 0.029856
! 475:
! 476: which is closer to the results. The approximation is probably useful
! 477: only for very small intervals and we don't have enough experience to
! 478: know if you will speed up the convergence or not.
! 479: </pre>
! 480:
! 481: <pre> -ln(12)= -2.484
! 482: -ln(6/1)=-ln(6)= -1.791
! 483: -ln(3/1)=-ln(3)= -1.0986
! 484: -ln(12/6)=-ln(2)= -0.693
! 485: </pre>
! 486:
! 487: <h4><font color="#FF0000">Guess values for computing variances</font></h4>
! 488:
! 489: <p>This is an output if <a href="#mle">mle</a>=1. But it can be
! 490: used as an input to get the various output data files (Health
! 491: expectancies, stationary prevalence etc.) and figures without
! 492: rerunning the rather long maximisation phase (mle=0). </p>
! 493:
! 494: <p>The scales are small values for the evaluation of numerical
! 495: derivatives. These derivatives are used to compute the hessian
! 496: matrix of the parameters, that is the inverse of the covariance
! 497: matrix, and the variances of health expectancies. Each line
! 498: consists in indices "ij" followed by the initial scales
! 499: (zero to simplify) associated with aij and bij. </p>
! 500:
! 501: <ul>
! 502: <li>If mle=1 you can enter zeros:</li>
! 503: <li><blockquote>
! 504: <pre># Scales (for hessian or gradient estimation)
! 505: 12 0. 0.
! 506: 13 0. 0.
! 507: 21 0. 0.
! 508: 23 0. 0. </pre>
! 509: </blockquote>
! 510: </li>
! 511: <li>If mle=0 you must enter a covariance matrix (usually
! 512: obtained from an earlier run).</li>
! 513: </ul>
! 514:
! 515: <h4><font color="#FF0000">Covariance matrix of parameters</font></h4>
! 516:
! 517: <p>This is an output if <a href="#mle">mle</a>=1. But it can be
! 518: used as an input to get the various output data files (Health
! 519: expectancies, stationary prevalence etc.) and figures without
! 520: rerunning the rather long maximisation phase (mle=0). <br>
! 521: Each line starts with indices "ijk" followed by the
! 522: covariances between aij and bij:<br>
! 523: </p>
! 524:
! 525: <pre>
! 526: 121 Var(a12)
! 527: 122 Cov(b12,a12) Var(b12)
! 528: ...
! 529: 232 Cov(b23,a12) Cov(b23,b12) ... Var (b23) </pre>
! 530:
! 531: <ul>
! 532: <li>If mle=1 you can enter zeros. </li>
! 533: <li><pre># Covariance matrix
! 534: 121 0.
! 535: 122 0. 0.
! 536: 131 0. 0. 0.
! 537: 132 0. 0. 0. 0.
! 538: 211 0. 0. 0. 0. 0.
! 539: 212 0. 0. 0. 0. 0. 0.
! 540: 231 0. 0. 0. 0. 0. 0. 0.
! 541: 232 0. 0. 0. 0. 0. 0. 0. 0.</pre>
! 542: </li>
! 543: <li>If mle=0 you must enter a covariance matrix (usually
! 544: obtained from an earlier run). </li>
! 545: </ul>
! 546:
! 547: <h4><font color="#FF0000">Age range for calculation of stationary
! 548: prevalences and health expectancies</font></h4>
! 549:
! 550: <pre>agemin=70 agemax=100 bage=50 fage=100</pre>
! 551:
! 552: <pre>
! 553: Once we obtained the estimated parameters, the program is able
! 554: to calculated stationary prevalence, transitions probabilities
! 555: and life expectancies at any age. Choice of age range is useful
! 556: for extrapolation. In our data file, ages varies from age 70 to
! 557: 102. It is possible to get extrapolated stationary prevalence by
! 558: age ranging from agemin to agemax.
! 559:
! 560:
! 561: Setting bage=50 (begin age) and fage=100 (final age), makes
! 562: the program computing life expectancy from age 'bage' to age
! 563: 'fage'. As we use a model, we can interessingly compute life
! 564: expectancy on a wider age range than the age range from the data.
! 565: But the model can be rather wrong on much larger intervals.
! 566: Program is limited to around 120 for upper age!
! 567: </pre>
! 568:
! 569: <ul>
! 570: <li><b>agemin=</b> Minimum age for calculation of the
! 571: stationary prevalence </li>
! 572: <li><b>agemax=</b> Maximum age for calculation of the
! 573: stationary prevalence </li>
! 574: <li><b>bage=</b> Minimum age for calculation of the health
! 575: expectancies </li>
! 576: <li><b>fage=</b> Maximum age for calculation of the health
! 577: expectancies </li>
! 578: </ul>
! 579:
! 580: <h4><a name="Computing"><font color="#FF0000">Computing</font></a><font
! 581: color="#FF0000"> the observed prevalence</font></h4>
! 582:
! 583: <pre>begin-prev-date=1/1/1984 end-prev-date=1/6/1988 estepm=1</pre>
! 584:
! 585: <pre>
! 586: Statements 'begin-prev-date' and 'end-prev-date' allow to
! 587: select the period in which we calculate the observed prevalences
! 588: in each state. In this example, the prevalences are calculated on
! 589: data survey collected between 1 january 1984 and 1 june 1988.
! 590: </pre>
! 591:
! 592: <ul>
! 593: <li><strong>begin-prev-date= </strong>Starting date
! 594: (day/month/year)</li>
! 595: <li><strong>end-prev-date= </strong>Final date
! 596: (day/month/year)</li>
! 597: <li><strong>estepm= </strong>Unit (in months).We compute the
! 598: life expectancy from trapezoids spaced every estepm
! 599: months. This is mainly to measure the difference between
! 600: two models: for example if stepm=24 months pijx are given
! 601: only every 2 years and by summing them we are calculating
! 602: an estimate of the Life Expectancy assuming a linear
! 603: progression inbetween and thus overestimating or
! 604: underestimating according to the curvature of the
! 605: survival function. If, for the same date, we estimate the
! 606: model with stepm=1 month, we can keep estepm to 24 months
! 607: to compare the new estimate of Life expectancy with the
! 608: same linear hypothesis. A more precise result, taking
! 609: into account a more precise curvature will be obtained if
! 610: estepm is as small as stepm.</li>
! 611: </ul>
! 612:
! 613: <h4><font color="#FF0000">Population- or status-based health
! 614: expectancies</font></h4>
! 615:
! 616: <pre>pop_based=0</pre>
! 617:
! 618: <p>The program computes status-based health expectancies, i.e
! 619: health expectancies which depends on your initial health state.
! 620: If you are healthy your healthy life expectancy (e11) is higher
! 621: than if you were disabled (e21, with e11 > e21).<br>
! 622: To compute a healthy life expectancy independant of the initial
! 623: status we have to weight e11 and e21 according to the probability
! 624: to be in each state at initial age or, with other word, according
! 625: to the proportion of people in each state.<br>
! 626: We prefer computing a 'pure' period healthy life expectancy based
! 627: only on the transtion forces. Then the weights are simply the
! 628: stationnary prevalences or 'implied' prevalences at the initial
! 629: age.<br>
! 630: Some other people would like to use the cross-sectional
! 631: prevalences (the "Sullivan prevalences") observed at
! 632: the initial age during a period of time <a href="#Computing">defined
! 633: just above</a>. <br>
! 634: </p>
! 635:
! 636: <ul>
! 637: <li><strong>popbased= 0 </strong>Health expectancies are
! 638: computed at each age from stationary prevalences
! 639: 'expected' at this initial age.</li>
! 640: <li><strong>popbased= 1 </strong>Health expectancies are
! 641: computed at each age from cross-sectional 'observed'
! 642: prevalence at this initial age. As all the population is
! 643: not observed at the same exact date we define a short
! 644: period were the observed prevalence is computed.</li>
! 645: </ul>
! 646:
! 647: <h4><font color="#FF0000">Prevalence forecasting ( Experimental)</font></h4>
! 648:
! 649: <pre>starting-proj-date=1/1/1989 final-proj-date=1/1/1992 mov_average=0 </pre>
! 650:
! 651: <p>Prevalence and population projections are only available if
! 652: the interpolation unit is a month, i.e. stepm=1 and if there are
! 653: no covariate. The programme estimates the prevalence in each
! 654: state at a precise date expressed in day/month/year. The
! 655: programme computes one forecasted prevalence a year from a
! 656: starting date (1 january of 1989 in this example) to a final date
! 657: (1 january 1992). The statement mov_average allows to compute
! 658: smoothed forecasted prevalences with a five-age moving average
! 659: centered at the mid-age of the five-age period. <br>
! 660: </p>
! 661:
! 662: <ul>
! 663: <li><strong>starting-proj-date</strong>= starting date
! 664: (day/month/year) of forecasting</li>
! 665: <li><strong>final-proj-date= </strong>final date
! 666: (day/month/year) of forecasting</li>
! 667: <li><strong>mov_average</strong>= smoothing with a five-age
! 668: moving average centered at the mid-age of the five-age
! 669: period. The command<strong> mov_average</strong> takes
! 670: value 1 if the prevalences are smoothed and 0 otherwise.</li>
! 671: </ul>
! 672:
! 673: <h4><font color="#FF0000">Last uncommented line : Population
! 674: forecasting </font></h4>
! 675:
! 676: <pre>popforecast=0 popfile=pyram.txt popfiledate=1/1/1989 last-popfiledate=1/1/1992</pre>
! 677:
! 678: <p>This command is available if the interpolation unit is a
! 679: month, i.e. stepm=1 and if popforecast=1. From a data file
! 680: including age and number of persons alive at the precise date
! 681: ‘popfiledate’, you can forecast the number of persons
! 682: in each state until date ‘last-popfiledate’. In this
! 683: example, the popfile <a href="pyram.txt"><b>pyram.txt</b></a>
! 684: includes real data which are the Japanese population in 1989.<br>
! 685: </p>
! 686:
! 687: <ul type="disc">
! 688: <li class="MsoNormal"
! 689: style="TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-list: l10 level1 lfo36; tab-stops: list 36.0pt"><b>popforecast=
! 690: 0 </b>Option for population forecasting. If
! 691: popforecast=1, the programme does the forecasting<b>.</b></li>
! 692: <li class="MsoNormal"
! 693: style="TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-list: l10 level1 lfo36; tab-stops: list 36.0pt"><b>popfile=
! 694: </b>name of the population file</li>
! 695: <li class="MsoNormal"
! 696: style="TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-list: l10 level1 lfo36; tab-stops: list 36.0pt"><b>popfiledate=</b>
! 697: date of the population population</li>
! 698: <li class="MsoNormal"
! 699: style="TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-list: l10 level1 lfo36; tab-stops: list 36.0pt"><b>last-popfiledate</b>=
! 700: date of the last population projection </li>
! 701: </ul>
! 702:
! 703: <hr>
! 704:
! 705: <h2><a name="running"></a><font color="#00006A">Running Imach
! 706: with this example</font></h2>
! 707:
! 708: <pre>We assume that you typed in your <a href="biaspar.imach">1st_example
! 709: parameter file</a> as explained <a href="#biaspar">above</a>.
! 710:
! 711: To run the program you should either:
! 712: </pre>
! 713:
! 714: <ul>
! 715: <li>click on the imach.exe icon and enter the name of the
! 716: parameter file which is for example <a
! 717: href="C:\usr\imach\mle\biaspar.imach">C:\usr\imach\mle\biaspar.imach</a>
! 718: </li>
! 719: <li>You also can locate the biaspar.imach icon in <a
! 720: href="C:\usr\imach\mle">C:\usr\imach\mle</a> with your
! 721: mouse and drag it with the mouse on the imach window). </li>
! 722: <li>With latest version (0.7 and higher) if you setup windows
! 723: in order to understand ".imach" extension you
! 724: can right click the biaspar.imach icon and either edit
! 725: with notepad the parameter file or execute it with imach
! 726: or whatever. </li>
! 727: </ul>
! 728:
! 729: <pre>The time to converge depends on the step unit that you used (1
! 730: month is cpu consuming), on the number of cases, and on the
! 731: number of variables.
! 732:
! 733:
! 734: The program outputs many files. Most of them are files which
! 735: will be plotted for better understanding.
! 736:
! 737: </pre>
! 738:
! 739: <hr>
! 740:
! 741: <h2><a name="output"><font color="#00006A">Output of the program
! 742: and graphs</font> </a></h2>
! 743:
! 744: <p>Once the optimization is finished, some graphics can be made
! 745: with a grapher. We use Gnuplot which is an interactive plotting
! 746: program copyrighted but freely distributed. A gnuplot reference
! 747: manual is available <a href="http://www.gnuplot.info/">here</a>. <br>
! 748: When the running is finished, the user should enter a caracter
! 749: for plotting and output editing. <br>
! 750: These caracters are:<br>
! 751: </p>
! 752:
! 753: <ul>
! 754: <li>'c' to start again the program from the beginning.</li>
! 755: <li>'e' opens the <a href="biaspar.htm"><strong>biaspar.htm</strong></a>
! 756: file to edit the output files and graphs. </li>
! 757: <li>'g' to graph again</li>
! 758: <li>'q' for exiting.</li>
! 759: </ul>
! 760:
! 761: <h5><font size="4"><strong>Results files </strong></font><br>
! 762: <br>
! 763: <font color="#EC5E5E" size="3"><strong>- </strong></font><a
! 764: name="Observed prevalence in each state"><font color="#EC5E5E"
! 765: size="3"><strong>Observed prevalence in each state</strong></font></a><font
! 766: color="#EC5E5E" size="3"><strong> (and at first pass)</strong></font><b>:
! 767: </b><a href="prbiaspar.txt"><b>prbiaspar.txt</b></a><br>
! 768: </h5>
! 769:
! 770: <p>The first line is the title and displays each field of the
! 771: file. The first column is age. The fields 2 and 6 are the
! 772: proportion of individuals in states 1 and 2 respectively as
! 773: observed during the first exam. Others fields are the numbers of
! 774: people in states 1, 2 or more. The number of columns increases if
! 775: the number of states is higher than 2.<br>
! 776: The header of the file is </p>
! 777:
! 778: <pre># Age Prev(1) N(1) N Age Prev(2) N(2) N
! 779: 70 1.00000 631 631 70 0.00000 0 631
! 780: 71 0.99681 625 627 71 0.00319 2 627
! 781: 72 0.97125 1115 1148 72 0.02875 33 1148 </pre>
! 782:
! 783: <p>It means that at age 70, the prevalence in state 1 is 1.000
! 784: and in state 2 is 0.00 . At age 71 the number of individuals in
! 785: state 1 is 625 and in state 2 is 2, hence the total number of
! 786: people aged 71 is 625+2=627. <br>
! 787: </p>
! 788:
! 789: <h5><font color="#EC5E5E" size="3"><b>- Estimated parameters and
! 790: covariance matrix</b></font><b>: </b><a href="rbiaspar.txt"><b>rbiaspar.imach</b></a></h5>
! 791:
! 792: <p>This file contains all the maximisation results: </p>
! 793:
! 794: <pre> -2 log likelihood= 21660.918613445392
! 795: Estimated parameters: a12 = -12.290174 b12 = 0.092161
! 796: a13 = -9.155590 b13 = 0.046627
! 797: a21 = -2.629849 b21 = -0.022030
! 798: a23 = -7.958519 b23 = 0.042614
! 799: Covariance matrix: Var(a12) = 1.47453e-001
! 800: Var(b12) = 2.18676e-005
! 801: Var(a13) = 2.09715e-001
! 802: Var(b13) = 3.28937e-005
! 803: Var(a21) = 9.19832e-001
! 804: Var(b21) = 1.29229e-004
! 805: Var(a23) = 4.48405e-001
! 806: Var(b23) = 5.85631e-005
! 807: </pre>
! 808:
! 809: <p>By substitution of these parameters in the regression model,
! 810: we obtain the elementary transition probabilities:</p>
! 811:
! 812: <p><img src="pebiaspar1.gif" width="400" height="300"></p>
! 813:
! 814: <h5><font color="#EC5E5E" size="3"><b>- Transition probabilities</b></font><b>:
! 815: </b><a href="pijrbiaspar.txt"><b>pijrbiaspar.txt</b></a></h5>
! 816:
! 817: <p>Here are the transitions probabilities Pij(x, x+nh) where nh
! 818: is a multiple of 2 years. The first column is the starting age x
! 819: (from age 50 to 100), the second is age (x+nh) and the others are
! 820: the transition probabilities p11, p12, p13, p21, p22, p23. For
! 821: example, line 5 of the file is: </p>
! 822:
! 823: <pre> 100 106 0.02655 0.17622 0.79722 0.01809 0.13678 0.84513 </pre>
! 824:
! 825: <p>and this means: </p>
! 826:
! 827: <pre>p11(100,106)=0.02655
! 828: p12(100,106)=0.17622
! 829: p13(100,106)=0.79722
! 830: p21(100,106)=0.01809
! 831: p22(100,106)=0.13678
! 832: p22(100,106)=0.84513 </pre>
! 833:
! 834: <h5><font color="#EC5E5E" size="3"><b>- </b></font><a
! 835: name="Stationary prevalence in each state"><font color="#EC5E5E"
! 836: size="3"><b>Stationary prevalence in each state</b></font></a><b>:
! 837: </b><a href="plrbiaspar.txt"><b>plrbiaspar.txt</b></a></h5>
! 838:
! 839: <pre>#Prevalence
! 840: #Age 1-1 2-2
! 841:
! 842: #************
! 843: 70 0.90134 0.09866
! 844: 71 0.89177 0.10823
! 845: 72 0.88139 0.11861
! 846: 73 0.87015 0.12985 </pre>
! 847:
! 848: <p>At age 70 the stationary prevalence is 0.90134 in state 1 and
! 849: 0.09866 in state 2. This stationary prevalence differs from
! 850: observed prevalence. Here is the point. The observed prevalence
! 851: at age 70 results from the incidence of disability, incidence of
! 852: recovery and mortality which occurred in the past of the cohort.
! 853: Stationary prevalence results from a simulation with actual
! 854: incidences and mortality (estimated from this cross-longitudinal
! 855: survey). It is the best predictive value of the prevalence in the
! 856: future if "nothing changes in the future". This is
! 857: exactly what demographers do with a Life table. Life expectancy
! 858: is the expected mean time to survive if observed mortality rates
! 859: (incidence of mortality) "remains constant" in the
! 860: future. </p>
! 861:
! 862: <h5><font color="#EC5E5E" size="3"><b>- Standard deviation of
! 863: stationary prevalence</b></font><b>: </b><a
! 864: href="vplrbiaspar.txt"><b>vplrbiaspar.txt</b></a></h5>
! 865:
! 866: <p>The stationary prevalence has to be compared with the observed
! 867: prevalence by age. But both are statistical estimates and
! 868: subjected to stochastic errors due to the size of the sample, the
! 869: design of the survey, and, for the stationary prevalence to the
! 870: model used and fitted. It is possible to compute the standard
! 871: deviation of the stationary prevalence at each age.</p>
! 872:
! 873: <h5><font color="#EC5E5E" size="3">-Observed and stationary
! 874: prevalence in state (2=disable) with confidence interval</font>:<b>
! 875: </b><a href="vbiaspar21.htm"><b>vbiaspar21.gif</b></a></h5>
! 876:
! 877: <p>This graph exhibits the stationary prevalence in state (2)
! 878: with the confidence interval in red. The green curve is the
! 879: observed prevalence (or proportion of individuals in state (2)).
! 880: Without discussing the results (it is not the purpose here), we
! 881: observe that the green curve is rather below the stationary
! 882: prevalence. It suggests an increase of the disability prevalence
! 883: in the future.</p>
! 884:
! 885: <p><img src="vbiaspar21.gif" width="400" height="300"></p>
! 886:
! 887: <h5><font color="#EC5E5E" size="3"><b>-Convergence to the
! 888: stationary prevalence of disability</b></font><b>: </b><a
! 889: href="pbiaspar11.gif"><b>pbiaspar11.gif</b></a><br>
! 890: <img src="pbiaspar11.gif" width="400" height="300"> </h5>
! 891:
! 892: <p>This graph plots the conditional transition probabilities from
! 893: an initial state (1=healthy in red at the bottom, or 2=disable in
! 894: green on top) at age <em>x </em>to the final state 2=disable<em> </em>at
! 895: age <em>x+h. </em>Conditional means at the condition to be alive
! 896: at age <em>x+h </em>which is <i>hP12x</i> + <em>hP22x</em>. The
! 897: curves <i>hP12x/(hP12x</i> + <em>hP22x) </em>and <i>hP22x/(hP12x</i>
! 898: + <em>hP22x) </em>converge with <em>h, </em>to the <em>stationary
! 899: prevalence of disability</em>. In order to get the stationary
! 900: prevalence at age 70 we should start the process at an earlier
! 901: age, i.e.50. If the disability state is defined by severe
! 902: disability criteria with only a few chance to recover, then the
! 903: incidence of recovery is low and the time to convergence is
! 904: probably longer. But we don't have experience yet.</p>
! 905:
! 906: <h5><font color="#EC5E5E" size="3"><b>- Life expectancies by age
! 907: and initial health status with standard deviation</b></font><b>: </b><a
! 908: href="erbiaspar.txt"><b>erbiaspar.txt</b></a></h5>
! 909:
! 910: <pre># Health expectancies
! 911: # Age 1-1 (SE) 1-2 (SE) 2-1 (SE) 2-2 (SE)
! 912: 70 10.4171 (0.1517) 3.0433 (0.4733) 5.6641 (0.1121) 5.6907 (0.3366)
! 913: 71 9.9325 (0.1409) 3.0495 (0.4234) 5.2627 (0.1107) 5.6384 (0.3129)
! 914: 72 9.4603 (0.1319) 3.0540 (0.3770) 4.8810 (0.1099) 5.5811 (0.2907)
! 915: 73 9.0009 (0.1246) 3.0565 (0.3345) 4.5188 (0.1098) 5.5187 (0.2702)
! 916: </pre>
! 917:
! 918: <pre>For example 70 10.4171 (0.1517) 3.0433 (0.4733) 5.6641 (0.1121) 5.6907 (0.3366) means:
! 919: e11=10.4171 e12=3.0433 e21=5.6641 e22=5.6907 </pre>
! 920:
! 921: <pre><img src="expbiaspar21.gif" width="400" height="300"><img
! 922: src="expbiaspar11.gif" width="400" height="300"></pre>
! 923:
! 924: <p>For example, life expectancy of a healthy individual at age 70
! 925: is 10.42 in the healthy state and 3.04 in the disability state
! 926: (=13.46 years). If he was disable at age 70, his life expectancy
! 927: will be shorter, 5.66 in the healthy state and 5.69 in the
! 928: disability state (=11.35 years). The total life expectancy is a
! 929: weighted mean of both, 13.46 and 11.35; weight is the proportion
! 930: of people disabled at age 70. In order to get a pure period index
! 931: (i.e. based only on incidences) we use the <a
! 932: href="#Stationary prevalence in each state">computed or
! 933: stationary prevalence</a> at age 70 (i.e. computed from
! 934: incidences at earlier ages) instead of the <a
! 935: href="#Observed prevalence in each state">observed prevalence</a>
! 936: (for example at first exam) (<a href="#Health expectancies">see
! 937: below</a>).</p>
! 938:
! 939: <h5><font color="#EC5E5E" size="3"><b>- Variances of life
! 940: expectancies by age and initial health status</b></font><b>: </b><a
! 941: href="vrbiaspar.txt"><b>vrbiaspar.txt</b></a></h5>
! 942:
! 943: <p>For example, the covariances of life expectancies Cov(ei,ej)
! 944: at age 50 are (line 3) </p>
! 945:
! 946: <pre> Cov(e1,e1)=0.4776 Cov(e1,e2)=0.0488=Cov(e2,e1) Cov(e2,e2)=0.0424</pre>
! 947:
! 948: <h5><font color="#EC5E5E" size="3"><b>-Variances of one-step
! 949: probabilities </b></font><b>: </b><a href="probrbiaspar.txt"><b>probrbiaspar.txt</b></a></h5>
! 950:
! 951: <p>For example, at age 65</p>
! 952:
! 953: <pre> p11=9.960e-001 standard deviation of p11=2.359e-004</pre>
! 954:
! 955: <h5><font color="#EC5E5E" size="3"><b>- </b></font><a
! 956: name="Health expectancies"><font color="#EC5E5E" size="3"><b>Health
! 957: expectancies</b></font></a><font color="#EC5E5E" size="3"><b>
! 958: with standard errors in parentheses</b></font><b>: </b><a
! 959: href="trbiaspar.txt"><font face="Courier New"><b>trbiaspar.txt</b></font></a></h5>
! 960:
! 961: <pre>#Total LEs with variances: e.. (std) e.1 (std) e.2 (std) </pre>
! 962:
! 963: <pre>70 13.26 (0.22) 9.95 (0.20) 3.30 (0.14) </pre>
! 964:
! 965: <p>Thus, at age 70 the total life expectancy, e..=13.26 years is
! 966: the weighted mean of e1.=13.46 and e2.=11.35 by the stationary
! 967: prevalence at age 70 which are 0.90134 in state 1 and 0.09866 in
! 968: state 2, respectively (the sum is equal to one). e.1=9.95 is the
! 969: Disability-free life expectancy at age 70 (it is again a weighted
! 970: mean of e11 and e21). e.2=3.30 is also the life expectancy at age
! 971: 70 to be spent in the disability state.</p>
! 972:
! 973: <h5><font color="#EC5E5E" size="3"><b>-Total life expectancy by
! 974: age and health expectancies in states (1=healthy) and (2=disable)</b></font><b>:
! 975: </b><a href="ebiaspar1.gif"><b>ebiaspar1.gif</b></a></h5>
! 976:
! 977: <p>This figure represents the health expectancies and the total
! 978: life expectancy with the confident interval in dashed curve. </p>
! 979:
! 980: <pre> <img src="ebiaspar1.gif" width="400" height="300"></pre>
! 981:
! 982: <p>Standard deviations (obtained from the information matrix of
! 983: the model) of these quantities are very useful.
! 984: Cross-longitudinal surveys are costly and do not involve huge
! 985: samples, generally a few thousands; therefore it is very
! 986: important to have an idea of the standard deviation of our
! 987: estimates. It has been a big challenge to compute the Health
! 988: Expectancy standard deviations. Don't be confuse: life expectancy
! 989: is, as any expected value, the mean of a distribution; but here
! 990: we are not computing the standard deviation of the distribution,
! 991: but the standard deviation of the estimate of the mean.</p>
! 992:
! 993: <p>Our health expectancies estimates vary according to the sample
! 994: size (and the standard deviations give confidence intervals of
! 995: the estimate) but also according to the model fitted. Let us
! 996: explain it in more details.</p>
! 997:
! 998: <p>Choosing a model means ar least two kind of choices. First we
! 999: have to decide the number of disability states. Second we have to
! 1000: design, within the logit model family, the model: variables,
! 1001: covariables, confonding factors etc. to be included.</p>
! 1002:
! 1003: <p>More disability states we have, better is our demographical
! 1004: approach of the disability process, but smaller are the number of
! 1005: transitions between each state and higher is the noise in the
! 1006: measurement. We do not have enough experiments of the various
! 1007: models to summarize the advantages and disadvantages, but it is
! 1008: important to say that even if we had huge and unbiased samples,
! 1009: the total life expectancy computed from a cross-longitudinal
! 1010: survey, varies with the number of states. If we define only two
! 1011: states, alive or dead, we find the usual life expectancy where it
! 1012: is assumed that at each age, people are at the same risk to die.
! 1013: If we are differentiating the alive state into healthy and
! 1014: disable, and as the mortality from the disability state is higher
! 1015: than the mortality from the healthy state, we are introducing
! 1016: heterogeneity in the risk of dying. The total mortality at each
! 1017: age is the weighted mean of the mortality in each state by the
! 1018: prevalence in each state. Therefore if the proportion of people
! 1019: at each age and in each state is different from the stationary
! 1020: equilibrium, there is no reason to find the same total mortality
! 1021: at a particular age. Life expectancy, even if it is a very useful
! 1022: tool, has a very strong hypothesis of homogeneity of the
! 1023: population. Our main purpose is not to measure differential
! 1024: mortality but to measure the expected time in a healthy or
! 1025: disability state in order to maximise the former and minimize the
! 1026: latter. But the differential in mortality complexifies the
! 1027: measurement.</p>
! 1028:
! 1029: <p>Incidences of disability or recovery are not affected by the
! 1030: number of states if these states are independant. But incidences
! 1031: estimates are dependant on the specification of the model. More
! 1032: covariates we added in the logit model better is the model, but
! 1033: some covariates are not well measured, some are confounding
! 1034: factors like in any statistical model. The procedure to "fit
! 1035: the best model' is similar to logistic regression which itself is
! 1036: similar to regression analysis. We haven't yet been sofar because
! 1037: we also have a severe limitation which is the speed of the
! 1038: convergence. On a Pentium III, 500 MHz, even the simplest model,
! 1039: estimated by month on 8,000 people may take 4 hours to converge.
! 1040: Also, the program is not yet a statistical package, which permits
! 1041: a simple writing of the variables and the model to take into
! 1042: account in the maximisation. The actual program allows only to
! 1043: add simple variables like age+sex or age+sex+ age*sex but will
! 1044: never be general enough. But what is to remember, is that
! 1045: incidences or probability of change from one state to another is
! 1046: affected by the variables specified into the model.</p>
! 1047:
! 1048: <p>Also, the age range of the people interviewed has a link with
! 1049: the age range of the life expectancy which can be estimated by
! 1050: extrapolation. If your sample ranges from age 70 to 95, you can
! 1051: clearly estimate a life expectancy at age 70 and trust your
! 1052: confidence interval which is mostly based on your sample size,
! 1053: but if you want to estimate the life expectancy at age 50, you
! 1054: should rely in your model, but fitting a logistic model on a age
! 1055: range of 70-95 and estimating probabilties of transition out of
! 1056: this age range, say at age 50 is very dangerous. At least you
! 1057: should remember that the confidence interval given by the
! 1058: standard deviation of the health expectancies, are under the
! 1059: strong assumption that your model is the 'true model', which is
! 1060: probably not the case.</p>
! 1061:
! 1062: <h5><font color="#EC5E5E" size="3"><b>- Copy of the parameter
! 1063: file</b></font><b>: </b><a href="orbiaspar.txt"><b>orbiaspar.txt</b></a></h5>
! 1064:
! 1065: <p>This copy of the parameter file can be useful to re-run the
! 1066: program while saving the old output files. </p>
! 1067:
! 1068: <h5><font color="#EC5E5E" size="3"><b>- Prevalence forecasting</b></font><b>:
! 1069: </b><a href="frbiaspar.txt"><b>frbiaspar.txt</b></a></h5>
! 1070:
! 1071: <p
! 1072: style="TEXT-ALIGN: justify; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt">First,
! 1073: we have estimated the observed prevalence between 1/1/1984 and
! 1074: 1/6/1988. The mean date of interview (weighed average of the
! 1075: interviews performed between1/1/1984 and 1/6/1988) is estimated
! 1076: to be 13/9/1985, as written on the top on the file. Then we
! 1077: forecast the probability to be in each state. </p>
! 1078:
! 1079: <p
! 1080: style="TEXT-ALIGN: justify; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt">Example,
! 1081: at date 1/1/1989 : </p>
! 1082:
! 1083: <pre class="MsoNormal"># StartingAge FinalAge P.1 P.2 P.3
! 1084: # Forecasting at date 1/1/1989
! 1085: 73 0.807 0.078 0.115</pre>
! 1086:
! 1087: <p
! 1088: style="TEXT-ALIGN: justify; tab-stops: 45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt">Since
! 1089: the minimum age is 70 on the 13/9/1985, the youngest forecasted
! 1090: age is 73. This means that at age a person aged 70 at 13/9/1989
! 1091: has a probability to enter state1 of 0.807 at age 73 on 1/1/1989.
! 1092: Similarly, the probability to be in state 2 is 0.078 and the
! 1093: probability to die is 0.115. Then, on the 1/1/1989, the
! 1094: prevalence of disability at age 73 is estimated to be 0.088.</p>
! 1095:
! 1096: <h5><font color="#EC5E5E" size="3"><b>- Population forecasting</b></font><b>:
! 1097: </b><a href="poprbiaspar.txt"><b>poprbiaspar.txt</b></a></h5>
! 1098:
! 1099: <pre># Age P.1 P.2 P.3 [Population]
! 1100: # Forecasting at date 1/1/1989
! 1101: 75 572685.22 83798.08
! 1102: 74 621296.51 79767.99
! 1103: 73 645857.70 69320.60 </pre>
! 1104:
! 1105: <pre># Forecasting at date 1/1/19909
! 1106: 76 442986.68 92721.14 120775.48
! 1107: 75 487781.02 91367.97 121915.51
! 1108: 74 512892.07 85003.47 117282.76 </pre>
! 1109:
! 1110: <p>From the population file, we estimate the number of people in
! 1111: each state. At age 73, 645857 persons are in state 1 and 69320
! 1112: are in state 2. One year latter, 512892 are still in state 1,
! 1113: 85003 are in state 2 and 117282 died before 1/1/1990.</p>
! 1114:
! 1115: <hr>
! 1116:
! 1117: <h2><a name="example"></a><font color="#00006A">Trying an example</font></h2>
! 1118:
! 1119: <p>Since you know how to run the program, it is time to test it
! 1120: on your own computer. Try for example on a parameter file named <a
! 1121: href="..\mytry\imachpar.imach">imachpar.imach</a> which is a copy
! 1122: of <font size="2" face="Courier New">mypar.imach</font> included
! 1123: in the subdirectory of imach, <font size="2" face="Courier New">mytry</font>.
! 1124: Edit it to change the name of the data file to <font size="2"
! 1125: face="Courier New">..\data\mydata.txt</font> if you don't want to
! 1126: copy it on the same directory. The file <font face="Courier New">mydata.txt</font>
! 1127: is a smaller file of 3,000 people but still with 4 waves. </p>
! 1128:
! 1129: <p>Click on the imach.exe icon to open a window. Answer to the
! 1130: question:'<strong>Enter the parameter file name:'</strong></p>
! 1131:
! 1132: <table border="1">
! 1133: <tr>
! 1134: <td width="100%"><strong>IMACH, Version 0.8a</strong><p><strong>Enter
! 1135: the parameter file name: ..\mytry\imachpar.imach</strong></p>
! 1136: </td>
! 1137: </tr>
! 1138: </table>
! 1139:
! 1140: <p>Most of the data files or image files generated, will use the
! 1141: 'imachpar' string into their name. The running time is about 2-3
! 1142: minutes on a Pentium III. If the execution worked correctly, the
! 1143: outputs files are created in the current directory, and should be
! 1144: the same as the mypar files initially included in the directory <font
! 1145: size="2" face="Courier New">mytry</font>.</p>
! 1146:
! 1147: <ul>
! 1148: <li><pre><u>Output on the screen</u> The output screen looks like <a
! 1149: href="imachrun.LOG">this Log file</a>
! 1150: #
! 1151:
! 1152: title=MLE datafile=..\data\mydata.txt lastobs=3000 firstpass=1 lastpass=3
! 1153: ftol=1.000000e-008 stepm=24 ncovcol=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre>
! 1154: </li>
! 1155: <li><pre>Total number of individuals= 2965, Agemin = 70.00, Agemax= 100.92
! 1156:
! 1157: Warning, no any valid information for:126 line=126
! 1158: Warning, no any valid information for:2307 line=2307
! 1159: Delay (in months) between two waves Min=21 Max=51 Mean=24.495826
! 1160: <font face="Times New Roman">These lines give some warnings on the data file and also some raw statistics on frequencies of transitions.</font>
! 1161: Age 70 1.=230 loss[1]=3.5% 2.=16 loss[2]=12.5% 1.=222 prev[1]=94.1% 2.=14
! 1162: prev[2]=5.9% 1-1=8 11=200 12=7 13=15 2-1=2 21=6 22=7 23=1
! 1163: Age 102 1.=0 loss[1]=NaNQ% 2.=0 loss[2]=NaNQ% 1.=0 prev[1]=NaNQ% 2.=0 </pre>
! 1164: </li>
! 1165: </ul>
! 1166:
! 1167: <p> </p>
! 1168:
! 1169: <ul>
! 1170: <li>Maximisation with the Powell algorithm. 8 directions are
! 1171: given corresponding to the 8 parameters. this can be
! 1172: rather long to get convergence.<br>
! 1173: <font size="1" face="Courier New"><br>
! 1174: Powell iter=1 -2*LL=11531.405658264877 1 0.000000000000 2
! 1175: 0.000000000000 3<br>
! 1176: 0.000000000000 4 0.000000000000 5 0.000000000000 6
! 1177: 0.000000000000 7 <br>
! 1178: 0.000000000000 8 0.000000000000<br>
! 1179: 1..........2.................3..........4.................5.........<br>
! 1180: 6................7........8...............<br>
! 1181: Powell iter=23 -2*LL=6744.954108371555 1 -12.967632334283
! 1182: <br>
! 1183: 2 0.135136681033 3 -7.402109728262 4 0.067844593326 <br>
! 1184: 5 -0.673601538129 6 -0.006615504377 7 -5.051341616718 <br>
! 1185: 8 0.051272038506<br>
! 1186: 1..............2...........3..............4...........<br>
! 1187: 5..........6................7...........8.........<br>
! 1188: #Number of iterations = 23, -2 Log likelihood =
! 1189: 6744.954042573691<br>
! 1190: # Parameters<br>
! 1191: 12 -12.966061 0.135117 <br>
! 1192: 13 -7.401109 0.067831 <br>
! 1193: 21 -0.672648 -0.006627 <br>
! 1194: 23 -5.051297 0.051271 </font><br>
! 1195: </li>
! 1196: <li><pre><font size="2">Calculation of the hessian matrix. Wait...
! 1197: 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
! 1198:
! 1199: Inverting the hessian to get the covariance matrix. Wait...
! 1200:
! 1201: #Hessian matrix#
! 1202: 3.344e+002 2.708e+004 -4.586e+001 -3.806e+003 -1.577e+000 -1.313e+002 3.914e-001 3.166e+001
! 1203: 2.708e+004 2.204e+006 -3.805e+003 -3.174e+005 -1.303e+002 -1.091e+004 2.967e+001 2.399e+003
! 1204: -4.586e+001 -3.805e+003 4.044e+002 3.197e+004 2.431e-002 1.995e+000 1.783e-001 1.486e+001
! 1205: -3.806e+003 -3.174e+005 3.197e+004 2.541e+006 2.436e+000 2.051e+002 1.483e+001 1.244e+003
! 1206: -1.577e+000 -1.303e+002 2.431e-002 2.436e+000 1.093e+002 8.979e+003 -3.402e+001 -2.843e+003
! 1207: -1.313e+002 -1.091e+004 1.995e+000 2.051e+002 8.979e+003 7.420e+005 -2.842e+003 -2.388e+005
! 1208: 3.914e-001 2.967e+001 1.783e-001 1.483e+001 -3.402e+001 -2.842e+003 1.494e+002 1.251e+004
! 1209: 3.166e+001 2.399e+003 1.486e+001 1.244e+003 -2.843e+003 -2.388e+005 1.251e+004 1.053e+006
! 1210: # Scales
! 1211: 12 1.00000e-004 1.00000e-006
! 1212: 13 1.00000e-004 1.00000e-006
! 1213: 21 1.00000e-003 1.00000e-005
! 1214: 23 1.00000e-004 1.00000e-005
! 1215: # Covariance
! 1216: 1 5.90661e-001
! 1217: 2 -7.26732e-003 8.98810e-005
! 1218: 3 8.80177e-002 -1.12706e-003 5.15824e-001
! 1219: 4 -1.13082e-003 1.45267e-005 -6.50070e-003 8.23270e-005
! 1220: 5 9.31265e-003 -1.16106e-004 6.00210e-004 -8.04151e-006 1.75753e+000
! 1221: 6 -1.15664e-004 1.44850e-006 -7.79995e-006 1.04770e-007 -2.12929e-002 2.59422e-004
! 1222: 7 1.35103e-003 -1.75392e-005 -6.38237e-004 7.85424e-006 4.02601e-001 -4.86776e-003 1.32682e+000
! 1223: 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
! 1224: # agemin agemax for lifexpectancy, bage fage (if mle==0 ie no data nor Max likelihood).
! 1225:
! 1226:
! 1227: agemin=70 agemax=100 bage=50 fage=100
! 1228: Computing prevalence limit: result on file 'plrmypar.txt'
! 1229: Computing pij: result on file 'pijrmypar.txt'
! 1230: Computing Health Expectancies: result on file 'ermypar.txt'
! 1231: Computing Variance-covariance of DFLEs: file 'vrmypar.txt'
! 1232: Computing Total LEs with variances: file 'trmypar.txt'
! 1233: Computing Variance-covariance of Prevalence limit: file 'vplrmypar.txt'
! 1234: End of Imach
! 1235: </font></pre>
! 1236: </li>
! 1237: </ul>
! 1238:
! 1239: <p><font size="3">Once the running is finished, the program
! 1240: requires a caracter:</font></p>
! 1241:
! 1242: <table border="1">
! 1243: <tr>
! 1244: <td width="100%"><strong>Type e to edit output files, g
! 1245: to graph again, c to start again, and q for exiting:</strong></td>
! 1246: </tr>
! 1247: </table>
! 1248:
! 1249: <p><font size="3">First you should enter <strong>e </strong>to
! 1250: edit the master file mypar.htm. </font></p>
! 1251:
! 1252: <ul>
! 1253: <li><u>Outputs files</u> <br>
! 1254: <br>
! 1255: - Copy of the parameter file: <a href="ormypar.txt">ormypar.txt</a><br>
! 1256: - Gnuplot file name: <a href="mypar.gp.txt">mypar.gp.txt</a><br>
! 1257: - Observed prevalence in each state: <a
! 1258: href="prmypar.txt">prmypar.txt</a> <br>
! 1259: - Stationary prevalence in each state: <a
! 1260: href="plrmypar.txt">plrmypar.txt</a> <br>
! 1261: - Transition probabilities: <a href="pijrmypar.txt">pijrmypar.txt</a><br>
! 1262: - Life expectancies by age and initial health status
! 1263: (estepm=24 months): <a href="ermypar.txt">ermypar.txt</a>
! 1264: <br>
! 1265: - Parameter file with estimated parameters and the
! 1266: covariance matrix: <a href="rmypar.txt">rmypar.txt</a> <br>
! 1267: - Variance of one-step probabilities: <a
! 1268: href="probrmypar.txt">probrmypar.txt</a> <br>
! 1269: - Variances of life expectancies by age and initial
! 1270: health status (estepm=24 months): <a href="vrmypar.txt">vrmypar.txt</a><br>
! 1271: - Health expectancies with their variances: <a
! 1272: href="trmypar.txt">trmypar.txt</a> <br>
! 1273: - Standard deviation of stationary prevalences: <a
! 1274: href="vplrmypar.txt">vplrmypar.txt</a> <br>
! 1275: No population forecast: popforecast = 0 (instead of 1) or
! 1276: stepm = 24 (instead of 1) or model=. (instead of .)<br>
! 1277: <br>
! 1278: </li>
! 1279: <li><u>Graphs</u> <br>
! 1280: <br>
! 1281: -<a href="../mytry/pemypar1.gif">One-step transition
! 1282: probabilities</a><br>
! 1283: -<a href="../mytry/pmypar11.gif">Convergence to the
! 1284: stationary prevalence</a><br>
! 1285: -<a href="..\mytry\vmypar11.gif">Observed and stationary
! 1286: prevalence in state (1) with the confident interval</a> <br>
! 1287: -<a href="..\mytry\vmypar21.gif">Observed and stationary
! 1288: prevalence in state (2) with the confident interval</a> <br>
! 1289: -<a href="..\mytry\expmypar11.gif">Health life
! 1290: expectancies by age and initial health state (1)</a> <br>
! 1291: -<a href="..\mytry\expmypar21.gif">Health life
! 1292: expectancies by age and initial health state (2)</a> <br>
! 1293: -<a href="..\mytry\emypar1.gif">Total life expectancy by
! 1294: age and health expectancies in states (1) and (2).</a> </li>
! 1295: </ul>
! 1296:
! 1297: <p>This software have been partly granted by <a
! 1298: href="http://euroreves.ined.fr">Euro-REVES</a>, a concerted
! 1299: action from the European Union. It will be copyrighted
! 1300: identically to a GNU software product, i.e. program and software
! 1301: can be distributed freely for non commercial use. Sources are not
! 1302: widely distributed today. You can get them by asking us with a
! 1303: simple justification (name, email, institute) <a
! 1304: href="mailto:brouard@ined.fr">mailto:brouard@ined.fr</a> and <a
! 1305: href="mailto:lievre@ined.fr">mailto:lievre@ined.fr</a> .</p>
! 1306:
! 1307: <p>Latest version (0.8a of May 2002) can be accessed at <a
! 1308: href="http://euroreves.ined.fr/imach">http://euroreves.ined.fr/imach</a><br>
! 1309: </p>
! 1310: </body>
! 1311: </html>
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