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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"> </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
! 32: 64b, May 2001</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 "Mortality,
43: Health and Epidemiology" 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: ("cross") where individuals from different ages are
132: interviewed on their health status or degree of disability. At
133: least a second wave of interviews ("longitudinal")
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), "New
192: Methods for Analyzing Active Life Expectancy". <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> </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:
1.3 ! lievre 276: <h2><a name="biaspar"></a>#Imach version 0.64b, May 2001,
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> </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> . <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>
390: Each of the four lines starts with indices "ij": <br>
391: <br>
392: <b>ij aij bij</b> </p>
393:
394: <blockquote>
395: <pre># Guess values of aij and bij in log (pij/pii) = aij + bij * age
396: 12 -14.155633 0.110794
397: 13 -7.925360 0.032091
398: 21 -1.890135 -0.029473
399: 23 -6.234642 0.022315 </pre>
400: </blockquote>
401:
402: <p>or, to simplify: </p>
403:
404: <blockquote>
405: <pre>12 0.0 0.0
406: 13 0.0 0.0
407: 21 0.0 0.0
408: 23 0.0 0.0</pre>
409: </blockquote>
410:
411: <h4><font color="#FF0000">Guess values for computing variances</font></h4>
412:
413: <p>This is an output if <a href="#mle">mle</a>=1. But it can be
414: used as an input to get the vairous output data files (Health
415: expectancies, stationary prevalence etc.) and figures without
416: rerunning the rather long maximisation phase (mle=0). </p>
417:
418: <p>The scales are small values for the evaluation of numerical
419: derivatives. These derivatives are used to compute the hessian
420: matrix of the parameters, that is the inverse of the covariance
421: matrix, and the variances of health expectancies. Each line
422: consists in indices "ij" followed by the initial scales
423: (zero to simplify) associated with aij and bij. </p>
424:
425: <ul>
426: <li>If mle=1 you can enter zeros:</li>
427: </ul>
428:
429: <blockquote>
430: <pre># Scales (for hessian or gradient estimation)
431: 12 0. 0.
432: 13 0. 0.
433: 21 0. 0.
434: 23 0. 0. </pre>
435: </blockquote>
436:
437: <ul>
438: <li>If mle=0 you must enter a covariance matrix (usually
439: obtained from an earlier run).</li>
440: </ul>
441:
442: <h4><font color="#FF0000">Covariance matrix of parameters</font></h4>
443:
444: <p>This is an output if <a href="#mle">mle</a>=1. But it can be
445: used as an input to get the vairous output data files (Health
446: expectancies, stationary prevalence etc.) and figures without
447: rerunning the rather long maximisation phase (mle=0). </p>
448:
449: <p>Each line starts with indices "ijk" followed by the
450: covariances between aij and bij: </p>
451:
452: <pre>
453: 121 Var(a12)
454: 122 Cov(b12,a12) Var(b12)
455: ...
456: 232 Cov(b23,a12) Cov(b23,b12) ... Var (b23) </pre>
457:
458: <ul>
459: <li>If mle=1 you can enter zeros. </li>
460: </ul>
461:
462: <blockquote>
463: <pre># Covariance matrix
464: 121 0.
465: 122 0. 0.
466: 131 0. 0. 0.
467: 132 0. 0. 0. 0.
468: 211 0. 0. 0. 0. 0.
469: 212 0. 0. 0. 0. 0. 0.
470: 231 0. 0. 0. 0. 0. 0. 0.
471: 232 0. 0. 0. 0. 0. 0. 0. 0.</pre>
472: </blockquote>
473:
474: <ul>
475: <li>If mle=0 you must enter a covariance matrix (usually
476: obtained from an earlier run).<br>
477: </li>
478: </ul>
479:
480: <h4><a name="biaspar-l"></a><font color="#FF0000">last
481: uncommented line</font></h4>
482:
483: <pre>agemin=70 agemax=100 bage=50 fage=100</pre>
484:
485: <p>Once we obtained the estimated parameters, the program is able
486: to calculated stationary prevalence, transitions probabilities
487: and life expectancies at any age. Choice of age ranges is useful
488: for extrapolation. In our data file, ages varies from age 70 to
489: 102. Setting bage=50 and fage=100, makes the program computing
490: life expectancy from age bage to age fage. As we use a model, we
491: can compute life expectancy on a wider age range than the age
492: range from the data. But the model can be rather wrong on big
493: intervals.</p>
494:
495: <p>Similarly, it is possible to get extrapolated stationary
496: prevalence by age raning from agemin to agemax. </p>
497:
498: <ul>
499: <li><b>agemin=</b> Minimum age for calculation of the
500: stationary prevalence </li>
501: <li><b>agemax=</b> Maximum age for calculation of the
502: stationary prevalence </li>
503: <li><b>bage=</b> Minimum age for calculation of the health
504: expectancies </li>
505: <li><b>fage=</b> Maximum ages for calculation of the health
506: expectancies </li>
507: </ul>
508:
509: <hr>
510:
511: <h2><a name="running"></a><font color="#00006A">Running Imach
512: with this example</font></h2>
513:
514: <p>We assume that you entered your <a href="biaspar.txt">1st_example
515: parameter file</a> as explained <a href="#biaspar">above</a>. To
516: run the program you should click on the imach.exe icon and enter
517: the name of the parameter file which is for example <a
518: href="C:\usr\imach\mle\biaspar.txt">C:\usr\imach\mle\biaspar.txt</a>
519: (you also can click on the biaspar.txt icon located in <br>
520: <a href="C:\usr\imach\mle">C:\usr\imach\mle</a> and put it with
521: the mouse on the imach window).<br>
522: </p>
523:
524: <p>The time to converge depends on the step unit that you used (1
525: month is cpu consuming), on the number of cases, and on the
526: number of variables.</p>
527:
528: <p>The program outputs many files. Most of them are files which
529: will be plotted for better understanding.</p>
530:
531: <hr>
532:
533: <h2><a name="output"><font color="#00006A">Output of the program
534: and graphs</font> </a></h2>
535:
536: <p>Once the optimization is finished, some graphics can be made
537: with a grapher. We use Gnuplot which is an interactive plotting
1.3 ! lievre 538: program copyrighted but freely distributed. A gnuplot reference
! 539: manual is available <a href="http://www.gnuplot.org/">here</a>. <br>
1.2 lievre 540: When the running is finished, the user should enter a caracter
541: for plotting and output editing. </p>
542:
543: <p>These caracters are:</p>
544:
545: <ul>
546: <li>'c' to start again the program from the beginning.</li>
1.3 ! lievre 547: <li>'e' opens the <a href="biaspar.htm"><strong>biaspar.htm</strong></a>
! 548: file to edit the output files and graphs. </li>
1.2 lievre 549: <li>'q' for exiting.</li>
550: </ul>
551:
552: <h5><font size="4"><strong>Results files </strong></font><br>
553: <br>
554: <font color="#EC5E5E" size="3"><strong>- </strong></font><a
555: name="Observed prevalence in each state"><font color="#EC5E5E"
556: size="3"><strong>Observed prevalence in each state</strong></font></a><font
557: color="#EC5E5E" size="3"><strong> (and at first pass)</strong></font><b>:
558: </b><a href="prbiaspar.txt"><b>prbiaspar.txt</b></a><br>
559: </h5>
560:
561: <p>The first line is the title and displays each field of the
562: file. The first column is age. The fields 2 and 6 are the
563: proportion of individuals in states 1 and 2 respectively as
564: observed during the first exam. Others fields are the numbers of
565: people in states 1, 2 or more. The number of columns increases if
566: the number of states is higher than 2.<br>
567: The header of the file is </p>
568:
569: <pre># Age Prev(1) N(1) N Age Prev(2) N(2) N
570: 70 1.00000 631 631 70 0.00000 0 631
571: 71 0.99681 625 627 71 0.00319 2 627
572: 72 0.97125 1115 1148 72 0.02875 33 1148 </pre>
573:
574: <p>It means that at age 70, the prevalence in state 1 is 1.000
575: and in state 2 is 0.00 . At age 71 the number of individuals in
576: state 1 is 625 and in state 2 is 2, hence the total number of
577: people aged 71 is 625+2=627. <br>
578: </p>
579:
580: <h5><font color="#EC5E5E" size="3"><b>- Estimated parameters and
581: covariance matrix</b></font><b>: </b><a href="rbiaspar.txt"><b>rbiaspar.txt</b></a></h5>
582:
583: <p>This file contains all the maximisation results: </p>
584:
1.3 ! lievre 585: <pre> -2 log likelihood= 21660.918613445392
! 586: Estimated parameters: a12 = -12.290174 b12 = 0.092161
! 587: a13 = -9.155590 b13 = 0.046627
! 588: a21 = -2.629849 b21 = -0.022030
! 589: a23 = -7.958519 b23 = 0.042614
! 590: Covariance matrix: Var(a12) = 1.47453e-001
! 591: Var(b12) = 2.18676e-005
! 592: Var(a13) = 2.09715e-001
! 593: Var(b13) = 3.28937e-005
! 594: Var(a21) = 9.19832e-001
! 595: Var(b21) = 1.29229e-004
! 596: Var(a23) = 4.48405e-001
! 597: Var(b23) = 5.85631e-005
1.2 lievre 598: </pre>
599:
1.3 ! lievre 600: <p>By substitution of these parameters in the regression model,
! 601: we obtain the elementary transition probabilities:</p>
! 602:
! 603: <p><img src="pebiaspar1.gif" width="400" height="300"></p>
! 604:
1.2 lievre 605: <h5><font color="#EC5E5E" size="3"><b>- Transition probabilities</b></font><b>:
606: </b><a href="pijrbiaspar.txt"><b>pijrbiaspar.txt</b></a></h5>
607:
608: <p>Here are the transitions probabilities Pij(x, x+nh) where nh
609: is a multiple of 2 years. The first column is the starting age x
610: (from age 50 to 100), the second is age (x+nh) and the others are
611: the transition probabilities p11, p12, p13, p21, p22, p23. For
612: example, line 5 of the file is: </p>
613:
1.3 ! lievre 614: <pre> 100 106 0.02655 0.17622 0.79722 0.01809 0.13678 0.84513 </pre>
1.2 lievre 615:
616: <p>and this means: </p>
617:
1.3 ! lievre 618: <pre>p11(100,106)=0.02655
! 619: p12(100,106)=0.17622
! 620: p13(100,106)=0.79722
! 621: p21(100,106)=0.01809
! 622: p22(100,106)=0.13678
! 623: p22(100,106)=0.84513 </pre>
1.2 lievre 624:
625: <h5><font color="#EC5E5E" size="3"><b>- </b></font><a
626: name="Stationary prevalence in each state"><font color="#EC5E5E"
627: size="3"><b>Stationary prevalence in each state</b></font></a><b>:
628: </b><a href="plrbiaspar.txt"><b>plrbiaspar.txt</b></a></h5>
629:
1.3 ! lievre 630: <pre>#Prevalence
! 631: #Age 1-1 2-2
! 632:
! 633: #************
! 634: 70 0.90134 0.09866
! 635: 71 0.89177 0.10823
! 636: 72 0.88139 0.11861
! 637: 73 0.87015 0.12985 </pre>
1.2 lievre 638:
1.3 ! lievre 639: <p>At age 70 the stationary prevalence is 0.90134 in state 1 and
! 640: 0.09866 in state 2. This stationary prevalence differs from
1.2 lievre 641: observed prevalence. Here is the point. The observed prevalence
642: at age 70 results from the incidence of disability, incidence of
643: recovery and mortality which occurred in the past of the cohort.
644: Stationary prevalence results from a simulation with actual
645: incidences and mortality (estimated from this cross-longitudinal
646: survey). It is the best predictive value of the prevalence in the
647: future if "nothing changes in the future". This is
648: exactly what demographers do with a Life table. Life expectancy
649: is the expected mean time to survive if observed mortality rates
650: (incidence of mortality) "remains constant" in the
651: future. </p>
652:
653: <h5><font color="#EC5E5E" size="3"><b>- Standard deviation of
654: stationary prevalence</b></font><b>: </b><a
655: href="vplrbiaspar.txt"><b>vplrbiaspar.txt</b></a></h5>
656:
657: <p>The stationary prevalence has to be compared with the observed
658: prevalence by age. But both are statistical estimates and
659: subjected to stochastic errors due to the size of the sample, the
660: design of the survey, and, for the stationary prevalence to the
661: model used and fitted. It is possible to compute the standard
662: deviation of the stationary prevalence at each age.</p>
663:
1.3 ! lievre 664: <h5><font color="#EC5E5E" size="3">-Observed and stationary
1.2 lievre 665: prevalence in state (2=disable) with the confident interval</font>:<b>
1.3 ! lievre 666: </b><a href="vbiaspar21.htm"><b>vbiaspar21.gif</b></a></h5>
1.2 lievre 667:
1.3 ! lievre 668: <p>This graph exhibits the stationary prevalence in state (2)
! 669: with the confidence interval in red. The green curve is the
! 670: observed prevalence (or proportion of individuals in state (2)).
! 671: Without discussing the results (it is not the purpose here), we
! 672: observe that the green curve is rather below the stationary
! 673: prevalence. It suggests an increase of the disability prevalence
! 674: in the future.</p>
! 675:
! 676: <p><img src="vbiaspar21.gif" width="400" height="300"></p>
! 677:
! 678: <h5><font color="#EC5E5E" size="3"><b>-Convergence to the
! 679: stationary prevalence of disability</b></font><b>: </b><a
! 680: href="pbiaspar11.gif"><b>pbiaspar11.gif</b></a><br>
! 681: <img src="pbiaspar11.gif" width="400" height="300"> </h5>
1.2 lievre 682:
683: <p>This graph plots the conditional transition probabilities from
684: an initial state (1=healthy in red at the bottom, or 2=disable in
685: green on top) at age <em>x </em>to the final state 2=disable<em> </em>at
686: age <em>x+h. </em>Conditional means at the condition to be alive
687: at age <em>x+h </em>which is <i>hP12x</i> + <em>hP22x</em>. The
688: curves <i>hP12x/(hP12x</i> + <em>hP22x) </em>and <i>hP22x/(hP12x</i>
689: + <em>hP22x) </em>converge with <em>h, </em>to the <em>stationary
690: prevalence of disability</em>. In order to get the stationary
691: prevalence at age 70 we should start the process at an earlier
692: age, i.e.50. If the disability state is defined by severe
693: disability criteria with only a few chance to recover, then the
694: incidence of recovery is low and the time to convergence is
695: probably longer. But we don't have experience yet.</p>
696:
697: <h5><font color="#EC5E5E" size="3"><b>- Life expectancies by age
698: and initial health status</b></font><b>: </b><a
699: href="erbiaspar.txt"><b>erbiaspar.txt</b></a></h5>
700:
701: <pre># Health expectancies
702: # Age 1-1 1-2 2-1 2-2
1.3 ! lievre 703: 70 10.9226 3.0401 5.6488 6.2122
! 704: 71 10.4384 3.0461 5.2477 6.1599
! 705: 72 9.9667 3.0502 4.8663 6.1025
! 706: 73 9.5077 3.0524 4.5044 6.0401 </pre>
1.2 lievre 707:
1.3 ! lievre 708: <pre>For example 70 10.9226 3.0401 5.6488 6.2122 means:
! 709: e11=10.9226 e12=3.0401 e21=5.6488 e22=6.2122</pre>
1.2 lievre 710:
1.3 ! lievre 711: <pre><img src="expbiaspar21.gif" width="400" height="300"><img
! 712: src="expbiaspar11.gif" width="400" height="300"></pre>
1.2 lievre 713:
714: <p>For example, life expectancy of a healthy individual at age 70
1.3 ! lievre 715: is 10.92 in the healthy state and 3.04 in the disability state
! 716: (=13.96 years). If he was disable at age 70, his life expectancy
! 717: will be shorter, 5.64 in the healthy state and 6.21 in the
! 718: disability state (=11.85 years). The total life expectancy is a
! 719: weighted mean of both, 13.96 and 11.85; weight is the proportion
1.2 lievre 720: of people disabled at age 70. In order to get a pure period index
721: (i.e. based only on incidences) we use the <a
722: href="#Stationary prevalence in each state">computed or
723: stationary prevalence</a> at age 70 (i.e. computed from
724: incidences at earlier ages) instead of the <a
725: href="#Observed prevalence in each state">observed prevalence</a>
726: (for example at first exam) (<a href="#Health expectancies">see
727: below</a>).</p>
728:
729: <h5><font color="#EC5E5E" size="3"><b>- Variances of life
730: expectancies by age and initial health status</b></font><b>: </b><a
731: href="vrbiaspar.txt"><b>vrbiaspar.txt</b></a></h5>
732:
733: <p>For example, the covariances of life expectancies Cov(ei,ej)
734: at age 50 are (line 3) </p>
735:
1.3 ! lievre 736: <pre> Cov(e1,e1)=0.4776 Cov(e1,e2)=0.0488=Cov(e2,e1) Cov(e2,e2)=0.0424</pre>
1.2 lievre 737:
738: <h5><font color="#EC5E5E" size="3"><b>- </b></font><a
739: name="Health expectancies"><font color="#EC5E5E" size="3"><b>Health
740: expectancies</b></font></a><font color="#EC5E5E" size="3"><b>
741: with standard errors in parentheses</b></font><b>: </b><a
742: href="trbiaspar.txt"><font face="Courier New"><b>trbiaspar.txt</b></font></a></h5>
743:
744: <pre>#Total LEs with variances: e.. (std) e.1 (std) e.2 (std) </pre>
745:
1.3 ! lievre 746: <pre>70 13.76 (0.22) 10.40 (0.20) 3.35 (0.14) </pre>
1.2 lievre 747:
1.3 ! lievre 748: <p>Thus, at age 70 the total life expectancy, e..=13.76years is
! 749: the weighted mean of e1.=13.96 and e2.=11.85 by the stationary
! 750: prevalence at age 70 which are 0.90134 in state 1 and 0.09866 in
! 751: state 2, respectively (the sum is equal to one). e.1=10.40 is the
1.2 lievre 752: Disability-free life expectancy at age 70 (it is again a weighted
1.3 ! lievre 753: mean of e11 and e21). e.2=3.35 is also the life expectancy at age
1.2 lievre 754: 70 to be spent in the disability state.</p>
755:
1.3 ! lievre 756: <h5><font color="#EC5E5E" size="3"><b>-Total life expectancy by
1.2 lievre 757: age and health expectancies in states (1=healthy) and (2=disable)</b></font><b>:
1.3 ! lievre 758: </b><a href="ebiaspar1.gif"><b>ebiaspar1.gif</b></a></h5>
1.2 lievre 759:
760: <p>This figure represents the health expectancies and the total
761: life expectancy with the confident interval in dashed curve. </p>
762:
1.3 ! lievre 763: <pre> <img src="ebiaspar1.gif" width="400" height="300"></pre>
1.2 lievre 764:
765: <p>Standard deviations (obtained from the information matrix of
766: the model) of these quantities are very useful.
767: Cross-longitudinal surveys are costly and do not involve huge
768: samples, generally a few thousands; therefore it is very
769: important to have an idea of the standard deviation of our
770: estimates. It has been a big challenge to compute the Health
771: Expectancy standard deviations. Don't be confuse: life expectancy
772: is, as any expected value, the mean of a distribution; but here
773: we are not computing the standard deviation of the distribution,
774: but the standard deviation of the estimate of the mean.</p>
775:
776: <p>Our health expectancies estimates vary according to the sample
777: size (and the standard deviations give confidence intervals of
778: the estimate) but also according to the model fitted. Let us
779: explain it in more details.</p>
780:
781: <p>Choosing a model means ar least two kind of choices. First we
782: have to decide the number of disability states. Second we have to
783: design, within the logit model family, the model: variables,
784: covariables, confonding factors etc. to be included.</p>
785:
786: <p>More disability states we have, better is our demographical
787: approach of the disability process, but smaller are the number of
788: transitions between each state and higher is the noise in the
789: measurement. We do not have enough experiments of the various
790: models to summarize the advantages and disadvantages, but it is
791: important to say that even if we had huge and unbiased samples,
792: the total life expectancy computed from a cross-longitudinal
793: survey, varies with the number of states. If we define only two
794: states, alive or dead, we find the usual life expectancy where it
795: is assumed that at each age, people are at the same risk to die.
796: If we are differentiating the alive state into healthy and
797: disable, and as the mortality from the disability state is higher
798: than the mortality from the healthy state, we are introducing
799: heterogeneity in the risk of dying. The total mortality at each
800: age is the weighted mean of the mortality in each state by the
801: prevalence in each state. Therefore if the proportion of people
802: at each age and in each state is different from the stationary
803: equilibrium, there is no reason to find the same total mortality
804: at a particular age. Life expectancy, even if it is a very useful
805: tool, has a very strong hypothesis of homogeneity of the
806: population. Our main purpose is not to measure differential
807: mortality but to measure the expected time in a healthy or
808: disability state in order to maximise the former and minimize the
809: latter. But the differential in mortality complexifies the
810: measurement.</p>
811:
812: <p>Incidences of disability or recovery are not affected by the
813: number of states if these states are independant. But incidences
814: estimates are dependant on the specification of the model. More
815: covariates we added in the logit model better is the model, but
816: some covariates are not well measured, some are confounding
817: factors like in any statistical model. The procedure to "fit
818: the best model' is similar to logistic regression which itself is
819: similar to regression analysis. We haven't yet been sofar because
820: we also have a severe limitation which is the speed of the
821: convergence. On a Pentium III, 500 MHz, even the simplest model,
822: estimated by month on 8,000 people may take 4 hours to converge.
823: Also, the program is not yet a statistical package, which permits
824: a simple writing of the variables and the model to take into
825: account in the maximisation. The actual program allows only to
1.3 ! lievre 826: add simple variables like age+sex or age+sex+ age*sex but will
1.2 lievre 827: never be general enough. But what is to remember, is that
828: incidences or probability of change from one state to another is
829: affected by the variables specified into the model.</p>
830:
831: <p>Also, the age range of the people interviewed has a link with
832: the age range of the life expectancy which can be estimated by
833: extrapolation. If your sample ranges from age 70 to 95, you can
834: clearly estimate a life expectancy at age 70 and trust your
835: confidence interval which is mostly based on your sample size,
836: but if you want to estimate the life expectancy at age 50, you
837: should rely in your model, but fitting a logistic model on a age
838: range of 70-95 and estimating probabilties of transition out of
839: this age range, say at age 50 is very dangerous. At least you
840: should remember that the confidence interval given by the
841: standard deviation of the health expectancies, are under the
842: strong assumption that your model is the 'true model', which is
843: probably not the case.</p>
844:
845: <h5><font color="#EC5E5E" size="3"><b>- Copy of the parameter
846: file</b></font><b>: </b><a href="orbiaspar.txt"><b>orbiaspar.txt</b></a></h5>
847:
848: <p>This copy of the parameter file can be useful to re-run the
849: program while saving the old output files. </p>
850:
851: <hr>
852:
853: <h2><a name="example" </a><font color="#00006A">Trying an example</font></a></h2>
854:
855: <p>Since you know how to run the program, it is time to test it
856: on your own computer. Try for example on a parameter file named <a
1.3 ! lievre 857: href="..\mytry\imachpar.txt">imachpar.txt</a> which is a copy of <font
! 858: size="2" face="Courier New">mypar.txt</font> included in the
! 859: subdirectory of imach, <font size="2" face="Courier New">mytry</font>.
! 860: Edit it to change the name of the data file to <font size="2"
! 861: face="Courier New">..\data\mydata.txt</font> if you don't want to
! 862: copy it on the same directory. The file <font face="Courier New">mydata.txt</font>
! 863: is a smaller file of 3,000 people but still with 4 waves. </p>
1.2 lievre 864:
865: <p>Click on the imach.exe icon to open a window. Answer to the
866: question:'<strong>Enter the parameter file name:'</strong></p>
867:
868: <table border="1">
869: <tr>
1.3 ! lievre 870: <td width="100%"><strong>IMACH, Version 0.64b</strong><p><strong>Enter
1.2 lievre 871: the parameter file name: ..\mytry\imachpar.txt</strong></p>
872: </td>
873: </tr>
874: </table>
875:
876: <p>Most of the data files or image files generated, will use the
877: 'imachpar' string into their name. The running time is about 2-3
878: minutes on a Pentium III. If the execution worked correctly, the
879: outputs files are created in the current directory, and should be
880: the same as the mypar files initially included in the directory <font
881: size="2" face="Courier New">mytry</font>.</p>
882:
883: <ul>
884: <li><pre><u>Output on the screen</u> The output screen looks like <a
885: href="imachrun.LOG">this Log file</a>
886: #
887:
888: title=MLE datafile=..\data\mydata.txt lastobs=3000 firstpass=1 lastpass=3
889: ftol=1.000000e-008 stepm=24 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre>
890: </li>
891: <li><pre>Total number of individuals= 2965, Agemin = 70.00, Agemax= 100.92
892:
893: Warning, no any valid information for:126 line=126
894: Warning, no any valid information for:2307 line=2307
895: Delay (in months) between two waves Min=21 Max=51 Mean=24.495826
896: <font face="Times New Roman">These lines give some warnings on the data file and also some raw statistics on frequencies of transitions.</font>
897: Age 70 1.=230 loss[1]=3.5% 2.=16 loss[2]=12.5% 1.=222 prev[1]=94.1% 2.=14
898: prev[2]=5.9% 1-1=8 11=200 12=7 13=15 2-1=2 21=6 22=7 23=1
899: Age 102 1.=0 loss[1]=NaNQ% 2.=0 loss[2]=NaNQ% 1.=0 prev[1]=NaNQ% 2.=0 </pre>
900: </li>
901: </ul>
902:
903: <p> </p>
904:
905: <ul>
906: <li>Maximisation with the Powell algorithm. 8 directions are
907: given corresponding to the 8 parameters. this can be
908: rather long to get convergence.<br>
909: <font size="1" face="Courier New"><br>
910: Powell iter=1 -2*LL=11531.405658264877 1 0.000000000000 2
911: 0.000000000000 3<br>
912: 0.000000000000 4 0.000000000000 5 0.000000000000 6
913: 0.000000000000 7 <br>
914: 0.000000000000 8 0.000000000000<br>
915: 1..........2.................3..........4.................5.........<br>
916: 6................7........8...............<br>
917: Powell iter=23 -2*LL=6744.954108371555 1 -12.967632334283
918: <br>
919: 2 0.135136681033 3 -7.402109728262 4 0.067844593326 <br>
920: 5 -0.673601538129 6 -0.006615504377 7 -5.051341616718 <br>
921: 8 0.051272038506<br>
922: 1..............2...........3..............4...........<br>
923: 5..........6................7...........8.........<br>
924: #Number of iterations = 23, -2 Log likelihood =
925: 6744.954042573691<br>
926: # Parameters<br>
927: 12 -12.966061 0.135117 <br>
928: 13 -7.401109 0.067831 <br>
929: 21 -0.672648 -0.006627 <br>
930: 23 -5.051297 0.051271 </font><br>
931: </li>
932: <li><pre><font size="2">Calculation of the hessian matrix. Wait...
933: 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
934:
935: Inverting the hessian to get the covariance matrix. Wait...
936:
937: #Hessian matrix#
938: 3.344e+002 2.708e+004 -4.586e+001 -3.806e+003 -1.577e+000 -1.313e+002 3.914e-001 3.166e+001
939: 2.708e+004 2.204e+006 -3.805e+003 -3.174e+005 -1.303e+002 -1.091e+004 2.967e+001 2.399e+003
940: -4.586e+001 -3.805e+003 4.044e+002 3.197e+004 2.431e-002 1.995e+000 1.783e-001 1.486e+001
941: -3.806e+003 -3.174e+005 3.197e+004 2.541e+006 2.436e+000 2.051e+002 1.483e+001 1.244e+003
942: -1.577e+000 -1.303e+002 2.431e-002 2.436e+000 1.093e+002 8.979e+003 -3.402e+001 -2.843e+003
943: -1.313e+002 -1.091e+004 1.995e+000 2.051e+002 8.979e+003 7.420e+005 -2.842e+003 -2.388e+005
944: 3.914e-001 2.967e+001 1.783e-001 1.483e+001 -3.402e+001 -2.842e+003 1.494e+002 1.251e+004
945: 3.166e+001 2.399e+003 1.486e+001 1.244e+003 -2.843e+003 -2.388e+005 1.251e+004 1.053e+006
946: # Scales
947: 12 1.00000e-004 1.00000e-006
948: 13 1.00000e-004 1.00000e-006
949: 21 1.00000e-003 1.00000e-005
950: 23 1.00000e-004 1.00000e-005
951: # Covariance
952: 1 5.90661e-001
953: 2 -7.26732e-003 8.98810e-005
954: 3 8.80177e-002 -1.12706e-003 5.15824e-001
955: 4 -1.13082e-003 1.45267e-005 -6.50070e-003 8.23270e-005
956: 5 9.31265e-003 -1.16106e-004 6.00210e-004 -8.04151e-006 1.75753e+000
957: 6 -1.15664e-004 1.44850e-006 -7.79995e-006 1.04770e-007 -2.12929e-002 2.59422e-004
958: 7 1.35103e-003 -1.75392e-005 -6.38237e-004 7.85424e-006 4.02601e-001 -4.86776e-003 1.32682e+000
959: 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
960: # agemin agemax for lifexpectancy, bage fage (if mle==0 ie no data nor Max likelihood).
961:
962:
963: agemin=70 agemax=100 bage=50 fage=100
964: Computing prevalence limit: result on file 'plrmypar.txt'
965: Computing pij: result on file 'pijrmypar.txt'
966: Computing Health Expectancies: result on file 'ermypar.txt'
967: Computing Variance-covariance of DFLEs: file 'vrmypar.txt'
968: Computing Total LEs with variances: file 'trmypar.txt'
969: Computing Variance-covariance of Prevalence limit: file 'vplrmypar.txt'
970: End of Imach
971: </font></pre>
972: </li>
973: </ul>
974:
975: <p><font size="3">Once the running is finished, the program
976: requires a caracter:</font></p>
977:
978: <table border="1">
979: <tr>
1.3 ! lievre 980: <td width="100%"><strong>Type e to edit output files, c
! 981: to start again, and q for exiting:</strong></td>
1.2 lievre 982: </tr>
983: </table>
984:
1.3 ! lievre 985: <p><font size="3">First you should enter <strong>e </strong>to
! 986: edit the master file mypar.htm. </font></p>
1.2 lievre 987:
988: <ul>
989: <li><u>Outputs files</u> <br>
1.3 ! lievre 990: <br>
1.2 lievre 991: - Observed prevalence in each state: <a
1.3 ! lievre 992: href="..\mytry\prmypar.txt">pmypar.txt</a> <br>
1.2 lievre 993: - Estimated parameters and the covariance matrix: <a
994: href="..\mytry\rmypar.txt">rmypar.txt</a> <br>
995: - Stationary prevalence in each state: <a
996: href="..\mytry\plrmypar.txt">plrmypar.txt</a> <br>
997: - Transition probabilities: <a
998: href="..\mytry\pijrmypar.txt">pijrmypar.txt</a> <br>
999: - Copy of the parameter file: <a
1000: href="..\mytry\ormypar.txt">ormypar.txt</a> <br>
1001: - Life expectancies by age and initial health status: <a
1002: href="..\mytry\ermypar.txt">ermypar.txt</a> <br>
1003: - Variances of life expectancies by age and initial
1004: health status: <a href="..\mytry\vrmypar.txt">vrmypar.txt</a>
1005: <br>
1006: - Health expectancies with their variances: <a
1007: href="..\mytry\trmypar.txt">trmypar.txt</a> <br>
1008: - Standard deviation of stationary prevalence: <a
1009: href="..\mytry\vplrmypar.txt">vplrmypar.txt</a> <br>
1010: <br>
1011: </li>
1012: <li><u>Graphs</u> <br>
1013: <br>
1.3 ! lievre 1014: -<a href="../mytry/pemypar1.gif">One-step transition
! 1015: probabilities</a><br>
! 1016: -<a href="../mytry/pmypar11.gif">Convergence to the
! 1017: stationary prevalence</a><br>
! 1018: -<a href="..\mytry\vmypar11.gif">Observed and stationary
1.2 lievre 1019: prevalence in state (1) with the confident interval</a> <br>
1.3 ! lievre 1020: -<a href="..\mytry\vmypar21.gif">Observed and stationary
1.2 lievre 1021: prevalence in state (2) with the confident interval</a> <br>
1.3 ! lievre 1022: -<a href="..\mytry\expmypar11.gif">Health life
! 1023: expectancies by age and initial health state (1)</a> <br>
! 1024: -<a href="..\mytry\expmypar21.gif">Health life
! 1025: expectancies by age and initial health state (2)</a> <br>
! 1026: -<a href="..\mytry\emypar1.gif">Total life expectancy by
1.2 lievre 1027: age and health expectancies in states (1) and (2).</a> </li>
1028: </ul>
1029:
1030: <p>This software have been partly granted by <a
1031: href="http://euroreves.ined.fr">Euro-REVES</a>, a concerted
1032: action from the European Union. It will be copyrighted
1033: identically to a GNU software product, i.e. program and software
1034: can be distributed freely for non commercial use. Sources are not
1035: widely distributed today. You can get them by asking us with a
1036: simple justification (name, email, institute) <a
1037: href="mailto:brouard@ined.fr">mailto:brouard@ined.fr</a> and <a
1038: href="mailto:lievre@ined.fr">mailto:lievre@ined.fr</a> .</p>
1039:
1.3 ! lievre 1040: <p>Latest version (0.64b of may 2001) can be accessed at <a
1.2 lievre 1041: href="http://euroeves.ined.fr/imach">http://euroreves.ined.fr/imach</a><br>
1042: </p>
1043: </body>
1044: </html>
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