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19:
20: <h1 align="center"><font color="#00006A">Computing Health
21: Expectancies using IMaCh</font></h1>
22:
23: <h1 align="center"><font color="#00006A" size="5">(a Maximum
24: Likelihood Computer Program using Interpolation of Markov Chains)</font></h1>
25:
26: <p align="center"> </p>
27:
28: <p align="center"><a href="http://www.ined.fr/"><img
29: src="logo-ined.gif" border="0" width="151" height="76"></a><img
30: src="euroreves2.gif" width="151" height="75"></p>
31:
32: <h3 align="center"><a href="http://www.ined.fr/"><font
33: color="#00006A">INED</font></a><font color="#00006A"> and </font><a
34: href="http://euroreves.ined.fr"><font color="#00006A">EUROREVES</font></a></h3>
35:
36: <p align="center"><font color="#00006A" size="4"><strong>Version
1.2 brouard 37: 0.97, June 2004</strong></font></p>
1.1 brouard 38:
39: <hr size="3" color="#EC5E5E">
40:
41: <p align="center"><font color="#00006A"><strong>Authors of the
42: program: </strong></font><a href="http://sauvy.ined.fr/brouard"><font
43: color="#00006A"><strong>Nicolas Brouard</strong></font></a><font
44: color="#00006A"><strong>, senior researcher at the </strong></font><a
45: href="http://www.ined.fr"><font color="#00006A"><strong>Institut
46: National d'Etudes Démographiques</strong></font></a><font
47: color="#00006A"><strong> (INED, Paris) in the "Mortality,
48: Health and Epidemiology" Research Unit </strong></font></p>
49:
50: <p align="center"><font color="#00006A"><strong>and Agnès
51: Lièvre<br clear="left">
52: </strong></font></p>
53:
54: <h4><font color="#00006A">Contribution to the mathematics: C. R.
55: Heathcote </font><font color="#00006A" size="2">(Australian
56: National University, Canberra).</font></h4>
57:
58: <h4><font color="#00006A">Contact: Agnès Lièvre (</font><a
59: href="mailto:lievre@ined.fr"><font color="#00006A"><i>lievre@ined.fr</i></font></a><font
60: color="#00006A">) </font></h4>
61:
62: <hr>
63:
64: <ul>
65: <li><a href="#intro">Introduction</a> </li>
66: <li><a href="#data">On what kind of data can it be used?</a></li>
67: <li><a href="#datafile">The data file</a> </li>
68: <li><a href="#biaspar">The parameter file</a> </li>
69: <li><a href="#running">Running Imach</a> </li>
70: <li><a href="#output">Output files and graphs</a> </li>
71: <li><a href="#example">Exemple</a> </li>
72: </ul>
73:
74: <hr>
75:
76: <h2><a name="intro"><font color="#00006A">Introduction</font></a></h2>
77:
78: <p>This program computes <b>Healthy Life Expectancies</b> from <b>cross-longitudinal
79: data</b> using the methodology pioneered by Laditka and Wolf (1).
80: Within the family of Health Expectancies (HE), Disability-free
81: life expectancy (DFLE) is probably the most important index to
82: monitor. In low mortality countries, there is a fear that when
83: mortality declines, the increase in DFLE is not proportionate to
84: the increase in total Life expectancy. This case is called the <em>Expansion
85: of morbidity</em>. Most of the data collected today, in
86: particular by the international <a href="http://www.reves.org">REVES</a>
87: network on Health expectancy, and most HE indices based on these
88: data, are <em>cross-sectional</em>. It means that the information
89: collected comes from a single cross-sectional survey: people from
90: various ages (but mostly old people) are surveyed on their health
91: status at a single date. Proportion of people disabled at each
92: age, can then be measured at that date. This age-specific
93: prevalence curve is then used to distinguish, within the
94: stationary population (which, by definition, is the life table
95: estimated from the vital statistics on mortality at the same
96: date), the disable population from the disability-free
97: population. Life expectancy (LE) (or total population divided by
98: the yearly number of births or deaths of this stationary
99: population) is then decomposed into DFLE and DLE. This method of
100: computing HE is usually called the Sullivan method (from the name
101: of the author who first described it).</p>
102:
1.2 brouard 103: <p>Age-specific proportions of people disabled (prevalence of
104: disability) are dependent on the historical flows from entering
105: disability and recovering in the past until today. The age-specific
106: forces (or incidence rates), estimated over a recent period of time
107: (like for period forces of mortality), of entering disability or
108: recovering a good health, are reflecting current conditions and
109: therefore can be used at each age to forecast the future of this
110: cohort<em>if nothing changes in the future</em>, i.e to forecast the
111: prevalence of disability of each cohort. Our finding (2) is that the period
112: prevalence of disability (computed from period incidences) is lower
113: than the cross-sectional prevalence. For example if a country is
114: improving its technology of prosthesis, the incidence of recovering
115: the ability to walk will be higher at each (old) age, but the
116: prevalence of disability will only slightly reflect an improve because
117: the prevalence is mostly affected by the history of the cohort and not
118: by recent period effects. To measure the period improvement we have to
119: simulate the future of a cohort of new-borns entering or leaving at
120: each age the disability state or dying according to the incidence
121: rates measured today on different cohorts. The proportion of people
122: disabled at each age in this simulated cohort will be much lower that
123: the proportions observed at each age in a cross-sectional survey. This
124: new prevalence curve introduced in a life table will give a more
125: realistic HE level than the Sullivan method which mostly measured the
126: History of health conditions in this country.</p>
1.1 brouard 127:
128: <p>Therefore, the main question is how to measure incidence rates
129: from cross-longitudinal surveys? This is the goal of the IMaCH
130: program. From your data and using IMaCH you can estimate period
131: HE and not only Sullivan's HE. Also the standard errors of the HE
132: are computed.</p>
133:
134: <p>A cross-longitudinal survey consists in a first survey
135: ("cross") where individuals from different ages are
136: interviewed on their health status or degree of disability. At
137: least a second wave of interviews ("longitudinal")
138: should measure each new individual health status. Health
139: expectancies are computed from the transitions observed between
140: waves and are computed for each degree of severity of disability
141: (number of life states). More degrees you consider, more time is
142: necessary to reach the Maximum Likelihood of the parameters
143: involved in the model. Considering only two states of disability
144: (disable and healthy) is generally enough but the computer
145: program works also with more health statuses.<br>
146: <br>
147: The simplest model is the multinomial logistic model where <i>pij</i>
148: is the probability to be observed in state <i>j</i> at the second
149: wave conditional to be observed in state <em>i</em> at the first
150: wave. Therefore a simple model is: log<em>(pij/pii)= aij +
151: bij*age+ cij*sex,</em> where '<i>age</i>' is age and '<i>sex</i>'
152: is a covariate. The advantage that this computer program claims,
153: comes from that if the delay between waves is not identical for
154: each individual, or if some individual missed an interview, the
155: information is not rounded or lost, but taken into account using
156: an interpolation or extrapolation. <i>hPijx</i> is the
157: probability to be observed in state <i>i</i> at age <i>x+h</i>
158: conditional to the observed state <i>i</i> at age <i>x</i>. The
159: delay '<i>h</i>' can be split into an exact number (<i>nh*stepm</i>)
160: of unobserved intermediate states. This elementary transition (by
161: month or quarter trimester, semester or year) is modeled as a
162: multinomial logistic. The <i>hPx</i> matrix is simply the matrix
163: product of <i>nh*stepm</i> elementary matrices and the
164: contribution of each individual to the likelihood is simply <i>hPijx</i>.
165: <br>
166: </p>
167:
168: <p>The program presented in this manual is a quite general
169: program named <strong>IMaCh</strong> (for <strong>I</strong>nterpolated
170: <strong>MA</strong>rkov <strong>CH</strong>ain), designed to
171: analyse transition data from longitudinal surveys. The first step
172: is the parameters estimation of a transition probabilities model
173: between an initial status and a final status. From there, the
174: computer program produces some indicators such as observed and
175: stationary prevalence, life expectancies and their variances and
176: graphs. Our transition model consists in absorbing and
177: non-absorbing states with the possibility of return across the
178: non-absorbing states. The main advantage of this package,
179: compared to other programs for the analysis of transition data
180: (For example: Proc Catmod of SAS<sup>®</sup>) is that the whole
181: individual information is used even if an interview is missing, a
182: status or a date is unknown or when the delay between waves is
183: not identical for each individual. The program can be executed
184: according to parameters: selection of a sub-sample, number of
185: absorbing and non-absorbing states, number of waves taken in
186: account (the user inputs the first and the last interview), a
187: tolerance level for the maximization function, the periodicity of
188: the transitions (we can compute annual, quarterly or monthly
189: transitions), covariates in the model. It works on Windows or on
190: Unix.<br>
191: </p>
192:
193: <hr>
194:
195: <p>(1) Laditka, Sarah B. and Wolf, Douglas A. (1998), "New
196: Methods for Analyzing Active Life Expectancy". <i>Journal of
197: Aging and Health</i>. Vol 10, No. 2. </p>
1.2 brouard 198: <p>(2) <a href=http://taylorandfrancis.metapress.com/app/home/contribution.asp?wasp=1f99bwtvmk5yrb7hlhw3&referrer=parent&backto=issue,1,2;journal,2,5;linkingpublicationresults,1:300265,1
199: >Lièvre A., Brouard N. and Heathcote Ch. (2003) Estimating Health Expectancies
200: from Cross-longitudinal surveys. <em>Mathematical Population Studies</em>.- 10(4), pp. 211-248</a>
1.1 brouard 201:
202: <hr>
203:
204: <h2><a name="data"><font color="#00006A">On what kind of data can
205: it be used?</font></a></h2>
206:
207: <p>The minimum data required for a transition model is the
208: recording of a set of individuals interviewed at a first date and
209: interviewed again at least one another time. From the
210: observations of an individual, we obtain a follow-up over time of
211: the occurrence of a specific event. In this documentation, the
212: event is related to health status at older ages, but the program
213: can be applied on a lot of longitudinal studies in different
214: contexts. To build the data file explained into the next section,
215: you must have the month and year of each interview and the
216: corresponding health status. But in order to get age, date of
217: birth (month and year) is required (missing values is allowed for
218: month). Date of death (month and year) is an important
219: information also required if the individual is dead. Shorter
220: steps (i.e. a month) will more closely take into account the
221: survival time after the last interview.</p>
222:
223: <hr>
224:
225: <h2><a name="datafile"><font color="#00006A">The data file</font></a></h2>
226:
227: <p>In this example, 8,000 people have been interviewed in a
1.2 brouard 228: cross-longitudinal survey of 4 waves (1984, 1986, 1988, 1990). Some
229: people missed 1, 2 or 3 interviews. Health statuses are healthy (1)
230: and disable (2). The survey is not a real one. It is a simulation of
231: the American Longitudinal Survey on Aging. The disability state is
232: defined if the individual missed one of four ADL (Activity of daily
233: living, like bathing, eating, walking). Therefore, even if the
234: individuals interviewed in the sample are virtual, the information
235: brought with this sample is close to the situation of the United
236: States. Sex is not recorded is this sample. The LSOA survey is biased
237: in the sense that people living in an institution were not surveyed at
238: first pass in 1984. Thus the prevalence of disability in 1984 is
239: biased downwards at old ages. But when people left their household to
240: an institution, they have been surveyed in their institution in 1986,
241: 1988 or 1990. Thus incidences are not biased. But cross-sectional
242: prevalences of disability at old ages are thus artificially increasing
243: in 1986, 1988 and 1990 because of a higher weight of people
244: institutionalized in the sample. Our article shows the
245: opposite: the period prevalence is lower at old ages than the
246: adjusted cross-sectional prevalence proving important current progress
247: against disability.</p>
1.1 brouard 248:
249: <p>Each line of the data set (named <a href="data1.txt">data1.txt</a>
1.2 brouard 250: in this first example) is an individual record. Fields are separated
251: by blanks: </p>
1.1 brouard 252:
253: <ul>
254: <li><b>Index number</b>: positive number (field 1) </li>
255: <li><b>First covariate</b> positive number (field 2) </li>
256: <li><b>Second covariate</b> positive number (field 3) </li>
257: <li><a name="Weight"><b>Weight</b></a>: positive number
258: (field 4) . In most surveys individuals are weighted
259: according to the stratification of the sample.</li>
260: <li><b>Date of birth</b>: coded as mm/yyyy. Missing dates are
261: coded as 99/9999 (field 5) </li>
262: <li><b>Date of death</b>: coded as mm/yyyy. Missing dates are
263: coded as 99/9999 (field 6) </li>
264: <li><b>Date of first interview</b>: coded as mm/yyyy. Missing
265: dates are coded as 99/9999 (field 7) </li>
266: <li><b>Status at first interview</b>: positive number.
267: Missing values ar coded -1. (field 8) </li>
268: <li><b>Date of second interview</b>: coded as mm/yyyy.
269: Missing dates are coded as 99/9999 (field 9) </li>
270: <li><strong>Status at second interview</strong> positive
271: number. Missing values ar coded -1. (field 10) </li>
272: <li><b>Date of third interview</b>: coded as mm/yyyy. Missing
273: dates are coded as 99/9999 (field 11) </li>
274: <li><strong>Status at third interview</strong> positive
275: number. Missing values ar coded -1. (field 12) </li>
276: <li><b>Date of fourth interview</b>: coded as mm/yyyy.
277: Missing dates are coded as 99/9999 (field 13) </li>
278: <li><strong>Status at fourth interview</strong> positive
279: number. Missing values are coded -1. (field 14) </li>
280: <li>etc</li>
281: </ul>
282:
283: <p> </p>
284:
1.4 ! brouard 285: <p>If your longitudinal survey does not include information about
1.1 brouard 286: weights or covariates, you must fill the column with a number
287: (e.g. 1) because a missing field is not allowed.</p>
288:
289: <hr>
290:
291: <h2><font color="#00006A">Your first example parameter file</font><a
292: href="http://euroreves.ined.fr/imach"></a><a name="uio"></a></h2>
293:
1.2 brouard 294: <h2><a name="biaspar"></a>#Imach version 0.97b, June 2004,
1.1 brouard 295: INED-EUROREVES </h2>
296:
1.2 brouard 297: <p>This first line was a comment. Comments line start with a '#'.</p>
1.1 brouard 298:
299: <h4><font color="#FF0000">First uncommented line</font></h4>
300:
301: <pre>title=1st_example datafile=data1.txt lastobs=8600 firstpass=1 lastpass=4</pre>
302:
303: <ul>
304: <li><b>title=</b> 1st_example is title of the run. </li>
305: <li><b>datafile=</b> data1.txt is the name of the data set.
306: Our example is a six years follow-up survey. It consists
307: in a baseline followed by 3 reinterviews. </li>
308: <li><b>lastobs=</b> 8600 the program is able to run on a
309: subsample where the last observation number is lastobs.
310: It can be set a bigger number than the real number of
311: observations (e.g. 100000). In this example, maximisation
312: will be done on the 8600 first records. </li>
313: <li><b>firstpass=1</b> , <b>lastpass=4 </b>In case of more
314: than two interviews in the survey, the program can be run
315: on selected transitions periods. firstpass=1 means the
316: first interview included in the calculation is the
317: baseline survey. lastpass=4 means that the information
318: brought by the 4th interview is taken into account.</li>
319: </ul>
320:
321: <p> </p>
322:
323: <h4><a name="biaspar-2"><font color="#FF0000">Second uncommented
324: line</font></a></h4>
325:
326: <pre>ftol=1.e-08 stepm=1 ncovcol=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre>
327:
328: <ul>
329: <li><b>ftol=1e-8</b> Convergence tolerance on the function
330: value in the maximisation of the likelihood. Choosing a
331: correct value for ftol is difficult. 1e-8 is a correct
332: value for a 32 bits computer.</li>
333: <li><b>stepm=1</b> Time unit in months for interpolation.
334: Examples:<ul>
335: <li>If stepm=1, the unit is a month </li>
336: <li>If stepm=4, the unit is a trimester</li>
337: <li>If stepm=12, the unit is a year </li>
338: <li>If stepm=24, the unit is two years</li>
339: <li>... </li>
340: </ul>
341: </li>
1.2 brouard 342: <li><b>ncovcol=2</b> Number of covariate columns included in the
343: datafile before the column of the date of birth. You can have
344: covariates that won't necessary be used during the
1.1 brouard 345: run. It is not the number of covariates that will be
1.2 brouard 346: specified by the model. The 'model' syntax describes the
347: covariates to be taken into account during the run. </li>
1.1 brouard 348: <li><b>nlstate=2</b> Number of non-absorbing (alive) states.
349: Here we have two alive states: disability-free is coded 1
350: and disability is coded 2. </li>
351: <li><b>ndeath=1</b> Number of absorbing states. The absorbing
352: state death is coded 3. </li>
353: <li><b>maxwav=4</b> Number of waves in the datafile.</li>
354: <li><a name="mle"><b>mle</b></a><b>=1</b> Option for the
355: Maximisation Likelihood Estimation. <ul>
356: <li>If mle=1 the program does the maximisation and
357: the calculation of health expectancies </li>
358: <li>If mle=0 the program only does the calculation of
1.2 brouard 359: the health expectancies and other indices and graphs
360: but without the maximization.. </li>
361: There also other possible values:
362: <ul>
363: <li>If mle=-1 you get a template which can be useful if
364: your model is complex with many covariates.</li>
365: <li> If mle=-3 IMaCh computes the mortality but without
366: any health status (May 2004)</li> <li>If mle=2 IMach
367: likelihood corresponds to a linear interpolation</li> <li>
368: If mle=3 IMach likelihood corresponds to an exponential
369: inter-extrapolation</li>
370: <li> If mle=4 IMach likelihood
371: corresponds to no inter-extrapolation, and thus biasing
372: the results. </li>
373: <li> If mle=5 IMach likelihood
374: corresponds to no inter-extrapolation, and before the
375: correction of the Jackson's bug (avoid this).</li>
376: </ul>
1.1 brouard 377: </ul>
378: </li>
379: <li><b>weight=0</b> Possibility to add weights. <ul>
380: <li>If weight=0 no weights are included </li>
381: <li>If weight=1 the maximisation integrates the
382: weights which are in field <a href="#Weight">4</a></li>
383: </ul>
384: </li>
385: </ul>
386:
387: <h4><font color="#FF0000">Covariates</font></h4>
388:
389: <p>Intercept and age are systematically included in the model.
390: Additional covariates can be included with the command: </p>
391:
392: <pre>model=<em>list of covariates</em></pre>
393:
394: <ul>
395: <li>if<strong> model=. </strong>then no covariates are
396: included</li>
397: <li>if <strong>model=V1</strong> the model includes the first
398: covariate (field 2)</li>
399: <li>if <strong>model=V2 </strong>the model includes the
400: second covariate (field 3)</li>
401: <li>if <strong>model=V1+V2 </strong>the model includes the
402: first and the second covariate (fields 2 and 3)</li>
403: <li>if <strong>model=V1*V2 </strong>the model includes the
404: product of the first and the second covariate (fields 2
405: and 3)</li>
406: <li>if <strong>model=V1+V1*age</strong> the model includes
407: the product covariate*age</li>
408: </ul>
409:
410: <p>In this example, we have two covariates in the data file
411: (fields 2 and 3). The number of covariates included in the data
412: file between the id and the date of birth is ncovcol=2 (it was
413: named ncov in version prior to 0.8). If you have 3 covariates in
414: the datafile (fields 2, 3 and 4), you will set ncovcol=3. Then
415: you can run the programme with a new parametrisation taking into
416: account the third covariate. For example, <strong>model=V1+V3 </strong>estimates
417: a model with the first and third covariates. More complicated
418: models can be used, but it will takes more time to converge. With
419: a simple model (no covariates), the programme estimates 8
420: parameters. Adding covariates increases the number of parameters
421: : 12 for <strong>model=V1, </strong>16 for <strong>model=V1+V1*age
422: </strong>and 20 for <strong>model=V1+V2+V3.</strong></p>
423:
424: <h4><font color="#FF0000">Guess values for optimization</font><font
425: color="#00006A"> </font></h4>
426:
427: <p>You must write the initial guess values of the parameters for
428: optimization. The number of parameters, <em>N</em> depends on the
429: number of absorbing states and non-absorbing states and on the
430: number of covariates. <br>
431: <em>N</em> is given by the formula <em>N</em>=(<em>nlstate</em> +
432: <em>ndeath</em>-1)*<em>nlstate</em>*<em>ncovmodel</em> . <br>
433: <br>
434: Thus in the simple case with 2 covariates (the model is log
435: (pij/pii) = aij + bij * age where intercept and age are the two
436: covariates), and 2 health degrees (1 for disability-free and 2
437: for disability) and 1 absorbing state (3), you must enter 8
438: initials values, a12, b12, a13, b13, a21, b21, a23, b23. You can
439: start with zeros as in this example, but if you have a more
440: precise set (for example from an earlier run) you can enter it
441: and it will speed up them<br>
442: Each of the four lines starts with indices "ij": <b>ij
443: aij bij</b> </p>
444:
445: <blockquote>
446: <pre># Guess values of aij and bij in log (pij/pii) = aij + bij * age
447: 12 -14.155633 0.110794
448: 13 -7.925360 0.032091
449: 21 -1.890135 -0.029473
450: 23 -6.234642 0.022315 </pre>
451: </blockquote>
452:
453: <p>or, to simplify (in most of cases it converges but there is no
454: warranty!): </p>
455:
456: <blockquote>
457: <pre>12 0.0 0.0
458: 13 0.0 0.0
459: 21 0.0 0.0
460: 23 0.0 0.0</pre>
461: </blockquote>
462:
463: <p>In order to speed up the convergence you can make a first run
464: with a large stepm i.e stepm=12 or 24 and then decrease the stepm
465: until stepm=1 month. If newstepm is the new shorter stepm and
466: stepm can be expressed as a multiple of newstepm, like newstepm=n
467: stepm, then the following approximation holds: </p>
468:
469: <pre>aij(stepm) = aij(n . stepm) - ln(n)
470: </pre>
471:
472: <p>and </p>
473:
474: <pre>bij(stepm) = bij(n . stepm) .</pre>
475:
476: <p>For example if you already ran for a 6 months interval and
477: got:<br>
478: </p>
479:
480: <pre># Parameters
481: 12 -13.390179 0.126133
482: 13 -7.493460 0.048069
483: 21 0.575975 -0.041322
484: 23 -4.748678 0.030626
485: </pre>
486:
487: <p>If you now want to get the monthly estimates, you can guess
488: the aij by substracting ln(6)= 1,7917<br>
489: and running<br>
490: </p>
491:
492: <pre>12 -15.18193847 0.126133
493: 13 -9.285219469 0.048069
494: 21 -1.215784469 -0.041322
495: 23 -6.540437469 0.030626
496: </pre>
497:
498: <p>and get<br>
499: </p>
500:
501: <pre>12 -15.029768 0.124347
502: 13 -8.472981 0.036599
503: 21 -1.472527 -0.038394
504: 23 -6.553602 0.029856
505:
506: which is closer to the results. The approximation is probably useful
507: only for very small intervals and we don't have enough experience to
508: know if you will speed up the convergence or not.
509: </pre>
510:
511: <pre> -ln(12)= -2.484
512: -ln(6/1)=-ln(6)= -1.791
513: -ln(3/1)=-ln(3)= -1.0986
514: -ln(12/6)=-ln(2)= -0.693
515: </pre>
516:
1.2 brouard 517: In version 0.9 and higher you can still have valuable results even if
518: your stepm parameter is bigger than a month. The idea is to run with
519: bigger stepm in order to have a quicker convergence at the price of a
520: small bias. Once you know which model you want to fit, you can put
521: stepm=1 and wait hours or days to get the convergence!
522:
523: To get unbiased results even with large stepm we introduce the idea of
524: pseudo likelihood by interpolating two exact likelihoods. Let us
525: detail this:
526: <p>
527: If the interval of <em>d</em> months between two waves is not a
528: mutliple of 'stepm', but is comprised between <em>(n-1) stepm</em> and
529: <em>n stepm</em> then both exact likelihoods are computed (the
530: contribution to the likelihood at <em>n stepm</em> requires one matrix
531: product more) (let us remember that we are modelling the probability
532: to be observed in a particular state after <em>d</em> months being
533: observed at a particular state at 0). The distance, (<em>bh</em> in
534: the program), from the month of interview to the rounded date of <em>n
535: stepm</em> is computed. It can be negative (interview occurs before
536: <em>n stepm</em>) or positive if the interview occurs after <em>n
537: stepm</em> (and before <em>(n+1)stepm</em>).
538: <br>
539: Then the final contribution to the total likelihood is a weighted
540: average of these two exact likelihoods at <em>n stepm</em> (out) and
541: at <em>(n-1)stepm</em>(savm). We did not want to compute the third
542: likelihood at <em>(n+1)stepm</em> because it is too costly in time, so
543: we used an extrapolation if <em>bh</em> is positive. <br> Formula of
544: inter/extrapolation may vary according to the value of parameter mle:
545: <pre>
546: mle=1 lli= log((1.+bbh)*out[s1][s2]- bbh*savm[s1][s2]); /* linear interpolation */
547:
548: mle=2 lli= (savm[s1][s2]>(double)1.e-8 ? \
549: log((1.+bbh)*out[s1][s2]- bbh*(savm[s1][s2])): \
550: log((1.+bbh)*out[s1][s2])); /* linear interpolation */
551: mle=3 lli= (savm[s1][s2]>1.e-8 ? \
552: (1.+bbh)*log(out[s1][s2])- bbh*log(savm[s1][s2]): \
553: log((1.+bbh)*out[s1][s2])); /* exponential inter-extrapolation */
554:
555: mle=4 lli=log(out[s[mw[mi][i]][i]][s[mw[mi+1][i]][i]]); /* No interpolation */
556: no need to save previous likelihood into memory.
557: </pre>
558: <p>
559: If the death occurs between first and second pass, and for example
560: more precisely between <em>n stepm</em> and <em>(n+1)stepm</em> the
561: contribution of this people to the likelihood is simply the difference
562: between the probability of dying before <em>n stepm</em> and the
563: probability of dying before <em>(n+1)stepm</em>. There was a bug in
564: version 0.8 and death was treated as any other state, i.e. as if it
565: was an observed death at second pass. This was not precise but
566: correct, but when information on the precise month of death came
567: (death occuring prior to second pass) we did not change the likelihood
568: accordingly. Thanks to Chris Jackson for correcting us. In earlier
569: versions (fortunately before first publication) the total mortality
570: was overestimated (people were dying too early) of about 10%. Version
571: 0.95 and higher are correct.
572:
573: <p> Our suggested choice is mle=1 . If stepm=1 there is no difference
574: between various mle options (methods of interpolation). If stepm is
575: big, like 12 or 24 or 48 and mle=4 (no interpolation) the bias may be
576: very important if the mean duration between two waves is not a
577: multiple of stepm. See the appendix in our main publication concerning
578: the sine curve of biases.
579:
580:
1.1 brouard 581: <h4><font color="#FF0000">Guess values for computing variances</font></h4>
582:
1.2 brouard 583: <p>These values are output by the maximisation of the likelihood <a
584: href="#mle">mle</a>=1. These valuse can be used as an input of a
585: second run in order to get the various output data files (Health
586: expectancies, period prevalence etc.) and figures without rerunning
587: the long maximisation phase (mle=0). </p>
588:
589: <p>These 'scales' are small values needed for the computing of
590: numerical derivatives. These derivatives are used to compute the
591: hessian matrix of the parameters, that is the inverse of the
592: covariance matrix. They are often used for estimating variances and
593: confidence intervals. Each line consists in indices "ij"
594: followed by the initial scales (zero to simplify) associated with aij
595: and bij. </p>
1.1 brouard 596:
597: <ul>
598: <li>If mle=1 you can enter zeros:</li>
599: <li><blockquote>
600: <pre># Scales (for hessian or gradient estimation)
601: 12 0. 0.
602: 13 0. 0.
603: 21 0. 0.
604: 23 0. 0. </pre>
605: </blockquote>
606: </li>
1.2 brouard 607: <li>If mle=0 (no maximisation of Likelihood) you must enter a covariance matrix (usually
1.1 brouard 608: obtained from an earlier run).</li>
609: </ul>
610:
611: <h4><font color="#FF0000">Covariance matrix of parameters</font></h4>
612:
1.2 brouard 613: <p>The covariance matrix is output if <a href="#mle">mle</a>=1. But it can be
614: also used as an input to get the various output data files (Health
615: expectancies, period prevalence etc.) and figures without
616: rerunning the maximisation phase (mle=0). <br>
1.1 brouard 617: Each line starts with indices "ijk" followed by the
618: covariances between aij and bij:<br>
619: </p>
620:
621: <pre>
622: 121 Var(a12)
623: 122 Cov(b12,a12) Var(b12)
624: ...
625: 232 Cov(b23,a12) Cov(b23,b12) ... Var (b23) </pre>
626:
627: <ul>
628: <li>If mle=1 you can enter zeros. </li>
629: <li><pre># Covariance matrix
630: 121 0.
631: 122 0. 0.
632: 131 0. 0. 0.
633: 132 0. 0. 0. 0.
634: 211 0. 0. 0. 0. 0.
635: 212 0. 0. 0. 0. 0. 0.
636: 231 0. 0. 0. 0. 0. 0. 0.
637: 232 0. 0. 0. 0. 0. 0. 0. 0.</pre>
638: </li>
639: <li>If mle=0 you must enter a covariance matrix (usually
640: obtained from an earlier run). </li>
641: </ul>
642:
643: <h4><font color="#FF0000">Age range for calculation of stationary
644: prevalences and health expectancies</font></h4>
645:
646: <pre>agemin=70 agemax=100 bage=50 fage=100</pre>
647:
1.2 brouard 648: <p>
1.1 brouard 649: Once we obtained the estimated parameters, the program is able
1.2 brouard 650: to calculate period prevalence, transitions probabilities
1.1 brouard 651: and life expectancies at any age. Choice of age range is useful
1.2 brouard 652: for extrapolation. In this example, age of people interviewed varies
653: from 69 to 102 and the model is estimated using their exact ages. But
654: if you are interested in the age-specific period prevalence you can
655: start the simulation at an exact age like 70 and stop at 100. Then the
656: program will draw at least two curves describing the forecasted
657: prevalences of two cohorts, one for healthy people at age 70 and the second
658: for disabled people at the same initial age. And according to the
659: mixing property (ergodicity) and because of recovery, both prevalences
660: will tend to be identical at later ages. Thus if you want to compute
661: the prevalence at age 70, you should enter a lower agemin value.
662:
663: <p>
664: Setting bage=50 (begin age) and fage=100 (final age), let
665: the program compute life expectancy from age 'bage' to age
1.1 brouard 666: 'fage'. As we use a model, we can interessingly compute life
667: expectancy on a wider age range than the age range from the data.
668: But the model can be rather wrong on much larger intervals.
669: Program is limited to around 120 for upper age!
670: </pre>
671:
672: <ul>
673: <li><b>agemin=</b> Minimum age for calculation of the
1.2 brouard 674: period prevalence </li>
1.1 brouard 675: <li><b>agemax=</b> Maximum age for calculation of the
1.2 brouard 676: period prevalence </li>
1.1 brouard 677: <li><b>bage=</b> Minimum age for calculation of the health
678: expectancies </li>
679: <li><b>fage=</b> Maximum age for calculation of the health
680: expectancies </li>
681: </ul>
682:
683: <h4><a name="Computing"><font color="#FF0000">Computing</font></a><font
1.2 brouard 684: color="#FF0000"> the cross-sectional prevalence</font></h4>
1.1 brouard 685:
686: <pre>begin-prev-date=1/1/1984 end-prev-date=1/6/1988 estepm=1</pre>
687:
1.2 brouard 688: <p>
1.1 brouard 689: Statements 'begin-prev-date' and 'end-prev-date' allow to
690: select the period in which we calculate the observed prevalences
691: in each state. In this example, the prevalences are calculated on
692: data survey collected between 1 january 1984 and 1 june 1988.
1.2 brouard 693: </p>
1.1 brouard 694:
695: <ul>
696: <li><strong>begin-prev-date= </strong>Starting date
697: (day/month/year)</li>
698: <li><strong>end-prev-date= </strong>Final date
699: (day/month/year)</li>
700: <li><strong>estepm= </strong>Unit (in months).We compute the
701: life expectancy from trapezoids spaced every estepm
702: months. This is mainly to measure the difference between
703: two models: for example if stepm=24 months pijx are given
704: only every 2 years and by summing them we are calculating
705: an estimate of the Life Expectancy assuming a linear
706: progression inbetween and thus overestimating or
707: underestimating according to the curvature of the
708: survival function. If, for the same date, we estimate the
709: model with stepm=1 month, we can keep estepm to 24 months
710: to compare the new estimate of Life expectancy with the
711: same linear hypothesis. A more precise result, taking
712: into account a more precise curvature will be obtained if
713: estepm is as small as stepm.</li>
714: </ul>
715:
716: <h4><font color="#FF0000">Population- or status-based health
717: expectancies</font></h4>
718:
719: <pre>pop_based=0</pre>
720:
1.2 brouard 721: <p>The program computes status-based health expectancies, i.e health
722: expectancies which depend on the initial health state. If you are
723: healthy, your healthy life expectancy (e11) is higher than if you were
724: disabled (e21, with e11 > e21).<br> To compute a healthy life
725: expectancy 'independent' of the initial status we have to weight e11
726: and e21 according to the probability to be in each state at initial
727: age which are corresponding to the proportions of people in each health
728: state (cross-sectional prevalences).<p>
729:
730: We could also compute e12 and e12 and get e.2 by weighting them
731: according to the observed cross-sectional prevalences at initial age.
732: <p> In a similar way we could compute the total life expectancy by
733: summing e.1 and e.2 .
734: <br>
735: The main difference between 'population based' and 'implied' or
736: 'period' consists in the weights used. 'Usually', cross-sectional
737: prevalences of disability are higher than period prevalences
738: particularly at old ages. This is true if the country is improving its
739: health system by teaching people how to prevent disability as by
740: promoting better screening, for example of people needing cataracts
741: surgeryand for many unknown reasons that this program may help to
742: discover. Then the proportion of disabled people at age 90 will be
743: lower than the current observed proportion.
744: <p>
745: Thus a better Health Expectancy and even a better Life Expectancy
746: value is given by forecasting not only the current lower mortality at
747: all ages but also a lower incidence of disability and higher recovery.
748: <br> Using the period prevalences as weight instead of the
749: cross-sectional prevalences we are computing indices which are more
750: specific to the current situations and therefore more useful to
751: predict improvements or regressions in the future as to compare
752: different policies in various countries.
1.1 brouard 753:
754: <ul>
1.2 brouard 755: <li><strong>popbased= 0 </strong>Health expectancies are computed
756: at each age from period prevalences 'expected' at this initial
757: age.</li>
1.1 brouard 758: <li><strong>popbased= 1 </strong>Health expectancies are
1.2 brouard 759: computed at each age from cross-sectional 'observed' prevalence at
760: this initial age. As all the population is not observed at the
761: same exact date we define a short period were the observed
762: prevalence can be computed.<br>
763:
764: We simply sum all people surveyed within these two exact dates
765: who belong to a particular age group (single year) at the date of
766: interview and being in a particular health state. Then it is easy to
767: get the proportion of people of a particular health status among all
768: people of the same age group.<br>
769:
770: If both dates are spaced and are covering two waves or more, people
771: being interviewed twice or more are counted twice or more. The program
772: takes into account the selection of individuals interviewed between
773: firstpass and lastpass too (we don't know if it can be useful).
774: </li>
1.1 brouard 775: </ul>
776:
1.2 brouard 777: <h4><font color="#FF0000">Prevalence forecasting (Experimental)</font></h4>
1.1 brouard 778:
779: <pre>starting-proj-date=1/1/1989 final-proj-date=1/1/1992 mov_average=0 </pre>
780:
781: <p>Prevalence and population projections are only available if
782: the interpolation unit is a month, i.e. stepm=1 and if there are
783: no covariate. The programme estimates the prevalence in each
784: state at a precise date expressed in day/month/year. The
785: programme computes one forecasted prevalence a year from a
786: starting date (1 january of 1989 in this example) to a final date
787: (1 january 1992). The statement mov_average allows to compute
788: smoothed forecasted prevalences with a five-age moving average
789: centered at the mid-age of the five-age period. <br>
790: </p>
791:
1.2 brouard 792: <h4><font color="#FF0000">Population forecasting (Experimental)</font></h4>
793:
1.1 brouard 794: <ul>
795: <li><strong>starting-proj-date</strong>= starting date
796: (day/month/year) of forecasting</li>
797: <li><strong>final-proj-date= </strong>final date
798: (day/month/year) of forecasting</li>
799: <li><strong>mov_average</strong>= smoothing with a five-age
800: moving average centered at the mid-age of the five-age
801: period. The command<strong> mov_average</strong> takes
802: value 1 if the prevalences are smoothed and 0 otherwise.</li>
803: </ul>
804:
805:
806: <ul type="disc">
1.2 brouard 807: <li><b>popforecast=
1.1 brouard 808: 0 </b>Option for population forecasting. If
809: popforecast=1, the programme does the forecasting<b>.</b></li>
1.2 brouard 810: <li><b>popfile=
1.1 brouard 811: </b>name of the population file</li>
1.2 brouard 812: <li><b>popfiledate=</b>
1.1 brouard 813: date of the population population</li>
1.2 brouard 814: <li><b>last-popfiledate</b>=
1.1 brouard 815: date of the last population projection </li>
816: </ul>
817:
818: <hr>
819:
820: <h2><a name="running"></a><font color="#00006A">Running Imach
821: with this example</font></h2>
822:
1.2 brouard 823: <p>We assume that you already typed your <a href="biaspar.imach">1st_example
1.1 brouard 824: parameter file</a> as explained <a href="#biaspar">above</a>.
825:
1.2 brouard 826: To run the program under Windows you should either:
827: </p>
1.1 brouard 828:
829: <ul>
1.2 brouard 830: <li>click on the imach.exe icon and either:
831: <ul>
832: <li>enter the name of the
833: parameter file which is for example <tt>
834: C:\home\myname\lsoa\biaspar.imach"</tt></li>
835: <li>or locate the biaspar.imach icon in your folder such as
836: <tt>C:\home\myname\lsoa</tt>
837: and drag it, with your mouse, on the already open imach window. </li>
838: </ul>
839:
840: <li>With version (0.97b) if you ran setup at installation, Windows is
841: supposed to understand the ".imach" extension and you can
842: right click the biaspar.imach icon and either edit with wordpad
843: (better than notepad) the parameter file or execute it with
844: IMaCh. </li>
1.1 brouard 845: </ul>
846:
1.2 brouard 847: <p>The time to converge depends on the step unit that you used (1
848: month is more precise but more cpu consuming), on the number of cases,
849: and on the number of variables (covariates).
850:
851: <p>
852: The program outputs many files. Most of them are files which will be
853: plotted for better understanding.
1.1 brouard 854:
1.2 brouard 855: </p>
856: To run under Linux it is mostly the same.
857: <p>
858: It is neither more difficult to run it under a MacIntosh.
1.1 brouard 859: <hr>
860:
861: <h2><a name="output"><font color="#00006A">Output of the program
862: and graphs</font> </a></h2>
863:
1.2 brouard 864: <p>Once the optimization is finished (once the convergence is
865: reached), many tables and graphics are produced.<p>
866: The IMaCh program will create a subdirectory of the same name as your
867: parameter file (here mypar) where all the tables and figures will be
868: stored.<br>
869:
870: Important files like the log file and the output parameter file (which
871: contains the estimates of the maximisation) are stored at the main
872: level not in this subdirectory. File with extension .log and .txt can
873: be edited with a standard editor like wordpad or notepad or even can be
874: viewed with a browser like Internet Explorer or Mozilla.
875:
876: <p> The main html file is also named with the same name <a
877: href="biaspar.htm">biaspar.htm</a>. You can click on it by holding
878: your shift key in order to open it in another window (Windows).
879: <p>
880: Our grapher is Gnuplot, it is an interactive plotting program (GPL) which
881: can also work in batch. A gnuplot reference manual is available <a
882: href="http://www.gnuplot.info/">here</a>. <br> When the run is
883: finished, and in order that the window doesn't disappear, the user
884: should enter a character like <tt>q</tt> for quitting. <br> These
885: characters are:<br>
1.1 brouard 886: </p>
887: <ul>
1.2 brouard 888: <li>'e' for opening the main result html file <a
889: href="biaspar.htm"><strong>biaspar.htm</strong></a> file to edit
890: the output files and graphs. </li>
891: <li>'g' to graph again</li>
1.1 brouard 892: <li>'c' to start again the program from the beginning.</li>
893: <li>'q' for exiting.</li>
894: </ul>
895:
1.2 brouard 896: The main gnuplot file is named <tt>biaspar.gp</tt> and can be edited (right
897: click) and run again.
898: <p>Gnuplot is easy and you can use it to make more complex
899: graphs. Just click on gnuplot and type plot sin(x) to see how easy it
900: is.
901:
902:
1.1 brouard 903: <h5><font size="4"><strong>Results files </strong></font><br>
904: <br>
905: <font color="#EC5E5E" size="3"><strong>- </strong></font><a
1.2 brouard 906: name="cross-sectional prevalence in each state"><font color="#EC5E5E"
907: size="3"><strong>cross-sectional prevalence in each state</strong></font></a><font
1.1 brouard 908: color="#EC5E5E" size="3"><strong> (and at first pass)</strong></font><b>:
1.2 brouard 909: </b><a href="biaspar/prbiaspar.txt"><b>biaspar/prbiaspar.txt</b></a><br>
1.1 brouard 910: </h5>
911:
912: <p>The first line is the title and displays each field of the
1.2 brouard 913: file. First column corresponds to age. Fields 2 and 6 are the
1.1 brouard 914: proportion of individuals in states 1 and 2 respectively as
1.2 brouard 915: observed at first exam. Others fields are the numbers of
1.1 brouard 916: people in states 1, 2 or more. The number of columns increases if
917: the number of states is higher than 2.<br>
918: The header of the file is </p>
919:
920: <pre># Age Prev(1) N(1) N Age Prev(2) N(2) N
921: 70 1.00000 631 631 70 0.00000 0 631
922: 71 0.99681 625 627 71 0.00319 2 627
923: 72 0.97125 1115 1148 72 0.02875 33 1148 </pre>
924:
1.2 brouard 925: <p>It means that at age 70 (between 70 and 71), the prevalence in state 1 is 1.000
1.1 brouard 926: and in state 2 is 0.00 . At age 71 the number of individuals in
927: state 1 is 625 and in state 2 is 2, hence the total number of
928: people aged 71 is 625+2=627. <br>
929: </p>
930:
931: <h5><font color="#EC5E5E" size="3"><b>- Estimated parameters and
932: covariance matrix</b></font><b>: </b><a href="rbiaspar.txt"><b>rbiaspar.imach</b></a></h5>
933:
934: <p>This file contains all the maximisation results: </p>
935:
936: <pre> -2 log likelihood= 21660.918613445392
937: Estimated parameters: a12 = -12.290174 b12 = 0.092161
938: a13 = -9.155590 b13 = 0.046627
939: a21 = -2.629849 b21 = -0.022030
940: a23 = -7.958519 b23 = 0.042614
941: Covariance matrix: Var(a12) = 1.47453e-001
942: Var(b12) = 2.18676e-005
943: Var(a13) = 2.09715e-001
944: Var(b13) = 3.28937e-005
945: Var(a21) = 9.19832e-001
946: Var(b21) = 1.29229e-004
947: Var(a23) = 4.48405e-001
948: Var(b23) = 5.85631e-005
949: </pre>
950:
951: <p>By substitution of these parameters in the regression model,
952: we obtain the elementary transition probabilities:</p>
953:
1.2 brouard 954: <p><img src="biaspar/pebiaspar11.png" width="400" height="300"></p>
1.1 brouard 955:
956: <h5><font color="#EC5E5E" size="3"><b>- Transition probabilities</b></font><b>:
1.2 brouard 957: </b><a href="biaspar/pijrbiaspar.txt"><b>biaspar/pijrbiaspar.txt</b></a></h5>
1.1 brouard 958:
1.2 brouard 959: <p>Here are the transitions probabilities Pij(x, x+nh). The second
960: column is the starting age x (from age 95 to 65), the third is age
961: (x+nh) and the others are the transition probabilities p11, p12, p13,
962: p21, p22, p23. The first column indicates the value of the covariate
963: (without any other variable than age it is equal to 1) For example, line 5 of the file
964: is: </p>
1.1 brouard 965:
1.2 brouard 966: <pre>1 100 106 0.02655 0.17622 0.79722 0.01809 0.13678 0.84513 </pre>
1.1 brouard 967:
968: <p>and this means: </p>
969:
970: <pre>p11(100,106)=0.02655
971: p12(100,106)=0.17622
972: p13(100,106)=0.79722
973: p21(100,106)=0.01809
974: p22(100,106)=0.13678
975: p22(100,106)=0.84513 </pre>
976:
977: <h5><font color="#EC5E5E" size="3"><b>- </b></font><a
1.2 brouard 978: name="Period prevalence in each state"><font color="#EC5E5E"
979: size="3"><b>Period prevalence in each state</b></font></a><b>:
980: </b><a href="biaspar/plrbiaspar.txt"><b>biaspar/plrbiaspar.txt</b></a></h5>
1.1 brouard 981:
982: <pre>#Prevalence
983: #Age 1-1 2-2
984:
985: #************
986: 70 0.90134 0.09866
987: 71 0.89177 0.10823
988: 72 0.88139 0.11861
989: 73 0.87015 0.12985 </pre>
990:
1.2 brouard 991: <p>At age 70 the period prevalence is 0.90134 in state 1 and 0.09866
992: in state 2. This period prevalence differs from the cross-sectional
993: prevalence. Here is the point. The cross-sectional prevalence at age
994: 70 results from the incidence of disability, incidence of recovery and
995: mortality which occurred in the past of the cohort. Period prevalence
996: results from a simulation with current incidences of disability,
997: recovery and mortality estimated from this cross-longitudinal
998: survey. It is a good predictin of the prevalence in the
999: future if "nothing changes in the future". This is exactly
1000: what demographers do with a period life table. Life expectancy is the
1001: expected mean survival time if current mortality rates (age-specific incidences
1002: of mortality) "remain constant" in the future. </p>
1.1 brouard 1003:
1004: <h5><font color="#EC5E5E" size="3"><b>- Standard deviation of
1.2 brouard 1005: period prevalence</b></font><b>: </b><a
1006: href="biaspar/vplrbiaspar.txt"><b>biaspar/vplrbiaspar.txt</b></a></h5>
1.1 brouard 1007:
1.2 brouard 1008: <p>The period prevalence has to be compared with the cross-sectional
1009: prevalence. But both are statistical estimates and therefore
1010: have confidence intervals.
1.3 brouard 1011: <br>For the cross-sectional prevalence we generally need information on
1.2 brouard 1012: the design of the surveys. It is usually not enough to consider the
1013: number of people surveyed at a particular age and to estimate a
1014: Bernouilli confidence interval based on the prevalence at that
1015: age. But you can do it to have an idea of the randomness. At least you
1016: can get a visual appreciation of the randomness by looking at the
1017: fluctuation over ages.
1018:
1019: <p> For the period prevalence it is possible to estimate the
1020: confidence interval from the Hessian matrix (see the publication for
1021: details). We are supposing that the design of the survey will only
1022: alter the weight of each individual. IMaCh is scaling the weights of
1023: individuals-waves contributing to the likelihood by making the sum of
1024: the weights equal to the sum of individuals-waves contributing: a
1025: weighted survey doesn't increase or decrease the size of the survey,
1026: it only give more weights to some individuals and thus less to the
1027: others.
1.1 brouard 1028:
1.2 brouard 1029: <h5><font color="#EC5E5E" size="3">-cross-sectional and period
1.1 brouard 1030: prevalence in state (2=disable) with confidence interval</font>:<b>
1.2 brouard 1031: </b><a href="biaspar/vbiaspar21.htm"><b>biaspar/vbiaspar21.png</b></a></h5>
1.1 brouard 1032:
1.2 brouard 1033: <p>This graph exhibits the period prevalence in state (2) with the
1034: confidence interval in red. The green curve is the observed prevalence
1035: (or proportion of individuals in state (2)). Without discussing the
1036: results (it is not the purpose here), we observe that the green curve
1037: is rather below the period prevalence. It the data where not biased by
1038: the non inclusion of people living in institutions we would have
1039: concluded that the prevalence of disability will increase in the
1040: future (see the main publication if you are interested in real data
1041: and results which are opposite).</p>
1.1 brouard 1042:
1.2 brouard 1043: <p><img src="biaspar/vbiaspar21.png" width="400" height="300"></p>
1.1 brouard 1044:
1045: <h5><font color="#EC5E5E" size="3"><b>-Convergence to the
1.2 brouard 1046: period prevalence of disability</b></font><b>: </b><a
1047: href="biaspar/pbiaspar11.png"><b>biaspar/pbiaspar11.png</b></a><br>
1048: <img src="biaspar/pbiaspar11.png" width="400" height="300"> </h5>
1.1 brouard 1049:
1050: <p>This graph plots the conditional transition probabilities from
1051: an initial state (1=healthy in red at the bottom, or 2=disable in
1052: green on top) at age <em>x </em>to the final state 2=disable<em> </em>at
1053: age <em>x+h. </em>Conditional means at the condition to be alive
1054: at age <em>x+h </em>which is <i>hP12x</i> + <em>hP22x</em>. The
1055: curves <i>hP12x/(hP12x</i> + <em>hP22x) </em>and <i>hP22x/(hP12x</i>
1.2 brouard 1056: + <em>hP22x) </em>converge with <em>h, </em>to the <em>period
1057: prevalence of disability</em>. In order to get the period
1.1 brouard 1058: prevalence at age 70 we should start the process at an earlier
1059: age, i.e.50. If the disability state is defined by severe
1060: disability criteria with only a few chance to recover, then the
1061: incidence of recovery is low and the time to convergence is
1062: probably longer. But we don't have experience yet.</p>
1063:
1064: <h5><font color="#EC5E5E" size="3"><b>- Life expectancies by age
1065: and initial health status with standard deviation</b></font><b>: </b><a
1.2 brouard 1066: href="biaspar/erbiaspar.txt"><b>biaspar/erbiaspar.txt</b></a></h5>
1.1 brouard 1067:
1068: <pre># Health expectancies
1069: # Age 1-1 (SE) 1-2 (SE) 2-1 (SE) 2-2 (SE)
1.2 brouard 1070: 70 11.0180 (0.1277) 3.1950 (0.3635) 4.6500 (0.0871) 4.4807 (0.2187)
1071: 71 10.4786 (0.1184) 3.2093 (0.3212) 4.3384 (0.0875) 4.4820 (0.2076)
1072: 72 9.9551 (0.1103) 3.2236 (0.2827) 4.0426 (0.0885) 4.4827 (0.1966)
1073: 73 9.4476 (0.1035) 3.2379 (0.2478) 3.7621 (0.0899) 4.4825 (0.1858)
1074: 74 8.9564 (0.0980) 3.2522 (0.2165) 3.4966 (0.0920) 4.4815 (0.1754)
1075: 75 8.4815 (0.0937) 3.2665 (0.1887) 3.2457 (0.0946) 4.4798 (0.1656)
1076: 76 8.0230 (0.0905) 3.2806 (0.1645) 3.0090 (0.0979) 4.4772 (0.1565)
1077: 77 7.5810 (0.0884) 3.2946 (0.1438) 2.7860 (0.1017) 4.4738 (0.1484)
1078: 78 7.1554 (0.0871) 3.3084 (0.1264) 2.5763 (0.1062) 4.4696 (0.1416)
1079: 79 6.7464 (0.0867) 3.3220 (0.1124) 2.3794 (0.1112) 4.4646 (0.1364)
1080: 80 6.3538 (0.0868) 3.3354 (0.1014) 2.1949 (0.1168) 4.4587 (0.1331)
1081: 81 5.9775 (0.0873) 3.3484 (0.0933) 2.0222 (0.1230) 4.4520 (0.1320)
1.1 brouard 1082: </pre>
1083:
1.2 brouard 1084: <pre>For example 70 11.0180 (0.1277) 3.1950 (0.3635) 4.6500 (0.0871) 4.4807 (0.2187)
1085: means
1086: e11=11.0180 e12=3.1950 e21=4.6500 e22=4.4807 </pre>
1.1 brouard 1087:
1.2 brouard 1088: <pre><img src="biaspar/expbiaspar21.png" width="400" height="300"><img
1089: src="biaspar/expbiaspar11.png" width="400" height="300"></pre>
1.1 brouard 1090:
1091: <p>For example, life expectancy of a healthy individual at age 70
1.2 brouard 1092: is 11.0 in the healthy state and 3.2 in the disability state
1093: (total of 14.2 years). If he was disable at age 70, his life expectancy
1094: will be shorter, 4.65 years in the healthy state and 4.5 in the
1095: disability state (=9.15 years). The total life expectancy is a
1096: weighted mean of both, 14.2 and 9.15. The weight is the proportion
1097: of people disabled at age 70. In order to get a period index
1.1 brouard 1098: (i.e. based only on incidences) we use the <a
1.2 brouard 1099: href="#Period prevalence in each state">stable or
1100: period prevalence</a> at age 70 (i.e. computed from
1.1 brouard 1101: incidences at earlier ages) instead of the <a
1.2 brouard 1102: href="#cross-sectional prevalence in each state">cross-sectional prevalence</a>
1103: (observed for example at first medical exam) (<a href="#Health expectancies">see
1.1 brouard 1104: below</a>).</p>
1105:
1106: <h5><font color="#EC5E5E" size="3"><b>- Variances of life
1107: expectancies by age and initial health status</b></font><b>: </b><a
1.2 brouard 1108: href="biaspar/vrbiaspar.txt"><b>biaspar/vrbiaspar.txt</b></a></h5>
1.1 brouard 1109:
1110: <p>For example, the covariances of life expectancies Cov(ei,ej)
1111: at age 50 are (line 3) </p>
1112:
1113: <pre> Cov(e1,e1)=0.4776 Cov(e1,e2)=0.0488=Cov(e2,e1) Cov(e2,e2)=0.0424</pre>
1114:
1115: <h5><font color="#EC5E5E" size="3"><b>-Variances of one-step
1.2 brouard 1116: probabilities </b></font><b>: </b><a href="biaspar/probrbiaspar.txt"><b>biaspar/probrbiaspar.txt</b></a></h5>
1.1 brouard 1117:
1118: <p>For example, at age 65</p>
1119:
1120: <pre> p11=9.960e-001 standard deviation of p11=2.359e-004</pre>
1121:
1122: <h5><font color="#EC5E5E" size="3"><b>- </b></font><a
1123: name="Health expectancies"><font color="#EC5E5E" size="3"><b>Health
1124: expectancies</b></font></a><font color="#EC5E5E" size="3"><b>
1125: with standard errors in parentheses</b></font><b>: </b><a
1.2 brouard 1126: href="biaspar/trbiaspar.txt"><font face="Courier New"><b>biaspar/trbiaspar.txt</b></font></a></h5>
1.1 brouard 1127:
1128: <pre>#Total LEs with variances: e.. (std) e.1 (std) e.2 (std) </pre>
1129:
1130: <pre>70 13.26 (0.22) 9.95 (0.20) 3.30 (0.14) </pre>
1131:
1132: <p>Thus, at age 70 the total life expectancy, e..=13.26 years is
1.2 brouard 1133: the weighted mean of e1.=13.46 and e2.=11.35 by the period
1134: prevalences at age 70 which are 0.90134 in state 1 and 0.09866 in
1135: state 2 respectively (the sum is equal to one). e.1=9.95 is the
1.1 brouard 1136: Disability-free life expectancy at age 70 (it is again a weighted
1137: mean of e11 and e21). e.2=3.30 is also the life expectancy at age
1138: 70 to be spent in the disability state.</p>
1139:
1140: <h5><font color="#EC5E5E" size="3"><b>-Total life expectancy by
1141: age and health expectancies in states (1=healthy) and (2=disable)</b></font><b>:
1.2 brouard 1142: </b><a href="biaspar/ebiaspar1.png"><b>biaspar/ebiaspar1.png</b></a></h5>
1.1 brouard 1143:
1144: <p>This figure represents the health expectancies and the total
1.2 brouard 1145: life expectancy with a confidence interval (dashed line). </p>
1.1 brouard 1146:
1.2 brouard 1147: <pre> <img src="biaspar/ebiaspar1.png" width="400" height="300"></pre>
1.1 brouard 1148:
1149: <p>Standard deviations (obtained from the information matrix of
1150: the model) of these quantities are very useful.
1151: Cross-longitudinal surveys are costly and do not involve huge
1152: samples, generally a few thousands; therefore it is very
1153: important to have an idea of the standard deviation of our
1154: estimates. It has been a big challenge to compute the Health
1155: Expectancy standard deviations. Don't be confuse: life expectancy
1156: is, as any expected value, the mean of a distribution; but here
1157: we are not computing the standard deviation of the distribution,
1158: but the standard deviation of the estimate of the mean.</p>
1159:
1160: <p>Our health expectancies estimates vary according to the sample
1161: size (and the standard deviations give confidence intervals of
1.2 brouard 1162: the estimates) but also according to the model fitted. Let us
1.1 brouard 1163: explain it in more details.</p>
1164:
1.2 brouard 1165: <p>Choosing a model means at least two kind of choices. At first we
1166: have to decide the number of disability states. And at second we have to
1167: design, within the logit model family, the model itself: variables,
1168: covariables, confounding factors etc. to be included.</p>
1.1 brouard 1169:
1170: <p>More disability states we have, better is our demographical
1171: approach of the disability process, but smaller are the number of
1172: transitions between each state and higher is the noise in the
1173: measurement. We do not have enough experiments of the various
1174: models to summarize the advantages and disadvantages, but it is
1175: important to say that even if we had huge and unbiased samples,
1176: the total life expectancy computed from a cross-longitudinal
1177: survey, varies with the number of states. If we define only two
1178: states, alive or dead, we find the usual life expectancy where it
1179: is assumed that at each age, people are at the same risk to die.
1180: If we are differentiating the alive state into healthy and
1181: disable, and as the mortality from the disability state is higher
1182: than the mortality from the healthy state, we are introducing
1183: heterogeneity in the risk of dying. The total mortality at each
1184: age is the weighted mean of the mortality in each state by the
1185: prevalence in each state. Therefore if the proportion of people
1.2 brouard 1186: at each age and in each state is different from the period
1.1 brouard 1187: equilibrium, there is no reason to find the same total mortality
1188: at a particular age. Life expectancy, even if it is a very useful
1189: tool, has a very strong hypothesis of homogeneity of the
1190: population. Our main purpose is not to measure differential
1191: mortality but to measure the expected time in a healthy or
1192: disability state in order to maximise the former and minimize the
1193: latter. But the differential in mortality complexifies the
1194: measurement.</p>
1195:
1.2 brouard 1196: <p>Incidences of disability or recovery are not affected by the number
1197: of states if these states are independent. But incidences estimates
1198: are dependent on the specification of the model. More covariates we
1199: added in the logit model better is the model, but some covariates are
1200: not well measured, some are confounding factors like in any
1201: statistical model. The procedure to "fit the best model' is
1202: similar to logistic regression which itself is similar to regression
1203: analysis. We haven't yet been sofar because we also have a severe
1204: limitation which is the speed of the convergence. On a Pentium III,
1205: 500 MHz, even the simplest model, estimated by month on 8,000 people
1206: may take 4 hours to converge. Also, the IMaCh program is not a
1207: statistical package, and does not allow sophisticated design
1208: variables. If you need sophisticated design variable you have to them
1209: your self and and add them as ordinary variables. IMaCX allows up to 8
1210: variables. The current version of this program allows only to add
1211: simple variables like age+sex or age+sex+ age*sex but will never be
1212: general enough. But what is to remember, is that incidences or
1213: probability of change from one state to another is affected by the
1214: variables specified into the model.</p>
1.1 brouard 1215:
1.2 brouard 1216: <p>Also, the age range of the people interviewed is linked
1.1 brouard 1217: the age range of the life expectancy which can be estimated by
1218: extrapolation. If your sample ranges from age 70 to 95, you can
1219: clearly estimate a life expectancy at age 70 and trust your
1.2 brouard 1220: confidence interval because it is mostly based on your sample size,
1.1 brouard 1221: but if you want to estimate the life expectancy at age 50, you
1.2 brouard 1222: should rely in the design of your model. Fitting a logistic model on a age
1223: range of 70 to 95 and estimating probabilties of transition out of
1224: this age range, say at age 50, is very dangerous. At least you
1.1 brouard 1225: should remember that the confidence interval given by the
1226: standard deviation of the health expectancies, are under the
1227: strong assumption that your model is the 'true model', which is
1.2 brouard 1228: probably not the case outside the age range of your sample.</p>
1.1 brouard 1229:
1230: <h5><font color="#EC5E5E" size="3"><b>- Copy of the parameter
1231: file</b></font><b>: </b><a href="orbiaspar.txt"><b>orbiaspar.txt</b></a></h5>
1232:
1233: <p>This copy of the parameter file can be useful to re-run the
1234: program while saving the old output files. </p>
1235:
1236: <h5><font color="#EC5E5E" size="3"><b>- Prevalence forecasting</b></font><b>:
1.2 brouard 1237: </b><a href="biaspar/frbiaspar.txt"><b>biaspar/frbiaspar.txt</b></a></h5>
1.1 brouard 1238:
1.2 brouard 1239: <p>
1240:
1241: First,
1.1 brouard 1242: we have estimated the observed prevalence between 1/1/1984 and
1.2 brouard 1243: 1/6/1988 (June, European syntax of dates). The mean date of all interviews (weighted average of the
1244: interviews performed between 1/1/1984 and 1/6/1988) is estimated
1.1 brouard 1245: to be 13/9/1985, as written on the top on the file. Then we
1246: forecast the probability to be in each state. </p>
1247:
1.2 brouard 1248: <p>
1249: For example on 1/1/1989 : </p>
1.1 brouard 1250:
1251: <pre class="MsoNormal"># StartingAge FinalAge P.1 P.2 P.3
1252: # Forecasting at date 1/1/1989
1253: 73 0.807 0.078 0.115</pre>
1254:
1.2 brouard 1255: <p>
1256:
1257: Since the minimum age is 70 on the 13/9/1985, the youngest forecasted
1258: age is 73. This means that at age a person aged 70 at 13/9/1989 has a
1259: probability to enter state1 of 0.807 at age 73 on 1/1/1989.
1.1 brouard 1260: Similarly, the probability to be in state 2 is 0.078 and the
1.2 brouard 1261: probability to die is 0.115. Then, on the 1/1/1989, the prevalence of
1262: disability at age 73 is estimated to be 0.088.</p>
1.1 brouard 1263:
1264: <h5><font color="#EC5E5E" size="3"><b>- Population forecasting</b></font><b>:
1.2 brouard 1265: </b><a href="biaspar/poprbiaspar.txt"><b>biaspar/poprbiaspar.txt</b></a></h5>
1.1 brouard 1266:
1267: <pre># Age P.1 P.2 P.3 [Population]
1268: # Forecasting at date 1/1/1989
1269: 75 572685.22 83798.08
1270: 74 621296.51 79767.99
1271: 73 645857.70 69320.60 </pre>
1272:
1273: <pre># Forecasting at date 1/1/19909
1274: 76 442986.68 92721.14 120775.48
1275: 75 487781.02 91367.97 121915.51
1276: 74 512892.07 85003.47 117282.76 </pre>
1277:
1278: <p>From the population file, we estimate the number of people in
1279: each state. At age 73, 645857 persons are in state 1 and 69320
1280: are in state 2. One year latter, 512892 are still in state 1,
1281: 85003 are in state 2 and 117282 died before 1/1/1990.</p>
1282:
1283: <hr>
1284:
1285: <h2><a name="example"></a><font color="#00006A">Trying an example</font></h2>
1286:
1287: <p>Since you know how to run the program, it is time to test it
1288: on your own computer. Try for example on a parameter file named <a
1.2 brouard 1289: href="imachpar.imach">imachpar.imach</a> which is a copy
1.1 brouard 1290: of <font size="2" face="Courier New">mypar.imach</font> included
1291: in the subdirectory of imach, <font size="2" face="Courier New">mytry</font>.
1.2 brouard 1292: Edit it and change the name of the data file to <font size="2"
1293: face="Courier New">mydata.txt</font> if you don't want to
1.1 brouard 1294: copy it on the same directory. The file <font face="Courier New">mydata.txt</font>
1295: is a smaller file of 3,000 people but still with 4 waves. </p>
1296:
1.2 brouard 1297: <p>Right click on the .imach file and a window will popup with the
1298: string '<strong>Enter the parameter file name:'</strong></p>
1.1 brouard 1299:
1300: <table border="1">
1301: <tr>
1.2 brouard 1302: <td width="100%"><strong>IMACH, Version 0.97b</strong><p><strong>Enter
1303: the parameter file name: imachpar.imach</strong></p>
1.1 brouard 1304: </td>
1305: </tr>
1306: </table>
1307:
1308: <p>Most of the data files or image files generated, will use the
1309: 'imachpar' string into their name. The running time is about 2-3
1310: minutes on a Pentium III. If the execution worked correctly, the
1311: outputs files are created in the current directory, and should be
1312: the same as the mypar files initially included in the directory <font
1313: size="2" face="Courier New">mytry</font>.</p>
1314:
1315: <ul>
1316: <li><pre><u>Output on the screen</u> The output screen looks like <a
1.2 brouard 1317: href="biaspar.log">biaspar.log</a>
1.1 brouard 1318: #
1.2 brouard 1319: title=MLE datafile=mydaiata.txt lastobs=3000 firstpass=1 lastpass=3
1.1 brouard 1320: ftol=1.000000e-008 stepm=24 ncovcol=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre>
1321: </li>
1322: <li><pre>Total number of individuals= 2965, Agemin = 70.00, Agemax= 100.92
1323:
1324: Warning, no any valid information for:126 line=126
1325: Warning, no any valid information for:2307 line=2307
1326: Delay (in months) between two waves Min=21 Max=51 Mean=24.495826
1327: <font face="Times New Roman">These lines give some warnings on the data file and also some raw statistics on frequencies of transitions.</font>
1328: Age 70 1.=230 loss[1]=3.5% 2.=16 loss[2]=12.5% 1.=222 prev[1]=94.1% 2.=14
1329: prev[2]=5.9% 1-1=8 11=200 12=7 13=15 2-1=2 21=6 22=7 23=1
1330: Age 102 1.=0 loss[1]=NaNQ% 2.=0 loss[2]=NaNQ% 1.=0 prev[1]=NaNQ% 2.=0 </pre>
1331: </li>
1332: </ul>
1.2 brouard 1333: It includes some warnings or errors which are very important for
1334: you. Be careful with such warnings because your results may be biased
1335: if, for example, you have people who accepted to be interviewed at
1336: first pass but never after. Or if you don't have the exact month of
1337: death. In such cases IMaCh doesn't take any initiative, it does only
1338: warn you. It is up to you to decide what to do with these
1339: people. Excluding them is usually a wrong decision. It is better to
1340: decide that the month of death is at the mid-interval between the last
1341: two waves for example.<p>
1342:
1343: If you survey suffers from severe attrition, you have to analyse the
1344: characteristics of the lost people and overweight people with same
1345: characteristics for example.
1346: <p>
1347: By default, IMaCH warns and excludes these problematic people, but you
1348: have to be careful with such results.
1.1 brouard 1349:
1350: <p> </p>
1351:
1352: <ul>
1353: <li>Maximisation with the Powell algorithm. 8 directions are
1354: given corresponding to the 8 parameters. this can be
1355: rather long to get convergence.<br>
1356: <font size="1" face="Courier New"><br>
1357: Powell iter=1 -2*LL=11531.405658264877 1 0.000000000000 2
1358: 0.000000000000 3<br>
1359: 0.000000000000 4 0.000000000000 5 0.000000000000 6
1360: 0.000000000000 7 <br>
1361: 0.000000000000 8 0.000000000000<br>
1362: 1..........2.................3..........4.................5.........<br>
1363: 6................7........8...............<br>
1364: Powell iter=23 -2*LL=6744.954108371555 1 -12.967632334283
1365: <br>
1366: 2 0.135136681033 3 -7.402109728262 4 0.067844593326 <br>
1367: 5 -0.673601538129 6 -0.006615504377 7 -5.051341616718 <br>
1368: 8 0.051272038506<br>
1369: 1..............2...........3..............4...........<br>
1370: 5..........6................7...........8.........<br>
1371: #Number of iterations = 23, -2 Log likelihood =
1372: 6744.954042573691<br>
1373: # Parameters<br>
1374: 12 -12.966061 0.135117 <br>
1375: 13 -7.401109 0.067831 <br>
1376: 21 -0.672648 -0.006627 <br>
1377: 23 -5.051297 0.051271 </font><br>
1378: </li>
1379: <li><pre><font size="2">Calculation of the hessian matrix. Wait...
1380: 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
1381:
1382: Inverting the hessian to get the covariance matrix. Wait...
1383:
1384: #Hessian matrix#
1385: 3.344e+002 2.708e+004 -4.586e+001 -3.806e+003 -1.577e+000 -1.313e+002 3.914e-001 3.166e+001
1386: 2.708e+004 2.204e+006 -3.805e+003 -3.174e+005 -1.303e+002 -1.091e+004 2.967e+001 2.399e+003
1387: -4.586e+001 -3.805e+003 4.044e+002 3.197e+004 2.431e-002 1.995e+000 1.783e-001 1.486e+001
1388: -3.806e+003 -3.174e+005 3.197e+004 2.541e+006 2.436e+000 2.051e+002 1.483e+001 1.244e+003
1389: -1.577e+000 -1.303e+002 2.431e-002 2.436e+000 1.093e+002 8.979e+003 -3.402e+001 -2.843e+003
1390: -1.313e+002 -1.091e+004 1.995e+000 2.051e+002 8.979e+003 7.420e+005 -2.842e+003 -2.388e+005
1391: 3.914e-001 2.967e+001 1.783e-001 1.483e+001 -3.402e+001 -2.842e+003 1.494e+002 1.251e+004
1392: 3.166e+001 2.399e+003 1.486e+001 1.244e+003 -2.843e+003 -2.388e+005 1.251e+004 1.053e+006
1393: # Scales
1394: 12 1.00000e-004 1.00000e-006
1395: 13 1.00000e-004 1.00000e-006
1396: 21 1.00000e-003 1.00000e-005
1397: 23 1.00000e-004 1.00000e-005
1398: # Covariance
1399: 1 5.90661e-001
1400: 2 -7.26732e-003 8.98810e-005
1401: 3 8.80177e-002 -1.12706e-003 5.15824e-001
1402: 4 -1.13082e-003 1.45267e-005 -6.50070e-003 8.23270e-005
1403: 5 9.31265e-003 -1.16106e-004 6.00210e-004 -8.04151e-006 1.75753e+000
1404: 6 -1.15664e-004 1.44850e-006 -7.79995e-006 1.04770e-007 -2.12929e-002 2.59422e-004
1405: 7 1.35103e-003 -1.75392e-005 -6.38237e-004 7.85424e-006 4.02601e-001 -4.86776e-003 1.32682e+000
1406: 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
1407: # agemin agemax for lifexpectancy, bage fage (if mle==0 ie no data nor Max likelihood).
1408:
1409:
1410: agemin=70 agemax=100 bage=50 fage=100
1411: Computing prevalence limit: result on file 'plrmypar.txt'
1412: Computing pij: result on file 'pijrmypar.txt'
1413: Computing Health Expectancies: result on file 'ermypar.txt'
1414: Computing Variance-covariance of DFLEs: file 'vrmypar.txt'
1415: Computing Total LEs with variances: file 'trmypar.txt'
1416: Computing Variance-covariance of Prevalence limit: file 'vplrmypar.txt'
1417: End of Imach
1418: </font></pre>
1419: </li>
1420: </ul>
1421:
1422: <p><font size="3">Once the running is finished, the program
1.2 brouard 1423: requires a character:</font></p>
1.1 brouard 1424:
1425: <table border="1">
1426: <tr>
1427: <td width="100%"><strong>Type e to edit output files, g
1428: to graph again, c to start again, and q for exiting:</strong></td>
1429: </tr>
1430: </table>
1431:
1.2 brouard 1432: In order to have an idea of the time needed to reach convergence,
1433: IMaCh gives an estimation if the convergence needs 10, 20 or 30
1434: iterations. It might be useful.
1435:
1.1 brouard 1436: <p><font size="3">First you should enter <strong>e </strong>to
1437: edit the master file mypar.htm. </font></p>
1438:
1439: <ul>
1440: <li><u>Outputs files</u> <br>
1441: <br>
1442: - Copy of the parameter file: <a href="ormypar.txt">ormypar.txt</a><br>
1443: - Gnuplot file name: <a href="mypar.gp.txt">mypar.gp.txt</a><br>
1.2 brouard 1444: - Cross-sectional prevalence in each state: <a
1.1 brouard 1445: href="prmypar.txt">prmypar.txt</a> <br>
1.2 brouard 1446: - Period prevalence in each state: <a
1.1 brouard 1447: href="plrmypar.txt">plrmypar.txt</a> <br>
1448: - Transition probabilities: <a href="pijrmypar.txt">pijrmypar.txt</a><br>
1449: - Life expectancies by age and initial health status
1450: (estepm=24 months): <a href="ermypar.txt">ermypar.txt</a>
1451: <br>
1452: - Parameter file with estimated parameters and the
1453: covariance matrix: <a href="rmypar.txt">rmypar.txt</a> <br>
1454: - Variance of one-step probabilities: <a
1455: href="probrmypar.txt">probrmypar.txt</a> <br>
1456: - Variances of life expectancies by age and initial
1457: health status (estepm=24 months): <a href="vrmypar.txt">vrmypar.txt</a><br>
1458: - Health expectancies with their variances: <a
1459: href="trmypar.txt">trmypar.txt</a> <br>
1.2 brouard 1460: - Standard deviation of period prevalences: <a
1.1 brouard 1461: href="vplrmypar.txt">vplrmypar.txt</a> <br>
1462: No population forecast: popforecast = 0 (instead of 1) or
1463: stepm = 24 (instead of 1) or model=. (instead of .)<br>
1464: <br>
1465: </li>
1466: <li><u>Graphs</u> <br>
1467: <br>
1468: -<a href="../mytry/pemypar1.gif">One-step transition
1469: probabilities</a><br>
1470: -<a href="../mytry/pmypar11.gif">Convergence to the
1.2 brouard 1471: period prevalence</a><br>
1472: -<a href="..\mytry\vmypar11.gif">Cross-sectional and period
1.1 brouard 1473: prevalence in state (1) with the confident interval</a> <br>
1.2 brouard 1474: -<a href="..\mytry\vmypar21.gif">Cross-sectional and period
1.1 brouard 1475: prevalence in state (2) with the confident interval</a> <br>
1476: -<a href="..\mytry\expmypar11.gif">Health life
1477: expectancies by age and initial health state (1)</a> <br>
1478: -<a href="..\mytry\expmypar21.gif">Health life
1479: expectancies by age and initial health state (2)</a> <br>
1480: -<a href="..\mytry\emypar1.gif">Total life expectancy by
1481: age and health expectancies in states (1) and (2).</a> </li>
1482: </ul>
1483:
1484: <p>This software have been partly granted by <a
1.4 ! brouard 1485: href="http://euroreves.ined.fr">Euro-REVES</a>, a concerted action
! 1486: from the European Union. Since 2003 it is also partly granted by the
! 1487: French Institute on Longevity. It will be copyrighted identically to a
! 1488: GNU software product, i.e. program and software can be distributed
! 1489: freely for non commercial use. Sources are not widely distributed
! 1490: today because some part of the codes are copyrighted by Numerical
! 1491: Recipes in C. You can get our GPL codes by asking us with a simple
! 1492: justification (name, email, institute) <a
1.1 brouard 1493: href="mailto:brouard@ined.fr">mailto:brouard@ined.fr</a> and <a
1494: href="mailto:lievre@ined.fr">mailto:lievre@ined.fr</a> .</p>
1495:
1.2 brouard 1496: <p>Latest version (0.97b of June 2004) can be accessed at <a
1.1 brouard 1497: href="http://euroreves.ined.fr/imach">http://euroreves.ined.fr/imach</a><br>
1498: </p>
1499: </body>
1500: </html>
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