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1:  10: </head> 11: 12: <body bgcolor="#FFFFFF"> 13: 14: <hr size="3" color="#EC5E5E"> 15: 16: <h1 align="center">Computing Health 17: Expectancies using IMaCh</h1> 18: 19: <h1 align="center">(a Maximum 20: Likelihood Computer Program using Interpolation of Markov Chains)</h1> 21: 22:   23: 24: <a href="http://www.ined.fr/"><img 25: src="logo-ined.gif" border="0" width="151" height="76"></a><img 26: src="euroreves2.gif" width="151" height="75"> 27: 28: <h3 align="center"><a href="http://www.ined.fr/">INED</a> and <a 30: href="http://euroreves.ined.fr">EUROREVES</a></h3> 31: 32: Version 33: 0.71a, March 2002 34: 35: <hr size="3" color="#EC5E5E"> 36: 37: Authors of the 38: program: <a href="http://sauvy.ined.fr/brouard">Nicolas Brouard</a>, senior researcher at the <a 41: href="http://www.ined.fr">Institut 42: National d'Etudes Démographiques</a> (INED, Paris) in the "Mortality, 44: Health and Epidemiology" Research Unit 45: 46: and Agnès 47: Lièvre 48: 49: 50: <h4>Contribution to the mathematics: C. R. 51: Heathcote (Australian 52: National University, Canberra).</h4> 53: 54: <h4>Contact: Agnès Lièvre (<a 55: href="mailto:lievre@ined.fr">lievre@ined.fr</a>) </h4> 57: 58: <hr> 59: 60: <ul> 61: <li><a href="#intro">Introduction</a> </li> 62: <li><a href="#data">On what kind of data can it be used?</a></li> 63: <li><a href="#datafile">The data file</a> </li> 64: <li><a href="#biaspar">The parameter file</a> </li> 65: <li><a href="#running">Running Imach</a> </li> 66: <li><a href="#output">Output files and graphs</a> </li> 67: <li><a href="#example">Exemple</a> </li> 68: </ul> 69: 70: <hr> 71: 72: <h2><a name="intro">Introduction</a></h2> 73: 74: This program computes Healthy Life Expectancies from cross-longitudinal 75: data using the methodology pioneered by Laditka and Wolf (1). 76: Within the family of Health Expectancies (HE), Disability-free 77: life expectancy (DFLE) is probably the most important index to 78: monitor. In low mortality countries, there is a fear that when 79: mortality declines, the increase in DFLE is not proportionate to 80: the increase in total Life expectancy. This case is called the Expansion 81: of morbidity. Most of the data collected today, in 82: particular by the international <a href="http://www.reves.org">REVES</a> 83: network on Health expectancy, and most HE indices based on these 84: data, are cross-sectional. It means that the information 85: collected comes from a single cross-sectional survey: people from 86: various ages (but mostly old people) are surveyed on their health 87: status at a single date. Proportion of people disabled at each 88: age, can then be measured at that date. This age-specific 89: prevalence curve is then used to distinguish, within the 90: stationary population (which, by definition, is the life table 91: estimated from the vital statistics on mortality at the same 92: date), the disable population from the disability-free 93: population. Life expectancy (LE) (or total population divided by 94: the yearly number of births or deaths of this stationary 95: population) is then decomposed into DFLE and DLE. This method of 96: computing HE is usually called the Sullivan method (from the name 97: of the author who first described it). 98: 99: Age-specific proportions of people disable are very difficult 100: to forecast because each proportion corresponds to historical 101: conditions of the cohort and it is the result of the historical 102: flows from entering disability and recovering in the past until 103: today. The age-specific intensities (or incidence rates) of 104: entering disability or recovering a good health, are reflecting 105: actual conditions and therefore can be used at each age to 106: forecast the future of this cohort. For example if a country is 107: improving its technology of prosthesis, the incidence of 108: recovering the ability to walk will be higher at each (old) age, 109: but the prevalence of disability will only slightly reflect an 110: improve because the prevalence is mostly affected by the history 111: of the cohort and not by recent period effects. To measure the 112: period improvement we have to simulate the future of a cohort of 113: new-borns entering or leaving at each age the disability state or 114: dying according to the incidence rates measured today on 115: different cohorts. The proportion of people disabled at each age 116: in this simulated cohort will be much lower (using the exemple of 117: an improvement) that the proportions observed at each age in a 118: cross-sectional survey. This new prevalence curve introduced in a 119: life table will give a much more actual and realistic HE level 120: than the Sullivan method which mostly measured the History of 121: health conditions in this country. 122: 123: Therefore, the main question is how to measure incidence rates 124: from cross-longitudinal surveys? This is the goal of the IMaCH 125: program. From your data and using IMaCH you can estimate period 126: HE and not only Sullivan's HE. Also the standard errors of the HE 127: are computed. 128: 129: A cross-longitudinal survey consists in a first survey 130: ("cross") where individuals from different ages are 131: interviewed on their health status or degree of disability. At 132: least a second wave of interviews ("longitudinal") 133: should measure each new individual health status. Health 134: expectancies are computed from the transitions observed between 135: waves and are computed for each degree of severity of disability 136: (number of life states). More degrees you consider, more time is 137: necessary to reach the Maximum Likelihood of the parameters 138: involved in the model. Considering only two states of disability 139: (disable and healthy) is generally enough but the computer 140: program works also with more health statuses. 141: 142: The simplest model is the multinomial logistic model where pij 143: is the probability to be observed in state j at the second 144: wave conditional to be observed in state i at the first 145: wave. Therefore a simple model is: log(pij/pii)= aij + 146: bij*age+ cij*sex, where 'age' is age and 'sex' 147: is a covariate. The advantage that this computer program claims, 148: comes from that if the delay between waves is not identical for 149: each individual, or if some individual missed an interview, the 150: information is not rounded or lost, but taken into account using 151: an interpolation or extrapolation. hPijx is the 152: probability to be observed in state i at age x+h 153: conditional to the observed state i at age x. The 154: delay 'h' can be split into an exact number (nh*stepm) 155: of unobserved intermediate states. This elementary transition (by 156: month or quarter trimester, semester or year) is modeled as a 157: multinomial logistic. The hPx matrix is simply the matrix 158: product of nh*stepm elementary matrices and the 159: contribution of each individual to the likelihood is simply hPijx. 160: 161: 162: 163: The program presented in this manual is a quite general 164: program named IMaCh (for Interpolated 165: MArkov CHain), designed to 166: analyse transition data from longitudinal surveys. The first step 167: is the parameters estimation of a transition probabilities model 168: between an initial status and a final status. From there, the 169: computer program produces some indicators such as observed and 170: stationary prevalence, life expectancies and their variances and 171: graphs. Our transition model consists in absorbing and 172: non-absorbing states with the possibility of return across the 173: non-absorbing states. The main advantage of this package, 174: compared to other programs for the analysis of transition data 175: (For example: Proc Catmod of SAS®) is that the whole 176: individual information is used even if an interview is missing, a 177: status or a date is unknown or when the delay between waves is 178: not identical for each individual. The program can be executed 179: according to parameters: selection of a sub-sample, number of 180: absorbing and non-absorbing states, number of waves taken in 181: account (the user inputs the first and the last interview), a 182: tolerance level for the maximization function, the periodicity of 183: the transitions (we can compute annual, quarterly or monthly 184: transitions), covariates in the model. It works on Windows or on 185: Unix. 186: 187: 188: <hr> 189: 190: (1) Laditka, Sarah B. and Wolf, Douglas A. (1998), "New 191: Methods for Analyzing Active Life Expectancy". Journal of 192: Aging and Health. Vol 10, No. 2. 193: 194: <hr> 195: 196: <h2><a name="data">On what kind of data can 197: it be used?</a></h2> 198: 199: The minimum data required for a transition model is the 200: recording of a set of individuals interviewed at a first date and 201: interviewed again at least one another time. From the 202: observations of an individual, we obtain a follow-up over time of 203: the occurrence of a specific event. In this documentation, the 204: event is related to health status at older ages, but the program 205: can be applied on a lot of longitudinal studies in different 206: contexts. To build the data file explained into the next section, 207: you must have the month and year of each interview and the 208: corresponding health status. But in order to get age, date of 209: birth (month and year) is required (missing values is allowed for 210: month). Date of death (month and year) is an important 211: information also required if the individual is dead. Shorter 212: steps (i.e. a month) will more closely take into account the 213: survival time after the last interview. 214: 215: <hr> 216: 217: <h2><a name="datafile">The data file</a></h2> 218: 219: In this example, 8,000 people have been interviewed in a 220: cross-longitudinal survey of 4 waves (1984, 1986, 1988, 1990). 221: Some people missed 1, 2 or 3 interviews. Health statuses are 222: healthy (1) and disable (2). The survey is not a real one. It is 223: a simulation of the American Longitudinal Survey on Aging. The 224: disability state is defined if the individual missed one of four 225: ADL (Activity of daily living, like bathing, eating, walking). 226: Therefore, even is the individuals interviewed in the sample are 227: virtual, the information brought with this sample is close to the 228: situation of the United States. Sex is not recorded is this 229: sample. 230: 231: Each line of the data set (named <a href="data1.txt">data1.txt</a> 232: in this first example) is an individual record which fields are: 233: 234: <ul> 235: <li>Index number: positive number (field 1) </li> 236: <li>First covariate positive number (field 2) </li> 237: <li>Second covariate positive number (field 3) </li> 238: <li><a name="Weight">Weight</a>: positive number 239: (field 4) . In most surveys individuals are weighted 240: according to the stratification of the sample.</li> 241: <li>Date of birth: coded as mm/yyyy. Missing dates are 242: coded as 99/9999 (field 5) </li> 243: <li>Date of death: coded as mm/yyyy. Missing dates are 244: coded as 99/9999 (field 6) </li> 245: <li>Date of first interview: coded as mm/yyyy. Missing 246: dates are coded as 99/9999 (field 7) </li> 247: <li>Status at first interview: positive number. 248: Missing values ar coded -1. (field 8) </li> 249: <li>Date of second interview: coded as mm/yyyy. 250: Missing dates are coded as 99/9999 (field 9) </li> 251: <li>Status at second interview positive 252: number. Missing values ar coded -1. (field 10) </li> 253: <li>Date of third interview: coded as mm/yyyy. Missing 254: dates are coded as 99/9999 (field 11) </li> 255: <li>Status at third interview positive 256: number. Missing values ar coded -1. (field 12) </li> 257: <li>Date of fourth interview: coded as mm/yyyy. 258: Missing dates are coded as 99/9999 (field 13) </li> 259: <li>Status at fourth interview positive 260: number. Missing values are coded -1. (field 14) </li> 261: <li>etc</li> 262: </ul> 263: 264:   265: 266: If your longitudinal survey do not include information about 267: weights or covariates, you must fill the column with a number 268: (e.g. 1) because a missing field is not allowed. 269: 270: <hr> 271: 272: <h2>Your first example parameter file<a 273: href="http://euroreves.ined.fr/imach"></a><a name="uio"></a></h2> 274: 275: <h2><a name="biaspar"></a>#Imach version 0.71a, March 2002, 276: INED-EUROREVES </h2> 277: 278: This is a comment. Comments start with a '#'. 279: 280: <h4>First uncommented line</h4> 281: 282: <pre>title=1st_example datafile=data1.txt lastobs=8600 firstpass=1 lastpass=4</pre> 283: 284: <ul> 285: <li>title= 1st_example is title of the run. </li> 286: <li>datafile=data1.txt is the name of the data set. 287: Our example is a six years follow-up survey. It consists 288: in a baseline followed by 3 reinterviews. </li> 289: <li>lastobs= 8600 the program is able to run on a 290: subsample where the last observation number is lastobs. 291: It can be set a bigger number than the real number of 292: observations (e.g. 100000). In this example, maximisation 293: will be done on the 8600 first records. </li> 294: <li>firstpass=1 , lastpass=4 In case of more 295: than two interviews in the survey, the program can be run 296: on selected transitions periods. firstpass=1 means the 297: first interview included in the calculation is the 298: baseline survey. lastpass=4 means that the information 299: brought by the 4th interview is taken into account.</li> 300: </ul> 301: 302:   303: 304: <h4><a name="biaspar-2">Second uncommented 305: line</a></h4> 306: 307: <pre>ftol=1.e-08 stepm=1 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre> 308: 309: <ul> 310: <li>ftol=1e-8 Convergence tolerance on the function 311: value in the maximisation of the likelihood. Choosing a 312: correct value for ftol is difficult. 1e-8 is a correct 313: value for a 32 bits computer.</li> 314: <li>stepm=1 Time unit in months for interpolation. 315: Examples:<ul> 316: <li>If stepm=1, the unit is a month </li> 317: <li>If stepm=4, the unit is a trimester</li> 318: <li>If stepm=12, the unit is a year </li> 319: <li>If stepm=24, the unit is two years</li> 320: <li>... </li> 321: </ul> 322: </li> 323: <li>ncov=2 Number of covariates in the datafile. The 324: intercept and the age parameter are counting for 2 325: covariates.</li> 326: <li>nlstate=2 Number of non-absorbing (alive) states. 327: Here we have two alive states: disability-free is coded 1 328: and disability is coded 2. </li> 329: <li>ndeath=1 Number of absorbing states. The absorbing 330: state death is coded 3. </li> 331: <li>maxwav=4 Number of waves in the datafile.</li> 332: <li><a name="mle">mle</a>=1 Option for the 333: Maximisation Likelihood Estimation. <ul> 334: <li>If mle=1 the program does the maximisation and 335: the calculation of health expectancies </li> 336: <li>If mle=0 the program only does the calculation of 337: the health expectancies. </li> 338: </ul> 339: </li> 340: <li>weight=0 Possibility to add weights. <ul> 341: <li>If weight=0 no weights are included </li> 342: <li>If weight=1 the maximisation integrates the 343: weights which are in field <a href="#Weight">4</a></li> 344: </ul> 345: </li> 346: </ul> 347: 348: <h4>Covariates</h4> 349: 350: Intercept and age are systematically included in the model. 351: Additional covariates (actually two) can be included with the command: 352: 353: <pre>model=list of covariates</pre> 354: 355: <ul> 356: <li>if model=. then no covariates are 357: included</li> 358: <li>if model=V1 the model includes the first 359: covariate (field 2)</li> 360: <li>if model=V2 the model includes the 361: second covariate (field 3)</li> 362: <li>if model=V1+V2 the model includes the 363: first and the second covariate (fields 2 and 3)</li> 364: <li>if model=V1*V2 the model includes the 365: product of the first and the second covariate (fields 2 366: and 3)</li> 367: <li>if model=V1+V1*age the model includes 368: the product covariate*age</li> 369: </ul> 370: 371: <h4>Guess values for optimization </h4> 373: 374: You must write the initial guess values of the parameters for 375: optimization. The number of parameters, N depends on the 376: number of absorbing states and non-absorbing states and on the 377: number of covariates. 378: N is given by the formula N=(nlstate + 379: ndeath-1)*nlstate*ncov . 380: 381: Thus in the simple case with 2 covariates (the model is log 382: (pij/pii) = aij + bij * age where intercept and age are the two 383: covariates), and 2 health degrees (1 for disability-free and 2 384: for disability) and 1 absorbing state (3), you must enter 8 385: initials values, a12, b12, a13, b13, a21, b21, a23, b23. You can 386: start with zeros as in this example, but if you have a more 387: precise set (for example from an earlier run) you can enter it 388: and it will speed up them 389: Each of the four lines starts with indices "ij": ij 390: aij bij 391: 392: <blockquote> 393: <pre># Guess values of aij and bij in log (pij/pii) = aij + bij * age 394: 12 -14.155633 0.110794 395: 13 -7.925360 0.032091 396: 21 -1.890135 -0.029473 397: 23 -6.234642 0.022315 </pre> 398: </blockquote> 399: 400: or, to simplify (in most of cases it converges but there is no warranty!): 401: 402: <blockquote> 403: <pre>12 0.0 0.0 404: 13 0.0 0.0 405: 21 0.0 0.0 406: 23 0.0 0.0</pre> 407: </blockquote> 408: 409: <h4>Guess values for computing variances</h4> 410: 411: This is an output if <a href="#mle">mle</a>=1. But it can be 412: used as an input to get the various output data files (Health 413: expectancies, stationary prevalence etc.) and figures without 414: rerunning the rather long maximisation phase (mle=0). 415: 416: The scales are small values for the evaluation of numerical 417: derivatives. These derivatives are used to compute the hessian 418: matrix of the parameters, that is the inverse of the covariance 419: matrix, and the variances of health expectancies. Each line 420: consists in indices "ij" followed by the initial scales 421: (zero to simplify) associated with aij and bij. 422: 423: <ul> 424: <li>If mle=1 you can enter zeros:</li> 425: </ul> 426: 427: <blockquote> 428: <pre># Scales (for hessian or gradient estimation) 429: 12 0. 0. 430: 13 0. 0. 431: 21 0. 0. 432: 23 0. 0. </pre> 433: </blockquote> 434: 435: <ul> 436: <li>If mle=0 you must enter a covariance matrix (usually 437: obtained from an earlier run).</li> 438: </ul> 439: 440: <h4>Covariance matrix of parameters</h4> 441: 442: This is an output if <a href="#mle">mle</a>=1. But it can be 443: used as an input to get the various output data files (Health 444: expectancies, stationary prevalence etc.) and figures without 445: rerunning the rather long maximisation phase (mle=0). 446: 447: Each line starts with indices "ijk" followed by the 448: covariances between aij and bij: 449: 450: <pre> 451: 121 Var(a12) 452: 122 Cov(b12,a12) Var(b12) 453: ... 454: 232 Cov(b23,a12) Cov(b23,b12) ... Var (b23) </pre> 455: 456: <ul> 457: <li>If mle=1 you can enter zeros. </li> 458: </ul> 459: 460: <blockquote> 461: <pre># Covariance matrix 462: 121 0. 463: 122 0. 0. 464: 131 0. 0. 0. 465: 132 0. 0. 0. 0. 466: 211 0. 0. 0. 0. 0. 467: 212 0. 0. 0. 0. 0. 0. 468: 231 0. 0. 0. 0. 0. 0. 0. 469: 232 0. 0. 0. 0. 0. 0. 0. 0.</pre> 470: </blockquote> 471: 472: <ul> 473: <li>If mle=0 you must enter a covariance matrix (usually 474: obtained from an earlier run). 475: </li> 476: </ul> 477: 478: <h4>Age range for calculation of stationary 479: prevalences and health expectancies</h4> 480: 481: <pre>agemin=70 agemax=100 bage=50 fage=100</pre> 482: 483: Once we obtained the estimated parameters, the program is able 484: to calculated stationary prevalence, transitions probabilities 485: and life expectancies at any age. Choice of age range is useful 486: for extrapolation. In our data file, ages varies from age 70 to 487: 102. It is possible to get extrapolated stationary 488: prevalence by age ranging from agemin to agemax. 489: 490: 491: Setting bage=50 (begin age) and fage=100 (final age), makes the program computing 492: life expectancy from age 'bage' to age 'fage'. As we use a model, we 493: can interessingly compute life expectancy on a wider age range than the age 494: range from the data. But the model can be rather wrong on much larger 495: intervals. Program is limited to around 120 for upper age! 496: 497: <ul> 498: <li>agemin= Minimum age for calculation of the 499: stationary prevalence </li> 500: <li>agemax= Maximum age for calculation of the 501: stationary prevalence </li> 502: <li>bage= Minimum age for calculation of the health 503: expectancies </li> 504: <li>fage= Maximum age for calculation of the health 505: expectancies </li> 506: </ul> 507: 508: <h4><a name="Computing">Computing</a> the observed prevalence</h4> 510: 511: <pre>begin-prev-date=1/1/1984 end-prev-date=1/6/1988 </pre> 512: 513: Statements 'begin-prev-date' and 'end-prev-date' allow to 514: select the period in which we calculate the observed prevalences 515: in each state. In this example, the prevalences are calculated on 516: data survey collected between 1 january 1984 and 1 june 1988. 517: 518: <ul> 519: <li>begin-prev-date= Starting date 520: (day/month/year)</li> 521: <li>end-prev-date= Final date 522: (day/month/year)</li> 523: </ul> 524: 525: <h4>Population- or status-based health 526: expectancies</h4> 527: 528: <pre>pop_based=0</pre> 529: 530: The program computes status-based health expectancies, i.e health 531: expectancies which depends on your initial health state. If you are 532: healthy your healthy life expectancy (e11) is higher than if you were 533: disabled (e21, with e11 > e21). 534: To compute a healthy life expectancy independant of the initial status 535: we have to weight e11 and e21 according to the probability to be in 536: each state at initial age or, with other word, according to the 537: proportion of people in each state. 538: 539: We prefer computing a 'pure' period healthy life expectancy based only 540: on the transtion forces. Then the weights are simply the stationnary 541: prevalences or 'implied' prevalences at the initial age. 542: 543: Some other people would like to use the cross-sectional prevalences 544: (the "Sullivan prevalences") observed at the initial age during a 545: period of time <a href="#Computing">defined just above</a>. 546: 547: <ul> 548: <li>popbased= 0 Health expectancies are computed 549: at each age from stationary prevalences 'expected' at this initial age.</li> 550: <li>popbased= 1 Health expectancies are computed 551: at each age from cross-sectional 'observed' prevalence at this 552: initial age. As all the population is not observed at the same exact date we 553: define a short period were the observed prevalence is computed.</li> 554: </ul> 555: 556: 557: 558: <h4>Prevalence forecasting ( Experimental)</h4> 559: 560: <pre>starting-proj-date=1/1/1989 final-proj-date=1/1/1992 mov_average=0 </pre> 561: 562: Prevalence and population projections are only available if the 563: interpolation unit is a month, i.e. stepm=1 and if there are no 564: covariate. The programme estimates the prevalence in each state at a 565: precise date expressed in day/month/year. The programme computes one 566: forecasted prevalence a year from a starting date (1 january of 1989 567: in this example) to a final date (1 january 1992). The statement 568: mov_average allows to compute smoothed forecasted prevalences with a 569: five-age moving average centered at the mid-age of the five-age 570: period. 571: 572: <ul> 573: <li>starting-proj-date= starting date 574: (day/month/year) of forecasting</li> 575: <li>final-proj-date= final date 576: (day/month/year) of forecasting</li> 577: <li>mov_average= smoothing with a five-age 578: moving average centered at the mid-age of the five-age 579: period. The command mov_average takes 580: value 1 if the prevalences are smoothed and 0 otherwise.</li> 581: </ul> 582: 583: <h4>Last uncommented line : Population 584: forecasting </h4> 585: 586: <pre>popforecast=0 popfile=pyram.txt popfiledate=1/1/1989 last-popfiledate=1/1/1992</pre> 587: 588: This command is available if the interpolation unit is a 589: month, i.e. stepm=1 and if popforecast=1. From a data file 590: including age and number of persons alive at the precise date 591: popfiledate, you can forecast the number of persons 592: in each state until date last-popfiledate. In this 593: example, the popfile <a href="pyram.txt">pyram.txt</a> 594: includes real data which are the Japanese population in 1989. 595: 596: <ul type="disc"> 597: <li class="MsoNormal" 598: style="TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-list: l10 level1 lfo36; tab-stops: list 36.0pt">popforecast= 599: 0 Option for population forecasting. If 600: popforecast=1, the programme does the forecasting.</li> 601: <li class="MsoNormal" 602: style="TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-list: l10 level1 lfo36; tab-stops: list 36.0pt">popfile= 603: name of the population file</li> 604: <li class="MsoNormal" 605: style="TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-list: l10 level1 lfo36; tab-stops: list 36.0pt">popfiledate= 606: date of the population population</li> 607: <li class="MsoNormal" 608: style="TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-list: l10 level1 lfo36; tab-stops: list 36.0pt">last-popfiledate= 609: date of the last population projection </li> 610: </ul> 611: 612: <hr> 613: 614: <h2><a name="running"></a>Running Imach 615: with this example</h2> 616: 617: We assume that you entered your <a href="biaspar.imach">1st_example 618: parameter file</a> as explained <a href="#biaspar">above</a>. To 619: run the program you should click on the imach.exe icon and enter 620: the name of the parameter file which is for example <a 621: href="C:\usr\imach\mle\biaspar.txt">C:\usr\imach\mle\biaspar.txt</a> 622: (you also can click on the biaspar.txt icon located in 623: <a href="C:\usr\imach\mle">C:\usr\imach\mle</a> and put it with 624: the mouse on the imach window). 625: 626: 627: The time to converge depends on the step unit that you used (1 628: month is cpu consuming), on the number of cases, and on the 629: number of variables. 630: 631: The program outputs many files. Most of them are files which 632: will be plotted for better understanding. 633: 634: <hr> 635: 636: <h2><a name="output">Output of the program 637: and graphs </a></h2> 638: 639: Once the optimization is finished, some graphics can be made 640: with a grapher. We use Gnuplot which is an interactive plotting 641: program copyrighted but freely distributed. A gnuplot reference 642: manual is available <a href="http://www.gnuplot.info/">here</a>. 643: When the running is finished, the user should enter a caracter 644: for plotting and output editing. 645: 646: These caracters are: 647: 648: <ul> 649: <li>'c' to start again the program from the beginning.</li> 650: <li>'e' opens the <a href="biaspar.htm">biaspar.htm</a> 651: file to edit the output files and graphs. </li> 652: <li>'q' for exiting.</li> 653: </ul> 654: 655: <h5>Results files 656: 657: - <a 658: name="Observed prevalence in each state">Observed prevalence in each state</a> (and at first pass): 661: <a href="prbiaspar.txt">prbiaspar.txt</a> 662: </h5> 663: 664: The first line is the title and displays each field of the 665: file. The first column is age. The fields 2 and 6 are the 666: proportion of individuals in states 1 and 2 respectively as 667: observed during the first exam. Others fields are the numbers of 668: people in states 1, 2 or more. The number of columns increases if 669: the number of states is higher than 2. 670: The header of the file is 671: 672: <pre># Age Prev(1) N(1) N Age Prev(2) N(2) N 673: 70 1.00000 631 631 70 0.00000 0 631 674: 71 0.99681 625 627 71 0.00319 2 627 675: 72 0.97125 1115 1148 72 0.02875 33 1148 </pre> 676: 677: It means that at age 70, the prevalence in state 1 is 1.000 678: and in state 2 is 0.00 . At age 71 the number of individuals in 679: state 1 is 625 and in state 2 is 2, hence the total number of 680: people aged 71 is 625+2=627. 681: 682: 683: <h5>- Estimated parameters and 684: covariance matrix: <a href="rbiaspar.txt">rbiaspar.txt</a></h5> 685: 686: This file contains all the maximisation results: 687: 688: <pre> -2 log likelihood= 21660.918613445392 689: Estimated parameters: a12 = -12.290174 b12 = 0.092161 690: a13 = -9.155590 b13 = 0.046627 691: a21 = -2.629849 b21 = -0.022030 692: a23 = -7.958519 b23 = 0.042614 693: Covariance matrix: Var(a12) = 1.47453e-001 694: Var(b12) = 2.18676e-005 695: Var(a13) = 2.09715e-001 696: Var(b13) = 3.28937e-005 697: Var(a21) = 9.19832e-001 698: Var(b21) = 1.29229e-004 699: Var(a23) = 4.48405e-001 700: Var(b23) = 5.85631e-005 701: </pre> 702: 703: By substitution of these parameters in the regression model, 704: we obtain the elementary transition probabilities: 705: 706: <img src="pebiaspar1.gif" width="400" height="300"> 707: 708: <h5>- Transition probabilities: 709: <a href="pijrbiaspar.txt">pijrbiaspar.txt</a></h5> 710: 711: Here are the transitions probabilities Pij(x, x+nh) where nh 712: is a multiple of 2 years. The first column is the starting age x 713: (from age 50 to 100), the second is age (x+nh) and the others are 714: the transition probabilities p11, p12, p13, p21, p22, p23. For 715: example, line 5 of the file is: 716: 717: <pre> 100 106 0.02655 0.17622 0.79722 0.01809 0.13678 0.84513 </pre> 718: 719: and this means: 720: 721: <pre>p11(100,106)=0.02655 722: p12(100,106)=0.17622 723: p13(100,106)=0.79722 724: p21(100,106)=0.01809 725: p22(100,106)=0.13678 726: p22(100,106)=0.84513 </pre> 727: 728: <h5>- <a 729: name="Stationary prevalence in each state">Stationary prevalence in each state</a>: 731: <a href="plrbiaspar.txt">plrbiaspar.txt</a></h5> 732: 733: <pre>#Prevalence 734: #Age 1-1 2-2 735: 736: #************ 737: 70 0.90134 0.09866 738: 71 0.89177 0.10823 739: 72 0.88139 0.11861 740: 73 0.87015 0.12985 </pre> 741: 742: At age 70 the stationary prevalence is 0.90134 in state 1 and 743: 0.09866 in state 2. This stationary prevalence differs from 744: observed prevalence. Here is the point. The observed prevalence 745: at age 70 results from the incidence of disability, incidence of 746: recovery and mortality which occurred in the past of the cohort. 747: Stationary prevalence results from a simulation with actual 748: incidences and mortality (estimated from this cross-longitudinal 749: survey). It is the best predictive value of the prevalence in the 750: future if "nothing changes in the future". This is 751: exactly what demographers do with a Life table. Life expectancy 752: is the expected mean time to survive if observed mortality rates 753: (incidence of mortality) "remains constant" in the 754: future. 755: 756: <h5>- Standard deviation of 757: stationary prevalence: <a 758: href="vplrbiaspar.txt">vplrbiaspar.txt</a></h5> 759: 760: The stationary prevalence has to be compared with the observed 761: prevalence by age. But both are statistical estimates and 762: subjected to stochastic errors due to the size of the sample, the 763: design of the survey, and, for the stationary prevalence to the 764: model used and fitted. It is possible to compute the standard 765: deviation of the stationary prevalence at each age. 766: 767: <h5>-Observed and stationary 768: prevalence in state (2=disable) with the confident interval: 769: <a href="vbiaspar21.htm">vbiaspar21.gif</a></h5> 770: 771: This graph exhibits the stationary prevalence in state (2) 772: with the confidence interval in red. The green curve is the 773: observed prevalence (or proportion of individuals in state (2)). 774: Without discussing the results (it is not the purpose here), we 775: observe that the green curve is rather below the stationary 776: prevalence. It suggests an increase of the disability prevalence 777: in the future. 778: 779: <img src="vbiaspar21.gif" width="400" height="300"> 780: 781: <h5>-Convergence to the 782: stationary prevalence of disability: <a 783: href="pbiaspar11.gif">pbiaspar11.gif</a> 784: <img src="pbiaspar11.gif" width="400" height="300"> </h5> 785: 786: This graph plots the conditional transition probabilities from 787: an initial state (1=healthy in red at the bottom, or 2=disable in 788: green on top) at age x to the final state 2=disable at 789: age x+h. Conditional means at the condition to be alive 790: at age x+h which is hP12x + hP22x. The 791: curves hP12x/(hP12x + hP22x) and hP22x/(hP12x 792: + hP22x) converge with h, to the stationary 793: prevalence of disability. In order to get the stationary 794: prevalence at age 70 we should start the process at an earlier 795: age, i.e.50. If the disability state is defined by severe 796: disability criteria with only a few chance to recover, then the 797: incidence of recovery is low and the time to convergence is 798: probably longer. But we don't have experience yet. 799: 800: <h5>- Life expectancies by age 801: and initial health status: <a 802: href="erbiaspar.txt">erbiaspar.txt</a></h5> 803: 804: <pre># Health expectancies 805: # Age 1-1 1-2 2-1 2-2 806: 70 10.9226 3.0401 5.6488 6.2122 807: 71 10.4384 3.0461 5.2477 6.1599 808: 72 9.9667 3.0502 4.8663 6.1025 809: 73 9.5077 3.0524 4.5044 6.0401 </pre> 810: 811: <pre>For example 70 10.4227 3.0402 5.6488 5.7123 means: 812: e11=10.4227 e12=3.0402 e21=5.6488 e22=5.7123</pre> 813: 814: <pre><img src="expbiaspar21.gif" width="400" height="300"><img 815: src="expbiaspar11.gif" width="400" height="300"></pre> 816: 817: For example, life expectancy of a healthy individual at age 70 818: is 10.42 in the healthy state and 3.04 in the disability state 819: (=13.46 years). If he was disable at age 70, his life expectancy 820: will be shorter, 5.64 in the healthy state and 5.71 in the 821: disability state (=11.35 years). The total life expectancy is a 822: weighted mean of both, 13.46 and 11.35; weight is the proportion 823: of people disabled at age 70. In order to get a pure period index 824: (i.e. based only on incidences) we use the <a 825: href="#Stationary prevalence in each state">computed or 826: stationary prevalence</a> at age 70 (i.e. computed from 827: incidences at earlier ages) instead of the <a 828: href="#Observed prevalence in each state">observed prevalence</a> 829: (for example at first exam) (<a href="#Health expectancies">see 830: below</a>). 831: 832: <h5>- Variances of life 833: expectancies by age and initial health status: <a 834: href="vrbiaspar.txt">vrbiaspar.txt</a></h5> 835: 836: For example, the covariances of life expectancies Cov(ei,ej) 837: at age 50 are (line 3) 838: 839: <pre> Cov(e1,e1)=0.4776 Cov(e1,e2)=0.0488=Cov(e2,e1) Cov(e2,e2)=0.0424</pre> 840: 841: <h5>- <a 842: name="Health expectancies">Health 843: expectancies</a> 844: with standard errors in parentheses: <a 845: href="trbiaspar.txt">trbiaspar.txt</a></h5> 846: 847: <pre>#Total LEs with variances: e.. (std) e.1 (std) e.2 (std) </pre> 848: 849: <pre>70 13.26 (0.22) 9.95 (0.20) 3.30 (0.14) </pre> 850: 851: Thus, at age 70 the total life expectancy, e..=13.26 years is 852: the weighted mean of e1.=13.46 and e2.=11.35 by the stationary 853: prevalence at age 70 which are 0.90134 in state 1 and 0.09866 in 854: state 2, respectively (the sum is equal to one). e.1=9.95 is the 855: Disability-free life expectancy at age 70 (it is again a weighted 856: mean of e11 and e21). e.2=3.30 is also the life expectancy at age 857: 70 to be spent in the disability state. 858: 859: <h5>-Total life expectancy by 860: age and health expectancies in states (1=healthy) and (2=disable): 861: <a href="ebiaspar1.gif">ebiaspar1.gif</a></h5> 862: 863: This figure represents the health expectancies and the total 864: life expectancy with the confident interval in dashed curve. 865: 866: <pre> <img src="ebiaspar1.gif" width="400" height="300"></pre> 867: 868: Standard deviations (obtained from the information matrix of 869: the model) of these quantities are very useful. 870: Cross-longitudinal surveys are costly and do not involve huge 871: samples, generally a few thousands; therefore it is very 872: important to have an idea of the standard deviation of our 873: estimates. It has been a big challenge to compute the Health 874: Expectancy standard deviations. Don't be confuse: life expectancy 875: is, as any expected value, the mean of a distribution; but here 876: we are not computing the standard deviation of the distribution, 877: but the standard deviation of the estimate of the mean. 878: 879: Our health expectancies estimates vary according to the sample 880: size (and the standard deviations give confidence intervals of 881: the estimate) but also according to the model fitted. Let us 882: explain it in more details. 883: 884: Choosing a model means ar least two kind of choices. First we 885: have to decide the number of disability states. Second we have to 886: design, within the logit model family, the model: variables, 887: covariables, confonding factors etc. to be included. 888: 889: More disability states we have, better is our demographical 890: approach of the disability process, but smaller are the number of 891: transitions between each state and higher is the noise in the 892: measurement. We do not have enough experiments of the various 893: models to summarize the advantages and disadvantages, but it is 894: important to say that even if we had huge and unbiased samples, 895: the total life expectancy computed from a cross-longitudinal 896: survey, varies with the number of states. If we define only two 897: states, alive or dead, we find the usual life expectancy where it 898: is assumed that at each age, people are at the same risk to die. 899: If we are differentiating the alive state into healthy and 900: disable, and as the mortality from the disability state is higher 901: than the mortality from the healthy state, we are introducing 902: heterogeneity in the risk of dying. The total mortality at each 903: age is the weighted mean of the mortality in each state by the 904: prevalence in each state. Therefore if the proportion of people 905: at each age and in each state is different from the stationary 906: equilibrium, there is no reason to find the same total mortality 907: at a particular age. Life expectancy, even if it is a very useful 908: tool, has a very strong hypothesis of homogeneity of the 909: population. Our main purpose is not to measure differential 910: mortality but to measure the expected time in a healthy or 911: disability state in order to maximise the former and minimize the 912: latter. But the differential in mortality complexifies the 913: measurement. 914: 915: Incidences of disability or recovery are not affected by the 916: number of states if these states are independant. But incidences 917: estimates are dependant on the specification of the model. More 918: covariates we added in the logit model better is the model, but 919: some covariates are not well measured, some are confounding 920: factors like in any statistical model. The procedure to "fit 921: the best model' is similar to logistic regression which itself is 922: similar to regression analysis. We haven't yet been sofar because 923: we also have a severe limitation which is the speed of the 924: convergence. On a Pentium III, 500 MHz, even the simplest model, 925: estimated by month on 8,000 people may take 4 hours to converge. 926: Also, the program is not yet a statistical package, which permits 927: a simple writing of the variables and the model to take into 928: account in the maximisation. The actual program allows only to 929: add simple variables like age+sex or age+sex+ age*sex but will 930: never be general enough. But what is to remember, is that 931: incidences or probability of change from one state to another is 932: affected by the variables specified into the model. 933: 934: Also, the age range of the people interviewed has a link with 935: the age range of the life expectancy which can be estimated by 936: extrapolation. If your sample ranges from age 70 to 95, you can 937: clearly estimate a life expectancy at age 70 and trust your 938: confidence interval which is mostly based on your sample size, 939: but if you want to estimate the life expectancy at age 50, you 940: should rely in your model, but fitting a logistic model on a age 941: range of 70-95 and estimating probabilties of transition out of 942: this age range, say at age 50 is very dangerous. At least you 943: should remember that the confidence interval given by the 944: standard deviation of the health expectancies, are under the 945: strong assumption that your model is the 'true model', which is 946: probably not the case. 947: 948: <h5>- Copy of the parameter 949: file: <a href="orbiaspar.txt">orbiaspar.txt</a></h5> 950: 951: This copy of the parameter file can be useful to re-run the 952: program while saving the old output files. 953: 954: <h5>- Prevalence forecasting: 955: <a href="frbiaspar.txt">frbiaspar.txt</a></h5> 956: 957: First, 959: we have estimated the observed prevalence between 1/1/1984 and 960: 1/6/1988. The mean date of interview (weighed average of the 961: interviews performed between1/1/1984 and 1/6/1988) is estimated 962: to be 13/9/1985, as written on the top on the file. Then we 963: forecast the probability to be in each state. 964: 965: Example, 967: at date 1/1/1989 : 968: 969: <pre class="MsoNormal"># StartingAge FinalAge P.1 P.2 P.3 970: # Forecasting at date 1/1/1989 971: 73 0.807 0.078 0.115</pre> 972: 973: Since 975: the minimum age is 70 on the 13/9/1985, the youngest forecasted 976: age is 73. This means that at age a person aged 70 at 13/9/1989 977: has a probability to enter state1 of 0.807 at age 73 on 1/1/1989. 978: Similarly, the probability to be in state 2 is 0.078 and the 979: probability to die is 0.115. Then, on the 1/1/1989, the 980: prevalence of disability at age 73 is estimated to be 0.088. 981: 982: <h5>- Population forecasting: 983: <a href="poprbiaspar.txt">poprbiaspar.txt</a></h5> 984: 985: <pre># Age P.1 P.2 P.3 [Population] 986: # Forecasting at date 1/1/1989 987: 75 572685.22 83798.08 988: 74 621296.51 79767.99 989: 73 645857.70 69320.60 </pre> 990: 991: <pre># Forecasting at date 1/1/19909 992: 76 442986.68 92721.14 120775.48 993: 75 487781.02 91367.97 121915.51 994: 74 512892.07 85003.47 117282.76 </pre> 995: 996: From the population file, we estimate the number of people in 997: each state. At age 73, 645857 persons are in state 1 and 69320 998: are in state 2. One year latter, 512892 are still in state 1, 999: 85003 are in state 2 and 117282 died before 1/1/1990. 1000: 1001: <hr> 1002: 1003: <h2><a name="example"> </a>Trying an example</a></h2> 1004: 1005: Since you know how to run the program, it is time to test it 1006: on your own computer. Try for example on a parameter file named <a 1007: href="..\mytry\imachpar.txt">imachpar.txt</a> which is a copy of mypar.txt included in the 1009: subdirectory of imach, mytry. 1010: Edit it to change the name of the data file to ..\data\mydata.txt if you don't want to 1012: copy it on the same directory. The file mydata.txt 1013: is a smaller file of 3,000 people but still with 4 waves. 1014: 1015: Click on the imach.exe icon to open a window. Answer to the 1016: question:'Enter the parameter file name:' 1017: 1018: <table border="1"> 1019: <tr> 1020: <td width="100%">IMACH, Version 0.71Enter 1021: the parameter file name: ..\mytry\imachpar.txt 1022: </td> 1023: </tr> 1024: </table> 1025: 1026: Most of the data files or image files generated, will use the 1027: 'imachpar' string into their name. The running time is about 2-3 1028: minutes on a Pentium III. If the execution worked correctly, the 1029: outputs files are created in the current directory, and should be 1030: the same as the mypar files initially included in the directory mytry. 1032: 1033: <ul> 1034: <li><pre>Output on the screen The output screen looks like <a 1035: href="imachrun.LOG">this Log file</a> 1036: # 1037: 1038: title=MLE datafile=..\data\mydata.txt lastobs=3000 firstpass=1 lastpass=3 1039: ftol=1.000000e-008 stepm=24 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre> 1040: </li> 1041: <li><pre>Total number of individuals= 2965, Agemin = 70.00, Agemax= 100.92 1042: 1043: Warning, no any valid information for:126 line=126 1044: Warning, no any valid information for:2307 line=2307 1045: Delay (in months) between two waves Min=21 Max=51 Mean=24.495826 1046: These lines give some warnings on the data file and also some raw statistics on frequencies of transitions. 1047: Age 70 1.=230 loss[1]=3.5% 2.=16 loss[2]=12.5% 1.=222 prev[1]=94.1% 2.=14 1048: prev[2]=5.9% 1-1=8 11=200 12=7 13=15 2-1=2 21=6 22=7 23=1 1049: Age 102 1.=0 loss[1]=NaNQ% 2.=0 loss[2]=NaNQ% 1.=0 prev[1]=NaNQ% 2.=0 </pre> 1050: </li> 1051: </ul> 1052: 1053:   1054: 1055: <ul> 1056: <li>Maximisation with the Powell algorithm. 8 directions are 1057: given corresponding to the 8 parameters. this can be 1058: rather long to get convergence. 1059: 1060: Powell iter=1 -2*LL=11531.405658264877 1 0.000000000000 2 1061: 0.000000000000 3 1062: 0.000000000000 4 0.000000000000 5 0.000000000000 6 1063: 0.000000000000 7 1064: 0.000000000000 8 0.000000000000 1065: 1..........2.................3..........4.................5......... 1066: 6................7........8............... 1067: Powell iter=23 -2*LL=6744.954108371555 1 -12.967632334283 1068: 1069: 2 0.135136681033 3 -7.402109728262 4 0.067844593326 1070: 5 -0.673601538129 6 -0.006615504377 7 -5.051341616718 1071: 8 0.051272038506 1072: 1..............2...........3..............4........... 1073: 5..........6................7...........8......... 1074: #Number of iterations = 23, -2 Log likelihood = 1075: 6744.954042573691 1076: # Parameters 1077: 12 -12.966061 0.135117 1078: 13 -7.401109 0.067831 1079: 21 -0.672648 -0.006627 1080: 23 -5.051297 0.051271 1081: </li> 1082: <li><pre>Calculation of the hessian matrix. Wait... 1083: 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 1084: 1085: Inverting the hessian to get the covariance matrix. Wait... 1086: 1087: #Hessian matrix# 1088: 3.344e+002 2.708e+004 -4.586e+001 -3.806e+003 -1.577e+000 -1.313e+002 3.914e-001 3.166e+001 1089: 2.708e+004 2.204e+006 -3.805e+003 -3.174e+005 -1.303e+002 -1.091e+004 2.967e+001 2.399e+003 1090: -4.586e+001 -3.805e+003 4.044e+002 3.197e+004 2.431e-002 1.995e+000 1.783e-001 1.486e+001 1091: -3.806e+003 -3.174e+005 3.197e+004 2.541e+006 2.436e+000 2.051e+002 1.483e+001 1.244e+003 1092: -1.577e+000 -1.303e+002 2.431e-002 2.436e+000 1.093e+002 8.979e+003 -3.402e+001 -2.843e+003 1093: -1.313e+002 -1.091e+004 1.995e+000 2.051e+002 8.979e+003 7.420e+005 -2.842e+003 -2.388e+005 1094: 3.914e-001 2.967e+001 1.783e-001 1.483e+001 -3.402e+001 -2.842e+003 1.494e+002 1.251e+004 1095: 3.166e+001 2.399e+003 1.486e+001 1.244e+003 -2.843e+003 -2.388e+005 1.251e+004 1.053e+006 1096: # Scales 1097: 12 1.00000e-004 1.00000e-006 1098: 13 1.00000e-004 1.00000e-006 1099: 21 1.00000e-003 1.00000e-005 1100: 23 1.00000e-004 1.00000e-005 1101: # Covariance 1102: 1 5.90661e-001 1103: 2 -7.26732e-003 8.98810e-005 1104: 3 8.80177e-002 -1.12706e-003 5.15824e-001 1105: 4 -1.13082e-003 1.45267e-005 -6.50070e-003 8.23270e-005 1106: 5 9.31265e-003 -1.16106e-004 6.00210e-004 -8.04151e-006 1.75753e+000 1107: 6 -1.15664e-004 1.44850e-006 -7.79995e-006 1.04770e-007 -2.12929e-002 2.59422e-004 1108: 7 1.35103e-003 -1.75392e-005 -6.38237e-004 7.85424e-006 4.02601e-001 -4.86776e-003 1.32682e+000 1109: 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 1110: # agemin agemax for lifexpectancy, bage fage (if mle==0 ie no data nor Max likelihood). 1111: 1112: 1113: agemin=70 agemax=100 bage=50 fage=100 1114: Computing prevalence limit: result on file 'plrmypar.txt' 1115: Computing pij: result on file 'pijrmypar.txt' 1116: Computing Health Expectancies: result on file 'ermypar.txt' 1117: Computing Variance-covariance of DFLEs: file 'vrmypar.txt' 1118: Computing Total LEs with variances: file 'trmypar.txt' 1119: Computing Variance-covariance of Prevalence limit: file 'vplrmypar.txt' 1120: End of Imach 1121: </pre> 1122: </li> 1123: </ul> 1124: 1125: Once the running is finished, the program 1126: requires a caracter: 1127: 1128: <table border="1"> 1129: <tr> 1130: <td width="100%">Type e to edit output files, c 1131: to start again, and q for exiting:</td> 1132: </tr> 1133: </table> 1134: 1135: First you should enter e to 1136: edit the master file mypar.htm. 1137: 1138: <ul> 1139: <li>Outputs files 1140: 1141: - Observed prevalence in each state: <a 1142: href="..\mytry\prmypar.txt">pmypar.txt</a> 1143: - Estimated parameters and the covariance matrix: <a 1144: href="..\mytry\rmypar.txt">rmypar.txt</a> 1145: - Stationary prevalence in each state: <a 1146: href="..\mytry\plrmypar.txt">plrmypar.txt</a> 1147: - Transition probabilities: <a 1148: href="..\mytry\pijrmypar.txt">pijrmypar.txt</a> 1149: - Copy of the parameter file: <a 1150: href="..\mytry\ormypar.txt">ormypar.txt</a> 1151: - Life expectancies by age and initial health status: <a 1152: href="..\mytry\ermypar.txt">ermypar.txt</a> 1153: - Variances of life expectancies by age and initial 1154: health status: <a href="..\mytry\vrmypar.txt">vrmypar.txt</a> 1155: 1156: - Health expectancies with their variances: <a 1157: href="..\mytry\trmypar.txt">trmypar.txt</a> 1158: - Standard deviation of stationary prevalence: <a 1159: href="..\mytry\vplrmypar.txt">vplrmypar.txt</a> 1160: - Prevalences forecasting: <a href="frmypar.txt">frmypar.txt</a> 1161: 1162: - Population forecasting (if popforecast=1): <a 1163: href="poprmypar.txt">poprmypar.txt</a> 1164: </li> 1165: <li>Graphs 1166: 1167: -<a href="../mytry/pemypar1.gif">One-step transition 1168: probabilities</a> 1169: -<a href="../mytry/pmypar11.gif">Convergence to the 1170: stationary prevalence</a> 1171: -<a href="..\mytry\vmypar11.gif">Observed and stationary 1172: prevalence in state (1) with the confident interval</a> 1173: -<a href="..\mytry\vmypar21.gif">Observed and stationary 1174: prevalence in state (2) with the confident interval</a> 1175: -<a href="..\mytry\expmypar11.gif">Health life 1176: expectancies by age and initial health state (1)</a> 1177: -<a href="..\mytry\expmypar21.gif">Health life 1178: expectancies by age and initial health state (2)</a> 1179: -<a href="..\mytry\emypar1.gif">Total life expectancy by 1180: age and health expectancies in states (1) and (2).</a> </li> 1181: </ul> 1182: 1183: This software have been partly granted by <a 1184: href="http://euroreves.ined.fr">Euro-REVES</a>, a concerted 1185: action from the European Union. It will be copyrighted 1186: identically to a GNU software product, i.e. program and software 1187: can be distributed freely for non commercial use. Sources are not 1188: widely distributed today. You can get them by asking us with a 1189: simple justification (name, email, institute) <a 1190: href="mailto:brouard@ined.fr">mailto:brouard@ined.fr</a> and <a 1191: href="mailto:lievre@ined.fr">mailto:lievre@ined.fr</a> . 1192: 1193: Latest version (0.71a of March 2002) can be accessed at <a 1194: href="http://euroeves.ined.fr/imach">http://euroreves.ined.fr/imach</a> 1195: 1196: </body> 1197: </html>