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1: <html> 2: 3: <head> 4: <meta http-equiv="Content-Type" 5: content="text/html; charset=iso-8859-1"> 6: <meta name="GENERATOR" content="Microsoft FrontPage Express 2.0"> 7: <title>Computing Health Expectancies using IMaCh</title> 8: </head> 9: 10: <body bgcolor="#FFFFFF"> 11: 12: <hr size="3" color="#EC5E5E"> 13: 14: <h1 align="center">Computing Health 15: Expectancies using IMaCh</h1> 16: 17: <h1 align="center">(a Maximum 18: Likelihood Computer Program using Interpolation of Markov Chains)</h1> 19: 20:   21: 22: <a href="http://www.ined.fr/"><img 23: src="logo-ined.gif" border="0" width="151" height="76"></a><img 24: src="euroreves2.gif" width="151" height="75"> 25: 26: <h3 align="center"><a href="http://www.ined.fr/">INED</a> and <a 28: href="http://euroreves.ined.fr">EUROREVES</a></h3> 29: 30: March 31: 2000 32: 33: <hr size="3" color="#EC5E5E"> 34: 35: Authors of the 36: program: <a href="http://sauvy.ined.fr/brouard">Nicolas Brouard</a>, senior researcher at the <a 39: href="http://www.ined.fr">Institut 40: National d'Etudes Démographiques</a> (INED, Paris) in the "Mortality, 42: Health and Epidemiology" Research Unit 43: 44: and Agnès 45: Lièvre 46: 47: 48: <h4>Contribution to the mathematics: C. R. 49: Heathcote (Australian 50: National University, Canberra).</h4> 51: 52: <h4>Contact: Agnès Lièvre (<a 53: href="mailto:lievre@ined.fr">lievre@ined.fr</a>) </h4> 55: 56: <hr> 57: 58: <ul> 59: <li><a href="#intro">Introduction</a> </li> 60: <li>The detailed statistical model (<a href="docmath.pdf">PDF 61: version</a>),(<a href="docmath.ps">ps version</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. Within the family of Health Expectancies (HE), 76: Disability-free life expectancy (DFLE) is probably the most 77: important index to monitor. In low mortality countries, there is 78: a fear that when mortality declines, the increase in DFLE is not 79: proportionate to the increase in total Life expectancy. This case 80: is called the Expansion of morbidity. Most of the data 81: collected today, in particular by the international <a 82: href="http://euroreves/reves">REVES</a> network on Health 83: expectancy, and most HE indices based on these data, are cross-sectional. 84: It means that the information collected comes from a single 85: cross-sectional survey: people from various ages (but mostly old 86: people) are surveyed on their health status at a single date. 87: Proportion of people disabled at each age, can then be measured 88: at that date. This age-specific prevalence curve is then used to 89: distinguish, within the stationary population (which, by 90: definition, is the life table estimated from the vital statistics 91: on mortality at the same date), the disable population from the 92: disability-free population. Life expectancy (LE) (or total 93: population divided by the yearly number of births or deaths of 94: this stationary population) is then decomposed into DFLE and DLE. 95: This method of computing HE is usually called the Sullivan method 96: (from the name of the author who first described it). 97: 98: Age-specific proportions of people disable are very difficult 99: to forecast because each proportion corresponds to historical 100: conditions of the cohort and it is the result of the historical 101: flows from entering disability and recovering in the past until 102: today. The age-specific intensities (or incidence rates) of 103: entering disability or recovering a good health, are reflecting 104: actual conditions and therefore can be used at each age to 105: forecast the future of this cohort. For example if a country is 106: improving its technology of prosthesis, the incidence of 107: recovering the ability to walk will be higher at each (old) age, 108: but the prevalence of disability will only slightly reflect an 109: improve because the prevalence is mostly affected by the history 110: of the cohort and not by recent period effects. To measure the 111: period improvement we have to simulate the future of a cohort of 112: new-borns entering or leaving at each age the disability state or 113: dying according to the incidence rates measured today on 114: different cohorts. The proportion of people disabled at each age 115: in this simulated cohort will be much lower (using the exemple of 116: an improvement) that the proportions observed at each age in a 117: cross-sectional survey. This new prevalence curve introduced in a 118: life table will give a much more actual and realistic HE level 119: than the Sullivan method which mostly measured the History of 120: health conditions in this country. 121: 122: Therefore, the main question is how to measure incidence rates 123: from cross-longitudinal surveys? This is the goal of the IMaCH 124: program. From your data and using IMaCH you can estimate period 125: HE and not only Sullivan's HE. Also the standard errors of the HE 126: are computed. 127: 128: A cross-longitudinal survey consists in a first survey 129: ("cross") where individuals from different ages are 130: interviewed on their health status or degree of disability. At 131: least a second wave of interviews ("longitudinal") 132: should measure each new individual health status. Health 133: expectancies are computed from the transitions observed between 134: waves and are computed for each degree of severity of disability 135: (number of life states). More degrees you consider, more time is 136: necessary to reach the Maximum Likelihood of the parameters 137: involved in the model. Considering only two states of disability 138: (disable and healthy) is generally enough but the computer 139: program works also with more health statuses. 140: 141: The simplest model is the multinomial logistic model where pij 142: is the probability to be observed in state j at the second 143: wave conditional to be observed in state i at the first 144: wave. Therefore a simple model is: log(pij/pii)= aij + 145: bij*age+ cij*sex, where 'age' is age and 'sex' 146: is a covariate. The advantage that this computer program claims, 147: comes from that if the delay between waves is not identical for 148: each individual, or if some individual missed an interview, the 149: information is not rounded or lost, but taken into account using 150: an interpolation or extrapolation. hPijx is the 151: probability to be observed in state i at age x+h 152: conditional to the observed state i at age x. The 153: delay 'h' can be split into an exact number (nh*stepm) 154: of unobserved intermediate states. This elementary transition (by 155: month or quarter trimester, semester or year) is modeled as a 156: multinomial logistic. The hPx matrix is simply the matrix 157: product of nh*stepm elementary matrices and the 158: contribution of each individual to the likelihood is simply hPijx. 159: 160: 161: 162: The program presented in this manual is a quite general 163: program named IMaCh (for Interpolated 164: MArkov CHain), designed to 165: analyse transition data from longitudinal surveys. The first step 166: is the parameters estimation of a transition probabilities model 167: between an initial status and a final status. From there, the 168: computer program produces some indicators such as observed and 169: stationary prevalence, life expectancies and their variances and 170: graphs. Our transition model consists in absorbing and 171: non-absorbing states with the possibility of return across the 172: non-absorbing states. The main advantage of this package, 173: compared to other programs for the analysis of transition data 174: (For example: Proc Catmod of SAS®) is that the whole 175: individual information is used even if an interview is missing, a 176: status or a date is unknown or when the delay between waves is 177: not identical for each individual. The program can be executed 178: according to parameters: selection of a sub-sample, number of 179: absorbing and non-absorbing states, number of waves taken in 180: account (the user inputs the first and the last interview), a 181: tolerance level for the maximization function, the periodicity of 182: the transitions (we can compute annual, quaterly or monthly 183: transitions), covariates in the model. It works on Windows or on 184: Unix. 185: 186: 187: <hr> 188: 189: <h2><a name="data">On what kind of data can 190: it be used?</a></h2> 191: 192: The minimum data required for a transition model is the 193: recording of a set of individuals interviewed at a first date and 194: interviewed again at least one another time. From the 195: observations of an individual, we obtain a follow-up over time of 196: the occurrence of a specific event. In this documentation, the 197: event is related to health status at older ages, but the program 198: can be applied on a lot of longitudinal studies in different 199: contexts. To build the data file explained into the next section, 200: you must have the month and year of each interview and the 201: corresponding health status. But in order to get age, date of 202: birth (month and year) is required (missing values is allowed for 203: month). Date of death (month and year) is an important 204: information also required if the individual is dead. Shorter 205: steps (i.e. a month) will more closely take into account the 206: survival time after the last interview. 207: 208: <hr> 209: 210: <h2><a name="datafile">The data file</a></h2> 211: 212: In this example, 8,000 people have been interviewed in a 213: cross-longitudinal survey of 4 waves (1984, 1986, 1988, 1990). 214: Some people missed 1, 2 or 3 interviews. Health statuses are 215: healthy (1) and disable (2). The survey is not a real one. It is 216: a simulation of the American Longitudinal Survey on Aging. The 217: disability state is defined if the individual missed one of four 218: ADL (Activity of daily living, like bathing, eating, walking). 219: Therefore, even is the individuals interviewed in the sample are 220: virtual, the information brought with this sample is close to the 221: situation of the United States. Sex is not recorded is this 222: sample. 223: 224: Each line of the data set (named <a href="data1.txt">data1.txt</a> 225: in this first example) is an individual record which fields are: 226: 227: <ul> 228: <li>Index number: positive number (field 1) </li> 229: <li>First covariate positive number (field 2) </li> 230: <li>Second covariate positive number (field 3) </li> 231: <li><a name="Weight">Weight</a>: positive number 232: (field 4) . In most surveys individuals are weighted 233: according to the stratification of the sample.</li> 234: <li>Date of birth: coded as mm/yyyy. Missing dates are 235: coded as 99/9999 (field 5) </li> 236: <li>Date of death: coded as mm/yyyy. Missing dates are 237: coded as 99/9999 (field 6) </li> 238: <li>Date of first interview: coded as mm/yyyy. Missing 239: dates are coded as 99/9999 (field 7) </li> 240: <li>Status at first interview: positive number. 241: Missing values ar coded -1. (field 8) </li> 242: <li>Date of second interview: coded as mm/yyyy. 243: Missing dates are coded as 99/9999 (field 9) </li> 244: <li>Status at second interview positive 245: number. Missing values ar coded -1. (field 10) </li> 246: <li>Date of third interview: coded as mm/yyyy. Missing 247: dates are coded as 99/9999 (field 11) </li> 248: <li>Status at third interview positive 249: number. Missing values ar coded -1. (field 12) </li> 250: <li>Date of fourth interview: coded as mm/yyyy. 251: Missing dates are coded as 99/9999 (field 13) </li> 252: <li>Status at fourth interview positive 253: number. Missing values are coded -1. (field 14) </li> 254: <li>etc</li> 255: </ul> 256: 257:   258: 259: If your longitudinal survey do not include information about 260: weights or covariates, you must fill the column with a number 261: (e.g. 1) because a missing field is not allowed. 262: 263: <hr> 264: 265: <h2>Your first example parameter file<a 266: href="http://euroreves.ined.fr/imach"></a><a name="uio"></a></h2> 267: 268: <h2><a name="biaspar"></a>#Imach version 0.63, February 2000, 269: INED-EUROREVES </h2> 270: 271: This is a comment. Comments start with a '#'. 272: 273: <h4>First uncommented line</h4> 274: 275: <pre>title=1st_example datafile=data1.txt lastobs=8600 firstpass=1 lastpass=4</pre> 276: 277: <ul> 278: <li>title= 1st_example is title of the run. </li> 279: <li>datafile=data1.txt is the name of the data set. 280: Our example is a six years follow-up survey. It consists 281: in a baseline followed by 3 reinterviews. </li> 282: <li>lastobs= 8600 the program is able to run on a 283: subsample where the last observation number is lastobs. 284: It can be set a bigger number than the real number of 285: observations (e.g. 100000). In this example, maximisation 286: will be done on the 8600 first records. </li> 287: <li>firstpass=1 , lastpass=4 In case of more 288: than two interviews in the survey, the program can be run 289: on selected transitions periods. firstpass=1 means the 290: first interview included in the calculation is the 291: baseline survey. lastpass=4 means that the information 292: brought by the 4th interview is taken into account.</li> 293: </ul> 294: 295:   296: 297: <h4><a name="biaspar-2">Second uncommented 298: line</a></h4> 299: 300: <pre>ftol=1.e-08 stepm=1 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre> 301: 302: <ul> 303: <li>ftol=1e-8 Convergence tolerance on the function 304: value in the maximisation of the likelihood. Choosing a 305: correct value for ftol is difficult. 1e-8 is a correct 306: value for a 32 bits computer.</li> 307: <li>stepm=1 Time unit in months for interpolation. 308: Examples:<ul> 309: <li>If stepm=1, the unit is a month </li> 310: <li>If stepm=4, the unit is a trimester</li> 311: <li>If stepm=12, the unit is a year </li> 312: <li>If stepm=24, the unit is two years</li> 313: <li>... </li> 314: </ul> 315: </li> 316: <li>ncov=2 Number of covariates to be add to the 317: model. The intercept and the age parameter are counting 318: for 2 covariates. For example, if you want to add gender 319: in the covariate vector you must write ncov=3 else 320: ncov=2. </li> 321: <li>nlstate=2 Number of non-absorbing (live) states. 322: Here we have two alive states: disability-free is coded 1 323: and disability is coded 2. </li> 324: <li>ndeath=1 Number of absorbing states. The absorbing 325: state death is coded 3. </li> 326: <li>maxwav=4 Maximum number of waves. The program can 327: not include more than 4 interviews. </li> 328: <li><a name="mle">mle</a>=1 Option for the 329: Maximisation Likelihood Estimation. <ul> 330: <li>If mle=1 the program does the maximisation and 331: the calculation of heath expectancies </li> 332: <li>If mle=0 the program only does the calculation of 333: the health expectancies. </li> 334: </ul> 335: </li> 336: <li>weight=0 Possibility to add weights. <ul> 337: <li>If weight=0 no weights are included </li> 338: <li>If weight=1 the maximisation integrates the 339: weights which are in field <a href="#Weight">4</a></li> 340: </ul> 341: </li> 342: </ul> 343: 344: <h4>Guess values for optimization </h4> 346: 347: You must write the initial guess values of the parameters for 348: optimization. The number of parameters, N depends on the 349: number of absorbing states and non-absorbing states and on the 350: number of covariates. 351: N is given by the formula N=(nlstate + 352: ndeath-1)*nlstate*ncov . 353: 354: Thus in the simple case with 2 covariates (the model is log 355: (pij/pii) = aij + bij * age where intercept and age are the two 356: covariates), and 2 health degrees (1 for disability-free and 2 357: for disability) and 1 absorbing state (3), you must enter 8 358: initials values, a12, b12, a13, b13, a21, b21, a23, b23. You can 359: start with zeros as in this example, but if you have a more 360: precise set (for example from an earlier run) you can enter it 361: and it will speed up them 362: Each of the four lines starts with indices "ij": 363: 364: ij aij bij 365: 366: <blockquote> 367: <pre># Guess values of aij and bij in log (pij/pii) = aij + bij * age 368: 12 -14.155633 0.110794 369: 13 -7.925360 0.032091 370: 21 -1.890135 -0.029473 371: 23 -6.234642 0.022315 </pre> 372: </blockquote> 373: 374: or, to simplify: 375: 376: <blockquote> 377: <pre>12 0.0 0.0 378: 13 0.0 0.0 379: 21 0.0 0.0 380: 23 0.0 0.0</pre> 381: </blockquote> 382: 383: <h4>Guess values for computing variances</h4> 384: 385: This is an output if <a href="#mle">mle</a>=1. But it can be 386: used as an input to get the vairous output data files (Health 387: expectancies, stationary prevalence etc.) and figures without 388: rerunning the rather long maximisation phase (mle=0). 389: 390: The scales are small values for the evaluation of numerical 391: derivatives. These derivatives are used to compute the hessian 392: matrix of the parameters, that is the inverse of the covariance 393: matrix, and the variances of health expectancies. Each line 394: consists in indices "ij" followed by the initial scales 395: (zero to simplify) associated with aij and bij. 396: 397: <ul> 398: <li>If mle=1 you can enter zeros:</li> 399: </ul> 400: 401: <blockquote> 402: <pre># Scales (for hessian or gradient estimation) 403: 12 0. 0. 404: 13 0. 0. 405: 21 0. 0. 406: 23 0. 0. </pre> 407: </blockquote> 408: 409: <ul> 410: <li>If mle=0 you must enter a covariance matrix (usually 411: obtained from an earlier run).</li> 412: </ul> 413: 414: <h4>Covariance matrix of parameters</h4> 415: 416: This is an output if <a href="#mle">mle</a>=1. But it can be 417: used as an input to get the vairous output data files (Health 418: expectancies, stationary prevalence etc.) and figures without 419: rerunning the rather long maximisation phase (mle=0). 420: 421: Each line starts with indices "ijk" followed by the 422: covariances between aij and bij: 423: 424: <pre> 425: 121 Var(a12) 426: 122 Cov(b12,a12) Var(b12) 427: ... 428: 232 Cov(b23,a12) Cov(b23,b12) ... Var (b23) </pre> 429: 430: <ul> 431: <li>If mle=1 you can enter zeros. </li> 432: </ul> 433: 434: <blockquote> 435: <pre># Covariance matrix 436: 121 0. 437: 122 0. 0. 438: 131 0. 0. 0. 439: 132 0. 0. 0. 0. 440: 211 0. 0. 0. 0. 0. 441: 212 0. 0. 0. 0. 0. 0. 442: 231 0. 0. 0. 0. 0. 0. 0. 443: 232 0. 0. 0. 0. 0. 0. 0. 0.</pre> 444: </blockquote> 445: 446: <ul> 447: <li>If mle=0 you must enter a covariance matrix (usually 448: obtained from an earlier run). 449: </li> 450: </ul> 451: 452: <h4><a name="biaspar-l"></a>last 453: uncommented line</h4> 454: 455: <pre>agemin=70 agemax=100 bage=50 fage=100</pre> 456: 457: Once we obtained the estimated parameters, the program is able 458: to calculated stationary prevalence, transitions probabilities 459: and life expectancies at any age. Choice of age ranges is useful 460: for extrapolation. In our data file, ages varies from age 70 to 461: 102. Setting bage=50 and fage=100, makes the program computing 462: life expectancy from age bage to age fage. As we use a model, we 463: can compute life expectancy on a wider age range than the age 464: range from the data. But the model can be rather wrong on big 465: intervals. 466: 467: Similarly, it is possible to get extrapolated stationary 468: prevalence by age raning from agemin to agemax. 469: 470: <ul> 471: <li>agemin= Minimum age for calculation of the 472: stationary prevalence </li> 473: <li>agemax= Maximum age for calculation of the 474: stationary prevalence </li> 475: <li>bage= Minimum age for calculation of the health 476: expectancies </li> 477: <li>fage= Maximum ages for calculation of the health 478: expectancies </li> 479: </ul> 480: 481: <hr> 482: 483: <h2><a name="running"></a>Running Imach 484: with this example</h2> 485: 486: We assume that you entered your <a href="biaspar.txt">1st_example 487: parameter file</a> as explained <a href="#biaspar">above</a>. To 488: run the program you should click on the imach.exe icon and enter 489: the name of the parameter file which is for example <a 490: href="C:\usr\imach\mle\biaspar.txt">C:\usr\imach\mle\biaspar.txt</a> 491: (you also can click on the biaspar.txt icon located in 492: <a href="C:\usr\imach\mle">C:\usr\imach\mle</a> and put it with 493: the mouse on the imach window). 494: 495: 496: The time to converge depends on the step unit that you used (1 497: month is cpu consuming), on the number of cases, and on the 498: number of variables. 499: 500: The program outputs many files. Most of them are files which 501: will be plotted for better understanding. 502: 503: <hr> 504: 505: <h2><a name="output">Output of the program 506: and graphs </a></h2> 507: 508: Once the optimization is finished, some graphics can be made 509: with a grapher. We use Gnuplot which is an interactive plotting 510: program copyrighted but freely distributed. Imach outputs the 511: source of a gnuplot file, named 'graph.gp', which can be directly 512: input into gnuplot. 513: When the running is finished, the user should enter a caracter 514: for plotting and output editing. 515: 516: These caracters are: 517: 518: <ul> 519: <li>'c' to start again the program from the beginning.</li> 520: <li>'g' to made graphics. The output graphs are in GIF format 521: and you have no control over which is produced. If you 522: want to modify the graphics or make another one, you 523: should modify the parameters in the file graph.gp 524: located in imach\bin. A gnuplot reference manual is 525: available <a 526: href="http://www.cs.dartmouth.edu/gnuplot/gnuplot.html">here</a>. 527: </li> 528: <li>'e' opens the index.htm file to edit the 529: output files and graphs. </li> 530: <li>'q' for exiting.</li> 531: </ul> 532: 533: <h5>Results files 534: 535: - <a 536: name="Observed prevalence in each state">Observed prevalence in each state</a> (and at first pass): 539: <a href="prbiaspar.txt">prbiaspar.txt</a> 540: </h5> 541: 542: The first line is the title and displays each field of the 543: file. The first column is age. The fields 2 and 6 are the 544: proportion of individuals in states 1 and 2 respectively as 545: observed during the first exam. Others fields are the numbers of 546: people in states 1, 2 or more. The number of columns increases if 547: the number of states is higher than 2. 548: The header of the file is 549: 550: <pre># Age Prev(1) N(1) N Age Prev(2) N(2) N 551: 70 1.00000 631 631 70 0.00000 0 631 552: 71 0.99681 625 627 71 0.00319 2 627 553: 72 0.97125 1115 1148 72 0.02875 33 1148 </pre> 554: 555: <pre># Age Prev(1) N(1) N Age Prev(2) N(2) N 556: 70 0.95721 604 631 70 0.04279 27 631</pre> 557: 558: It means that at age 70, the prevalence in state 1 is 1.000 559: and in state 2 is 0.00 . At age 71 the number of individuals in 560: state 1 is 625 and in state 2 is 2, hence the total number of 561: people aged 71 is 625+2=627. 562: 563: 564: <h5>- Estimated parameters and 565: covariance matrix: <a href="rbiaspar.txt">rbiaspar.txt</a></h5> 566: 567: This file contains all the maximisation results: 568: 569: <pre> Number of iterations=47 570: -2 log likelihood=46553.005854373667 571: Estimated parameters: a12 = -12.691743 b12 = 0.095819 572: a13 = -7.815392 b13 = 0.031851 573: a21 = -1.809895 b21 = -0.030470 574: a23 = -7.838248 b23 = 0.039490 575: Covariance matrix: Var(a12) = 1.03611e-001 576: Var(b12) = 1.51173e-005 577: Var(a13) = 1.08952e-001 578: Var(b13) = 1.68520e-005 579: Var(a21) = 4.82801e-001 580: Var(b21) = 6.86392e-005 581: Var(a23) = 2.27587e-001 582: Var(b23) = 3.04465e-005 583: </pre> 584: 585: <h5>- Transition probabilities: 586: <a href="pijrbiaspar.txt">pijrbiaspar.txt</a></h5> 587: 588: Here are the transitions probabilities Pij(x, x+nh) where nh 589: is a multiple of 2 years. The first column is the starting age x 590: (from age 50 to 100), the second is age (x+nh) and the others are 591: the transition probabilities p11, p12, p13, p21, p22, p23. For 592: example, line 5 of the file is: 593: 594: <pre> 100 106 0.03286 0.23512 0.73202 0.02330 0.19210 0.78460 </pre> 595: 596: and this means: 597: 598: <pre>p11(100,106)=0.03286 599: p12(100,106)=0.23512 600: p13(100,106)=0.73202 601: p21(100,106)=0.02330 602: p22(100,106)=0.19210 603: p22(100,106)=0.78460 </pre> 604: 605: <h5>- <a 606: name="Stationary prevalence in each state">Stationary prevalence in each state</a>: 608: <a href="plrbiaspar.txt">plrbiaspar.txt</a></h5> 609: 610: <pre>#Age 1-1 2-2 611: 70 0.92274 0.07726 612: 71 0.91420 0.08580 613: 72 0.90481 0.09519 614: 73 0.89453 0.10547</pre> 615: 616: At age 70 the stationary prevalence is 0.92274 in state 1 and 617: 0.07726 in state 2. This stationary prevalence differs from 618: observed prevalence. Here is the point. The observed prevalence 619: at age 70 results from the incidence of disability, incidence of 620: recovery and mortality which occurred in the past of the cohort. 621: Stationary prevalence results from a simulation with actual 622: incidences and mortality (estimated from this cross-longitudinal 623: survey). It is the best predictive value of the prevalence in the 624: future if "nothing changes in the future". This is 625: exactly what demographers do with a Life table. Life expectancy 626: is the expected mean time to survive if observed mortality rates 627: (incidence of mortality) "remains constant" in the 628: future. 629: 630: <h5>- Standard deviation of 631: stationary prevalence: <a 632: href="vplrbiaspar.txt">vplrbiaspar.txt</a></h5> 633: 634: The stationary prevalence has to be compared with the observed 635: prevalence by age. But both are statistical estimates and 636: subjected to stochastic errors due to the size of the sample, the 637: design of the survey, and, for the stationary prevalence to the 638: model used and fitted. It is possible to compute the standard 639: deviation of the stationary prevalence at each age. 640: 641: <h6>Observed and stationary 642: prevalence in state (2=disable) with the confident interval: 643: vbiaspar2.gif</h6> 644: 645: 646: This graph exhibits the stationary prevalence in state (2) with 647: the confidence interval in red. The green curve is the observed 648: prevalence (or proportion of individuals in state (2)). Without 649: discussing the results (it is not the purpose here), we observe 650: that the green curve is rather below the stationary prevalence. 651: It suggests an increase of the disability prevalence in the 652: future. 653: 654: <img src="vbiaspar2.gif" width="400" height="300"> 655: 656: <h6>Convergence to the 657: stationary prevalence of disability: pbiaspar1.gif 658: <img src="pbiaspar1.gif" width="400" height="300"> </h6> 659: 660: This graph plots the conditional transition probabilities from 661: an initial state (1=healthy in red at the bottom, or 2=disable in 662: green on top) at age x to the final state 2=disable at 663: age x+h. Conditional means at the condition to be alive 664: at age x+h which is hP12x + hP22x. The 665: curves hP12x/(hP12x + hP22x) and hP22x/(hP12x 666: + hP22x) converge with h, to the stationary 667: prevalence of disability. In order to get the stationary 668: prevalence at age 70 we should start the process at an earlier 669: age, i.e.50. If the disability state is defined by severe 670: disability criteria with only a few chance to recover, then the 671: incidence of recovery is low and the time to convergence is 672: probably longer. But we don't have experience yet. 673: 674: <h5>- Life expectancies by age 675: and initial health status: <a 676: href="erbiaspar.txt">erbiaspar.txt</a></h5> 677: 678: <pre># Health expectancies 679: # Age 1-1 1-2 2-1 2-2 680: 70 10.7297 2.7809 6.3440 5.9813 681: 71 10.3078 2.8233 5.9295 5.9959 682: 72 9.8927 2.8643 5.5305 6.0033 683: 73 9.4848 2.9036 5.1474 6.0035 </pre> 684: 685: <pre>For example 70 10.7297 2.7809 6.3440 5.9813 means: 686: e11=10.7297 e12=2.7809 e21=6.3440 e22=5.9813</pre> 687: 688: <pre><img src="exbiaspar1.gif" width="400" height="300"><img 689: src="exbiaspar2.gif" width="400" height="300"></pre> 690: 691: For example, life expectancy of a healthy individual at age 70 692: is 10.73 in the healthy state and 2.78 in the disability state 693: (=13.51 years). If he was disable at age 70, his life expectancy 694: will be shorter, 6.34 in the healthy state and 5.98 in the 695: disability state (=12.32 years). The total life expectancy is a 696: weighted mean of both, 13.51 and 12.32; weight is the proportion 697: of people disabled at age 70. In order to get a pure period index 698: (i.e. based only on incidences) we use the <a 699: href="#Stationary prevalence in each state">computed or 700: stationary prevalence</a> at age 70 (i.e. computed from 701: incidences at earlier ages) instead of the <a 702: href="#Observed prevalence in each state">observed prevalence</a> 703: (for example at first exam) (<a href="#Health expectancies">see 704: below</a>). 705: 706: <h5>- Variances of life 707: expectancies by age and initial health status: <a 708: href="vrbiaspar.txt">vrbiaspar.txt</a></h5> 709: 710: For example, the covariances of life expectancies Cov(ei,ej) 711: at age 50 are (line 3) 712: 713: <pre> Cov(e1,e1)=0.4667 Cov(e1,e2)=0.0605=Cov(e2,e1) Cov(e2,e2)=0.0183</pre> 714: 715: <h5>- <a 716: name="Health expectancies">Health 717: expectancies</a> 718: with standard errors in parentheses: <a 719: href="trbiaspar.txt">trbiaspar.txt</a></h5> 720: 721: <pre>#Total LEs with variances: e.. (std) e.1 (std) e.2 (std) </pre> 722: 723: <pre>70 13.42 (0.18) 10.39 (0.15) 3.03 (0.10)70 13.81 (0.18) 11.28 (0.14) 2.53 (0.09) </pre> 724: 725: Thus, at age 70 the total life expectancy, e..=13.42 years is 726: the weighted mean of e1.=13.51 and e2.=12.32 by the stationary 727: prevalence at age 70 which are 0.92274 in state 1 and 0.07726 in 728: state 2, respectively (the sum is equal to one). e.1=10.39 is the 729: Disability-free life expectancy at age 70 (it is again a weighted 730: mean of e11 and e21). e.2=3.03 is also the life expectancy at age 731: 70 to be spent in the disability state. 732: 733: <h6>Total life expectancy by 734: age and health expectancies in states (1=healthy) and (2=disable): 735: ebiaspar.gif</h6> 736: 737: This figure represents the health expectancies and the total 738: life expectancy with the confident interval in dashed curve. 739: 740: <pre> <img src="ebiaspar.gif" width="400" height="300"></pre> 741: 742: Standard deviations (obtained from the information matrix of 743: the model) of these quantities are very useful. 744: Cross-longitudinal surveys are costly and do not involve huge 745: samples, generally a few thousands; therefore it is very 746: important to have an idea of the standard deviation of our 747: estimates. It has been a big challenge to compute the Health 748: Expectancy standard deviations. Don't be confuse: life expectancy 749: is, as any expected value, the mean of a distribution; but here 750: we are not computing the standard deviation of the distribution, 751: but the standard deviation of the estimate of the mean. 752: 753: Our health expectancies estimates vary according to the sample 754: size (and the standard deviations give confidence intervals of 755: the estimate) but also according to the model fitted. Let us 756: explain it in more details. 757: 758: Choosing a model means ar least two kind of choices. First we 759: have to decide the number of disability states. Second we have to 760: design, within the logit model family, the model: variables, 761: covariables, confonding factors etc. to be included. 762: 763: More disability states we have, better is our demographical 764: approach of the disability process, but smaller are the number of 765: transitions between each state and higher is the noise in the 766: measurement. We do not have enough experiments of the various 767: models to summarize the advantages and disadvantages, but it is 768: important to say that even if we had huge and unbiased samples, 769: the total life expectancy computed from a cross-longitudinal 770: survey, varies with the number of states. If we define only two 771: states, alive or dead, we find the usual life expectancy where it 772: is assumed that at each age, people are at the same risk to die. 773: If we are differentiating the alive state into healthy and 774: disable, and as the mortality from the disability state is higher 775: than the mortality from the healthy state, we are introducing 776: heterogeneity in the risk of dying. The total mortality at each 777: age is the weighted mean of the mortality in each state by the 778: prevalence in each state. Therefore if the proportion of people 779: at each age and in each state is different from the stationary 780: equilibrium, there is no reason to find the same total mortality 781: at a particular age. Life expectancy, even if it is a very useful 782: tool, has a very strong hypothesis of homogeneity of the 783: population. Our main purpose is not to measure differential 784: mortality but to measure the expected time in a healthy or 785: disability state in order to maximise the former and minimize the 786: latter. But the differential in mortality complexifies the 787: measurement. 788: 789: Incidences of disability or recovery are not affected by the 790: number of states if these states are independant. But incidences 791: estimates are dependant on the specification of the model. More 792: covariates we added in the logit model better is the model, but 793: some covariates are not well measured, some are confounding 794: factors like in any statistical model. The procedure to "fit 795: the best model' is similar to logistic regression which itself is 796: similar to regression analysis. We haven't yet been sofar because 797: we also have a severe limitation which is the speed of the 798: convergence. On a Pentium III, 500 MHz, even the simplest model, 799: estimated by month on 8,000 people may take 4 hours to converge. 800: Also, the program is not yet a statistical package, which permits 801: a simple writing of the variables and the model to take into 802: account in the maximisation. The actual program allows only to 803: add simple variables without covariations, like age+sex but 804: without age+sex+ age*sex . This can be done from the source code 805: (you have to change three lines in the source code) but will 806: never be general enough. But what is to remember, is that 807: incidences or probability of change from one state to another is 808: affected by the variables specified into the model. 809: 810: Also, the age range of the people interviewed has a link with 811: the age range of the life expectancy which can be estimated by 812: extrapolation. If your sample ranges from age 70 to 95, you can 813: clearly estimate a life expectancy at age 70 and trust your 814: confidence interval which is mostly based on your sample size, 815: but if you want to estimate the life expectancy at age 50, you 816: should rely in your model, but fitting a logistic model on a age 817: range of 70-95 and estimating probabilties of transition out of 818: this age range, say at age 50 is very dangerous. At least you 819: should remember that the confidence interval given by the 820: standard deviation of the health expectancies, are under the 821: strong assumption that your model is the 'true model', which is 822: probably not the case. 823: 824: <h5>- Copy of the parameter 825: file: <a href="orbiaspar.txt">orbiaspar.txt</a></h5> 826: 827: This copy of the parameter file can be useful to re-run the 828: program while saving the old output files. 829: 830: <hr> 831: 832: <h2><a name="example" </a>Trying an example</a></h2> 833: 834: Since you know how to run the program, it is time to test it 835: on your own computer. Try for example on a parameter file named <a 836: href="file://../mytry/imachpar.txt">imachpar.txt</a> which is a 837: copy of mypar.txt 838: included in the subdirectory of imach, mytry. Edit it to change the name of 840: the data file to ..\data\mydata.txt 841: if you don't want to copy it on the same directory. The file mydata.txt is a smaller file of 3,000 843: people but still with 4 waves. 844: 845: Click on the imach.exe icon to open a window. Answer to the 846: question:'Enter the parameter file name:' 847: 848: <table border="1"> 849: <tr> 850: <td width="100%">IMACH, Version 0.63Enter 851: the parameter file name: ..\mytry\imachpar.txt 852: </td> 853: </tr> 854: </table> 855: 856: Most of the data files or image files generated, will use the 857: 'imachpar' string into their name. The running time is about 2-3 858: minutes on a Pentium III. If the execution worked correctly, the 859: outputs files are created in the current directory, and should be 860: the same as the mypar files initially included in the directory mytry. 862: 863: <ul> 864: <li><pre>Output on the screen The output screen looks like <a 865: href="imachrun.LOG">this Log file</a> 866: # 867: 868: title=MLE datafile=..\data\mydata.txt lastobs=3000 firstpass=1 lastpass=3 869: ftol=1.000000e-008 stepm=24 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre> 870: </li> 871: <li><pre>Total number of individuals= 2965, Agemin = 70.00, Agemax= 100.92 872: 873: Warning, no any valid information for:126 line=126 874: Warning, no any valid information for:2307 line=2307 875: Delay (in months) between two waves Min=21 Max=51 Mean=24.495826 876: These lines give some warnings on the data file and also some raw statistics on frequencies of transitions. 877: Age 70 1.=230 loss[1]=3.5% 2.=16 loss[2]=12.5% 1.=222 prev[1]=94.1% 2.=14 878: prev[2]=5.9% 1-1=8 11=200 12=7 13=15 2-1=2 21=6 22=7 23=1 879: Age 102 1.=0 loss[1]=NaNQ% 2.=0 loss[2]=NaNQ% 1.=0 prev[1]=NaNQ% 2.=0 </pre> 880: </li> 881: </ul> 882: 883:   884: 885: <ul> 886: <li>Maximisation with the Powell algorithm. 8 directions are 887: given corresponding to the 8 parameters. this can be 888: rather long to get convergence. 889: 890: Powell iter=1 -2*LL=11531.405658264877 1 0.000000000000 2 891: 0.000000000000 3 892: 0.000000000000 4 0.000000000000 5 0.000000000000 6 893: 0.000000000000 7 894: 0.000000000000 8 0.000000000000 895: 1..........2.................3..........4.................5......... 896: 6................7........8............... 897: Powell iter=23 -2*LL=6744.954108371555 1 -12.967632334283 898: 899: 2 0.135136681033 3 -7.402109728262 4 0.067844593326 900: 5 -0.673601538129 6 -0.006615504377 7 -5.051341616718 901: 8 0.051272038506 902: 1..............2...........3..............4........... 903: 5..........6................7...........8......... 904: #Number of iterations = 23, -2 Log likelihood = 905: 6744.954042573691 906: # Parameters 907: 12 -12.966061 0.135117 908: 13 -7.401109 0.067831 909: 21 -0.672648 -0.006627 910: 23 -5.051297 0.051271 911: </li> 912: <li><pre>Calculation of the hessian matrix. Wait... 913: 12345678.12.13.14.15.16.17.18.23.24.25.26.27.28.34.35.36.37.38.45.46.47.48.56.57.58.67.68.78 914: 915: Inverting the hessian to get the covariance matrix. Wait... 916: 917: #Hessian matrix# 918: 3.344e+002 2.708e+004 -4.586e+001 -3.806e+003 -1.577e+000 -1.313e+002 3.914e-001 3.166e+001 919: 2.708e+004 2.204e+006 -3.805e+003 -3.174e+005 -1.303e+002 -1.091e+004 2.967e+001 2.399e+003 920: -4.586e+001 -3.805e+003 4.044e+002 3.197e+004 2.431e-002 1.995e+000 1.783e-001 1.486e+001 921: -3.806e+003 -3.174e+005 3.197e+004 2.541e+006 2.436e+000 2.051e+002 1.483e+001 1.244e+003 922: -1.577e+000 -1.303e+002 2.431e-002 2.436e+000 1.093e+002 8.979e+003 -3.402e+001 -2.843e+003 923: -1.313e+002 -1.091e+004 1.995e+000 2.051e+002 8.979e+003 7.420e+005 -2.842e+003 -2.388e+005 924: 3.914e-001 2.967e+001 1.783e-001 1.483e+001 -3.402e+001 -2.842e+003 1.494e+002 1.251e+004 925: 3.166e+001 2.399e+003 1.486e+001 1.244e+003 -2.843e+003 -2.388e+005 1.251e+004 1.053e+006 926: # Scales 927: 12 1.00000e-004 1.00000e-006 928: 13 1.00000e-004 1.00000e-006 929: 21 1.00000e-003 1.00000e-005 930: 23 1.00000e-004 1.00000e-005 931: # Covariance 932: 1 5.90661e-001 933: 2 -7.26732e-003 8.98810e-005 934: 3 8.80177e-002 -1.12706e-003 5.15824e-001 935: 4 -1.13082e-003 1.45267e-005 -6.50070e-003 8.23270e-005 936: 5 9.31265e-003 -1.16106e-004 6.00210e-004 -8.04151e-006 1.75753e+000 937: 6 -1.15664e-004 1.44850e-006 -7.79995e-006 1.04770e-007 -2.12929e-002 2.59422e-004 938: 7 1.35103e-003 -1.75392e-005 -6.38237e-004 7.85424e-006 4.02601e-001 -4.86776e-003 1.32682e+000 939: 8 -1.82421e-005 2.35811e-007 7.75503e-006 -9.58687e-008 -4.86589e-003 5.91641e-005 -1.57767e-002 1.88622e-004 940: # agemin agemax for lifexpectancy, bage fage (if mle==0 ie no data nor Max likelihood). 941: 942: 943: agemin=70 agemax=100 bage=50 fage=100 944: Computing prevalence limit: result on file 'plrmypar.txt' 945: Computing pij: result on file 'pijrmypar.txt' 946: Computing Health Expectancies: result on file 'ermypar.txt' 947: Computing Variance-covariance of DFLEs: file 'vrmypar.txt' 948: Computing Total LEs with variances: file 'trmypar.txt' 949: Computing Variance-covariance of Prevalence limit: file 'vplrmypar.txt' 950: End of Imach 951: </pre> 952: </li> 953: </ul> 954: 955: Once the running is finished, the program 956: requires a caracter: 957: 958: <table border="1"> 959: <tr> 960: <td width="100%">Type g for plotting (available 961: if mle=1), e to edit output files, c to start again,and 962: q for exiting: 963: </td> 964: </tr> 965: </table> 966: 967: First you should enter g to 968: make the figures and then you can edit all the results by typing e. 969: 970: 971: <ul> 972: <li>Outputs files 973: - index.htm, this file is the master file on which you 974: should click first. 975: - Observed prevalence in each state: <a 976: href="..\mytry\prmypar.txt">mypar.txt</a> 977: - Estimated parameters and the covariance matrix: <a 978: href="..\mytry\rmypar.txt">rmypar.txt</a> 979: - Stationary prevalence in each state: <a 980: href="..\mytry\plrmypar.txt">plrmypar.txt</a> 981: - Transition probabilities: <a 982: href="..\mytry\pijrmypar.txt">pijrmypar.txt</a> 983: - Copy of the parameter file: <a 984: href="..\mytry\ormypar.txt">ormypar.txt</a> 985: - Life expectancies by age and initial health status: <a 986: href="..\mytry\ermypar.txt">ermypar.txt</a> 987: - Variances of life expectancies by age and initial 988: health status: <a href="..\mytry\vrmypar.txt">vrmypar.txt</a> 989: 990: - Health expectancies with their variances: <a 991: href="..\mytry\trmypar.txt">trmypar.txt</a> 992: - Standard deviation of stationary prevalence: <a 993: href="..\mytry\vplrmypar.txt">vplrmypar.txt</a> 994: 995: </li> 996: <li>Graphs 997: 998: -<a href="..\mytry\vmypar1.gif">Observed and stationary 999: prevalence in state (1) with the confident interval</a> 1000: -<a href="..\mytry\vmypar2.gif">Observed and stationary 1001: prevalence in state (2) with the confident interval</a> 1002: -<a href="..\mytry\exmypar1.gif">Health life expectancies 1003: by age and initial health state (1)</a> 1004: -<a href="..\mytry\exmypar2.gif">Health life expectancies 1005: by age and initial health state (2)</a> 1006: -<a href="..\mytry\emypar.gif">Total life expectancy by 1007: age and health expectancies in states (1) and (2).</a> </li> 1008: </ul> 1009: 1010: This software have been partly granted by <a 1011: href="http://euroreves.ined.fr">Euro-REVES</a>, a concerted 1012: action from the European Union. It will be copyrighted 1013: identically to a GNU software product, i.e. program and software 1014: can be distributed freely for non commercial use. Sources are not 1015: widely distributed today. You can get them by asking us with a 1016: simple justification (name, email, institute) <a 1017: href="mailto:brouard@ined.fr">mailto:brouard@ined.fr</a> and <a 1018: href="mailto:lievre@ined.fr">mailto:lievre@ined.fr</a> . 1019: 1020: Latest version (0.63 of 16 march 2000) can be accessed at <a 1021: href="http://euroeves.ined.fr/imach">http://euroreves.ined.fr/imach</a> 1022: 1023: </body> 1024: </html>