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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:  9: </head> 10: 11: <body bgcolor="#FFFFFF"> 12: 13: <hr size="3" color="#EC5E5E"> 14: 15: <h1 align="center">Computing Health 16: Expectancies using IMaCh</h1> 17: 18: <h1 align="center">(a Maximum 19: Likelihood Computer Program using Interpolation of Markov Chains)</h1> 20: 21:   22: 23: <a href="http://www.ined.fr/"><img 24: src="logo-ined.gif" border="0" width="151" height="76"></a><img 25: src="euroreves2.gif" width="151" height="75"> 26: 27: <h3 align="center"><a href="http://www.ined.fr/">INED</a> and <a 29: href="http://euroreves.ined.fr">EUROREVES</a></h3> 30: 31: March 32: 2000 33: 34: <hr size="3" color="#EC5E5E"> 35: 36: Authors of the 37: program: <a href="http://sauvy.ined.fr/brouard">Nicolas Brouard</a>, senior researcher at the <a 40: href="http://www.ined.fr">Institut 41: National d'Etudes Démographiques</a> (INED, Paris) in the "Mortality, 43: Health and Epidemiology" Research Unit 44: 45: and Agnès 46: Lièvre 47: 48: 49: <h4>Contribution to the mathematics: C. R. 50: Heathcote (Australian 51: National University, Canberra).</h4> 52: 53: <h4>Contact: Agnès Lièvre (<a 54: href="mailto:lievre@ined.fr">lievre@ined.fr</a>) </h4> 56: 57: <hr> 58: 59: <ul> 60: <li><a href="#intro">Introduction</a> </li> 61: <li>The detailed statistical model (<a href="docmath.pdf">PDF 62: version</a>),(<a href="docmath.ps">ps version</a>) </li> 63: <li><a href="#data">On what kind of data can it be used?</a></li> 64: <li><a href="#datafile">The data file</a> </li> 65: <li><a href="#biaspar">The parameter file</a> </li> 66: <li><a href="#running">Running Imach</a> </li> 67: <li><a href="#output">Output files and graphs</a> </li> 68: <li><a href="#example">Exemple</a> </li> 69: </ul> 70: 71: <hr> 72: 73: <h2><a name="intro">Introduction</a></h2> 74: 75: This program computes Healthy Life Expectancies from cross-longitudinal 76: data using the methodology pioneered by Laditka and Wolf (1). 77: Within the family of Health Expectancies (HE), Disability-free 78: life expectancy (DFLE) is probably the most important index to 79: monitor. In low mortality countries, there is a fear that when 80: mortality declines, the increase in DFLE is not proportionate to 81: the increase in total Life expectancy. This case is called the Expansion 82: of morbidity. Most of the data collected today, in 83: particular by the international <a href="http://euroreves/reves">REVES</a> 84: network on Health expectancy, and most HE indices based on these 85: data, are cross-sectional. It means that the information 86: collected comes from a single cross-sectional survey: people from 87: various ages (but mostly old people) are surveyed on their health 88: status at a single date. Proportion of people disabled at each 89: age, can then be measured at that date. This age-specific 90: prevalence curve is then used to distinguish, within the 91: stationary population (which, by definition, is the life table 92: estimated from the vital statistics on mortality at the same 93: date), the disable population from the disability-free 94: population. Life expectancy (LE) (or total population divided by 95: the yearly number of births or deaths of this stationary 96: population) is then decomposed into DFLE and DLE. This method of 97: computing HE is usually called the Sullivan method (from the name 98: of the author who first described it). 99: 100: Age-specific proportions of people disable are very difficult 101: to forecast because each proportion corresponds to historical 102: conditions of the cohort and it is the result of the historical 103: flows from entering disability and recovering in the past until 104: today. The age-specific intensities (or incidence rates) of 105: entering disability or recovering a good health, are reflecting 106: actual conditions and therefore can be used at each age to 107: forecast the future of this cohort. For example if a country is 108: improving its technology of prosthesis, the incidence of 109: recovering the ability to walk will be higher at each (old) age, 110: but the prevalence of disability will only slightly reflect an 111: improve because the prevalence is mostly affected by the history 112: of the cohort and not by recent period effects. To measure the 113: period improvement we have to simulate the future of a cohort of 114: new-borns entering or leaving at each age the disability state or 115: dying according to the incidence rates measured today on 116: different cohorts. The proportion of people disabled at each age 117: in this simulated cohort will be much lower (using the exemple of 118: an improvement) that the proportions observed at each age in a 119: cross-sectional survey. This new prevalence curve introduced in a 120: life table will give a much more actual and realistic HE level 121: than the Sullivan method which mostly measured the History of 122: health conditions in this country. 123: 124: Therefore, the main question is how to measure incidence rates 125: from cross-longitudinal surveys? This is the goal of the IMaCH 126: program. From your data and using IMaCH you can estimate period 127: HE and not only Sullivan's HE. Also the standard errors of the HE 128: are computed. 129: 130: A cross-longitudinal survey consists in a first survey 131: ("cross") where individuals from different ages are 132: interviewed on their health status or degree of disability. At 133: least a second wave of interviews ("longitudinal") 134: should measure each new individual health status. Health 135: expectancies are computed from the transitions observed between 136: waves and are computed for each degree of severity of disability 137: (number of life states). More degrees you consider, more time is 138: necessary to reach the Maximum Likelihood of the parameters 139: involved in the model. Considering only two states of disability 140: (disable and healthy) is generally enough but the computer 141: program works also with more health statuses. 142: 143: The simplest model is the multinomial logistic model where pij 144: is the probability to be observed in state j at the second 145: wave conditional to be observed in state i at the first 146: wave. Therefore a simple model is: log(pij/pii)= aij + 147: bij*age+ cij*sex, where 'age' is age and 'sex' 148: is a covariate. The advantage that this computer program claims, 149: comes from that if the delay between waves is not identical for 150: each individual, or if some individual missed an interview, the 151: information is not rounded or lost, but taken into account using 152: an interpolation or extrapolation. hPijx is the 153: probability to be observed in state i at age x+h 154: conditional to the observed state i at age x. The 155: delay 'h' can be split into an exact number (nh*stepm) 156: of unobserved intermediate states. This elementary transition (by 157: month or quarter trimester, semester or year) is modeled as a 158: multinomial logistic. The hPx matrix is simply the matrix 159: product of nh*stepm elementary matrices and the 160: contribution of each individual to the likelihood is simply hPijx. 161: 162: 163: 164: The program presented in this manual is a quite general 165: program named IMaCh (for Interpolated 166: MArkov CHain), designed to 167: analyse transition data from longitudinal surveys. The first step 168: is the parameters estimation of a transition probabilities model 169: between an initial status and a final status. From there, the 170: computer program produces some indicators such as observed and 171: stationary prevalence, life expectancies and their variances and 172: graphs. Our transition model consists in absorbing and 173: non-absorbing states with the possibility of return across the 174: non-absorbing states. The main advantage of this package, 175: compared to other programs for the analysis of transition data 176: (For example: Proc Catmod of SAS®) is that the whole 177: individual information is used even if an interview is missing, a 178: status or a date is unknown or when the delay between waves is 179: not identical for each individual. The program can be executed 180: according to parameters: selection of a sub-sample, number of 181: absorbing and non-absorbing states, number of waves taken in 182: account (the user inputs the first and the last interview), a 183: tolerance level for the maximization function, the periodicity of 184: the transitions (we can compute annual, quaterly or monthly 185: transitions), covariates in the model. It works on Windows or on 186: Unix. 187: 188: 189: <hr> 190: 191: (1) Laditka, Sarah B. and Wolf, Douglas A. (1998), "New 192: Methods for Analyzing Active Life Expectancy". Journal of 193: Aging and Health. Vol 10, No. 2. 194: 195: <hr> 196: 197: <h2><a name="data">On what kind of data can 198: it be used?</a></h2> 199: 200: The minimum data required for a transition model is the 201: recording of a set of individuals interviewed at a first date and 202: interviewed again at least one another time. From the 203: observations of an individual, we obtain a follow-up over time of 204: the occurrence of a specific event. In this documentation, the 205: event is related to health status at older ages, but the program 206: can be applied on a lot of longitudinal studies in different 207: contexts. To build the data file explained into the next section, 208: you must have the month and year of each interview and the 209: corresponding health status. But in order to get age, date of 210: birth (month and year) is required (missing values is allowed for 211: month). Date of death (month and year) is an important 212: information also required if the individual is dead. Shorter 213: steps (i.e. a month) will more closely take into account the 214: survival time after the last interview. 215: 216: <hr> 217: 218: <h2><a name="datafile">The data file</a></h2> 219: 220: In this example, 8,000 people have been interviewed in a 221: cross-longitudinal survey of 4 waves (1984, 1986, 1988, 1990). 222: Some people missed 1, 2 or 3 interviews. Health statuses are 223: healthy (1) and disable (2). The survey is not a real one. It is 224: a simulation of the American Longitudinal Survey on Aging. The 225: disability state is defined if the individual missed one of four 226: ADL (Activity of daily living, like bathing, eating, walking). 227: Therefore, even is the individuals interviewed in the sample are 228: virtual, the information brought with this sample is close to the 229: situation of the United States. Sex is not recorded is this 230: sample. 231: 232: Each line of the data set (named <a href="data1.txt">data1.txt</a> 233: in this first example) is an individual record which fields are: 234: 235: <ul> 236: <li>Index number: positive number (field 1) </li> 237: <li>First covariate positive number (field 2) </li> 238: <li>Second covariate positive number (field 3) </li> 239: <li><a name="Weight">Weight</a>: positive number 240: (field 4) . In most surveys individuals are weighted 241: according to the stratification of the sample.</li> 242: <li>Date of birth: coded as mm/yyyy. Missing dates are 243: coded as 99/9999 (field 5) </li> 244: <li>Date of death: coded as mm/yyyy. Missing dates are 245: coded as 99/9999 (field 6) </li> 246: <li>Date of first interview: coded as mm/yyyy. Missing 247: dates are coded as 99/9999 (field 7) </li> 248: <li>Status at first interview: positive number. 249: Missing values ar coded -1. (field 8) </li> 250: <li>Date of second interview: coded as mm/yyyy. 251: Missing dates are coded as 99/9999 (field 9) </li> 252: <li>Status at second interview positive 253: number. Missing values ar coded -1. (field 10) </li> 254: <li>Date of third interview: coded as mm/yyyy. Missing 255: dates are coded as 99/9999 (field 11) </li> 256: <li>Status at third interview positive 257: number. Missing values ar coded -1. (field 12) </li> 258: <li>Date of fourth interview: coded as mm/yyyy. 259: Missing dates are coded as 99/9999 (field 13) </li> 260: <li>Status at fourth interview positive 261: number. Missing values are coded -1. (field 14) </li> 262: <li>etc</li> 263: </ul> 264: 265:   266: 267: If your longitudinal survey do not include information about 268: weights or covariates, you must fill the column with a number 269: (e.g. 1) because a missing field is not allowed. 270: 271: <hr> 272: 273: <h2>Your first example parameter file<a 274: href="http://euroreves.ined.fr/imach"></a><a name="uio"></a></h2> 275: 276: <h2><a name="biaspar"></a>#Imach version 0.63, February 2000, 277: INED-EUROREVES </h2> 278: 279: This is a comment. Comments start with a '#'. 280: 281: <h4>First uncommented line</h4> 282: 283: <pre>title=1st_example datafile=data1.txt lastobs=8600 firstpass=1 lastpass=4</pre> 284: 285: <ul> 286: <li>title= 1st_example is title of the run. </li> 287: <li>datafile=data1.txt is the name of the data set. 288: Our example is a six years follow-up survey. It consists 289: in a baseline followed by 3 reinterviews. </li> 290: <li>lastobs= 8600 the program is able to run on a 291: subsample where the last observation number is lastobs. 292: It can be set a bigger number than the real number of 293: observations (e.g. 100000). In this example, maximisation 294: will be done on the 8600 first records. </li> 295: <li>firstpass=1 , lastpass=4 In case of more 296: than two interviews in the survey, the program can be run 297: on selected transitions periods. firstpass=1 means the 298: first interview included in the calculation is the 299: baseline survey. lastpass=4 means that the information 300: brought by the 4th interview is taken into account.</li> 301: </ul> 302: 303:   304: 305: <h4><a name="biaspar-2">Second uncommented 306: line</a></h4> 307: 308: <pre>ftol=1.e-08 stepm=1 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre> 309: 310: <ul> 311: <li>ftol=1e-8 Convergence tolerance on the function 312: value in the maximisation of the likelihood. Choosing a 313: correct value for ftol is difficult. 1e-8 is a correct 314: value for a 32 bits computer.</li> 315: <li>stepm=1 Time unit in months for interpolation. 316: Examples:<ul> 317: <li>If stepm=1, the unit is a month </li> 318: <li>If stepm=4, the unit is a trimester</li> 319: <li>If stepm=12, the unit is a year </li> 320: <li>If stepm=24, the unit is two years</li> 321: <li>... </li> 322: </ul> 323: </li> 324: <li>ncov=2 Number of covariates in the datafile. The 325: intercept and the age parameter are counting for 2 326: covariates.</li> 327: <li>nlstate=2 Number of non-absorbing (alive) states. 328: Here we have two alive states: disability-free is coded 1 329: and disability is coded 2. </li> 330: <li>ndeath=1 Number of absorbing states. The absorbing 331: state death is coded 3. </li> 332: <li>maxwav=4 Number of waves in the datafile.</li> 333: <li><a name="mle">mle</a>=1 Option for the 334: Maximisation Likelihood Estimation. <ul> 335: <li>If mle=1 the program does the maximisation and 336: the calculation of health expectancies </li> 337: <li>If mle=0 the program only does the calculation of 338: the health expectancies. </li> 339: </ul> 340: </li> 341: <li>weight=0 Possibility to add weights. <ul> 342: <li>If weight=0 no weights are included </li> 343: <li>If weight=1 the maximisation integrates the 344: weights which are in field <a href="#Weight">4</a></li> 345: </ul> 346: </li> 347: </ul> 348: 349: <h4>Covariates</h4> 350: 351: Intercept and age are systematically included in the model. 352: Additional covariates can be included with the command 353: 354: <pre>model=list of covariates</pre> 355: 356: <ul> 357: <li>if model=. then no covariates are 358: included</li> 359: <li>if model=V1 the model includes the first 360: covariate (field 2)</li> 361: <li>if model=V2 the model includes the 362: second covariate (field 3)</li> 363: <li>if model=V1+V2 the model includes the 364: first and the second covariate (fields 2 and 3)</li> 365: <li>if model=V1*V2 the model includes the 366: product of the first and the second covariate (fields 2 367: and 3)</li> 368: </ul> 369: 370: <h4>Guess values for optimization </h4> 372: 373: You must write the initial guess values of the parameters for 374: optimization. The number of parameters, N depends on the 375: number of absorbing states and non-absorbing states and on the 376: number of covariates. 377: N is given by the formula N=(nlstate + 378: ndeath-1)*nlstate*ncov . 379: 380: Thus in the simple case with 2 covariates (the model is log 381: (pij/pii) = aij + bij * age where intercept and age are the two 382: covariates), and 2 health degrees (1 for disability-free and 2 383: for disability) and 1 absorbing state (3), you must enter 8 384: initials values, a12, b12, a13, b13, a21, b21, a23, b23. You can 385: start with zeros as in this example, but if you have a more 386: precise set (for example from an earlier run) you can enter it 387: and it will speed up them 388: Each of the four lines starts with indices "ij": 389: 390: ij 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: 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 vairous 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 vairous 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><a name="biaspar-l"></a>last 479: uncommented line</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 ranges is useful 486: for extrapolation. In our data file, ages varies from age 70 to 487: 102. Setting bage=50 and fage=100, makes the program computing 488: life expectancy from age bage to age fage. As we use a model, we 489: can compute life expectancy on a wider age range than the age 490: range from the data. But the model can be rather wrong on big 491: intervals. 492: 493: Similarly, it is possible to get extrapolated stationary 494: prevalence by age raning from agemin to agemax. 495: 496: <ul> 497: <li>agemin= Minimum age for calculation of the 498: stationary prevalence </li> 499: <li>agemax= Maximum age for calculation of the 500: stationary prevalence </li> 501: <li>bage= Minimum age for calculation of the health 502: expectancies </li> 503: <li>fage= Maximum ages for calculation of the health 504: expectancies </li> 505: </ul> 506: 507: <hr> 508: 509: <h2><a name="running"></a>Running Imach 510: with this example</h2> 511: 512: We assume that you entered your <a href="biaspar.txt">1st_example 513: parameter file</a> as explained <a href="#biaspar">above</a>. To 514: run the program you should click on the imach.exe icon and enter 515: the name of the parameter file which is for example <a 516: href="C:\usr\imach\mle\biaspar.txt">C:\usr\imach\mle\biaspar.txt</a> 517: (you also can click on the biaspar.txt icon located in 518: <a href="C:\usr\imach\mle">C:\usr\imach\mle</a> and put it with 519: the mouse on the imach window). 520: 521: 522: The time to converge depends on the step unit that you used (1 523: month is cpu consuming), on the number of cases, and on the 524: number of variables. 525: 526: The program outputs many files. Most of them are files which 527: will be plotted for better understanding. 528: 529: <hr> 530: 531: <h2><a name="output">Output of the program 532: and graphs </a></h2> 533: 534: Once the optimization is finished, some graphics can be made 535: with a grapher. We use Gnuplot which is an interactive plotting 536: program copyrighted but freely distributed. Imach outputs the 537: source of a gnuplot file, named 'graph.gp', which can be directly 538: input into gnuplot. 539: When the running is finished, the user should enter a caracter 540: for plotting and output editing. 541: 542: These caracters are: 543: 544: <ul> 545: <li>'c' to start again the program from the beginning.</li> 546: <li>'g' to made graphics. The output graphs are in GIF format 547: and you have no control over which is produced. If you 548: want to modify the graphics or make another one, you 549: should modify the parameters in the file graph.gp 550: located in imach\bin. A gnuplot reference manual is 551: available <a 552: href="http://www.cs.dartmouth.edu/gnuplot/gnuplot.html">here</a>. 553: </li> 554: <li>'e' opens the index.htm file to edit the 555: output files and graphs. </li> 556: <li>'q' for exiting.</li> 557: </ul> 558: 559: <h5>Results files 560: 561: - <a 562: name="Observed prevalence in each state">Observed prevalence in each state</a> (and at first pass): 565: <a href="prbiaspar.txt">prbiaspar.txt</a> 566: </h5> 567: 568: The first line is the title and displays each field of the 569: file. The first column is age. The fields 2 and 6 are the 570: proportion of individuals in states 1 and 2 respectively as 571: observed during the first exam. Others fields are the numbers of 572: people in states 1, 2 or more. The number of columns increases if 573: the number of states is higher than 2. 574: The header of the file is 575: 576: <pre># Age Prev(1) N(1) N Age Prev(2) N(2) N 577: 70 1.00000 631 631 70 0.00000 0 631 578: 71 0.99681 625 627 71 0.00319 2 627 579: 72 0.97125 1115 1148 72 0.02875 33 1148 </pre> 580: 581: <pre># Age Prev(1) N(1) N Age Prev(2) N(2) N 582: 70 0.95721 604 631 70 0.04279 27 631</pre> 583: 584: It means that at age 70, the prevalence in state 1 is 1.000 585: and in state 2 is 0.00 . At age 71 the number of individuals in 586: state 1 is 625 and in state 2 is 2, hence the total number of 587: people aged 71 is 625+2=627. 588: 589: 590: <h5>- Estimated parameters and 591: covariance matrix: <a href="rbiaspar.txt">rbiaspar.txt</a></h5> 592: 593: This file contains all the maximisation results: 594: 595: <pre> Number of iterations=47 596: -2 log likelihood=46553.005854373667 597: Estimated parameters: a12 = -12.691743 b12 = 0.095819 598: a13 = -7.815392 b13 = 0.031851 599: a21 = -1.809895 b21 = -0.030470 600: a23 = -7.838248 b23 = 0.039490 601: Covariance matrix: Var(a12) = 1.03611e-001 602: Var(b12) = 1.51173e-005 603: Var(a13) = 1.08952e-001 604: Var(b13) = 1.68520e-005 605: Var(a21) = 4.82801e-001 606: Var(b21) = 6.86392e-005 607: Var(a23) = 2.27587e-001 608: Var(b23) = 3.04465e-005 609: </pre> 610: 611: <h5>- Transition probabilities: 612: <a href="pijrbiaspar.txt">pijrbiaspar.txt</a></h5> 613: 614: Here are the transitions probabilities Pij(x, x+nh) where nh 615: is a multiple of 2 years. The first column is the starting age x 616: (from age 50 to 100), the second is age (x+nh) and the others are 617: the transition probabilities p11, p12, p13, p21, p22, p23. For 618: example, line 5 of the file is: 619: 620: <pre> 100 106 0.03286 0.23512 0.73202 0.02330 0.19210 0.78460 </pre> 621: 622: and this means: 623: 624: <pre>p11(100,106)=0.03286 625: p12(100,106)=0.23512 626: p13(100,106)=0.73202 627: p21(100,106)=0.02330 628: p22(100,106)=0.19210 629: p22(100,106)=0.78460 </pre> 630: 631: <h5>- <a 632: name="Stationary prevalence in each state">Stationary prevalence in each state</a>: 634: <a href="plrbiaspar.txt">plrbiaspar.txt</a></h5> 635: 636: <pre>#Age 1-1 2-2 637: 70 0.92274 0.07726 638: 71 0.91420 0.08580 639: 72 0.90481 0.09519 640: 73 0.89453 0.10547</pre> 641: 642: At age 70 the stationary prevalence is 0.92274 in state 1 and 643: 0.07726 in state 2. This stationary prevalence differs from 644: observed prevalence. Here is the point. The observed prevalence 645: at age 70 results from the incidence of disability, incidence of 646: recovery and mortality which occurred in the past of the cohort. 647: Stationary prevalence results from a simulation with actual 648: incidences and mortality (estimated from this cross-longitudinal 649: survey). It is the best predictive value of the prevalence in the 650: future if "nothing changes in the future". This is 651: exactly what demographers do with a Life table. Life expectancy 652: is the expected mean time to survive if observed mortality rates 653: (incidence of mortality) "remains constant" in the 654: future. 655: 656: <h5>- Standard deviation of 657: stationary prevalence: <a 658: href="vplrbiaspar.txt">vplrbiaspar.txt</a></h5> 659: 660: The stationary prevalence has to be compared with the observed 661: prevalence by age. But both are statistical estimates and 662: subjected to stochastic errors due to the size of the sample, the 663: design of the survey, and, for the stationary prevalence to the 664: model used and fitted. It is possible to compute the standard 665: deviation of the stationary prevalence at each age. 666: 667: <h6>Observed and stationary 668: prevalence in state (2=disable) with the confident interval: 669: vbiaspar2.gif</h6> 670: 671: 672: This graph exhibits the stationary prevalence in state (2) with 673: the confidence interval in red. The green curve is the observed 674: prevalence (or proportion of individuals in state (2)). Without 675: discussing the results (it is not the purpose here), we observe 676: that the green curve is rather below the stationary prevalence. 677: It suggests an increase of the disability prevalence in the 678: future. 679: 680: <img src="vbiaspar2.gif" width="400" height="300"> 681: 682: <h6>Convergence to the 683: stationary prevalence of disability: pbiaspar1.gif 684: <img src="pbiaspar1.gif" width="400" height="300"> </h6> 685: 686: This graph plots the conditional transition probabilities from 687: an initial state (1=healthy in red at the bottom, or 2=disable in 688: green on top) at age x to the final state 2=disable at 689: age x+h. Conditional means at the condition to be alive 690: at age x+h which is hP12x + hP22x. The 691: curves hP12x/(hP12x + hP22x) and hP22x/(hP12x 692: + hP22x) converge with h, to the stationary 693: prevalence of disability. In order to get the stationary 694: prevalence at age 70 we should start the process at an earlier 695: age, i.e.50. If the disability state is defined by severe 696: disability criteria with only a few chance to recover, then the 697: incidence of recovery is low and the time to convergence is 698: probably longer. But we don't have experience yet. 699: 700: <h5>- Life expectancies by age 701: and initial health status: <a 702: href="erbiaspar.txt">erbiaspar.txt</a></h5> 703: 704: <pre># Health expectancies 705: # Age 1-1 1-2 2-1 2-2 706: 70 10.7297 2.7809 6.3440 5.9813 707: 71 10.3078 2.8233 5.9295 5.9959 708: 72 9.8927 2.8643 5.5305 6.0033 709: 73 9.4848 2.9036 5.1474 6.0035 </pre> 710: 711: <pre>For example 70 10.7297 2.7809 6.3440 5.9813 means: 712: e11=10.7297 e12=2.7809 e21=6.3440 e22=5.9813</pre> 713: 714: <pre><img src="exbiaspar1.gif" width="400" height="300"><img 715: src="exbiaspar2.gif" width="400" height="300"></pre> 716: 717: For example, life expectancy of a healthy individual at age 70 718: is 10.73 in the healthy state and 2.78 in the disability state 719: (=13.51 years). If he was disable at age 70, his life expectancy 720: will be shorter, 6.34 in the healthy state and 5.98 in the 721: disability state (=12.32 years). The total life expectancy is a 722: weighted mean of both, 13.51 and 12.32; weight is the proportion 723: of people disabled at age 70. In order to get a pure period index 724: (i.e. based only on incidences) we use the <a 725: href="#Stationary prevalence in each state">computed or 726: stationary prevalence</a> at age 70 (i.e. computed from 727: incidences at earlier ages) instead of the <a 728: href="#Observed prevalence in each state">observed prevalence</a> 729: (for example at first exam) (<a href="#Health expectancies">see 730: below</a>). 731: 732: <h5>- Variances of life 733: expectancies by age and initial health status: <a 734: href="vrbiaspar.txt">vrbiaspar.txt</a></h5> 735: 736: For example, the covariances of life expectancies Cov(ei,ej) 737: at age 50 are (line 3) 738: 739: <pre> Cov(e1,e1)=0.4667 Cov(e1,e2)=0.0605=Cov(e2,e1) Cov(e2,e2)=0.0183</pre> 740: 741: <h5>- <a 742: name="Health expectancies">Health 743: expectancies</a> 744: with standard errors in parentheses: <a 745: href="trbiaspar.txt">trbiaspar.txt</a></h5> 746: 747: <pre>#Total LEs with variances: e.. (std) e.1 (std) e.2 (std) </pre> 748: 749: <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> 750: 751: Thus, at age 70 the total life expectancy, e..=13.42 years is 752: the weighted mean of e1.=13.51 and e2.=12.32 by the stationary 753: prevalence at age 70 which are 0.92274 in state 1 and 0.07726 in 754: state 2, respectively (the sum is equal to one). e.1=10.39 is the 755: Disability-free life expectancy at age 70 (it is again a weighted 756: mean of e11 and e21). e.2=3.03 is also the life expectancy at age 757: 70 to be spent in the disability state. 758: 759: <h6>Total life expectancy by 760: age and health expectancies in states (1=healthy) and (2=disable): 761: ebiaspar.gif</h6> 762: 763: This figure represents the health expectancies and the total 764: life expectancy with the confident interval in dashed curve. 765: 766: <pre> <img src="ebiaspar.gif" width="400" height="300"></pre> 767: 768: Standard deviations (obtained from the information matrix of 769: the model) of these quantities are very useful. 770: Cross-longitudinal surveys are costly and do not involve huge 771: samples, generally a few thousands; therefore it is very 772: important to have an idea of the standard deviation of our 773: estimates. It has been a big challenge to compute the Health 774: Expectancy standard deviations. Don't be confuse: life expectancy 775: is, as any expected value, the mean of a distribution; but here 776: we are not computing the standard deviation of the distribution, 777: but the standard deviation of the estimate of the mean. 778: 779: Our health expectancies estimates vary according to the sample 780: size (and the standard deviations give confidence intervals of 781: the estimate) but also according to the model fitted. Let us 782: explain it in more details. 783: 784: Choosing a model means ar least two kind of choices. First we 785: have to decide the number of disability states. Second we have to 786: design, within the logit model family, the model: variables, 787: covariables, confonding factors etc. to be included. 788: 789: More disability states we have, better is our demographical 790: approach of the disability process, but smaller are the number of 791: transitions between each state and higher is the noise in the 792: measurement. We do not have enough experiments of the various 793: models to summarize the advantages and disadvantages, but it is 794: important to say that even if we had huge and unbiased samples, 795: the total life expectancy computed from a cross-longitudinal 796: survey, varies with the number of states. If we define only two 797: states, alive or dead, we find the usual life expectancy where it 798: is assumed that at each age, people are at the same risk to die. 799: If we are differentiating the alive state into healthy and 800: disable, and as the mortality from the disability state is higher 801: than the mortality from the healthy state, we are introducing 802: heterogeneity in the risk of dying. The total mortality at each 803: age is the weighted mean of the mortality in each state by the 804: prevalence in each state. Therefore if the proportion of people 805: at each age and in each state is different from the stationary 806: equilibrium, there is no reason to find the same total mortality 807: at a particular age. Life expectancy, even if it is a very useful 808: tool, has a very strong hypothesis of homogeneity of the 809: population. Our main purpose is not to measure differential 810: mortality but to measure the expected time in a healthy or 811: disability state in order to maximise the former and minimize the 812: latter. But the differential in mortality complexifies the 813: measurement. 814: 815: Incidences of disability or recovery are not affected by the 816: number of states if these states are independant. But incidences 817: estimates are dependant on the specification of the model. More 818: covariates we added in the logit model better is the model, but 819: some covariates are not well measured, some are confounding 820: factors like in any statistical model. The procedure to "fit 821: the best model' is similar to logistic regression which itself is 822: similar to regression analysis. We haven't yet been sofar because 823: we also have a severe limitation which is the speed of the 824: convergence. On a Pentium III, 500 MHz, even the simplest model, 825: estimated by month on 8,000 people may take 4 hours to converge. 826: Also, the program is not yet a statistical package, which permits 827: a simple writing of the variables and the model to take into 828: account in the maximisation. The actual program allows only to 829: add simple variables without covariations, like age+sex but 830: without age+sex+ age*sex . This can be done from the source code 831: (you have to change three lines in the source code) but will 832: never be general enough. But what is to remember, is that 833: incidences or probability of change from one state to another is 834: affected by the variables specified into the model. 835: 836: Also, the age range of the people interviewed has a link with 837: the age range of the life expectancy which can be estimated by 838: extrapolation. If your sample ranges from age 70 to 95, you can 839: clearly estimate a life expectancy at age 70 and trust your 840: confidence interval which is mostly based on your sample size, 841: but if you want to estimate the life expectancy at age 50, you 842: should rely in your model, but fitting a logistic model on a age 843: range of 70-95 and estimating probabilties of transition out of 844: this age range, say at age 50 is very dangerous. At least you 845: should remember that the confidence interval given by the 846: standard deviation of the health expectancies, are under the 847: strong assumption that your model is the 'true model', which is 848: probably not the case. 849: 850: <h5>- Copy of the parameter 851: file: <a href="orbiaspar.txt">orbiaspar.txt</a></h5> 852: 853: This copy of the parameter file can be useful to re-run the 854: program while saving the old output files. 855: 856: <hr> 857: 858: <h2><a name="example" </a>Trying an example</a></h2> 859: 860: Since you know how to run the program, it is time to test it 861: on your own computer. Try for example on a parameter file named <a 862: href="file://../mytry/imachpar.txt">imachpar.txt</a> which is a 863: copy of mypar.txt 864: included in the subdirectory of imach, mytry. Edit it to change the name of 866: the data file to ..\data\mydata.txt 867: if you don't want to copy it on the same directory. The file mydata.txt is a smaller file of 3,000 869: people but still with 4 waves. 870: 871: Click on the imach.exe icon to open a window. Answer to the 872: question:'Enter the parameter file name:' 873: 874: <table border="1"> 875: <tr> 876: <td width="100%">IMACH, Version 0.63Enter 877: the parameter file name: ..\mytry\imachpar.txt 878: </td> 879: </tr> 880: </table> 881: 882: Most of the data files or image files generated, will use the 883: 'imachpar' string into their name. The running time is about 2-3 884: minutes on a Pentium III. If the execution worked correctly, the 885: outputs files are created in the current directory, and should be 886: the same as the mypar files initially included in the directory mytry. 888: 889: <ul> 890: <li><pre>Output on the screen The output screen looks like <a 891: href="imachrun.LOG">this Log file</a> 892: # 893: 894: title=MLE datafile=..\data\mydata.txt lastobs=3000 firstpass=1 lastpass=3 895: ftol=1.000000e-008 stepm=24 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre> 896: </li> 897: <li><pre>Total number of individuals= 2965, Agemin = 70.00, Agemax= 100.92 898: 899: Warning, no any valid information for:126 line=126 900: Warning, no any valid information for:2307 line=2307 901: Delay (in months) between two waves Min=21 Max=51 Mean=24.495826 902: These lines give some warnings on the data file and also some raw statistics on frequencies of transitions. 903: Age 70 1.=230 loss[1]=3.5% 2.=16 loss[2]=12.5% 1.=222 prev[1]=94.1% 2.=14 904: prev[2]=5.9% 1-1=8 11=200 12=7 13=15 2-1=2 21=6 22=7 23=1 905: Age 102 1.=0 loss[1]=NaNQ% 2.=0 loss[2]=NaNQ% 1.=0 prev[1]=NaNQ% 2.=0 </pre> 906: </li> 907: </ul> 908: 909:   910: 911: <ul> 912: <li>Maximisation with the Powell algorithm. 8 directions are 913: given corresponding to the 8 parameters. this can be 914: rather long to get convergence. 915: 916: Powell iter=1 -2*LL=11531.405658264877 1 0.000000000000 2 917: 0.000000000000 3 918: 0.000000000000 4 0.000000000000 5 0.000000000000 6 919: 0.000000000000 7 920: 0.000000000000 8 0.000000000000 921: 1..........2.................3..........4.................5......... 922: 6................7........8............... 923: Powell iter=23 -2*LL=6744.954108371555 1 -12.967632334283 924: 925: 2 0.135136681033 3 -7.402109728262 4 0.067844593326 926: 5 -0.673601538129 6 -0.006615504377 7 -5.051341616718 927: 8 0.051272038506 928: 1..............2...........3..............4........... 929: 5..........6................7...........8......... 930: #Number of iterations = 23, -2 Log likelihood = 931: 6744.954042573691 932: # Parameters 933: 12 -12.966061 0.135117 934: 13 -7.401109 0.067831 935: 21 -0.672648 -0.006627 936: 23 -5.051297 0.051271 937: </li> 938: <li><pre>Calculation of the hessian matrix. Wait... 939: 12345678.12.13.14.15.16.17.18.23.24.25.26.27.28.34.35.36.37.38.45.46.47.48.56.57.58.67.68.78 940: 941: Inverting the hessian to get the covariance matrix. Wait... 942: 943: #Hessian matrix# 944: 3.344e+002 2.708e+004 -4.586e+001 -3.806e+003 -1.577e+000 -1.313e+002 3.914e-001 3.166e+001 945: 2.708e+004 2.204e+006 -3.805e+003 -3.174e+005 -1.303e+002 -1.091e+004 2.967e+001 2.399e+003 946: -4.586e+001 -3.805e+003 4.044e+002 3.197e+004 2.431e-002 1.995e+000 1.783e-001 1.486e+001 947: -3.806e+003 -3.174e+005 3.197e+004 2.541e+006 2.436e+000 2.051e+002 1.483e+001 1.244e+003 948: -1.577e+000 -1.303e+002 2.431e-002 2.436e+000 1.093e+002 8.979e+003 -3.402e+001 -2.843e+003 949: -1.313e+002 -1.091e+004 1.995e+000 2.051e+002 8.979e+003 7.420e+005 -2.842e+003 -2.388e+005 950: 3.914e-001 2.967e+001 1.783e-001 1.483e+001 -3.402e+001 -2.842e+003 1.494e+002 1.251e+004 951: 3.166e+001 2.399e+003 1.486e+001 1.244e+003 -2.843e+003 -2.388e+005 1.251e+004 1.053e+006 952: # Scales 953: 12 1.00000e-004 1.00000e-006 954: 13 1.00000e-004 1.00000e-006 955: 21 1.00000e-003 1.00000e-005 956: 23 1.00000e-004 1.00000e-005 957: # Covariance 958: 1 5.90661e-001 959: 2 -7.26732e-003 8.98810e-005 960: 3 8.80177e-002 -1.12706e-003 5.15824e-001 961: 4 -1.13082e-003 1.45267e-005 -6.50070e-003 8.23270e-005 962: 5 9.31265e-003 -1.16106e-004 6.00210e-004 -8.04151e-006 1.75753e+000 963: 6 -1.15664e-004 1.44850e-006 -7.79995e-006 1.04770e-007 -2.12929e-002 2.59422e-004 964: 7 1.35103e-003 -1.75392e-005 -6.38237e-004 7.85424e-006 4.02601e-001 -4.86776e-003 1.32682e+000 965: 8 -1.82421e-005 2.35811e-007 7.75503e-006 -9.58687e-008 -4.86589e-003 5.91641e-005 -1.57767e-002 1.88622e-004 966: # agemin agemax for lifexpectancy, bage fage (if mle==0 ie no data nor Max likelihood). 967: 968: 969: agemin=70 agemax=100 bage=50 fage=100 970: Computing prevalence limit: result on file 'plrmypar.txt' 971: Computing pij: result on file 'pijrmypar.txt' 972: Computing Health Expectancies: result on file 'ermypar.txt' 973: Computing Variance-covariance of DFLEs: file 'vrmypar.txt' 974: Computing Total LEs with variances: file 'trmypar.txt' 975: Computing Variance-covariance of Prevalence limit: file 'vplrmypar.txt' 976: End of Imach 977: </pre> 978: </li> 979: </ul> 980: 981: Once the running is finished, the program 982: requires a caracter: 983: 984: <table border="1"> 985: <tr> 986: <td width="100%">Type g for plotting (available 987: if mle=1), e to edit output files, c to start again,and 988: q for exiting: 989: </td> 990: </tr> 991: </table> 992: 993: First you should enter g to 994: make the figures and then you can edit all the results by typing e. 995: 996: 997: <ul> 998: <li>Outputs files 999: - index.htm, this file is the master file on which you 1000: should click first. 1001: - Observed prevalence in each state: <a 1002: href="..\mytry\prmypar.txt">mypar.txt</a> 1003: - Estimated parameters and the covariance matrix: <a 1004: href="..\mytry\rmypar.txt">rmypar.txt</a> 1005: - Stationary prevalence in each state: <a 1006: href="..\mytry\plrmypar.txt">plrmypar.txt</a> 1007: - Transition probabilities: <a 1008: href="..\mytry\pijrmypar.txt">pijrmypar.txt</a> 1009: - Copy of the parameter file: <a 1010: href="..\mytry\ormypar.txt">ormypar.txt</a> 1011: - Life expectancies by age and initial health status: <a 1012: href="..\mytry\ermypar.txt">ermypar.txt</a> 1013: - Variances of life expectancies by age and initial 1014: health status: <a href="..\mytry\vrmypar.txt">vrmypar.txt</a> 1015: 1016: - Health expectancies with their variances: <a 1017: href="..\mytry\trmypar.txt">trmypar.txt</a> 1018: - Standard deviation of stationary prevalence: <a 1019: href="..\mytry\vplrmypar.txt">vplrmypar.txt</a> 1020: 1021: </li> 1022: <li>Graphs 1023: 1024: -<a href="..\mytry\vmypar1.gif">Observed and stationary 1025: prevalence in state (1) with the confident interval</a> 1026: -<a href="..\mytry\vmypar2.gif">Observed and stationary 1027: prevalence in state (2) with the confident interval</a> 1028: -<a href="..\mytry\exmypar1.gif">Health life expectancies 1029: by age and initial health state (1)</a> 1030: -<a href="..\mytry\exmypar2.gif">Health life expectancies 1031: by age and initial health state (2)</a> 1032: -<a href="..\mytry\emypar.gif">Total life expectancy by 1033: age and health expectancies in states (1) and (2).</a> </li> 1034: </ul> 1035: 1036: This software have been partly granted by <a 1037: href="http://euroreves.ined.fr">Euro-REVES</a>, a concerted 1038: action from the European Union. It will be copyrighted 1039: identically to a GNU software product, i.e. program and software 1040: can be distributed freely for non commercial use. Sources are not 1041: widely distributed today. You can get them by asking us with a 1042: simple justification (name, email, institute) <a 1043: href="mailto:brouard@ined.fr">mailto:brouard@ined.fr</a> and <a 1044: href="mailto:lievre@ined.fr">mailto:lievre@ined.fr</a> . 1045: 1046: Latest version (0.63 of 16 march 2000) can be accessed at <a 1047: href="http://euroeves.ined.fr/imach">http://euroreves.ined.fr/imach</a> 1048: 1049: </body> 1050: </html>