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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: Version 32: 64b, May 2001 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 0.64b, May 2001, 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: <li>if model=V1+V1*age the model includes 369: the product covariate*age</li> 370: </ul> 371: 372: <h4>Guess values for optimization </h4> 374: 375: You must write the initial guess values of the parameters for 376: optimization. The number of parameters, N depends on the 377: number of absorbing states and non-absorbing states and on the 378: number of covariates. 379: N is given by the formula N=(nlstate + 380: ndeath-1)*nlstate*ncov . 381: 382: Thus in the simple case with 2 covariates (the model is log 383: (pij/pii) = aij + bij * age where intercept and age are the two 384: covariates), and 2 health degrees (1 for disability-free and 2 385: for disability) and 1 absorbing state (3), you must enter 8 386: initials values, a12, b12, a13, b13, a21, b21, a23, b23. You can 387: start with zeros as in this example, but if you have a more 388: precise set (for example from an earlier run) you can enter it 389: and it will speed up them 390: Each of the four lines starts with indices "ij": 391: 392: ij aij bij 393: 394: <blockquote> 395: <pre># Guess values of aij and bij in log (pij/pii) = aij + bij * age 396: 12 -14.155633 0.110794 397: 13 -7.925360 0.032091 398: 21 -1.890135 -0.029473 399: 23 -6.234642 0.022315 </pre> 400: </blockquote> 401: 402: or, to simplify: 403: 404: <blockquote> 405: <pre>12 0.0 0.0 406: 13 0.0 0.0 407: 21 0.0 0.0 408: 23 0.0 0.0</pre> 409: </blockquote> 410: 411: <h4>Guess values for computing variances</h4> 412: 413: This is an output if <a href="#mle">mle</a>=1. But it can be 414: used as an input to get the vairous output data files (Health 415: expectancies, stationary prevalence etc.) and figures without 416: rerunning the rather long maximisation phase (mle=0). 417: 418: The scales are small values for the evaluation of numerical 419: derivatives. These derivatives are used to compute the hessian 420: matrix of the parameters, that is the inverse of the covariance 421: matrix, and the variances of health expectancies. Each line 422: consists in indices "ij" followed by the initial scales 423: (zero to simplify) associated with aij and bij. 424: 425: <ul> 426: <li>If mle=1 you can enter zeros:</li> 427: </ul> 428: 429: <blockquote> 430: <pre># Scales (for hessian or gradient estimation) 431: 12 0. 0. 432: 13 0. 0. 433: 21 0. 0. 434: 23 0. 0. </pre> 435: </blockquote> 436: 437: <ul> 438: <li>If mle=0 you must enter a covariance matrix (usually 439: obtained from an earlier run).</li> 440: </ul> 441: 442: <h4>Covariance matrix of parameters</h4> 443: 444: This is an output if <a href="#mle">mle</a>=1. But it can be 445: used as an input to get the vairous output data files (Health 446: expectancies, stationary prevalence etc.) and figures without 447: rerunning the rather long maximisation phase (mle=0). 448: 449: Each line starts with indices "ijk" followed by the 450: covariances between aij and bij: 451: 452: <pre> 453: 121 Var(a12) 454: 122 Cov(b12,a12) Var(b12) 455: ... 456: 232 Cov(b23,a12) Cov(b23,b12) ... Var (b23) </pre> 457: 458: <ul> 459: <li>If mle=1 you can enter zeros. </li> 460: </ul> 461: 462: <blockquote> 463: <pre># Covariance matrix 464: 121 0. 465: 122 0. 0. 466: 131 0. 0. 0. 467: 132 0. 0. 0. 0. 468: 211 0. 0. 0. 0. 0. 469: 212 0. 0. 0. 0. 0. 0. 470: 231 0. 0. 0. 0. 0. 0. 0. 471: 232 0. 0. 0. 0. 0. 0. 0. 0.</pre> 472: </blockquote> 473: 474: <ul> 475: <li>If mle=0 you must enter a covariance matrix (usually 476: obtained from an earlier run). 477: </li> 478: </ul> 479: 480: <h4><a name="biaspar-l"></a>last 481: uncommented line</h4> 482: 483: <pre>agemin=70 agemax=100 bage=50 fage=100</pre> 484: 485: Once we obtained the estimated parameters, the program is able 486: to calculated stationary prevalence, transitions probabilities 487: and life expectancies at any age. Choice of age ranges is useful 488: for extrapolation. In our data file, ages varies from age 70 to 489: 102. Setting bage=50 and fage=100, makes the program computing 490: life expectancy from age bage to age fage. As we use a model, we 491: can compute life expectancy on a wider age range than the age 492: range from the data. But the model can be rather wrong on big 493: intervals. 494: 495: Similarly, it is possible to get extrapolated stationary 496: prevalence by age raning from agemin to agemax. 497: 498: <ul> 499: <li>agemin= Minimum age for calculation of the 500: stationary prevalence </li> 501: <li>agemax= Maximum age for calculation of the 502: stationary prevalence </li> 503: <li>bage= Minimum age for calculation of the health 504: expectancies </li> 505: <li>fage= Maximum ages for calculation of the health 506: expectancies </li> 507: </ul> 508: 509: <hr> 510: 511: <h2><a name="running"></a>Running Imach 512: with this example</h2> 513: 514: We assume that you entered your <a href="biaspar.txt">1st_example 515: parameter file</a> as explained <a href="#biaspar">above</a>. To 516: run the program you should click on the imach.exe icon and enter 517: the name of the parameter file which is for example <a 518: href="C:\usr\imach\mle\biaspar.txt">C:\usr\imach\mle\biaspar.txt</a> 519: (you also can click on the biaspar.txt icon located in 520: <a href="C:\usr\imach\mle">C:\usr\imach\mle</a> and put it with 521: the mouse on the imach window). 522: 523: 524: The time to converge depends on the step unit that you used (1 525: month is cpu consuming), on the number of cases, and on the 526: number of variables. 527: 528: The program outputs many files. Most of them are files which 529: will be plotted for better understanding. 530: 531: <hr> 532: 533: <h2><a name="output">Output of the program 534: and graphs </a></h2> 535: 536: Once the optimization is finished, some graphics can be made 537: with a grapher. We use Gnuplot which is an interactive plotting 538: program copyrighted but freely distributed. A gnuplot reference 539: manual is available <a href="http://www.gnuplot.org/">here</a>. 540: When the running is finished, the user should enter a caracter 541: for plotting and output editing. 542: 543: These caracters are: 544: 545: <ul> 546: <li>'c' to start again the program from the beginning.</li> 547: <li>'e' opens the <a href="biaspar.htm">biaspar.htm</a> 548: file to edit the output files and graphs. </li> 549: <li>'q' for exiting.</li> 550: </ul> 551: 552: <h5>Results files 553: 554: - <a 555: name="Observed prevalence in each state">Observed prevalence in each state</a> (and at first pass): 558: <a href="prbiaspar.txt">prbiaspar.txt</a> 559: </h5> 560: 561: The first line is the title and displays each field of the 562: file. The first column is age. The fields 2 and 6 are the 563: proportion of individuals in states 1 and 2 respectively as 564: observed during the first exam. Others fields are the numbers of 565: people in states 1, 2 or more. The number of columns increases if 566: the number of states is higher than 2. 567: The header of the file is 568: 569: <pre># Age Prev(1) N(1) N Age Prev(2) N(2) N 570: 70 1.00000 631 631 70 0.00000 0 631 571: 71 0.99681 625 627 71 0.00319 2 627 572: 72 0.97125 1115 1148 72 0.02875 33 1148 </pre> 573: 574: It means that at age 70, the prevalence in state 1 is 1.000 575: and in state 2 is 0.00 . At age 71 the number of individuals in 576: state 1 is 625 and in state 2 is 2, hence the total number of 577: people aged 71 is 625+2=627. 578: 579: 580: <h5>- Estimated parameters and 581: covariance matrix: <a href="rbiaspar.txt">rbiaspar.txt</a></h5> 582: 583: This file contains all the maximisation results: 584: 585: <pre> -2 log likelihood= 21660.918613445392 586: Estimated parameters: a12 = -12.290174 b12 = 0.092161 587: a13 = -9.155590 b13 = 0.046627 588: a21 = -2.629849 b21 = -0.022030 589: a23 = -7.958519 b23 = 0.042614 590: Covariance matrix: Var(a12) = 1.47453e-001 591: Var(b12) = 2.18676e-005 592: Var(a13) = 2.09715e-001 593: Var(b13) = 3.28937e-005 594: Var(a21) = 9.19832e-001 595: Var(b21) = 1.29229e-004 596: Var(a23) = 4.48405e-001 597: Var(b23) = 5.85631e-005 598: </pre> 599: 600: By substitution of these parameters in the regression model, 601: we obtain the elementary transition probabilities: 602: 603: <img src="pebiaspar1.gif" width="400" height="300"> 604: 605: <h5>- Transition probabilities: 606: <a href="pijrbiaspar.txt">pijrbiaspar.txt</a></h5> 607: 608: Here are the transitions probabilities Pij(x, x+nh) where nh 609: is a multiple of 2 years. The first column is the starting age x 610: (from age 50 to 100), the second is age (x+nh) and the others are 611: the transition probabilities p11, p12, p13, p21, p22, p23. For 612: example, line 5 of the file is: 613: 614: <pre> 100 106 0.02655 0.17622 0.79722 0.01809 0.13678 0.84513 </pre> 615: 616: and this means: 617: 618: <pre>p11(100,106)=0.02655 619: p12(100,106)=0.17622 620: p13(100,106)=0.79722 621: p21(100,106)=0.01809 622: p22(100,106)=0.13678 623: p22(100,106)=0.84513 </pre> 624: 625: <h5>- <a 626: name="Stationary prevalence in each state">Stationary prevalence in each state</a>: 628: <a href="plrbiaspar.txt">plrbiaspar.txt</a></h5> 629: 630: <pre>#Prevalence 631: #Age 1-1 2-2 632: 633: #************ 634: 70 0.90134 0.09866 635: 71 0.89177 0.10823 636: 72 0.88139 0.11861 637: 73 0.87015 0.12985 </pre> 638: 639: At age 70 the stationary prevalence is 0.90134 in state 1 and 640: 0.09866 in state 2. This stationary prevalence differs from 641: observed prevalence. Here is the point. The observed prevalence 642: at age 70 results from the incidence of disability, incidence of 643: recovery and mortality which occurred in the past of the cohort. 644: Stationary prevalence results from a simulation with actual 645: incidences and mortality (estimated from this cross-longitudinal 646: survey). It is the best predictive value of the prevalence in the 647: future if "nothing changes in the future". This is 648: exactly what demographers do with a Life table. Life expectancy 649: is the expected mean time to survive if observed mortality rates 650: (incidence of mortality) "remains constant" in the 651: future. 652: 653: <h5>- Standard deviation of 654: stationary prevalence: <a 655: href="vplrbiaspar.txt">vplrbiaspar.txt</a></h5> 656: 657: The stationary prevalence has to be compared with the observed 658: prevalence by age. But both are statistical estimates and 659: subjected to stochastic errors due to the size of the sample, the 660: design of the survey, and, for the stationary prevalence to the 661: model used and fitted. It is possible to compute the standard 662: deviation of the stationary prevalence at each age. 663: 664: <h5>-Observed and stationary 665: prevalence in state (2=disable) with the confident interval: 666: <a href="vbiaspar21.htm">vbiaspar21.gif</a></h5> 667: 668: This graph exhibits the stationary prevalence in state (2) 669: with the confidence interval in red. The green curve is the 670: observed prevalence (or proportion of individuals in state (2)). 671: Without discussing the results (it is not the purpose here), we 672: observe that the green curve is rather below the stationary 673: prevalence. It suggests an increase of the disability prevalence 674: in the future. 675: 676: <img src="vbiaspar21.gif" width="400" height="300"> 677: 678: <h5>-Convergence to the 679: stationary prevalence of disability: <a 680: href="pbiaspar11.gif">pbiaspar11.gif</a> 681: <img src="pbiaspar11.gif" width="400" height="300"> </h5> 682: 683: This graph plots the conditional transition probabilities from 684: an initial state (1=healthy in red at the bottom, or 2=disable in 685: green on top) at age x to the final state 2=disable at 686: age x+h. Conditional means at the condition to be alive 687: at age x+h which is hP12x + hP22x. The 688: curves hP12x/(hP12x + hP22x) and hP22x/(hP12x 689: + hP22x) converge with h, to the stationary 690: prevalence of disability. In order to get the stationary 691: prevalence at age 70 we should start the process at an earlier 692: age, i.e.50. If the disability state is defined by severe 693: disability criteria with only a few chance to recover, then the 694: incidence of recovery is low and the time to convergence is 695: probably longer. But we don't have experience yet. 696: 697: <h5>- Life expectancies by age 698: and initial health status: <a 699: href="erbiaspar.txt">erbiaspar.txt</a></h5> 700: 701: <pre># Health expectancies 702: # Age 1-1 1-2 2-1 2-2 703: 70 10.9226 3.0401 5.6488 6.2122 704: 71 10.4384 3.0461 5.2477 6.1599 705: 72 9.9667 3.0502 4.8663 6.1025 706: 73 9.5077 3.0524 4.5044 6.0401 </pre> 707: 708: <pre>For example 70 10.9226 3.0401 5.6488 6.2122 means: 709: e11=10.9226 e12=3.0401 e21=5.6488 e22=6.2122</pre> 710: 711: <pre><img src="expbiaspar21.gif" width="400" height="300"><img 712: src="expbiaspar11.gif" width="400" height="300"></pre> 713: 714: For example, life expectancy of a healthy individual at age 70 715: is 10.92 in the healthy state and 3.04 in the disability state 716: (=13.96 years). If he was disable at age 70, his life expectancy 717: will be shorter, 5.64 in the healthy state and 6.21 in the 718: disability state (=11.85 years). The total life expectancy is a 719: weighted mean of both, 13.96 and 11.85; weight is the proportion 720: of people disabled at age 70. In order to get a pure period index 721: (i.e. based only on incidences) we use the <a 722: href="#Stationary prevalence in each state">computed or 723: stationary prevalence</a> at age 70 (i.e. computed from 724: incidences at earlier ages) instead of the <a 725: href="#Observed prevalence in each state">observed prevalence</a> 726: (for example at first exam) (<a href="#Health expectancies">see 727: below</a>). 728: 729: <h5>- Variances of life 730: expectancies by age and initial health status: <a 731: href="vrbiaspar.txt">vrbiaspar.txt</a></h5> 732: 733: For example, the covariances of life expectancies Cov(ei,ej) 734: at age 50 are (line 3) 735: 736: <pre> Cov(e1,e1)=0.4776 Cov(e1,e2)=0.0488=Cov(e2,e1) Cov(e2,e2)=0.0424</pre> 737: 738: <h5>- <a 739: name="Health expectancies">Health 740: expectancies</a> 741: with standard errors in parentheses: <a 742: href="trbiaspar.txt">trbiaspar.txt</a></h5> 743: 744: <pre>#Total LEs with variances: e.. (std) e.1 (std) e.2 (std) </pre> 745: 746: <pre>70 13.76 (0.22) 10.40 (0.20) 3.35 (0.14) </pre> 747: 748: Thus, at age 70 the total life expectancy, e..=13.76years is 749: the weighted mean of e1.=13.96 and e2.=11.85 by the stationary 750: prevalence at age 70 which are 0.90134 in state 1 and 0.09866 in 751: state 2, respectively (the sum is equal to one). e.1=10.40 is the 752: Disability-free life expectancy at age 70 (it is again a weighted 753: mean of e11 and e21). e.2=3.35 is also the life expectancy at age 754: 70 to be spent in the disability state. 755: 756: <h5>-Total life expectancy by 757: age and health expectancies in states (1=healthy) and (2=disable): 758: <a href="ebiaspar1.gif">ebiaspar1.gif</a></h5> 759: 760: This figure represents the health expectancies and the total 761: life expectancy with the confident interval in dashed curve. 762: 763: <pre> <img src="ebiaspar1.gif" width="400" height="300"></pre> 764: 765: Standard deviations (obtained from the information matrix of 766: the model) of these quantities are very useful. 767: Cross-longitudinal surveys are costly and do not involve huge 768: samples, generally a few thousands; therefore it is very 769: important to have an idea of the standard deviation of our 770: estimates. It has been a big challenge to compute the Health 771: Expectancy standard deviations. Don't be confuse: life expectancy 772: is, as any expected value, the mean of a distribution; but here 773: we are not computing the standard deviation of the distribution, 774: but the standard deviation of the estimate of the mean. 775: 776: Our health expectancies estimates vary according to the sample 777: size (and the standard deviations give confidence intervals of 778: the estimate) but also according to the model fitted. Let us 779: explain it in more details. 780: 781: Choosing a model means ar least two kind of choices. First we 782: have to decide the number of disability states. Second we have to 783: design, within the logit model family, the model: variables, 784: covariables, confonding factors etc. to be included. 785: 786: More disability states we have, better is our demographical 787: approach of the disability process, but smaller are the number of 788: transitions between each state and higher is the noise in the 789: measurement. We do not have enough experiments of the various 790: models to summarize the advantages and disadvantages, but it is 791: important to say that even if we had huge and unbiased samples, 792: the total life expectancy computed from a cross-longitudinal 793: survey, varies with the number of states. If we define only two 794: states, alive or dead, we find the usual life expectancy where it 795: is assumed that at each age, people are at the same risk to die. 796: If we are differentiating the alive state into healthy and 797: disable, and as the mortality from the disability state is higher 798: than the mortality from the healthy state, we are introducing 799: heterogeneity in the risk of dying. The total mortality at each 800: age is the weighted mean of the mortality in each state by the 801: prevalence in each state. Therefore if the proportion of people 802: at each age and in each state is different from the stationary 803: equilibrium, there is no reason to find the same total mortality 804: at a particular age. Life expectancy, even if it is a very useful 805: tool, has a very strong hypothesis of homogeneity of the 806: population. Our main purpose is not to measure differential 807: mortality but to measure the expected time in a healthy or 808: disability state in order to maximise the former and minimize the 809: latter. But the differential in mortality complexifies the 810: measurement. 811: 812: Incidences of disability or recovery are not affected by the 813: number of states if these states are independant. But incidences 814: estimates are dependant on the specification of the model. More 815: covariates we added in the logit model better is the model, but 816: some covariates are not well measured, some are confounding 817: factors like in any statistical model. The procedure to "fit 818: the best model' is similar to logistic regression which itself is 819: similar to regression analysis. We haven't yet been sofar because 820: we also have a severe limitation which is the speed of the 821: convergence. On a Pentium III, 500 MHz, even the simplest model, 822: estimated by month on 8,000 people may take 4 hours to converge. 823: Also, the program is not yet a statistical package, which permits 824: a simple writing of the variables and the model to take into 825: account in the maximisation. The actual program allows only to 826: add simple variables like age+sex or age+sex+ age*sex but will 827: never be general enough. But what is to remember, is that 828: incidences or probability of change from one state to another is 829: affected by the variables specified into the model. 830: 831: Also, the age range of the people interviewed has a link with 832: the age range of the life expectancy which can be estimated by 833: extrapolation. If your sample ranges from age 70 to 95, you can 834: clearly estimate a life expectancy at age 70 and trust your 835: confidence interval which is mostly based on your sample size, 836: but if you want to estimate the life expectancy at age 50, you 837: should rely in your model, but fitting a logistic model on a age 838: range of 70-95 and estimating probabilties of transition out of 839: this age range, say at age 50 is very dangerous. At least you 840: should remember that the confidence interval given by the 841: standard deviation of the health expectancies, are under the 842: strong assumption that your model is the 'true model', which is 843: probably not the case. 844: 845: <h5>- Copy of the parameter 846: file: <a href="orbiaspar.txt">orbiaspar.txt</a></h5> 847: 848: This copy of the parameter file can be useful to re-run the 849: program while saving the old output files. 850: 851: <hr> 852: 853: <h2><a name="example" </a>Trying an example</a></h2> 854: 855: Since you know how to run the program, it is time to test it 856: on your own computer. Try for example on a parameter file named <a 857: href="..\mytry\imachpar.txt">imachpar.txt</a> which is a copy of mypar.txt included in the 859: subdirectory of imach, mytry. 860: Edit it to change the name of the data file to ..\data\mydata.txt if you don't want to 862: copy it on the same directory. The file mydata.txt 863: is a smaller file of 3,000 people but still with 4 waves. 864: 865: Click on the imach.exe icon to open a window. Answer to the 866: question:'Enter the parameter file name:' 867: 868: <table border="1"> 869: <tr> 870: <td width="100%">IMACH, Version 0.64bEnter 871: the parameter file name: ..\mytry\imachpar.txt 872: </td> 873: </tr> 874: </table> 875: 876: Most of the data files or image files generated, will use the 877: 'imachpar' string into their name. The running time is about 2-3 878: minutes on a Pentium III. If the execution worked correctly, the 879: outputs files are created in the current directory, and should be 880: the same as the mypar files initially included in the directory mytry. 882: 883: <ul> 884: <li><pre>Output on the screen The output screen looks like <a 885: href="imachrun.LOG">this Log file</a> 886: # 887: 888: title=MLE datafile=..\data\mydata.txt lastobs=3000 firstpass=1 lastpass=3 889: ftol=1.000000e-008 stepm=24 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</pre> 890: </li> 891: <li><pre>Total number of individuals= 2965, Agemin = 70.00, Agemax= 100.92 892: 893: Warning, no any valid information for:126 line=126 894: Warning, no any valid information for:2307 line=2307 895: Delay (in months) between two waves Min=21 Max=51 Mean=24.495826 896: These lines give some warnings on the data file and also some raw statistics on frequencies of transitions. 897: Age 70 1.=230 loss[1]=3.5% 2.=16 loss[2]=12.5% 1.=222 prev[1]=94.1% 2.=14 898: prev[2]=5.9% 1-1=8 11=200 12=7 13=15 2-1=2 21=6 22=7 23=1 899: Age 102 1.=0 loss[1]=NaNQ% 2.=0 loss[2]=NaNQ% 1.=0 prev[1]=NaNQ% 2.=0 </pre> 900: </li> 901: </ul> 902: 903:   904: 905: <ul> 906: <li>Maximisation with the Powell algorithm. 8 directions are 907: given corresponding to the 8 parameters. this can be 908: rather long to get convergence. 909: 910: Powell iter=1 -2*LL=11531.405658264877 1 0.000000000000 2 911: 0.000000000000 3 912: 0.000000000000 4 0.000000000000 5 0.000000000000 6 913: 0.000000000000 7 914: 0.000000000000 8 0.000000000000 915: 1..........2.................3..........4.................5......... 916: 6................7........8............... 917: Powell iter=23 -2*LL=6744.954108371555 1 -12.967632334283 918: 919: 2 0.135136681033 3 -7.402109728262 4 0.067844593326 920: 5 -0.673601538129 6 -0.006615504377 7 -5.051341616718 921: 8 0.051272038506 922: 1..............2...........3..............4........... 923: 5..........6................7...........8......... 924: #Number of iterations = 23, -2 Log likelihood = 925: 6744.954042573691 926: # Parameters 927: 12 -12.966061 0.135117 928: 13 -7.401109 0.067831 929: 21 -0.672648 -0.006627 930: 23 -5.051297 0.051271 931: </li> 932: <li><pre>Calculation of the hessian matrix. Wait... 933: 12345678.12.13.14.15.16.17.18.23.24.25.26.27.28.34.35.36.37.38.45.46.47.48.56.57.58.67.68.78 934: 935: Inverting the hessian to get the covariance matrix. Wait... 936: 937: #Hessian matrix# 938: 3.344e+002 2.708e+004 -4.586e+001 -3.806e+003 -1.577e+000 -1.313e+002 3.914e-001 3.166e+001 939: 2.708e+004 2.204e+006 -3.805e+003 -3.174e+005 -1.303e+002 -1.091e+004 2.967e+001 2.399e+003 940: -4.586e+001 -3.805e+003 4.044e+002 3.197e+004 2.431e-002 1.995e+000 1.783e-001 1.486e+001 941: -3.806e+003 -3.174e+005 3.197e+004 2.541e+006 2.436e+000 2.051e+002 1.483e+001 1.244e+003 942: -1.577e+000 -1.303e+002 2.431e-002 2.436e+000 1.093e+002 8.979e+003 -3.402e+001 -2.843e+003 943: -1.313e+002 -1.091e+004 1.995e+000 2.051e+002 8.979e+003 7.420e+005 -2.842e+003 -2.388e+005 944: 3.914e-001 2.967e+001 1.783e-001 1.483e+001 -3.402e+001 -2.842e+003 1.494e+002 1.251e+004 945: 3.166e+001 2.399e+003 1.486e+001 1.244e+003 -2.843e+003 -2.388e+005 1.251e+004 1.053e+006 946: # Scales 947: 12 1.00000e-004 1.00000e-006 948: 13 1.00000e-004 1.00000e-006 949: 21 1.00000e-003 1.00000e-005 950: 23 1.00000e-004 1.00000e-005 951: # Covariance 952: 1 5.90661e-001 953: 2 -7.26732e-003 8.98810e-005 954: 3 8.80177e-002 -1.12706e-003 5.15824e-001 955: 4 -1.13082e-003 1.45267e-005 -6.50070e-003 8.23270e-005 956: 5 9.31265e-003 -1.16106e-004 6.00210e-004 -8.04151e-006 1.75753e+000 957: 6 -1.15664e-004 1.44850e-006 -7.79995e-006 1.04770e-007 -2.12929e-002 2.59422e-004 958: 7 1.35103e-003 -1.75392e-005 -6.38237e-004 7.85424e-006 4.02601e-001 -4.86776e-003 1.32682e+000 959: 8 -1.82421e-005 2.35811e-007 7.75503e-006 -9.58687e-008 -4.86589e-003 5.91641e-005 -1.57767e-002 1.88622e-004 960: # agemin agemax for lifexpectancy, bage fage (if mle==0 ie no data nor Max likelihood). 961: 962: 963: agemin=70 agemax=100 bage=50 fage=100 964: Computing prevalence limit: result on file 'plrmypar.txt' 965: Computing pij: result on file 'pijrmypar.txt' 966: Computing Health Expectancies: result on file 'ermypar.txt' 967: Computing Variance-covariance of DFLEs: file 'vrmypar.txt' 968: Computing Total LEs with variances: file 'trmypar.txt' 969: Computing Variance-covariance of Prevalence limit: file 'vplrmypar.txt' 970: End of Imach 971: </pre> 972: </li> 973: </ul> 974: 975: Once the running is finished, the program 976: requires a caracter: 977: 978: <table border="1"> 979: <tr> 980: <td width="100%">Type e to edit output files, c 981: to start again, and q for exiting:</td> 982: </tr> 983: </table> 984: 985: First you should enter e to 986: edit the master file mypar.htm. 987: 988: <ul> 989: <li>Outputs files 990: 991: - Observed prevalence in each state: <a 992: href="..\mytry\prmypar.txt">pmypar.txt</a> 993: - Estimated parameters and the covariance matrix: <a 994: href="..\mytry\rmypar.txt">rmypar.txt</a> 995: - Stationary prevalence in each state: <a 996: href="..\mytry\plrmypar.txt">plrmypar.txt</a> 997: - Transition probabilities: <a 998: href="..\mytry\pijrmypar.txt">pijrmypar.txt</a> 999: - Copy of the parameter file: <a 1000: href="..\mytry\ormypar.txt">ormypar.txt</a> 1001: - Life expectancies by age and initial health status: <a 1002: href="..\mytry\ermypar.txt">ermypar.txt</a> 1003: - Variances of life expectancies by age and initial 1004: health status: <a href="..\mytry\vrmypar.txt">vrmypar.txt</a> 1005: 1006: - Health expectancies with their variances: <a 1007: href="..\mytry\trmypar.txt">trmypar.txt</a> 1008: - Standard deviation of stationary prevalence: <a 1009: href="..\mytry\vplrmypar.txt">vplrmypar.txt</a> 1010: 1011: </li> 1012: <li>Graphs 1013: 1014: -<a href="../mytry/pemypar1.gif">One-step transition 1015: probabilities</a> 1016: -<a href="../mytry/pmypar11.gif">Convergence to the 1017: stationary prevalence</a> 1018: -<a href="..\mytry\vmypar11.gif">Observed and stationary 1019: prevalence in state (1) with the confident interval</a> 1020: -<a href="..\mytry\vmypar21.gif">Observed and stationary 1021: prevalence in state (2) with the confident interval</a> 1022: -<a href="..\mytry\expmypar11.gif">Health life 1023: expectancies by age and initial health state (1)</a> 1024: -<a href="..\mytry\expmypar21.gif">Health life 1025: expectancies by age and initial health state (2)</a> 1026: -<a href="..\mytry\emypar1.gif">Total life expectancy by 1027: age and health expectancies in states (1) and (2).</a> </li> 1028: </ul> 1029: 1030: This software have been partly granted by <a 1031: href="http://euroreves.ined.fr">Euro-REVES</a>, a concerted 1032: action from the European Union. It will be copyrighted 1033: identically to a GNU software product, i.e. program and software 1034: can be distributed freely for non commercial use. Sources are not 1035: widely distributed today. You can get them by asking us with a 1036: simple justification (name, email, institute) <a 1037: href="mailto:brouard@ined.fr">mailto:brouard@ined.fr</a> and <a 1038: href="mailto:lievre@ined.fr">mailto:lievre@ined.fr</a> . 1039: 1040: Latest version (0.64b of may 2001) can be accessed at <a 1041: href="http://euroeves.ined.fr/imach">http://euroreves.ined.fr/imach</a> 1042: 1043: </body> 1044: </html>