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1: <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN"> 2:  3: <!-- $Id: imach.htm,v 1.7 2005/06/15 14:29:14 lievre Exp $ --!><HTML><HEAD><TITLE>Computing Health Expectancies using IMaCh</TITLE> 4: <META content="text/html; charset=iso-8859-1" http-equiv=Content-Type> 5: <META content="MSHTML 5.00.3315.2870" name=GENERATOR></HEAD> 6: <BODY bgColor=#ffffff> 7: <HR color=#ec5e5e SIZE=3> 8: 9: <H1 align=center>Computing Health Expectancies using 10: IMaCh</H1> 11: <H1 align=center>(a Maximum Likelihood Computer 12: Program using Interpolation of Markov Chains)</H1> 13:   14: <A href="http://www.ined.fr/"><IMG border=0 height=76 15: src="logo-ined.gif" 16: width=151></A><IMG height=75 17: src="euroreves2.gif" 18: width=151> 19: <H3 align=center><A href="http://www.ined.fr/">INED</A> and <A 21: href="http://euroreves.ined.fr/">EUROREVES</A></H3> 22: Version 0.97, June 23: 2004 24: <HR color=#ec5e5e SIZE=3> 25: 26: Authors of the program: 27: <A href="http://sauvy.ined.fr/brouard">Nicolas Brouard</A>, senior researcher at the <A 30: href="http://www.ined.fr/">Institut National 31: d'Etudes Démographiques</A> (INED, 32: Paris) in the "Mortality, Health and Epidemiology" Research Unit 33: 34: and Agnès Lièvre 36: <H4>Contribution to the mathematics: C. R. Heathcote 37: (Australian National University, 38: Canberra).</H4> 39: <H4>Contact: Agnès Lièvre (<A 40: href="mailto:lievre@ined.fr">lievre@ined.fr</A>) </H4> 42: <HR> 43: 44: <UL> 45: <LI><A 46: href="http://euroreves.ined.fr/imach/doc/imach.htm#intro">Introduction</A> 47: <LI><A href="http://euroreves.ined.fr/imach/doc/imach.htm#data">What kind 48: of data can is required?</A> 49: <LI><A href="http://euroreves.ined.fr/imach/doc/imach.htm#datafile">The data 50: file</A> 51: <LI><A href="http://euroreves.ined.fr/imach/doc/imach.htm#biaspar">The 52: parameter file</A> 53: <LI><A href="http://euroreves.ined.fr/imach/doc/imach.htm#running">Running 54: Imach</A> 55: <LI><A href="http://euroreves.ined.fr/imach/doc/imach.htm#output">Output files 56: and graphs</A> 57: <LI><A href="http://euroreves.ined.fr/imach/doc/imach.htm#example">Exemple</A> 58: </LI></UL> 59: <HR> 60: 61: <H2><A name=intro>Introduction</A></H2> 62: This program computes Healthy Life Expectancies from 63: cross-longitudinal data using the methodology pioneered by Laditka and 64: Wolf (1). Within the family of Health Expectancies (HE), disability-free life 65: expectancy (DFLE) is probably the most important index to monitor. In low 66: mortality countries, there is a fear that when mortality declines (and therefore total life expectancy improves), the increase will not be as great, leading to an Expansion of morbidity. Most of the data collected today, 67: in particular by the international <A href="http://www.reves.org/">REVES</A> 68: network on Health Expectancy and the disability process, and most HE indices based on these data, are 69: cross-sectional. This means that the information collected comes from a 70: single cross-sectional survey: people from a variety of ages (but often old people) 71: are surveyed on their health status at a single date. The proportion of people 72: disabled at each age can then be estimated at that date. This age-specific 73: prevalence curve is used to distinguish, within the stationary population 74: (which, by definition, is the life table estimated from the vital statistics on 75: mortality at the same date), the disabled population from the disability-free 76: population. Life expectancy (LE) (or total population divided by the yearly 77: number of births or deaths of this stationary population) is then decomposed 78: into disability-free life expectancy (DFLE) and disability life 79: expectancy (DLE). This method of computing HE is usually called the Sullivan 80: method (after the author who first described it). 81: The age-specific proportions of people disabled (prevalence of disability) are 82: dependent upon the historical flows from entering disability and recovering in the past. The age-specific forces (or incidence rates) of entering 83: disability or recovering a good health, estimated over a recent period of time (as period forces of mortality), are reflecting current conditions and 84: therefore can be used at each age to forecast the future of this cohort if 85: nothing changes in the future, i.e to forecast the prevalence of disability of each cohort. Our finding (2) is that the period prevalence of disability 86: (computed from period incidences) is lower than the cross-sectional prevalence. 87: For example if a country is improving its technology of prosthesis, the 88: incidence of recovering the ability to walk will be higher at each (old) age, 89: but the prevalence of disability will only slightly reflect an improvement because 90: the prevalence is mostly affected by the history of the cohort and not by recent 91: period effects. To measure the period improvement we have to simulate the future 92: of a cohort of new-borns entering or leaving the disability state or 93: dying at each age according to the incidence rates measured today on different cohorts. The 94: proportion of people disabled at each age in this simulated cohort will be much 95: lower that the proportions observed at each age in a cross-sectional survey. 96: This new prevalence curve introduced in a life table will give a more realistic 97: HE level than the Sullivan method which mostly reflects the history of health 98: conditions in a country. 99: Therefore, the main question is how can we measure incidence rates from 100: cross-longitudinal surveys? This is the goal of the IMaCH program. From your 101: data and using IMaCH you can estimate period HE as well as the Sullivan HE. In addition the standard errors of the HE are computed. 102: A cross-longitudinal survey consists of a first survey ("cross") where 103: individuals of different ages are interviewed about their health status or degree 104: of disability. At least a second wave of interviews ("longitudinal") should 105: measure each individual new health status. Health expectancies are computed from 106: the transitions observed between waves (interviews) and are computed for each degree of 107: severity of disability (number of health states). The more degrees of severity considered, the more 108: time is necessary to reach the Maximum Likelihood of the parameters involved in 109: the model. Considering only two states of disability (disabled and healthy) is 110: generally enough but the computer program works also with more health 111: states. The simplest model for the transition probabilities is the multinomial logistic model where 112: pij is the probability to be observed in state j at the second 113: wave conditional to be observed in state i at the first wave. Therefore 114: a simple model is: log(pij/pii)= aij + bij*age+ cij*sex, where 115: 'age' is age and 'sex' is a covariate. The advantage that this 116: computer program claims, is that if the delay between waves is not 117: identical for each individual, or if some individual missed an interview, the 118: information is not rounded or lost, but taken into account using an 119: interpolation or extrapolation. hPijx is the probability to be observed 120: in state i at age x+h conditional on the observed state i 121: at age x. The delay 'h' can be split into an exact number 122: (nh*stepm) of unobserved intermediate states. This elementary transition 123: (by month or quarter, trimester, semester or year) is modeled as the above multinomial 124: logistic. The hPx matrix is simply the matrix product of nh*stepm 125: elementary matrices and the contribution of each individual to the likelihood is 126: simply hPijx. 127: The program presented in this manual is a general program named 128: IMaCh (for Interpolated 129: MArkov CHain), designed to analyse transitions from longitudinal surveys. The first step is the estimation of the set of the parameters of a model for the 130: transition probabilities between an initial state and a final state. 131: From there, the computer program produces indicators such as the observed and 132: stationary prevalence, life expectancies and their variances both numerically and graphically. Our 133: transition model consists of absorbing and non-absorbing states assuming the 134: possibility of return across the non-absorbing states. The main advantage of 135: this package, compared to other programs for the analysis of transition data 136: (for example: Proc Catmod of SAS®) is that the whole individual 137: information is used even if an interview is missing, a state or a date is 138: unknown or when the delay between waves is not identical for each individual. 139: The program is dependent upon a set of parameters inputted by the user: selection of a sub-sample, 140: number of absorbing and non-absorbing states, number of waves to be taken in account , a tolerance level for the 141: maximization function, the periodicity of the transitions (we can compute 142: annual, quarterly or monthly transitions), covariates in the model. IMaCh works on 143: Windows or on Unix platform. 144: <HR> 145: 146: (1) Laditka S. B. and Wolf, D. (1998), New Methods for Analyzing 147: Active Life Expectancy. Journal of Aging and Health. Vol 10, No. 2. 148: (2) <A 149: href="http://taylorandfrancis.metapress.com/app/home/contribution.asp?wasp=1f99bwtvmk5yrb7hlhw3&referrer=parent&backto=issue,1,2;journal,2,5;linkingpublicationresults,1:300265,1">Lièvre 150: A., Brouard N. and Heathcote Ch. (2003) Estimating Health Expectancies from 151: Cross-longitudinal surveys. Mathematical Population Studies.- 10(4), 152: pp. 211-248</A> 153: <HR> 154: 155: <H2><A name=data>What kind of data is required?</A></H2> 156: The minimum data required for a transition model is the recording of a set of 157: individuals interviewed at a first date and interviewed once more. From the observations of an individual, we obtain a follow-up over 158: time of the occurrence of a specific event. In this documentation, the event is 159: related to health state, but the program can be applied to many 160: longitudinal studies with different contexts. To build the data file 161: as explained 162: in the next section, you must have the month and year of each interview and 163: the corresponding health state. In order to get age, date of birth (month 164: and year) are required (missing values are allowed for month). Date of death 165: (month and year) is an important information also required if the individual is 166: dead. Shorter steps (i.e. a month) will more closely take into account the 167: survival time after the last interview. 168: <HR> 169: 170: <H2><A name=datafile>The data file</A></H2> 171: In this example, 8,000 people have been interviewed in a cross-longitudinal 172: survey of 4 waves (1984, 1986, 1988, 1990). Some people missed 1, 2 or 3 173: interviews. Health states are healthy (1) and disabled (2). The survey is not a 174: real one but a simulation of the American Longitudinal Survey on Aging. The 175: disability state is defined as dependence in at least one of four ADLs (Activities 176: of daily living, like bathing, eating, walking). Therefore, even if the 177: individuals interviewed in the sample are virtual, the information in 178: this sample is close to reality for the United States. Sex is not recorded 179: is this sample. The LSOA survey is biased in the sense that people 180: living in an institution were not included in the first interview in 181: 1984. Thus the prevalence of disability observed in 1984 is lower than 182: the true prevalence at old ages. However when people moved into an 183: institution, they were interviewed there in 1986, 1988 and 1990. Thus 184: the incidences of disabilities are not biased. Cross-sectional 185: prevalences of disability at old ages are thus artificially increasing in 1986, 186: 1988 and 1990 because of a greater proportion of the sample 187: institutionalized. Our article (Lièvre A., Brouard N. and Heathcote 188: Ch. (2003)) shows the opposite: the period prevalence based on the 189: incidences is lower at old 190: ages than the adjusted cross-sectional prevalence illustrating that 191: there has been significant progress against disability. 192: Each line of the data set (named <A 193: href="http://euroreves.ined.fr/imach/doc/data1.txt">data1.txt</A> in this first 194: example) is an individual record. Fields are separated by blanks: 195: <UL> 196: <LI>Index number: positive number (field 1) 197: <LI>First covariate positive number (field 2) 198: <LI>Second covariate positive number (field 3) 199: <LI><A name=Weight>Weight</A>: positive number (field 4) . In most 200: surveys individuals are weighted to account for stratification of the 201: sample. 202: <LI>Date of birth: coded as mm/yyyy. Missing dates are coded as 99/9999 203: (field 5) 204: <LI>Date of death: coded as mm/yyyy. Missing dates are coded as 99/9999 205: (field 6) 206: <LI>Date of first interview: coded as mm/yyyy. Missing dates are coded 207: as 99/9999 (field 7) 208: <LI>Status at first interview: positive number. Missing values ar coded 209: -1. (field 8) 210: <LI>Date of second interview: coded as mm/yyyy. Missing dates are coded 211: as 99/9999 (field 9) 212: <LI>Status at second interview positive number. Missing 213: values ar coded -1. (field 10) 214: <LI>Date of third interview: coded as mm/yyyy. Missing dates are coded 215: as 99/9999 (field 11) 216: <LI>Status at third interview positive number. Missing values 217: ar coded -1. (field 12) 218: <LI>Date of fourth interview: coded as mm/yyyy. Missing dates are coded 219: as 99/9999 (field 13) 220: <LI>Status at fourth interview positive number. Missing 221: values are coded -1. (field 14) 222: <LI>etc </LI></UL> 223:   224: If you do not wish to include information on weights or 225: covariates, you must fill the column with a number (e.g. 1) since all 226: fields must be present. 227: <HR> 228: 229: <H2>Your first example parameter file<A 230: href="http://euroreves.ined.fr/imach"></A><A name=uio></A></H2> 231: <H2><A name=biaspar></A>#Imach version 0.97b, June 2004, INED-EUROREVES </H2> 232: This first line was a comment. Comments line start with a '#'. 233: <H4>First uncommented line</H4><PRE>title=1st_example datafile=data1.txt lastobs=8600 firstpass=1 lastpass=4</PRE> 234: <UL> 235: <LI>title= 1st_example is title of the run. 236: <LI>datafile= data1.txt is the name of the data set. Our example is a 237: six years follow-up survey. It consists of a baseline followed by 3 238: reinterviews. 239: <LI>lastobs= 8600 the program is able to run on a subsample where the 240: last observation number is lastobs. It can be set a bigger number than the 241: real number of observations (e.g. 100000). In this example, maximisation will 242: be done on the first 8600 records. 243: <LI>firstpass=1 , lastpass=4 If there are more than two interviews 244: in the survey, the program can be run on selected transitions periods. 245: firstpass=1 means the first interview included in the calculation is the 246: baseline survey. lastpass=4 means that the last interview to be 247: included will be by the 4th. </LI></UL> 248:   249: <H4><A name=biaspar-2>Second uncommented 250: line</A></H4><PRE>ftol=1.e-08 stepm=1 ncovcol=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</PRE> 251: <UL> 252: <LI>ftol=1e-8 Convergence tolerance on the function value in the 253: maximisation of the likelihood. Choosing a correct value for ftol is 254: difficult. 1e-8 is the correct value for a 32 bit computer. 255: <LI>stepm=1 The time unit in months for interpolation. Examples: 256: <UL> 257: <LI>If stepm=1, the unit is a month 258: <LI>If stepm=4, the unit is a trimester 259: <LI>If stepm=12, the unit is a year 260: <LI>If stepm=24, the unit is two years 261: <LI>... </LI></UL> 262: <LI>ncovcol=2 Number of covariate columns included in the datafile 263: before the column for the date of birth. You can include covariates 264: that will not be used in the model as this number is not the number of covariates that will 265: be specified by the model. The 'model' syntax describes the covariates to be 266: taken into account during the run. 267: <LI>nlstate=2 Number of non-absorbing (alive) states. Here we have two 268: alive states: disability-free is coded 1 and disability is coded 2. 269: <LI>ndeath=1 Number of absorbing states. The absorbing state death is 270: coded 3. 271: <LI>maxwav=4 Number of waves in the datafile. 272: <LI><A name=mle>mle</A>=1 Option for the Maximisation Likelihood 273: Estimation. 274: <UL> 275: <LI>If mle=1 the program does the maximisation and the calculation of health 276: expectancies 277: <LI>If mle=0 the program only does the calculation of the health 278: expectancies and other indices and graphs but without the maximization. 279: There are also other possible values: 280: <UL> 281: <LI>If mle=-1 you get a template for the number of parameters 282: and the size of the variance-covariance matrix. This is useful if the model is 283: complex with many covariates. 284: <LI>If mle=-3 IMaCh computes the mortality but without any health status 285: (May 2004) 286: <LI>If mle=2 IMach likelihood corresponds to a linear interpolation 287: <LI>If mle=3 IMach likelihood corresponds to an exponential 288: inter-extrapolation 289: <LI>If mle=4 IMach likelihood corresponds to no inter-extrapolation, thus biasing the results. 290: <LI>If mle=5 IMach likelihood corresponds to no inter-extrapolation, and 291: before the correction of the Jackson's bug (avoid this). </LI></UL></LI></UL> 292: <LI>weight=0 Provides the possibility of adding weights. 293: <UL> 294: <LI>If weight=0 no weights are included 295: <LI>If weight=1 the maximisation integrates the weights which are in field 296: <A href="http://euroreves.ined.fr/imach/doc/imach.htm#Weight">4</A> 297: </LI></UL></LI></UL> 298: <H4>Covariates</H4> 299: Intercept and age are automatically included in the model. Additional 300: covariates can be included with the command: <PRE>model=list of covariates</PRE> 301: <UL> 302: <LI>if model=. then no covariates are included 303: <LI>if model=V1 the model includes the first covariate (field 304: 2) 305: <LI>if model=V2 the model includes the second covariate 306: (field 3) 307: <LI>if model=V1+V2 the model includes the first and the 308: second covariate (fields 2 and 3) 309: <LI>if model=V1*V2 the model includes the product of the 310: first and the second covariate (fields 2 and 3) 311: <LI>if model=V1+V1*age the model includes the product 312: covariate*age </LI></UL> 313: In this example, we have two covariates in the data file (fields 2 and 3). 314: The number of covariates included in the data file between the id and the date 315: of birth is ncovcol=2 (it was named ncov in version prior to 0.8). If you have 3 316: covariates in the datafile (fields 2, 3 and 4), you will set ncovcol=3. Then you 317: can run the programme with a new parametrisation taking into account the third 318: covariate. For example, model=V1+V3 estimates a model with the 319: first and third covariates. More complicated models can be used, but this will 320: take more time to converge. With a simple model (no covariates), the programme 321: estimates 8 parameters. Adding covariates increases the number of parameters : 322: 12 for model=V1, 16 for model=V1+V1*age and 20 323: for model=V1+V2+V3. 324: <H4>Guess values for optimization 325: </H4> 326: You must write the initial guess values of the parameters for optimization. 327: The number of parameters, N depends on the number of absorbing states 328: and non-absorbing states and on the number of covariates in the model (ncovmodel). N is 329: given by the formula N=(nlstate + 330: ndeath-1)*nlstate*ncovmodel . Thus in 331: the simple case with 2 covariates in the model(the model is log (pij/pii) = aij + bij * age 332: where intercept and age are the two covariates), and 2 health states (1 for 333: disability-free and 2 for disability) and 1 absorbing state (3), you must enter 334: 8 initials values, a12, b12, a13, b13, a21, b21, a23, b23. You can start with 335: zeros as in this example, but if you have a more precise set (for example from 336: an earlier run) you can enter it and it will speed up the convergence Each of the four 337: lines starts with indices "ij": ij aij bij 338: <BLOCKQUOTE><PRE># Guess values of aij and bij in log (pij/pii) = aij + bij * age 339: 12 -14.155633 0.110794 340: 13 -7.925360 0.032091 341: 21 -1.890135 -0.029473 342: 23 -6.234642 0.022315 </PRE></BLOCKQUOTE> 343: or, to simplify (in most of cases it converges but there is no warranty!): 344: 345: <BLOCKQUOTE><PRE>12 0.0 0.0 346: 13 0.0 0.0 347: 21 0.0 0.0 348: 23 0.0 0.0</PRE></BLOCKQUOTE> 349: In order to speed up the convergence you can make a first run with a large 350: stepm i.e stepm=12 or 24 and then decrease the stepm until stepm=1 month. If 351: newstepm is the new shorter stepm and stepm can be expressed as a multiple of 352: newstepm, like newstepm=n stepm, then the following approximation holds: <PRE>aij(stepm) = aij(n . stepm) - ln(n) 353: </PRE> 354: and <PRE>bij(stepm) = bij(n . stepm) .</PRE> 355: For example if you already ran with stepm=6 (a 6 months interval) and got: <PRE># Parameters 356: 12 -13.390179 0.126133 357: 13 -7.493460 0.048069 358: 21 0.575975 -0.041322 359: 23 -4.748678 0.030626 360: </PRE> 361: Then you now want to get the monthly estimates, you can guess the aij by 362: subtracting ln(6)= 1.7917 and running using <PRE>12 -15.18193847 0.126133 363: 13 -9.285219469 0.048069 364: 21 -1.215784469 -0.041322 365: 23 -6.540437469 0.030626 366: </PRE> 367: and get <PRE>12 -15.029768 0.124347 368: 13 -8.472981 0.036599 369: 21 -1.472527 -0.038394 370: 23 -6.553602 0.029856 371: 372: which is closer to the results. The approximation is probably useful 373: only for very small intervals and we don't have enough experience to 374: know if you will speed up the convergence or not. 375: </PRE><PRE> -ln(12)= -2.484 376: -ln(6/1)=-ln(6)= -1.791 377: -ln(3/1)=-ln(3)= -1.0986 378: -ln(12/6)=-ln(2)= -0.693 379: </PRE>In version 0.9 and higher you can still have valuable results even if your 380: stepm parameter is bigger than a month. The idea is to run with bigger stepm in 381: order to have a quicker convergence at the price of a small bias. Once you know 382: which model you want to fit, you can put stepm=1 and wait hours or days to get 383: the convergence! To get unbiased results even with large stepm we introduce the 384: idea of pseudo likelihood by interpolating two exact likelihoods. In 385: more detail: 386: If the interval of d months between two waves is not a multiple of 387: 'stepm', but is between (n-1) stepm and n stepm then 388: both exact likelihoods are computed (the contribution to the likelihood at n 389: stepm requires one matrix product more) (let us remember that we are 390: modelling the probability to be observed in a particular state after d 391: months being observed at a particular state at 0). The distance, (bh in 392: the program), from the month of interview to the rounded date of n 393: stepm is computed. It can be negative (interview occurs before n 394: stepm) or positive if the interview occurs after n stepm (and 395: before (n+1)stepm). Then the final contribution to the total 396: likelihood is a weighted average of these two exact likelihoods at n 397: stepm (out) and at (n-1)stepm(savm). We did not want to compute 398: the third likelihood at (n+1)stepm because it is too costly in time, so 399: we used an extrapolation if bh is positive. The formula 400: for the inter/extrapolation may vary according to the value of parameter mle: <PRE>mle=1 lli= log((1.+bbh)*out[s1][s2]- bbh*savm[s1][s2]); /* linear interpolation */ 401: 402: mle=2 lli= (savm[s1][s2]>(double)1.e-8 ? \ 403: log((1.+bbh)*out[s1][s2]- bbh*(savm[s1][s2])): \ 404: log((1.+bbh)*out[s1][s2])); /* linear interpolation */ 405: mle=3 lli= (savm[s1][s2]>1.e-8 ? \ 406: (1.+bbh)*log(out[s1][s2])- bbh*log(savm[s1][s2]): \ 407: log((1.+bbh)*out[s1][s2])); /* exponential inter-extrapolation */ 408: 409: mle=4 lli=log(out[s[mw[mi][i]][i]][s[mw[mi+1][i]][i]]); /* No interpolation */ 410: no need to save previous likelihood into memory. 411: </PRE> 412: If the death occurs between the first and second pass, and for example more 413: precisely between n stepm and (n+1)stepm the contribution of 414: these people to the likelihood is simply the difference between the probability 415: of dying before n stepm and the probability of dying before 416: (n+1)stepm. There was a bug in version 0.8 and death was treated as any 417: other state, i.e. as if it was an observed death at second pass. This was not 418: precise but correct, although when information on the precise month of 419: death came (death occuring prior to second pass) we did not change the 420: likelihood accordingly. We thank Chris Jackson for correcting it. In earlier 421: versions (fortunately before first publication) the total mortality 422: was thus overestimated (people were dying too early) by about 10%. Version 423: 0.95 and higher are correct. 424: 425: Our suggested choice is mle=1 . If stepm=1 there is no difference between 426: various mle options (methods of interpolation). If stepm is big, like 12 or 24 427: or 48 and mle=4 (no interpolation) the bias may be very important if the mean 428: duration between two waves is not a multiple of stepm. See the appendix in our 429: main publication concerning the sine curve of biases. 430: <H4>Guess values for computing variances</H4> 431: These values are output by the maximisation of the likelihood <A 432: href="http://euroreves.ined.fr/imach/doc/imach.htm#mle">mle</A>=1 and 433: can be used as an input for a second run in order to get the various output data 434: files (Health expectancies, period prevalence etc.) and figures without 435: rerunning the long maximisation phase (mle=0). 436: The 'scales' are small values needed for the computing of numerical 437: derivatives. These derivatives are used to compute the hessian matrix of the 438: parameters, that is the inverse of the covariance matrix. They are often used 439: for estimating variances and confidence intervals. Each line consists of indices 440: "ij" followed by the initial scales (zero to simplify) associated with aij and 441: bij. 442: <UL> 443: <LI>If mle=1 you can enter zeros: 444: <LI> 445: <BLOCKQUOTE><PRE># Scales (for hessian or gradient estimation) 446: 12 0. 0. 447: 13 0. 0. 448: 21 0. 0. 449: 23 0. 0. </PRE></BLOCKQUOTE> 450: <LI>If mle=0 (no maximisation of Likelihood) you must enter a covariance 451: matrix (usually obtained from an earlier run). </LI></UL> 452: <H4>Covariance matrix of parameters</H4> 453: The covariance matrix is output if <A 454: href="http://euroreves.ined.fr/imach/doc/imach.htm#mle">mle</A>=1. But it can be 455: also be used as an input to get the various output data files (Health 456: expectancies, period prevalence etc.) and figures without rerunning 457: the maximisation phase (mle=0). Each line starts with indices 458: "ijk" followed by the covariances between aij and bij: 459: <PRE> 121 Var(a12) 460: 122 Cov(b12,a12) Var(b12) 461: ... 462: 232 Cov(b23,a12) Cov(b23,b12) ... Var (b23) </PRE> 463: <UL> 464: <LI>If mle=1 you can enter zeros. 465: <LI><PRE># Covariance matrix 466: 121 0. 467: 122 0. 0. 468: 131 0. 0. 0. 469: 132 0. 0. 0. 0. 470: 211 0. 0. 0. 0. 0. 471: 212 0. 0. 0. 0. 0. 0. 472: 231 0. 0. 0. 0. 0. 0. 0. 473: 232 0. 0. 0. 0. 0. 0. 0. 0.</PRE> 474: <LI>If mle=0 you must enter a covariance matrix (usually obtained from an 475: earlier run). </LI></UL> 476: <H4>Age range for calculation of stationary prevalences and 477: health expectancies</H4><PRE>agemin=70 agemax=100 bage=50 fage=100</PRE> 478: Once we obtained the estimated parameters, the program is able to calculate 479: period prevalence, transitions probabilities and life expectancies at any age. 480: Choice of the age range is useful for extrapolation. In this example, 481: the age of people interviewed varies from 69 to 102 and the model is 482: estimated using their exact ages. But if you are interested in the 483: age-specific period prevalence you can start the simulation at an 484: exact age like 70 and stop at 100. Then the program will draw at 485: least two curves describing the forecasted prevalences of two cohorts, 486: one for healthy people at age 70 and the second for disabled people at 487: the same initial age. And according to the mixing property 488: (ergodicity) and because of recovery, both prevalences will tend to be 489: identical at later ages. Thus if you want to compute the prevalence at 490: age 70, you should enter a lower agemin value. 491: Setting bage=50 (begin age) and fage=100 (final age), let the program compute 492: life expectancy from age 'bage' to age 'fage'. As we use a model, we can 493: interessingly compute life expectancy on a wider age range than the age range 494: from the data. But the model can be rather wrong on much larger intervals. 495: Program is limited to around 120 for upper age! <PRE></PRE> 496: <UL> 497: <LI>agemin= Minimum age for calculation of the period prevalence 498: <LI>agemax= Maximum age for calculation of the period prevalence 499: <LI>bage= Minimum age for calculation of the health expectancies 500: <LI>fage= Maximum age for calculation of the health expectancies 501: </LI></UL> 502: <H4><A name=Computing>Computing</A> the cross-sectional prevalence</H4><PRE>begin-prev-date=1/1/1984 end-prev-date=1/6/1988 estepm=1</PRE> 504: Statements 'begin-prev-date' and 'end-prev-date' allow the user to 505: select the period in which the observed prevalences in each state. In 506: this example, the prevalences are calculated on data survey collected 507: between 1 January 1984 and 1 June 1988. 508: <UL> 509: <LI>begin-prev-date= Starting date (day/month/year) 510: <LI>end-prev-date= Final date (day/month/year) 511: <LI>estepm= Unit (in months).We compute the life expectancy 512: from trapezoids spaced every estepm months. This is mainly to measure the 513: difference between two models: for example if stepm=24 months pijx are given 514: only every 2 years and by summing them we are calculating an estimate of the 515: Life Expectancy assuming a linear progression inbetween and thus 516: overestimating or underestimating according to the curvature of the survival 517: function. If, for the same date, we estimate the model with stepm=1 month, we 518: can keep estepm to 24 months to compare the new estimate of Life expectancy 519: with the same linear hypothesis. A more precise result, taking into account a 520: more precise curvature will be obtained if estepm is as small as stepm. 521: </LI></UL> 522: <H4>Population- or status-based health 523: expectancies</H4><PRE>pop_based=0</PRE> 524: The program computes status-based health expectancies, i.e health 525: expectancies which depend on the initial health state. If you are healthy, your 526: healthy life expectancy (e11) is higher than if you were disabled (e21, with e11 527: > e21). To compute a healthy life expectancy 'independent' of the initial 528: status we have to weight e11 and e21 according to the probability of 529: being in each state at initial age which correspond to the proportions 530: of people in each health state (cross-sectional prevalences). 531: We could also compute e12 and e12 and get e.2 by weighting them according to 532: the observed cross-sectional prevalences at initial age. 533: In a similar way we could compute the total life expectancy by summing e.1 534: and e.2 . The main difference between 'population based' and 'implied' or 535: 'period' is in the weights used. 'Usually', cross-sectional prevalences of 536: disability are higher than period prevalences particularly at old ages. This is 537: true if the country is improving its health system by teaching people how to 538: prevent disability by promoting better screening, for example of people 539: needing cataract surgery. Then the proportion of disabled people at 540: age 90 will be lower than the current observed proportion. 541: Thus a better Health Expectancy and even a better Life Expectancy value is 542: given by forecasting not only the current lower mortality at all ages but also a 543: lower incidence of disability and higher recovery. Using the period 544: prevalences as weight instead of the cross-sectional prevalences we are 545: computing indices which are more specific to the current situations and 546: therefore more useful to predict improvements or regressions in the future as to 547: compare different policies in various countries. 548: <UL> 549: <LI>popbased= 0 Health expectancies are computed at each age 550: from period prevalences 'expected' at this initial age. 551: <LI>popbased= 1 Health expectancies are computed at each age 552: from cross-sectional 'observed' prevalence at the initial age. As all the 553: population is not observed at the same exact date we define a short period 554: where the observed prevalence can be computed as follows: we simply sum all people 555: surveyed within these two exact dates who belong to a particular age group 556: (single year) at the date of interview and are in a particular health state. 557: Then it is easy to get the proportion of people in a particular 558: health state as a percentage of all people of the same age group. If both dates are spaced and are 559: covering two waves or more, people being interviewed twice or more are counted 560: twice or more. The program takes into account the selection of individuals 561: interviewed between firstpass and lastpass too (we don't know if 562: this is useful). </LI></UL> 563: <H4>Prevalence forecasting (Experimental)</H4><PRE>starting-proj-date=1/1/1989 final-proj-date=1/1/1992 mov_average=0 </PRE> 564: Prevalence and population projections are only available if the interpolation 565: unit is a month, i.e. stepm=1 and if there are no covariate. The programme 566: estimates the prevalence in each state at a precise date expressed in 567: day/month/year. The programme computes one forecasted prevalence a year from a 568: starting date (1 January 1989 in this example) to a final date (1 January 569: 1992). The statement mov_average allows computation of smoothed forecasted 570: prevalences with a five-age moving average centered at the mid-age of the 571: fiveyear-age period. 572: <H4>Population forecasting (Experimental)</H4> 573: <UL> 574: <LI>starting-proj-date= starting date (day/month/year) of 575: forecasting 576: <LI>final-proj-date= final date (day/month/year) of 577: forecasting 578: <LI>mov_average= smoothing with a five-age moving average 579: centered at the mid-age of the fiveyear-age period. The command 580: mov_average takes value 1 if the prevalences are smoothed and 0 581: otherwise. </LI></UL> 582: <UL type=disc> 583: <LI>popforecast= 0 Option for population forecasting. If popforecast=1, 584: the programme does the forecasting. 585: <LI>popfile= name of the population file 586: <LI>popfiledate= date of the population population 587: <LI>last-popfiledate= date of the last population projection  588: </LI></UL> 589: <HR> 590: 591: <H2><A name=running></A>Running Imach with this 592: example</H2> 593: We assume that you have already typed your <A 594: href="http://euroreves.ined.fr/imach/doc/biaspar.imach">1st_example parameter 595: file</A> as explained <A 596: href="http://euroreves.ined.fr/imach/doc/imach.htm#biaspar">above</A>. To run 597: the program under Windows you should either: 598: <UL> 599: <LI>click on the imach.exe icon and either: 600: <UL> 601: <LI>enter the name of the parameter file which is for example 602: <TT>C:\home\myname\lsoa\biaspar.imach</TT> 603: <LI>or locate the biaspar.imach icon in your folder such as 604: <TT>C:\home\myname\lsoa</TT> and drag it, with your mouse, on the already 605: open imach window. </LI></UL> 606: <LI>With version (0.97b) if you ran setup at installation, Windows is supposed 607: to understand the ".imach" extension and you can right click the biaspar.imach 608: icon and either edit with wordpad (better than notepad) the parameter file or 609: execute it with IMaCh. </LI></UL> 610: The time to converge depends on the step unit used (1 month is more 611: precise but more cpu time consuming), on the number of cases, and on the number of 612: variables (covariates). 613: The program outputs many files. Most of them are files which will be plotted 614: for better understanding. To run under Linux is mostly the same. 615: It is no more difficult to run IMaCh on a MacIntosh. 616: <HR> 617: 618: <H2><A name=output>Output of the program and graphs 619: </A></H2> 620: Once the optimization is finished (once the convergence is reached), many 621: tables and graphics are produced. 622: The IMaCh program will create a subdirectory with the same name as your 623: parameter file (here mypar) where all the tables and figures will be 624: stored. Important files like the log file and the output parameter file 625: (the latter contains the maximum likelihood estimates) are stored at 626: the main level not in this subdirectory. Files with extension .log and 627: .txt can be edited with a standard editor like wordpad or notepad or 628: even can be viewed with a browser like Internet Explorer or Mozilla. 629: The main html file is also named with the same name <A 630: href="http://euroreves.ined.fr/imach/doc/biaspar.htm">biaspar.htm</A>. You can 631: click on it by holding your shift key in order to open it in another window 632: (Windows). 633: Our grapher is Gnuplot, an interactive plotting program (GPL) which can 634: also work in batch mode. A gnuplot reference manual is available <A 635: href="http://www.gnuplot.info/">here</A>. When the run is finished, and in 636: order that the window doesn't disappear, the user should enter a character like 637: <TT>q</TT> for quitting. These characters are: 638: <UL> 639: <LI>'e' for opening the main result html file <A 640: href="http://euroreves.ined.fr/imach/doc/biaspar.htm">biaspar.htm</A> 641: file to edit the output files and graphs. 642: <LI>'g' to graph again 643: <LI>'c' to start again the program from the beginning. 644: <LI>'q' for exiting. </LI></UL>The main gnuplot file is named 645: <TT>biaspar.gp</TT> and can be edited (right click) and run again. 646: Gnuplot is easy and you can use it to make more complex graphs. Just click on 647: gnuplot and type plot sin(x) to see how easy it is. 648: <H5>Results files - <A 650: name="cross-sectional prevalence in each state">cross-sectional prevalence in each state</A> (and at first pass): <A 653: href="http://euroreves.ined.fr/imach/doc/biaspar/prbiaspar.txt">biaspar/prbiaspar.txt</A> </H5> 654: The first line is the title and displays each field of the file. First column 655: corresponds to age. Fields 2 and 6 are the proportion of individuals in states 1 656: and 2 respectively as observed at first exam. Others fields are the numbers of 657: people in states 1, 2 or more. The number of columns increases if the number of 658: states is higher than 2. The header of the file is <PRE># Age Prev(1) N(1) N Age Prev(2) N(2) N 659: 70 1.00000 631 631 70 0.00000 0 631 660: 71 0.99681 625 627 71 0.00319 2 627 661: 72 0.97125 1115 1148 72 0.02875 33 1148 </PRE> 662: It means that at age 70 (between 70 and 71), the prevalence in state 1 is 663: 1.000 and in state 2 is 0.00 . At age 71 the number of individuals in state 1 is 664: 625 and in state 2 is 2, hence the total number of people aged 71 is 625+2=627. 665: 666: <H5>- Estimated parameters and covariance 667: matrix: <A 668: href="http://euroreves.ined.fr/imach/doc/rbiaspar.txt">rbiaspar.imach</A></H5> 669: This file contains all the maximisation results: <PRE> -2 log likelihood= 21660.918613445392 670: Estimated parameters: a12 = -12.290174 b12 = 0.092161 671: a13 = -9.155590 b13 = 0.046627 672: a21 = -2.629849 b21 = -0.022030 673: a23 = -7.958519 b23 = 0.042614 674: Covariance matrix: Var(a12) = 1.47453e-001 675: Var(b12) = 2.18676e-005 676: Var(a13) = 2.09715e-001 677: Var(b13) = 3.28937e-005 678: Var(a21) = 9.19832e-001 679: Var(b21) = 1.29229e-004 680: Var(a23) = 4.48405e-001 681: Var(b23) = 5.85631e-005 682: </PRE> 683: By substitution of these parameters in the regression model, we obtain the 684: elementary transition probabilities: 685: <IMG height=300 686: src="biaspar/pebiaspar11.png" 687: width=400> 688: <H5>- Transition probabilities: 689: <A 690: href="http://euroreves.ined.fr/imach/doc/biaspar/pijrbiaspar.txt">biaspar/pijrbiaspar.txt</A></H5> 691: Here are the transitions probabilities Pij(x, x+nh). The second column is the 692: starting age x (from age 95 to 65), the third is age (x+nh) and the others are 693: the transition probabilities p11, p12, p13, p21, p22, p23. The first column 694: indicates the value of the covariate (without any other variable than age it is 695: equal to 1) For example, line 5 of the file is: <PRE>1 100 106 0.02655 0.17622 0.79722 0.01809 0.13678 0.84513 </PRE> 696: and this means: <PRE>p11(100,106)=0.02655 697: p12(100,106)=0.17622 698: p13(100,106)=0.79722 699: p21(100,106)=0.01809 700: p22(100,106)=0.13678 701: p22(100,106)=0.84513 </PRE> 702: <H5>- <A 703: name="Period prevalence in each state">Period 704: prevalence in each state</A>: <A 705: href="http://euroreves.ined.fr/imach/doc/biaspar/plrbiaspar.txt">biaspar/plrbiaspar.txt</A></H5><PRE>#Prevalence 706: #Age 1-1 2-2 707: 708: #************ 709: 70 0.90134 0.09866 710: 71 0.89177 0.10823 711: 72 0.88139 0.11861 712: 73 0.87015 0.12985 </PRE> 713: At age 70 the period prevalence is 0.90134 in state 1 and 0.09866 in state 2. 714: This period prevalence differs from the cross-sectional prevalence and 715: we explaining. The cross-sectional prevalence at age 70 results from 716: the incidence of disability, incidence of recovery and mortality which 717: occurred in the past for the cohort. Period prevalence results from a 718: simulation with current incidences of disability, recovery and 719: mortality estimated from this cross-longitudinal survey. It is a good 720: prediction of the prevalence in the future if "nothing changes in the 721: future". This is exactly what demographers do with a period life 722: table. Life expectancy is the expected mean survival time if current 723: mortality rates (age-specific incidences of mortality) "remain 724: constant" in the future. 725: 726: <H5>- Standard deviation of period 727: prevalence: <A 728: href="http://euroreves.ined.fr/imach/doc/biaspar/vplrbiaspar.txt">biaspar/vplrbiaspar.txt</A></H5> 729: The period prevalence has to be compared with the cross-sectional prevalence. 730: But both are statistical estimates and therefore have confidence intervals. 731: For the cross-sectional prevalence we generally need information on the 732: design of the surveys. It is usually not enough to consider the number of people 733: surveyed at a particular age and to estimate a Bernouilli confidence interval 734: based on the prevalence at that age. But you can do it to have an idea of the 735: randomness. At least you can get a visual appreciation of the randomness by 736: looking at the fluctuation over ages. 737: For the period prevalence it is possible to estimate the confidence interval 738: from the Hessian matrix (see the publication for details). We are supposing that 739: the design of the survey will only alter the weight of each individual. IMaCh 740: scales the weights of individuals-waves contributing to the likelihood by 741: making the sum of the weights equal to the sum of individuals-waves 742: contributing: a weighted survey doesn't increase or decrease the size of the 743: survey, it only give more weight to some individuals and thus less to the 744: others. 745: <H5>-cross-sectional and period prevalence in state 746: (2=disable) with confidence interval: <A 747: href="http://euroreves.ined.fr/imach/doc/biaspar/vbiaspar21.htm">biaspar/vbiaspar21.png</A></H5> 748: This graph exhibits the period prevalence in state (2) with the confidence 749: interval in red. The green curve is the observed prevalence (or proportion of 750: individuals in state (2)). Without discussing the results (it is not the purpose 751: here), we observe that the green curve is somewhat below the period 752: prevalence. If the data were not biased by the non inclusion of people 753: living in institutions we would have concluded that the prevalence of 754: disability will increase in the future (see the main publication if 755: you are interested in real data and results which are opposite). 756: <IMG height=300 757: src="biaspar/vbiaspar21.png" 758: width=400> 759: <H5>-Convergence to the period prevalence of 760: disability: <A 761: href="biaspar/pbiaspar11.png">biaspar/pbiaspar11.png</A> <IMG 762: height=300 763: src="biaspar/pbiaspar11.png" 764: width=400> </H5> 765: This graph plots the conditional transition probabilities from an initial 766: state (1=healthy in red at the bottom, or 2=disabled in green on the top) at age 767: x to the final state 2=disabled at age x+h 768: where conditional means conditional on being alive at age x+h which is 769: hP12x + hP22x. The curves hP12x/(hP12x + hP22x) 770: and hP22x/(hP12x + hP22x) converge with h, to the 771: period prevalence of disability. In order to get the period prevalence 772: at age 70 we should start the process at an earlier age, i.e.50. If the 773: disability state is defined by severe disability criteria with only a 774: small chance of recovering, then the incidence of recovery is low and the time to convergence is 775: probably longer. But we don't have experience of this yet. 776: <H5>- Life expectancies by age and initial health 777: status with standard deviation: <A 778: href="http://euroreves.ined.fr/imach/doc/biaspar/erbiaspar.txt">biaspar/erbiaspar.txt</A></H5><PRE># Health expectancies 779: # Age 1-1 (SE) 1-2 (SE) 2-1 (SE) 2-2 (SE) 780: 70 11.0180 (0.1277) 3.1950 (0.3635) 4.6500 (0.0871) 4.4807 (0.2187) 781: 71 10.4786 (0.1184) 3.2093 (0.3212) 4.3384 (0.0875) 4.4820 (0.2076) 782: 72 9.9551 (0.1103) 3.2236 (0.2827) 4.0426 (0.0885) 4.4827 (0.1966) 783: 73 9.4476 (0.1035) 3.2379 (0.2478) 3.7621 (0.0899) 4.4825 (0.1858) 784: 74 8.9564 (0.0980) 3.2522 (0.2165) 3.4966 (0.0920) 4.4815 (0.1754) 785: 75 8.4815 (0.0937) 3.2665 (0.1887) 3.2457 (0.0946) 4.4798 (0.1656) 786: 76 8.0230 (0.0905) 3.2806 (0.1645) 3.0090 (0.0979) 4.4772 (0.1565) 787: 77 7.5810 (0.0884) 3.2946 (0.1438) 2.7860 (0.1017) 4.4738 (0.1484) 788: 78 7.1554 (0.0871) 3.3084 (0.1264) 2.5763 (0.1062) 4.4696 (0.1416) 789: 79 6.7464 (0.0867) 3.3220 (0.1124) 2.3794 (0.1112) 4.4646 (0.1364) 790: 80 6.3538 (0.0868) 3.3354 (0.1014) 2.1949 (0.1168) 4.4587 (0.1331) 791: 81 5.9775 (0.0873) 3.3484 (0.0933) 2.0222 (0.1230) 4.4520 (0.1320) 792: </PRE><PRE>For example 70 11.0180 (0.1277) 3.1950 (0.3635) 4.6500 (0.0871) 4.4807 (0.2187) 793: means 794: e11=11.0180 e12=3.1950 e21=4.6500 e22=4.4807 </PRE><PRE><IMG height=300 src="biaspar/expbiaspar21.png" width=400><IMG height=300 src="biaspar/expbiaspar11.png" width=400></PRE> 795: For example, life expectancy of a healthy individual at age 70 is 11.0 in the 796: healthy state and 3.2 in the disability state (total of 14.2 years). If he was 797: disabled at age 70, his life expectancy will be shorter, 4.65 years in the 798: healthy state and 4.5 in the disability state (=9.15 years). The total life 799: expectancy is a weighted mean of both, 14.2 and 9.15. The weight is the 800: proportion of people disabled at age 70. In order to get a period index (i.e. 801: based only on incidences) we use the <A 802: href="http://euroreves.ined.fr/imach/doc/imach.htm#Period prevalence in each state">stable 803: or period prevalence</A> at age 70 (i.e. computed from incidences at earlier 804: ages) instead of the <A 805: href="http://euroreves.ined.fr/imach/doc/imach.htm#cross-sectional prevalence in each state">cross-sectional 806: prevalence</A> (observed for example at first interview) (<A 807: href="http://euroreves.ined.fr/imach/doc/imach.htm#Health expectancies">see 808: below</A>). 809: <H5>- Variances of life expectancies by age and 810: initial health status: <A 811: href="http://euroreves.ined.fr/imach/doc/biaspar/vrbiaspar.txt">biaspar/vrbiaspar.txt</A></H5> 812: For example, the covariances of life expectancies Cov(ei,ej) at age 50 are 813: (line 3) <PRE> Cov(e1,e1)=0.4776 Cov(e1,e2)=0.0488=Cov(e2,e1) Cov(e2,e2)=0.0424</PRE> 814: <H5>-Variances of one-step probabilities 815: : <A 816: href="http://euroreves.ined.fr/imach/doc/biaspar/probrbiaspar.txt">biaspar/probrbiaspar.txt</A></H5> 817: For example, at age 65<PRE> p11=9.960e-001 standard deviation of p11=2.359e-004</PRE> 818: <H5>- <A 819: name="Health expectancies">Health 820: expectancies</A> with standard errors 821: in parentheses: <A 822: href="http://euroreves.ined.fr/imach/doc/biaspar/trbiaspar.txt">biaspar/trbiaspar.txt</A></H5><PRE>#Total LEs with variances: e.. (std) e.1 (std) e.2 (std) </PRE><PRE>70 13.26 (0.22) 9.95 (0.20) 3.30 (0.14) </PRE> 824: Thus, at age 70 the total life expectancy, e..=13.26 years is the weighted 825: mean of e1.=13.46 and e2.=11.35 by the period prevalences at age 70 which are 826: 0.90134 in state 1 and 0.09866 in state 2 respectively (the sum is equal to 827: one). e.1=9.95 is the Disability-free life expectancy at age 70 (it is again a 828: weighted mean of e11 and e21). e.2=3.30 is also the life expectancy at age 70 to 829: be spent in the disability state. 830: <H5>-Total life expectancy by age and health 831: expectancies in states (1=healthy) and (2=disable): <A 832: href="biaspar/ebiaspar1.png">biaspar/ebiaspar1.png</A></H5> 833: This figure represents the health expectancies and the total life expectancy 834: with a confidence interval (dashed line). <PRE> <IMG height=300 src="biaspar/ebiaspar1.png" width=400></PRE> 835: Standard deviations (obtained from the information matrix of the model) of 836: these quantities are very useful. Cross-longitudinal surveys are costly and do 837: not involve huge samples, generally a few thousands; therefore it is very 838: important to have an idea of the standard deviation of our estimates. It has 839: been a big challenge to compute the Health Expectancy standard deviations. Don't 840: be confused: life expectancy is, as any expected value, the mean of a 841: distribution; but here we are not computing the standard deviation of the 842: distribution, but the standard deviation of the estimate of the mean. 843: Our health expectancy estimates vary according to the sample size (and the 844: standard deviations give confidence intervals of the estimates) but also 845: according to the model fitted. We explain this in more detail. 846: Choosing a model means at least two kind of choices. First we have to 847: decide the number of disability states. And second we have to design, within 848: the logit model family, the model itself: variables, covariates, confounding 849: factors etc. to be included. 850: The more disability states we have, the better is our demographical 851: approximation of the disability process, but the smaller the number of 852: transitions between each state and the higher the noise in the 853: measurement. We have not experimented enough with the various models 854: to summarize the advantages and disadvantages, but it is important to 855: note that even if we had huge unbiased samples, the total life 856: expectancy computed from a cross-longitudinal survey would vary with 857: the number of states. If we define only two states, alive or dead, we 858: find the usual life expectancy where it is assumed that at each age, 859: people are at the same risk of dying. If we are differentiating the 860: alive state into healthy and disabled, and as mortality from the 861: disabled state is higher than mortality from the healthy state, we are 862: introducing heterogeneity in the risk of dying. The total mortality at 863: each age is the weighted mean of the mortality from each state by the 864: prevalence of each state. Therefore if the proportion of people at each age and 865: in each state is different from the period equilibrium, there is no reason to 866: find the same total mortality at a particular age. Life expectancy, even if it 867: is a very useful tool, has a very strong hypothesis of homogeneity of the 868: population. Our main purpose is not to measure differential mortality but to 869: measure the expected time in a healthy or disabled state in order to maximise 870: the former and minimize the latter. But the differential in mortality 871: complicates the measurement. 872: Incidences of disability or recovery are not affected by the number of states 873: if these states are independent. But incidence estimates are dependent on the 874: specification of the model. The more covariates we add in the logit 875: model the better 876: is the model, but some covariates are not well measured, some are confounding 877: factors like in any statistical model. The procedure to "fit the best model' is 878: similar to logistic regression which itself is similar to regression analysis. 879: We haven't yet been sofar because we also have a severe limitation which is the 880: speed of the convergence. On a Pentium III, 500 MHz, even the simplest model, 881: estimated by month on 8,000 people may take 4 hours to converge. Also, the IMaCh 882: program is not a statistical package, and does not allow sophisticated design 883: variables. If you need sophisticated design variable you have to them your self 884: and and add them as ordinary variables. IMaCh allows up to 8 variables. The 885: current version of this program allows only to add simple variables like age+sex 886: or age+sex+ age*sex but will never be general enough. But what is to remember, 887: is that incidences or probability of change from one state to another is 888: affected by the variables specified into the model. 889: Also, the age range of the people interviewed is linked the age range of the 890: life expectancy which can be estimated by extrapolation. If your sample ranges 891: from age 70 to 95, you can clearly estimate a life expectancy at age 70 and 892: trust your confidence interval because it is mostly based on your sample size, 893: but if you want to estimate the life expectancy at age 50, you should rely in 894: the design of your model. Fitting a logistic model on a age range of 70 to 95 895: and estimating probabilties of transition out of this age range, say at age 50, 896: is very dangerous. At least you should remember that the confidence interval 897: given by the standard deviation of the health expectancies, are under the strong 898: assumption that your model is the 'true model', which is probably not the case 899: outside the age range of your sample. 900: <H5>- Copy of the parameter file: 901: <A 902: href="http://euroreves.ined.fr/imach/doc/orbiaspar.txt">orbiaspar.txt</A></H5> 903: This copy of the parameter file can be useful to re-run the program while 904: saving the old output files. 905: <H5>- Prevalence forecasting: <A 906: href="http://euroreves.ined.fr/imach/doc/biaspar/frbiaspar.txt">biaspar/frbiaspar.txt</A></H5> 907: First, we have estimated the observed prevalence between 1/1/1984 and 908: 1/6/1988 (June, European syntax of dates). The mean date of all interviews 909: (weighted average of the interviews performed between 1/1/1984 and 1/6/1988) is 910: estimated to be 13/9/1985, as written on the top on the file. Then we forecast 911: the probability to be in each state. 912: For example on 1/1/1989 : <PRE class=MsoNormal># StartingAge FinalAge P.1 P.2 P.3 913: # Forecasting at date 1/1/1989 914: 73 0.807 0.078 0.115</PRE> 915: Since the minimum age is 70 on the 13/9/1985, the youngest forecasted age is 916: 73. This means that at age a person aged 70 at 13/9/1989 has a probability to 917: enter state1 of 0.807 at age 73 on 1/1/1989. Similarly, the probability to be in 918: state 2 is 0.078 and the probability to die is 0.115. Then, on the 1/1/1989, the 919: prevalence of disability at age 73 is estimated to be 0.088. 920: <H5>- Population forecasting: <A 921: href="http://euroreves.ined.fr/imach/doc/biaspar/poprbiaspar.txt">biaspar/poprbiaspar.txt</A></H5><PRE># Age P.1 P.2 P.3 [Population] 922: # Forecasting at date 1/1/1989 923: 75 572685.22 83798.08 924: 74 621296.51 79767.99 925: 73 645857.70 69320.60 </PRE><PRE># Forecasting at date 1/1/19909 926: 76 442986.68 92721.14 120775.48 927: 75 487781.02 91367.97 121915.51 928: 74 512892.07 85003.47 117282.76 </PRE> 929: From the population file, we estimate the number of people in each state. At 930: age 73, 645857 persons are in state 1 and 69320 are in state 2. One year latter, 931: 512892 are still in state 1, 85003 are in state 2 and 117282 died before 932: 1/1/1990. 933: <HR> 934: 935: <H2><A name=example></A>Trying an example</H2> 936: Since you know how to run the program, it is time to test it on your own 937: computer. Try for example on a parameter file named <A 938: href="http://euroreves.ined.fr/imach/doc/imachpar.imach">imachpar.imach</A> 939: which is a copy of mypar.imach included 940: in the subdirectory of imach, mytry. Edit 941: it and change the name of the data file to mydata.txt if you don't want to copy it on the same directory. The 943: file mydata.txt is a smaller file of 3,000 944: people but still with 4 waves. 945: Right click on the .imach file and a window will popup with the string 946: 'Enter the parameter file name:' 947: <TABLE border=1> 948: <TBODY> 949: <TR> 950: <TD width="100%">IMACH, Version 0.97b 951: Enter the parameter file name: 952: imachpar.imach</TD></TR></TBODY></TABLE> 953: Most of the data files or image files generated, will use the 'imachpar' 954: string into their name. The running time is about 2-3 minutes on a Pentium III. 955: If the execution worked correctly, the outputs files are created in the current 956: directory, and should be the same as the mypar files initially included in the 957: directory mytry. 958: <UL> 959: <LI><PRE>Output on the screen The output screen looks like <A href="http://euroreves.ined.fr/imach/doc/biaspar.log">biaspar.log</A> 960: # 961: title=MLE datafile=mydaiata.txt lastobs=3000 firstpass=1 lastpass=3 962: ftol=1.000000e-008 stepm=24 ncovcol=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0</PRE> 963: <LI><PRE>Total number of individuals= 2965, Agemin = 70.00, Agemax= 100.92 964: 965: Warning, no any valid information for:126 line=126 966: Warning, no any valid information for:2307 line=2307 967: Delay (in months) between two waves Min=21 Max=51 Mean=24.495826 968: These lines give some warnings on the data file and also some raw statistics on frequencies of transitions. 969: Age 70 1.=230 loss[1]=3.5% 2.=16 loss[2]=12.5% 1.=222 prev[1]=94.1% 2.=14 970: prev[2]=5.9% 1-1=8 11=200 12=7 13=15 2-1=2 21=6 22=7 23=1 971: Age 102 1.=0 loss[1]=NaNQ% 2.=0 loss[2]=NaNQ% 1.=0 prev[1]=NaNQ% 2.=0 </PRE></LI></UL>It 972: includes some warnings or errors which are very important for you. Be careful 973: with such warnings because your results may be biased if, for example, you have 974: people who accepted to be interviewed at first pass but never after. Or if you 975: don't have the exact month of death. In such cases IMaCh doesn't take any 976: initiative, it does only warn you. It is up to you to decide what to do with 977: these people. Excluding them is usually a wrong decision. It is better to decide 978: that the month of death is at the mid-interval between the last two waves for 979: example. 980: If you survey suffers from severe attrition, you have to analyse the 981: characteristics of the lost people and overweight people with same 982: characteristics for example. 983: By default, IMaCH warns and excludes these problematic people, but you have 984: to be careful with such results. 985:   986: <UL> 987: <LI>Maximisation with the Powell algorithm. 8 directions are given 988: corresponding to the 8 parameters. this can be rather long to get 989: convergence. Powell iter=1 990: -2*LL=11531.405658264877 1 0.000000000000 2 0.000000000000 3 0.000000000000 991: 4 0.000000000000 5 0.000000000000 6 0.000000000000 7 0.000000000000 8 992: 0.000000000000 1..........2.................3..........4.................5......... 6................7........8............... Powell 993: iter=23 -2*LL=6744.954108371555 1 -12.967632334283 2 0.135136681033 3 994: -7.402109728262 4 0.067844593326 5 -0.673601538129 6 -0.006615504377 7 995: -5.051341616718 8 996: 0.051272038506 1..............2...........3..............4........... 5..........6................7...........8......... #Number 997: of iterations = 23, -2 Log likelihood = 6744.954042573691 # 998: Parameters 12 -12.966061 0.135117 13 -7.401109 0.067831 21 999: -0.672648 -0.006627 23 -5.051297 0.051271 1000: <LI><PRE>Calculation of the hessian matrix. Wait... 1001: 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 1002: 1003: Inverting the hessian to get the covariance matrix. Wait... 1004: 1005: #Hessian matrix# 1006: 3.344e+002 2.708e+004 -4.586e+001 -3.806e+003 -1.577e+000 -1.313e+002 3.914e-001 3.166e+001 1007: 2.708e+004 2.204e+006 -3.805e+003 -3.174e+005 -1.303e+002 -1.091e+004 2.967e+001 2.399e+003 1008: -4.586e+001 -3.805e+003 4.044e+002 3.197e+004 2.431e-002 1.995e+000 1.783e-001 1.486e+001 1009: -3.806e+003 -3.174e+005 3.197e+004 2.541e+006 2.436e+000 2.051e+002 1.483e+001 1.244e+003 1010: -1.577e+000 -1.303e+002 2.431e-002 2.436e+000 1.093e+002 8.979e+003 -3.402e+001 -2.843e+003 1011: -1.313e+002 -1.091e+004 1.995e+000 2.051e+002 8.979e+003 7.420e+005 -2.842e+003 -2.388e+005 1012: 3.914e-001 2.967e+001 1.783e-001 1.483e+001 -3.402e+001 -2.842e+003 1.494e+002 1.251e+004 1013: 3.166e+001 2.399e+003 1.486e+001 1.244e+003 -2.843e+003 -2.388e+005 1.251e+004 1.053e+006 1014: # Scales 1015: 12 1.00000e-004 1.00000e-006 1016: 13 1.00000e-004 1.00000e-006 1017: 21 1.00000e-003 1.00000e-005 1018: 23 1.00000e-004 1.00000e-005 1019: # Covariance 1020: 1 5.90661e-001 1021: 2 -7.26732e-003 8.98810e-005 1022: 3 8.80177e-002 -1.12706e-003 5.15824e-001 1023: 4 -1.13082e-003 1.45267e-005 -6.50070e-003 8.23270e-005 1024: 5 9.31265e-003 -1.16106e-004 6.00210e-004 -8.04151e-006 1.75753e+000 1025: 6 -1.15664e-004 1.44850e-006 -7.79995e-006 1.04770e-007 -2.12929e-002 2.59422e-004 1026: 7 1.35103e-003 -1.75392e-005 -6.38237e-004 7.85424e-006 4.02601e-001 -4.86776e-003 1.32682e+000 1027: 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 1028: # agemin agemax for lifexpectancy, bage fage (if mle==0 ie no data nor Max likelihood). 1029: 1030: 1031: agemin=70 agemax=100 bage=50 fage=100 1032: Computing prevalence limit: result on file 'plrmypar.txt' 1033: Computing pij: result on file 'pijrmypar.txt' 1034: Computing Health Expectancies: result on file 'ermypar.txt' 1035: Computing Variance-covariance of DFLEs: file 'vrmypar.txt' 1036: Computing Total LEs with variances: file 'trmypar.txt' 1037: Computing Variance-covariance of Prevalence limit: file 'vplrmypar.txt' 1038: End of Imach 1039: </PRE></LI></UL> 1040: Once the running is finished, the program requires a 1041: character: 1042: <TABLE border=1> 1043: <TBODY> 1044: <TR> 1045: <TD width="100%">Type e to edit output files, g to graph again, c 1046: to start again, and q for exiting:</TD></TR></TBODY></TABLE>In order to 1047: have an idea of the time needed to reach convergence, IMaCh gives an estimation 1048: if the convergence needs 10, 20 or 30 iterations. It might be useful. 1049: First you should enter e to edit the master 1050: file mypar.htm. 1051: <UL> 1052: <LI>Outputs files - Copy of the parameter file: <A 1053: href="http://euroreves.ined.fr/imach/doc/ormypar.txt">ormypar.txt</A> - 1054: Gnuplot file name: <A 1055: href="http://euroreves.ined.fr/imach/doc/mypar.gp.txt">mypar.gp.txt</A> - 1056: Cross-sectional prevalence in each state: <A 1057: href="http://euroreves.ined.fr/imach/doc/prmypar.txt">prmypar.txt</A> - 1058: Period prevalence in each state: <A 1059: href="http://euroreves.ined.fr/imach/doc/plrmypar.txt">plrmypar.txt</A> - 1060: Transition probabilities: <A 1061: href="http://euroreves.ined.fr/imach/doc/pijrmypar.txt">pijrmypar.txt</A> - 1062: Life expectancies by age and initial health status (estepm=24 months): <A 1063: href="http://euroreves.ined.fr/imach/doc/ermypar.txt">ermypar.txt</A> - 1064: Parameter file with estimated parameters and the covariance matrix: <A 1065: href="http://euroreves.ined.fr/imach/doc/rmypar.txt">rmypar.txt</A> - 1066: Variance of one-step probabilities: <A 1067: href="http://euroreves.ined.fr/imach/doc/probrmypar.txt">probrmypar.txt</A> 1068: - Variances of life expectancies by age and initial health status 1069: (estepm=24 months): <A 1070: href="http://euroreves.ined.fr/imach/doc/vrmypar.txt">vrmypar.txt</A> - 1071: Health expectancies with their variances: <A 1072: href="http://euroreves.ined.fr/imach/doc/trmypar.txt">trmypar.txt</A> - 1073: Standard deviation of period prevalences: <A 1074: href="http://euroreves.ined.fr/imach/doc/vplrmypar.txt">vplrmypar.txt</A> 1075: No population forecast: popforecast = 0 (instead of 1) or stepm = 24 1076: (instead of 1) or model=. (instead of .) 1077: <LI>Graphs -<A 1078: href="http://euroreves.ined.fr/imach/mytry/pemypar1.gif">One-step transition 1079: probabilities</A> -<A 1080: href="http://euroreves.ined.fr/imach/mytry/pmypar11.gif">Convergence to the 1081: period prevalence</A> -<A 1082: href="http://euroreves.ined.fr/imach/mytry/vmypar11.gif">Cross-sectional and 1083: period prevalence in state (1) with the confident interval</A> -<A 1084: href="http://euroreves.ined.fr/imach/mytry/vmypar21.gif">Cross-sectional and 1085: period prevalence in state (2) with the confident interval</A> -<A 1086: href="http://euroreves.ined.fr/imach/mytry/expmypar11.gif">Health life 1087: expectancies by age and initial health state (1)</A> -<A 1088: href="http://euroreves.ined.fr/imach/mytry/expmypar21.gif">Health life 1089: expectancies by age and initial health state (2)</A> -<A 1090: href="http://euroreves.ined.fr/imach/mytry/emypar1.gif">Total life expectancy 1091: by age and health expectancies in states (1) and (2).</A> </LI></UL> 1092: This software have been partly granted by <A 1093: href="http://euroreves.ined.fr/">Euro-REVES</A>, a concerted action from the 1094: European Union. Since 2003 it is also partly granted by the French 1095: Institute on Longevity. It will be copyrighted identically to a GNU software product, i.e. program and software can be distributed freely for non commercial use. 1096: Sources are not widely distributed today because some part of the codes are copyrighted by Numerical Recipes in C. You can get them by asking us with a 1097: simple justification (name, email, institute) <A 1098: href="mailto:brouard@ined.fr">mailto:brouard@ined.fr</A> and <A 1099: href="mailto:lievre@ined.fr">mailto:lievre@ined.fr</A> . 1100: Latest version (0.97b of June 2004) can be accessed at <A 1101: href="http://euroreves.ined.fr/imach">http://euroreves.ined.fr/imach</A> </BODY></HTML>