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9: <H1 align=center><FONT color=#00006a>Computing Health Expectancies using
10: IMaCh</FONT></H1>
11: <H1 align=center><FONT color=#00006a size=5>(a Maximum Likelihood Computer
12: Program using Interpolation of Markov Chains)</FONT></H1>
13: <P align=center> </P>
14: <P align=center><A href="http://www.ined.fr/"><IMG border=0 height=76
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19: <H3 align=center><A href="http://www.ined.fr/"><FONT
20: color=#00006a>INED</FONT></A><FONT color=#00006a> and </FONT><A
21: href="http://euroreves.ined.fr/"><FONT color=#00006a>EUROREVES</FONT></A></H3>
22: <P align=center><FONT color=#00006a size=4><STRONG>Version 0.97, June
23: 2004</STRONG></FONT></P>
24: <HR color=#ec5e5e SIZE=3>
25:
26: <P align=center><FONT color=#00006a><STRONG>Authors of the program:
27: </STRONG></FONT><A href="http://sauvy.ined.fr/brouard"><FONT
28: color=#00006a><STRONG>Nicolas Brouard</STRONG></FONT></A><FONT
29: color=#00006a><STRONG>, senior researcher at the </STRONG></FONT><A
30: href="http://www.ined.fr/"><FONT color=#00006a><STRONG>Institut National
31: d'Etudes Démographiques</STRONG></FONT></A><FONT color=#00006a><STRONG> (INED,
32: Paris) in the "Mortality, Health and Epidemiology" Research Unit
33: </STRONG></FONT></P>
34: <P align=center><FONT color=#00006a><STRONG>and Agnès Lièvre<BR
35: clear=left></STRONG></FONT></P>
36: <H4><FONT color=#00006a>Contribution to the mathematics: C. R. Heathcote
37: </FONT><FONT color=#00006a size=2>(Australian National University,
38: Canberra).</FONT></H4>
39: <H4><FONT color=#00006a>Contact: Agnès Lièvre (</FONT><A
40: href="mailto:lievre@ined.fr"><FONT
41: color=#00006a><I>lievre@ined.fr</I></FONT></A><FONT color=#00006a>) </FONT></H4>
42: <HR>
43:
44: <UL>
45: <LI><A
46: href="http://euroreves.ined.fr/imach/doc/imach.htm#intro">Introduction</A>
1.6 lievre 47: <LI><A href="http://euroreves.ined.fr/imach/doc/imach.htm#data">What kind
48: of data can is required?</A>
1.5 lievre 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><FONT color=#00006a>Introduction</FONT></A></H2>
62: <P>This program computes <B>Healthy Life Expectancies</B> from
63: <B>cross-longitudinal data</B> 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 <EM>Expansion of morbidity</EM>. 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: <EM>cross-sectional</EM>. 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).</P>
81: <P>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 <EM>if
85: nothing changes in the future</EM>, 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.</P>
99: <P>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.</P>
102: <P>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.<BR><BR>The simplest model for the transition probabilities is the multinomial logistic model where
112: <I>pij</I> is the probability to be observed in state <I>j</I> at the second
113: wave conditional to be observed in state <EM>i</EM> at the first wave. Therefore
114: a simple model is: log<EM>(pij/pii)= aij + bij*age+ cij*sex,</EM> where
115: '<I>age</I>' is age and '<I>sex</I>' 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. <I>hPijx</I> is the probability to be observed
120: in state <I>i</I> at age <I>x+h</I> conditional on the observed state <I>i</I>
121: at age <I>x</I>. The delay '<I>h</I>' can be split into an exact number
122: (<I>nh*stepm</I>) 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 <I>hPx</I> matrix is simply the matrix product of <I>nh*stepm</I>
125: elementary matrices and the contribution of each individual to the likelihood is
126: simply <I>hPijx</I>. <BR></P>
127: <P>The program presented in this manual is a general program named
128: <STRONG>IMaCh</STRONG> (for <STRONG>I</STRONG>nterpolated
129: <STRONG>MA</STRONG>rkov <STRONG>CH</STRONG>ain), 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<SUP>®</SUP>) 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.<BR></P>
144: <HR>
145:
146: <P>(1) Laditka S. B. and Wolf, D. (1998), New Methods for Analyzing
147: Active Life Expectancy. <I>Journal of Aging and Health</I>. Vol 10, No. 2. </P>
148: <P>(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. <EM>Mathematical Population Studies</EM>.- 10(4),
152: pp. 211-248</A>
153: <HR>
154:
155: <H2><A name=data><FONT color=#00006a>What kind of data is required?</FONT></A></H2>
156: <P>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.</P>
168: <HR>
169:
170: <H2><A name=datafile><FONT color=#00006a>The data file</FONT></A></H2>
171: <P>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.</P>
192: <P>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: </P>
195: <UL>
196: <LI><B>Index number</B>: positive number (field 1)
197: <LI><B>First covariate</B> positive number (field 2)
198: <LI><B>Second covariate</B> positive number (field 3)
199: <LI><A name=Weight><B>Weight</B></A>: positive number (field 4) . In most
200: surveys individuals are weighted to account for stratification of the
201: sample.
202: <LI><B>Date of birth</B>: coded as mm/yyyy. Missing dates are coded as 99/9999
203: (field 5)
204: <LI><B>Date of death</B>: coded as mm/yyyy. Missing dates are coded as 99/9999
205: (field 6)
206: <LI><B>Date of first interview</B>: coded as mm/yyyy. Missing dates are coded
207: as 99/9999 (field 7)
208: <LI><B>Status at first interview</B>: positive number. Missing values ar coded
209: -1. (field 8)
210: <LI><B>Date of second interview</B>: coded as mm/yyyy. Missing dates are coded
211: as 99/9999 (field 9)
212: <LI><STRONG>Status at second interview</STRONG> positive number. Missing
213: values ar coded -1. (field 10)
214: <LI><B>Date of third interview</B>: coded as mm/yyyy. Missing dates are coded
215: as 99/9999 (field 11)
216: <LI><STRONG>Status at third interview</STRONG> positive number. Missing values
217: ar coded -1. (field 12)
218: <LI><B>Date of fourth interview</B>: coded as mm/yyyy. Missing dates are coded
219: as 99/9999 (field 13)
220: <LI><STRONG>Status at fourth interview</STRONG> positive number. Missing
221: values are coded -1. (field 14)
222: <LI>etc </LI></UL>
223: <P> </P>
224: <P>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.</P>
227: <HR>
228:
229: <H2><FONT color=#00006a>Your first example parameter file</FONT><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: <P>This first line was a comment. Comments line start with a '#'.</P>
233: <H4><FONT color=#ff0000>First uncommented line</FONT></H4><PRE>title=1st_example datafile=data1.txt lastobs=8600 firstpass=1 lastpass=4</PRE>
234: <UL>
235: <LI><B>title=</B> 1st_example is title of the run.
236: <LI><B>datafile=</B> 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><B>lastobs=</B> 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><B>firstpass=1</B> , <B>lastpass=4 </B>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: <P> </P>
249: <H4><A name=biaspar-2><FONT color=#ff0000>Second uncommented
250: line</FONT></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><B>ftol=1e-8</B> 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><B>stepm=1</B> 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><B>ncovcol=2</B> 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><B>nlstate=2</B> 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><B>ndeath=1</B> Number of absorbing states. The absorbing state death is
270: coded 3.
271: <LI><B>maxwav=4</B> Number of waves in the datafile.
272: <LI><A name=mle><B>mle</B></A><B>=1</B> 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><B>weight=0</B> 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><FONT color=#ff0000>Covariates</FONT></H4>
299: <P>Intercept and age are automatically included in the model. Additional
300: covariates can be included with the command: </P><PRE>model=<EM>list of covariates</EM></PRE>
301: <UL>
302: <LI>if<STRONG> model=. </STRONG>then no covariates are included
303: <LI>if <STRONG>model=V1</STRONG> the model includes the first covariate (field
304: 2)
305: <LI>if <STRONG>model=V2 </STRONG>the model includes the second covariate
306: (field 3)
307: <LI>if <STRONG>model=V1+V2 </STRONG>the model includes the first and the
308: second covariate (fields 2 and 3)
309: <LI>if <STRONG>model=V1*V2 </STRONG>the model includes the product of the
310: first and the second covariate (fields 2 and 3)
311: <LI>if <STRONG>model=V1+V1*age</STRONG> the model includes the product
312: covariate*age </LI></UL>
313: <P>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, <STRONG>model=V1+V3 </STRONG>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 <STRONG>model=V1, </STRONG>16 for <STRONG>model=V1+V1*age </STRONG>and 20
323: for <STRONG>model=V1+V2+V3.</STRONG></P>
324: <H4><FONT color=#ff0000>Guess values for optimization</FONT><FONT color=#00006a>
325: </FONT></H4>
326: <P>You must write the initial guess values of the parameters for optimization.
327: The number of parameters, <EM>N</EM> depends on the number of absorbing states
328: and non-absorbing states and on the number of covariates in the model (ncovmodel). <BR><EM>N</EM> is
329: given by the formula <EM>N</EM>=(<EM>nlstate</EM> +
330: <EM>ndeath</EM>-1)*<EM>nlstate</EM>*<EM>ncovmodel</EM> . <BR><BR>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<BR>Each of the four
337: lines starts with indices "ij": <B>ij aij bij</B> </P>
338: <BLOCKQUOTE><PRE># Guess values of aij and bij in log (pij/pii) = aij + bij * age
1.1 brouard 339: 12 -14.155633 0.110794
340: 13 -7.925360 0.032091
341: 21 -1.890135 -0.029473
1.5 lievre 342: 23 -6.234642 0.022315 </PRE></BLOCKQUOTE>
343: <P>or, to simplify (in most of cases it converges but there is no warranty!):
344: </P>
345: <BLOCKQUOTE><PRE>12 0.0 0.0
1.1 brouard 346: 13 0.0 0.0
347: 21 0.0 0.0
1.5 lievre 348: 23 0.0 0.0</PRE></BLOCKQUOTE>
349: <P>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: </P><PRE>aij(stepm) = aij(n . stepm) - ln(n)
353: </PRE>
354: <P>and </P><PRE>bij(stepm) = bij(n . stepm) .</PRE>
355: <P>For example if you already ran with stepm=6 (a 6 months interval) and got:<BR></P><PRE># Parameters
1.1 brouard 356: 12 -13.390179 0.126133
357: 13 -7.493460 0.048069
358: 21 0.575975 -0.041322
359: 23 -4.748678 0.030626
1.5 lievre 360: </PRE>
361: <P>Then you now want to get the monthly estimates, you can guess the aij by
362: subtracting ln(6)= 1.7917<BR>and running using<BR></P><PRE>12 -15.18193847 0.126133
1.1 brouard 363: 13 -9.285219469 0.048069
364: 21 -1.215784469 -0.041322
365: 23 -6.540437469 0.030626
1.5 lievre 366: </PRE>
367: <P>and get<BR></P><PRE>12 -15.029768 0.124347
1.1 brouard 368: 13 -8.472981 0.036599
369: 21 -1.472527 -0.038394
370: 23 -6.553602 0.029856
371:
1.5 lievre 372: <P>which is closer to the results. The approximation is probably useful
1.1 brouard 373: only for very small intervals and we don't have enough experience to
1.5 lievre 374: know if you will speed up the convergence or not.<BR></P>
375: </PRE><PRE> -ln(12)= -2.484
1.1 brouard 376: -ln(6/1)=-ln(6)= -1.791
377: -ln(3/1)=-ln(3)= -1.0986
378: -ln(12/6)=-ln(2)= -0.693
1.5 lievre 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: <P>If the interval of <EM>d</EM> months between two waves is not a multiple of
387: 'stepm', but is between <EM>(n-1) stepm</EM> and <EM>n stepm</EM> then
388: both exact likelihoods are computed (the contribution to the likelihood at <EM>n
389: stepm</EM> requires one matrix product more) (let us remember that we are
390: modelling the probability to be observed in a particular state after <EM>d</EM>
391: months being observed at a particular state at 0). The distance, (<EM>bh</EM> in
392: the program), from the month of interview to the rounded date of <EM>n
393: stepm</EM> is computed. It can be negative (interview occurs before <EM>n
394: stepm</EM>) or positive if the interview occurs after <EM>n stepm</EM> (and
395: before <EM>(n+1)stepm</EM>). <BR>Then the final contribution to the total
396: likelihood is a weighted average of these two exact likelihoods at <EM>n
397: stepm</EM> (out) and at <EM>(n-1)stepm</EM>(savm). We did not want to compute
398: the third likelihood at <EM>(n+1)stepm</EM> because it is too costly in time, so
399: we used an extrapolation if <EM>bh</EM> is positive. <BR>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 ? \
1.2 brouard 403: log((1.+bbh)*out[s1][s2]- bbh*(savm[s1][s2])): \
404: log((1.+bbh)*out[s1][s2])); /* linear interpolation */
1.5 lievre 405: mle=3 lli= (savm[s1][s2]>1.e-8 ? \
1.2 brouard 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.
1.5 lievre 411: </PRE>
412: <P>If the death occurs between the first and second pass, and for example more
413: precisely between <EM>n stepm</EM> and <EM>(n+1)stepm</EM> the contribution of
414: these people to the likelihood is simply the difference between the probability
415: of dying before <EM>n stepm</EM> and the probability of dying before
416: <EM>(n+1)stepm</EM>. 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
1.2 brouard 421: versions (fortunately before first publication) the total mortality
1.5 lievre 422: was thus overestimated (people were dying too early) by about 10%. Version
423: 0.95 and higher are correct.
1.1 brouard 424:
1.5 lievre 425: <P>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><FONT color=#ff0000>Guess values for computing variances</FONT></H4>
431: <P>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). </P>
436: <P>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. </P>
442: <UL>
443: <LI>If mle=1 you can enter zeros:
444: <LI>
445: <BLOCKQUOTE><PRE># Scales (for hessian or gradient estimation)
1.1 brouard 446: 12 0. 0.
447: 13 0. 0.
448: 21 0. 0.
1.5 lievre 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><FONT color=#ff0000>Covariance matrix of parameters</FONT></H4>
453: <P>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). <BR>Each line starts with indices
458: "ijk" followed by the covariances between aij and bij:<BR>
459: </P><PRE> 121 Var(a12)
1.1 brouard 460: 122 Cov(b12,a12) Var(b12)
461: ...
1.5 lievre 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
1.1 brouard 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.
1.5 lievre 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><FONT color=#ff0000>Age range for calculation of stationary prevalences and
477: health expectancies</FONT></H4><PRE>agemin=70 agemax=100 bage=50 fage=100</PRE>
478: <P>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: <P>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><B>agemin=</B> Minimum age for calculation of the period prevalence
498: <LI><B>agemax=</B> Maximum age for calculation of the period prevalence
499: <LI><B>bage=</B> Minimum age for calculation of the health expectancies
500: <LI><B>fage=</B> Maximum age for calculation of the health expectancies
501: </LI></UL>
502: <H4><A name=Computing><FONT color=#ff0000>Computing</FONT></A><FONT
503: color=#ff0000> the cross-sectional prevalence</FONT></H4><PRE>begin-prev-date=1/1/1984 end-prev-date=1/6/1988 estepm=1</PRE>
504: <P>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. </P>
508: <UL>
509: <LI><STRONG>begin-prev-date= </STRONG>Starting date (day/month/year)
510: <LI><STRONG>end-prev-date= </STRONG>Final date (day/month/year)
511: <LI><STRONG>estepm= </STRONG>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><FONT color=#ff0000>Population- or status-based health
523: expectancies</FONT></H4><PRE>pop_based=0</PRE>
524: <P>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).<BR>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: <P>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: <P>In a similar way we could compute the total life expectancy by summing e.1
534: and e.2 . <BR>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: <P>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. <BR>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><STRONG>popbased= 0 </STRONG>Health expectancies are computed at each age
550: from period prevalences 'expected' at this initial age.
551: <LI><STRONG>popbased= 1 </STRONG>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:<BR>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.<BR>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><FONT color=#ff0000>Prevalence forecasting (Experimental)</FONT></H4><PRE>starting-proj-date=1/1/1989 final-proj-date=1/1/1992 mov_average=0 </PRE>
564: <P>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. <BR></P>
572: <H4><FONT color=#ff0000>Population forecasting (Experimental)</FONT></H4>
573: <UL>
574: <LI><STRONG>starting-proj-date</STRONG>= starting date (day/month/year) of
575: forecasting
576: <LI><STRONG>final-proj-date= </STRONG>final date (day/month/year) of
577: forecasting
578: <LI><STRONG>mov_average</STRONG>= smoothing with a five-age moving average
579: centered at the mid-age of the fiveyear-age period. The command<STRONG>
580: mov_average</STRONG> takes value 1 if the prevalences are smoothed and 0
581: otherwise. </LI></UL>
582: <UL type=disc>
583: <LI><B>popforecast= 0 </B>Option for population forecasting. If popforecast=1,
584: the programme does the forecasting<B>.</B>
585: <LI><B>popfile= </B>name of the population file
586: <LI><B>popfiledate=</B> date of the population population
587: <LI><B>last-popfiledate</B>= date of the last population projection
588: </LI></UL>
589: <HR>
590:
591: <H2><A name=running></A><FONT color=#00006a>Running Imach with this
592: example</FONT></H2>
593: <P>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: </P>
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: <P>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: <P>The program outputs many files. Most of them are files which will be plotted
614: for better understanding. </P>To run under Linux is mostly the same.
615: <P>It is no more difficult to run IMaCh on a MacIntosh.
616: <HR>
617:
618: <H2><A name=output><FONT color=#00006a>Output of the program and graphs</FONT>
619: </A></H2>
620: <P>Once the optimization is finished (once the convergence is reached), many
621: tables and graphics are produced.
622: <P>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.<BR>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: <P>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: <P>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>. <BR>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. <BR>These characters are:<BR></P>
638: <UL>
639: <LI>'e' for opening the main result html file <A
640: href="http://euroreves.ined.fr/imach/doc/biaspar.htm"><STRONG>biaspar.htm</STRONG></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: <P>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><FONT size=4><STRONG>Results files </STRONG></FONT><BR><BR><FONT
649: color=#ec5e5e size=3><STRONG>- </STRONG></FONT><A
650: name="cross-sectional prevalence in each state"><FONT color=#ec5e5e
651: size=3><STRONG>cross-sectional prevalence in each state</STRONG></FONT></A><FONT
652: color=#ec5e5e size=3><STRONG> (and at first pass)</STRONG></FONT><B>: </B><A
653: href="http://euroreves.ined.fr/imach/doc/biaspar/prbiaspar.txt"><B>biaspar/prbiaspar.txt</B></A><BR></H5>
654: <P>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.<BR>The header of the file is </P><PRE># Age Prev(1) N(1) N Age Prev(2) N(2) N
1.1 brouard 659: 70 1.00000 631 631 70 0.00000 0 631
660: 71 0.99681 625 627 71 0.00319 2 627
1.5 lievre 661: 72 0.97125 1115 1148 72 0.02875 33 1148 </PRE>
662: <P>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: <BR></P>
666: <H5><FONT color=#ec5e5e size=3><B>- Estimated parameters and covariance
667: matrix</B></FONT><B>: </B><A
668: href="http://euroreves.ined.fr/imach/doc/rbiaspar.txt"><B>rbiaspar.imach</B></A></H5>
669: <P>This file contains all the maximisation results: </P><PRE> -2 log likelihood= 21660.918613445392
1.1 brouard 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
1.5 lievre 682: </PRE>
683: <P>By substitution of these parameters in the regression model, we obtain the
684: elementary transition probabilities:</P>
685: <P><IMG height=300
1.6 lievre 686: src="biaspar/pebiaspar11.png"
1.5 lievre 687: width=400></P>
688: <H5><FONT color=#ec5e5e size=3><B>- Transition probabilities</B></FONT><B>:
689: </B><A
690: href="http://euroreves.ined.fr/imach/doc/biaspar/pijrbiaspar.txt"><B>biaspar/pijrbiaspar.txt</B></A></H5>
691: <P>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: </P><PRE>1 100 106 0.02655 0.17622 0.79722 0.01809 0.13678 0.84513 </PRE>
696: <P>and this means: </P><PRE>p11(100,106)=0.02655
1.1 brouard 697: p12(100,106)=0.17622
698: p13(100,106)=0.79722
699: p21(100,106)=0.01809
700: p22(100,106)=0.13678
1.5 lievre 701: p22(100,106)=0.84513 </PRE>
702: <H5><FONT color=#ec5e5e size=3><B>- </B></FONT><A
703: name="Period prevalence in each state"><FONT color=#ec5e5e size=3><B>Period
704: prevalence in each state</B></FONT></A><B>: </B><A
705: href="http://euroreves.ined.fr/imach/doc/biaspar/plrbiaspar.txt"><B>biaspar/plrbiaspar.txt</B></A></H5><PRE>#Prevalence
1.1 brouard 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
1.5 lievre 712: 73 0.87015 0.12985 </PRE>
713: <P>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: </P>
726: <H5><FONT color=#ec5e5e size=3><B>- Standard deviation of period
727: prevalence</B></FONT><B>: </B><A
728: href="http://euroreves.ined.fr/imach/doc/biaspar/vplrbiaspar.txt"><B>biaspar/vplrbiaspar.txt</B></A></H5>
729: <P>The period prevalence has to be compared with the cross-sectional prevalence.
730: But both are statistical estimates and therefore have confidence intervals.
731: <BR>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: <P>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><FONT color=#ec5e5e size=3>-cross-sectional and period prevalence in state
746: (2=disable) with confidence interval</FONT>:<B> </B><A
747: href="http://euroreves.ined.fr/imach/doc/biaspar/vbiaspar21.htm"><B>biaspar/vbiaspar21.png</B></A></H5>
748: <P>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).</P>
756: <P><IMG height=300
1.6 lievre 757: src="biaspar/vbiaspar21.png"
1.5 lievre 758: width=400></P>
759: <H5><FONT color=#ec5e5e size=3><B>-Convergence to the period prevalence of
760: disability</B></FONT><B>: </B><A
1.6 lievre 761: href="biaspar/pbiaspar11.png"><B>biaspar/pbiaspar11.png</B></A><BR><IMG
1.5 lievre 762: height=300
1.6 lievre 763: src="biaspar/pbiaspar11.png"
1.5 lievre 764: width=400> </H5>
765: <P>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: <EM>x </EM>to the final state 2=disabled<EM> </EM>at age <EM>x+h
768: </EM> where conditional means conditional on being alive at age <EM>x+h </EM>which is
769: <I>hP12x</I> + <EM>hP22x</EM>. The curves <I>hP12x/(hP12x</I> + <EM>hP22x)
770: </EM>and <I>hP22x/(hP12x</I> + <EM>hP22x) </EM>converge with <EM>h, </EM>to the
771: <EM>period prevalence of disability</EM>. 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.</P>
776: <H5><FONT color=#ec5e5e size=3><B>- Life expectancies by age and initial health
777: status with standard deviation</B></FONT><B>: </B><A
778: href="http://euroreves.ined.fr/imach/doc/biaspar/erbiaspar.txt"><B>biaspar/erbiaspar.txt</B></A></H5><PRE># Health expectancies
1.1 brouard 779: # Age 1-1 (SE) 1-2 (SE) 2-1 (SE) 2-2 (SE)
1.2 brouard 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)
1.5 lievre 792: </PRE><PRE>For example 70 11.0180 (0.1277) 3.1950 (0.3635) 4.6500 (0.0871) 4.4807 (0.2187)
1.2 brouard 793: means
1.6 lievre 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>
1.5 lievre 795: <P>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>).</P>
809: <H5><FONT color=#ec5e5e size=3><B>- Variances of life expectancies by age and
810: initial health status</B></FONT><B>: </B><A
811: href="http://euroreves.ined.fr/imach/doc/biaspar/vrbiaspar.txt"><B>biaspar/vrbiaspar.txt</B></A></H5>
812: <P>For example, the covariances of life expectancies Cov(ei,ej) at age 50 are
813: (line 3) </P><PRE> Cov(e1,e1)=0.4776 Cov(e1,e2)=0.0488=Cov(e2,e1) Cov(e2,e2)=0.0424</PRE>
814: <H5><FONT color=#ec5e5e size=3><B>-Variances of one-step probabilities
815: </B></FONT><B>: </B><A
816: href="http://euroreves.ined.fr/imach/doc/biaspar/probrbiaspar.txt"><B>biaspar/probrbiaspar.txt</B></A></H5>
817: <P>For example, at age 65</P><PRE> p11=9.960e-001 standard deviation of p11=2.359e-004</PRE>
818: <H5><FONT color=#ec5e5e size=3><B>- </B></FONT><A
819: name="Health expectancies"><FONT color=#ec5e5e size=3><B>Health
820: expectancies</B></FONT></A><FONT color=#ec5e5e size=3><B> with standard errors
821: in parentheses</B></FONT><B>: </B><A
822: href="http://euroreves.ined.fr/imach/doc/biaspar/trbiaspar.txt"><FONT
823: face="Courier New"><B>biaspar/trbiaspar.txt</B></FONT></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: <P>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.</P>
830: <H5><FONT color=#ec5e5e size=3><B>-Total life expectancy by age and health
831: expectancies in states (1=healthy) and (2=disable)</B></FONT><B>: </B><A
1.6 lievre 832: href="biaspar/ebiaspar1.png"><B>biaspar/ebiaspar1.png</B></A></H5>
1.5 lievre 833: <P>This figure represents the health expectancies and the total life expectancy
1.6 lievre 834: with a confidence interval (dashed line). </P><PRE> <IMG height=300 src="biaspar/ebiaspar1.png" width=400></PRE>
1.5 lievre 835: <P>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.</P>
843: <P>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.</P>
846: <P>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.</P>
850: <P>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.</P>
872: <P>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.</P>
889: <P>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.</P>
900: <H5><FONT color=#ec5e5e size=3><B>- Copy of the parameter file</B></FONT><B>:
901: </B><A
902: href="http://euroreves.ined.fr/imach/doc/orbiaspar.txt"><B>orbiaspar.txt</B></A></H5>
903: <P>This copy of the parameter file can be useful to re-run the program while
904: saving the old output files. </P>
905: <H5><FONT color=#ec5e5e size=3><B>- Prevalence forecasting</B></FONT><B>: </B><A
906: href="http://euroreves.ined.fr/imach/doc/biaspar/frbiaspar.txt"><B>biaspar/frbiaspar.txt</B></A></H5>
907: <P>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. </P>
912: <P>For example on 1/1/1989 : </P><PRE class=MsoNormal># StartingAge FinalAge P.1 P.2 P.3
1.1 brouard 913: # Forecasting at date 1/1/1989
1.5 lievre 914: 73 0.807 0.078 0.115</PRE>
915: <P>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.</P>
920: <H5><FONT color=#ec5e5e size=3><B>- Population forecasting</B></FONT><B>: </B><A
921: href="http://euroreves.ined.fr/imach/doc/biaspar/poprbiaspar.txt"><B>biaspar/poprbiaspar.txt</B></A></H5><PRE># Age P.1 P.2 P.3 [Population]
1.1 brouard 922: # Forecasting at date 1/1/1989
923: 75 572685.22 83798.08
924: 74 621296.51 79767.99
1.5 lievre 925: 73 645857.70 69320.60 </PRE><PRE># Forecasting at date 1/1/19909
1.1 brouard 926: 76 442986.68 92721.14 120775.48
927: 75 487781.02 91367.97 121915.51
1.5 lievre 928: 74 512892.07 85003.47 117282.76 </PRE>
929: <P>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.</P>
933: <HR>
934:
935: <H2><A name=example></A><FONT color=#00006a>Trying an example</FONT></H2>
936: <P>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 <FONT face="Courier New" size=2>mypar.imach</FONT> included
940: in the subdirectory of imach, <FONT face="Courier New" size=2>mytry</FONT>. Edit
941: it and change the name of the data file to <FONT face="Courier New"
942: size=2>mydata.txt</FONT> if you don't want to copy it on the same directory. The
943: file <FONT face="Courier New">mydata.txt</FONT> is a smaller file of 3,000
944: people but still with 4 waves. </P>
945: <P>Right click on the .imach file and a window will popup with the string
946: '<STRONG>Enter the parameter file name:'</STRONG></P>
947: <TABLE border=1>
948: <TBODY>
949: <TR>
950: <TD width="100%"><STRONG>IMACH, Version 0.97b</STRONG>
951: <P><STRONG>Enter the parameter file name:
952: imachpar.imach</STRONG></P></TD></TR></TBODY></TABLE>
953: <P>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 <FONT face="Courier New" size=2>mytry</FONT>.</P>
958: <UL>
959: <LI><PRE><U>Output on the screen</U> The output screen looks like <A href="http://euroreves.ined.fr/imach/doc/biaspar.log">biaspar.log</A>
1.1 brouard 960: #
1.2 brouard 961: title=MLE datafile=mydaiata.txt lastobs=3000 firstpass=1 lastpass=3
1.5 lievre 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
1.1 brouard 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
1.5 lievre 968: <FONT face="Times New Roman">These lines give some warnings on the data file and also some raw statistics on frequencies of transitions.</FONT>
1.1 brouard 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
1.5 lievre 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: <P>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: <P>By default, IMaCH warns and excludes these problematic people, but you have
984: to be careful with such results.
985: <P> </P>
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.<BR><FONT face="Courier New" size=1><BR>Powell iter=1
990: -2*LL=11531.405658264877 1 0.000000000000 2 0.000000000000 3<BR>0.000000000000
991: 4 0.000000000000 5 0.000000000000 6 0.000000000000 7 <BR>0.000000000000 8
992: 0.000000000000<BR>1..........2.................3..........4.................5.........<BR>6................7........8...............<BR>Powell
993: iter=23 -2*LL=6744.954108371555 1 -12.967632334283 <BR>2 0.135136681033 3
994: -7.402109728262 4 0.067844593326 <BR>5 -0.673601538129 6 -0.006615504377 7
995: -5.051341616718 <BR>8
996: 0.051272038506<BR>1..............2...........3..............4...........<BR>5..........6................7...........8.........<BR>#Number
997: of iterations = 23, -2 Log likelihood = 6744.954042573691<BR>#
998: Parameters<BR>12 -12.966061 0.135117 <BR>13 -7.401109 0.067831 <BR>21
999: -0.672648 -0.006627 <BR>23 -5.051297 0.051271 </FONT><BR>
1000: <LI><PRE><FONT size=2>Calculation of the hessian matrix. Wait...
1.1 brouard 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
1.5 lievre 1039: </FONT></PRE></LI></UL>
1040: <P><FONT size=3>Once the running is finished, the program requires a
1041: character:</FONT></P>
1042: <TABLE border=1>
1043: <TBODY>
1044: <TR>
1045: <TD width="100%"><STRONG>Type e to edit output files, g to graph again, c
1046: to start again, and q for exiting:</STRONG></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: <P><FONT size=3>First you should enter <STRONG>e </STRONG>to edit the master
1050: file mypar.htm. </FONT></P>
1051: <UL>
1052: <LI><U>Outputs files</U> <BR><BR>- Copy of the parameter file: <A
1053: href="http://euroreves.ined.fr/imach/doc/ormypar.txt">ormypar.txt</A><BR>-
1054: Gnuplot file name: <A
1055: href="http://euroreves.ined.fr/imach/doc/mypar.gp.txt">mypar.gp.txt</A><BR>-
1056: Cross-sectional prevalence in each state: <A
1057: href="http://euroreves.ined.fr/imach/doc/prmypar.txt">prmypar.txt</A> <BR>-
1058: Period prevalence in each state: <A
1059: href="http://euroreves.ined.fr/imach/doc/plrmypar.txt">plrmypar.txt</A> <BR>-
1060: Transition probabilities: <A
1061: href="http://euroreves.ined.fr/imach/doc/pijrmypar.txt">pijrmypar.txt</A><BR>-
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> <BR>-
1064: Parameter file with estimated parameters and the covariance matrix: <A
1065: href="http://euroreves.ined.fr/imach/doc/rmypar.txt">rmypar.txt</A> <BR>-
1066: Variance of one-step probabilities: <A
1067: href="http://euroreves.ined.fr/imach/doc/probrmypar.txt">probrmypar.txt</A>
1068: <BR>- 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><BR>-
1071: Health expectancies with their variances: <A
1072: href="http://euroreves.ined.fr/imach/doc/trmypar.txt">trmypar.txt</A> <BR>-
1073: Standard deviation of period prevalences: <A
1074: href="http://euroreves.ined.fr/imach/doc/vplrmypar.txt">vplrmypar.txt</A>
1075: <BR>No population forecast: popforecast = 0 (instead of 1) or stepm = 24
1076: (instead of 1) or model=. (instead of .)<BR><BR>
1077: <LI><U>Graphs</U> <BR><BR>-<A
1078: href="http://euroreves.ined.fr/imach/mytry/pemypar1.gif">One-step transition
1079: probabilities</A><BR>-<A
1080: href="http://euroreves.ined.fr/imach/mytry/pmypar11.gif">Convergence to the
1081: period prevalence</A><BR>-<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> <BR>-<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> <BR>-<A
1086: href="http://euroreves.ined.fr/imach/mytry/expmypar11.gif">Health life
1087: expectancies by age and initial health state (1)</A> <BR>-<A
1088: href="http://euroreves.ined.fr/imach/mytry/expmypar21.gif">Health life
1089: expectancies by age and initial health state (2)</A> <BR>-<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: <P>This software have been partly granted by <A
1093: href="http://euroreves.ined.fr/">Euro-REVES</A>, a concerted action from the
1.7 ! lievre 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
1.5 lievre 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> .</P>
1100: <P>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><BR></P></BODY></HTML>
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