--- imach/html/doc/imach.htm 2004/06/16 22:45:51 1.3 +++ imach/html/doc/imach.htm 2005/06/10 08:54:06 1.5 @@ -1,632 +1,468 @@ - - - - - -Computing Health Expectancies using IMaCh - - - - -IMaCh - - - - -
- -

Computing Health -Expectancies using IMaCh

- -

(a Maximum -Likelihood Computer Program using Interpolation of Markov Chains)

- -

 

- -

- -

INED and EUROREVES

- -

Version -0.97, June 2004

- -
- -

Authors of the -program: Nicolas Brouard, senior researcher at the Institut -National d'Etudes Démographiques (INED, Paris) in the "Mortality, -Health and Epidemiology" Research Unit

- -

and Agnès -Lièvre
-

- -

Contribution to the mathematics: C. R. -Heathcote (Australian -National University, Canberra).

- -

Contact: Agnès Lièvre (lievre@ined.fr)

- -
- - - -
- -

Introduction

- -

This program computes Healthy Life Expectancies from cross-longitudinal -data using the methodology pioneered by Laditka and Wolf (1). -Within the family of Health Expectancies (HE), Disability-free -life expectancy (DFLE) is probably the most important index to -monitor. In low mortality countries, there is a fear that when -mortality declines, the increase in DFLE is not proportionate to -the increase in total Life expectancy. This case is called the Expansion -of morbidity. Most of the data collected today, in -particular by the international REVES -network on Health expectancy, and most HE indices based on these -data, are cross-sectional. It means that the information -collected comes from a single cross-sectional survey: people from -various ages (but mostly old people) are surveyed on their health -status at a single date. Proportion of people disabled at each -age, can then be measured at that date. This age-specific -prevalence curve is then used to distinguish, within the -stationary population (which, by definition, is the life table -estimated from the vital statistics on mortality at the same -date), the disable population from the disability-free -population. Life expectancy (LE) (or total population divided by -the yearly number of births or deaths of this stationary -population) is then decomposed into DFLE and DLE. This method of -computing HE is usually called the Sullivan method (from the name -of the author who first described it).

- -

Age-specific proportions of people disabled (prevalence of -disability) are dependent on the historical flows from entering -disability and recovering in the past until today. The age-specific -forces (or incidence rates), estimated over a recent period of time -(like for period forces of mortality), of entering disability or -recovering a good health, are reflecting current conditions and -therefore can be used at each age to forecast the future of this -cohortif 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 (computed from period incidences) is lower -than the cross-sectional prevalence. For example if a country is -improving its technology of prosthesis, the incidence of recovering -the ability to walk will be higher at each (old) age, but the -prevalence of disability will only slightly reflect an improve because -the prevalence is mostly affected by the history of the cohort and not -by recent period effects. To measure the period improvement we have to -simulate the future of a cohort of new-borns entering or leaving at -each age the disability state or dying according to the incidence -rates measured today on different cohorts. The proportion of people -disabled at each age in this simulated cohort will be much lower that -the proportions observed at each age in a cross-sectional survey. This -new prevalence curve introduced in a life table will give a more -realistic HE level than the Sullivan method which mostly measured the -History of health conditions in this country.

- -

Therefore, the main question is how to measure incidence rates -from cross-longitudinal surveys? This is the goal of the IMaCH -program. From your data and using IMaCH you can estimate period -HE and not only Sullivan's HE. Also the standard errors of the HE -are computed.

- -

A cross-longitudinal survey consists in a first survey -("cross") where individuals from different ages are -interviewed on their health status or degree of disability. At -least a second wave of interviews ("longitudinal") -should measure each new individual health status. Health -expectancies are computed from the transitions observed between -waves and are computed for each degree of severity of disability -(number of life states). More degrees you consider, more time is -necessary to reach the Maximum Likelihood of the parameters -involved in the model. Considering only two states of disability -(disable and healthy) is generally enough but the computer -program works also with more health statuses.
-
-The simplest model is the multinomial logistic model where pij -is the probability to be observed in state j at the second -wave conditional to be observed in state i at the first -wave. Therefore a simple model is: log(pij/pii)= aij + -bij*age+ cij*sex, where 'age' is age and 'sex' -is a covariate. The advantage that this computer program claims, -comes from that if the delay between waves is not identical for -each individual, or if some individual missed an interview, the -information is not rounded or lost, but taken into account using -an interpolation or extrapolation. hPijx is the -probability to be observed in state i at age x+h -conditional to the observed state i at age x. The -delay 'h' can be split into an exact number (nh*stepm) -of unobserved intermediate states. This elementary transition (by -month or quarter trimester, semester or year) is modeled as a -multinomial logistic. The hPx matrix is simply the matrix -product of nh*stepm elementary matrices and the -contribution of each individual to the likelihood is simply hPijx. -
-

- -

The program presented in this manual is a quite general -program named IMaCh (for Interpolated -MArkov CHain), designed to -analyse transition data from longitudinal surveys. The first step -is the parameters estimation of a transition probabilities model -between an initial status and a final status. From there, the -computer program produces some indicators such as observed and -stationary prevalence, life expectancies and their variances and -graphs. Our transition model consists in absorbing and -non-absorbing states with the possibility of return across the -non-absorbing states. The main advantage of this package, -compared to other programs for the analysis of transition data -(For example: Proc Catmod of SAS®) is that the whole -individual information is used even if an interview is missing, a -status or a date is unknown or when the delay between waves is -not identical for each individual. The program can be executed -according to parameters: selection of a sub-sample, number of -absorbing and non-absorbing states, number of waves taken in -account (the user inputs the first and the last interview), a -tolerance level for the maximization function, the periodicity of -the transitions (we can compute annual, quarterly or monthly -transitions), covariates in the model. It works on Windows or on -Unix.
-

- -
- -

(1) Laditka, Sarah B. and Wolf, Douglas A. (1998), "New -Methods for Analyzing Active Life Expectancy". Journal of -Aging and Health. Vol 10, No. 2.

-

(2) Lièvre A., Brouard N. and Heathcote Ch. (2003) Estimating Health Expectancies -from Cross-longitudinal surveys. Mathematical Population Studies.- 10(4), pp. 211-248 - -


- -

On what kind of data can -it be used?

- -

The minimum data required for a transition model is the -recording of a set of individuals interviewed at a first date and -interviewed again at least one another time. From the -observations of an individual, we obtain a follow-up over time of -the occurrence of a specific event. In this documentation, the -event is related to health status at older ages, but the program -can be applied on a lot of longitudinal studies in different -contexts. To build the data file explained into the next section, -you must have the month and year of each interview and the -corresponding health status. But in order to get age, date of -birth (month and year) is required (missing values is allowed for -month). Date of death (month and year) is an important -information also required if the individual is dead. Shorter -steps (i.e. a month) will more closely take into account the -survival time after the last interview.

- -
- -

The data file

- -

In this example, 8,000 people have been interviewed in a -cross-longitudinal survey of 4 waves (1984, 1986, 1988, 1990). Some -people missed 1, 2 or 3 interviews. Health statuses are healthy (1) -and disable (2). The survey is not a real one. It is a simulation of -the American Longitudinal Survey on Aging. The disability state is -defined if the individual missed one of four ADL (Activity of daily -living, like bathing, eating, walking). Therefore, even if the -individuals interviewed in the sample are virtual, the information -brought with this sample is close to the situation of the United -States. Sex is not recorded is this sample. The LSOA survey is biased -in the sense that people living in an institution were not surveyed at -first pass in 1984. Thus the prevalence of disability in 1984 is -biased downwards at old ages. But when people left their household to -an institution, they have been surveyed in their institution in 1986, -1988 or 1990. Thus incidences are not biased. But cross-sectional -prevalences of disability at old ages are thus artificially increasing -in 1986, 1988 and 1990 because of a higher weight of people -institutionalized in the sample. Our article shows the -opposite: the period prevalence is lower at old ages than the -adjusted cross-sectional prevalence proving important current progress -against disability.

- -

Each line of the data set (named data1.txt -in this first example) is an individual record. Fields are separated -by blanks:

- - - -

 

- -

If your longitudinal survey do not include information about -weights or covariates, you must fill the column with a number -(e.g. 1) because a missing field is not allowed.

- -
- -

Your first example parameter file

- -

#Imach version 0.97b, June 2004, -INED-EUROREVES

- -

This first line was a comment. Comments line start with a '#'.

- -

First uncommented line

- -
title=1st_example datafile=data1.txt lastobs=8600 firstpass=1 lastpass=4
- - - -

 

- -

Second uncommented -line

- -
ftol=1.e-08 stepm=1 ncovcol=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0
- - - -

Covariates

- -

Intercept and age are systematically included in the model. -Additional covariates can be included with the command:

- -
model=list of covariates
- - - -

In this example, we have two covariates in the data file -(fields 2 and 3). The number of covariates included in the data -file between the id and the date of birth is ncovcol=2 (it was -named ncov in version prior to 0.8). If you have 3 covariates in -the datafile (fields 2, 3 and 4), you will set ncovcol=3. Then -you can run the programme with a new parametrisation taking into -account the third covariate. For example, model=V1+V3 estimates -a model with the first and third covariates. More complicated -models can be used, but it will takes more time to converge. With -a simple model (no covariates), the programme estimates 8 -parameters. Adding covariates increases the number of parameters -: 12 for model=V1, 16 for model=V1+V1*age -and 20 for model=V1+V2+V3.

- -

Guess values for optimization

- -

You must write the initial guess values of the parameters for -optimization. The number of parameters, N depends on the -number of absorbing states and non-absorbing states and on the -number of covariates.
-N is given by the formula N=(nlstate + -ndeath-1)*nlstate*ncovmodel .
-
-Thus in the simple case with 2 covariates (the model is log -(pij/pii) = aij + bij * age where intercept and age are the two -covariates), and 2 health degrees (1 for disability-free and 2 -for disability) and 1 absorbing state (3), you must enter 8 -initials values, a12, b12, a13, b13, a21, b21, a23, b23. You can -start with zeros as in this example, but if you have a more -precise set (for example from an earlier run) you can enter it -and it will speed up them
-Each of the four lines starts with indices "ij": ij -aij bij

- -
-
# Guess values of aij and bij in log (pij/pii) = aij + bij * age
+
+
+Computing Health Expectancies using IMaCh
+
+
+
+
+ +

Computing Health Expectancies using +IMaCh

+

(a Maximum Likelihood Computer +Program using Interpolation of Markov Chains)

+

 

+

+

INED and EUROREVES

+

Version 0.97, June +2004

+
+ +

Authors of the program: +Nicolas Brouard, senior researcher at the Institut National +d'Etudes Démographiques (INED, +Paris) in the "Mortality, Health and Epidemiology" Research Unit +

+

and Agnès Lièvre

+

Contribution to the mathematics: C. R. Heathcote +(Australian National University, +Canberra).

+

Contact: Agnès Lièvre (lievre@ined.fr)

+
+ + +
+ +

Introduction

+

This program computes Healthy Life Expectancies from +cross-longitudinal data using the methodology pioneered by Laditka and +Wolf (1). Within the family of Health Expectancies (HE), disability-free life +expectancy (DFLE) is probably the most important index to monitor. In low +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, +in particular by the international REVES +network on Health Expectancy and the disability process, and most HE indices based on these data, are +cross-sectional. This means that the information collected comes from a +single cross-sectional survey: people from a variety of ages (but often old people) +are surveyed on their health status at a single date. The proportion of people +disabled at each age can then be estimated at that date. This age-specific +prevalence curve is used to distinguish, within the stationary population +(which, by definition, is the life table estimated from the vital statistics on +mortality at the same date), the disabled population from the disability-free +population. Life expectancy (LE) (or total population divided by the yearly +number of births or deaths of this stationary population) is then decomposed +into disability-free life expectancy (DFLE) and disability life +expectancy (DLE). This method of computing HE is usually called the Sullivan +method (after the author who first described it).

+

The age-specific proportions of people disabled (prevalence of disability) are +dependent upon the historical flows from entering disability and recovering in the past. The age-specific forces (or incidence rates) of entering +disability or recovering a good health, estimated over a recent period of time (as period forces of mortality), are reflecting current conditions and +therefore can be used at each age to forecast the future of this cohort if +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 +(computed from period incidences) is lower than the cross-sectional prevalence. +For example if a country is improving its technology of prosthesis, the +incidence of recovering the ability to walk will be higher at each (old) age, +but the prevalence of disability will only slightly reflect an improvement because +the prevalence is mostly affected by the history of the cohort and not by recent +period effects. To measure the period improvement we have to simulate the future +of a cohort of new-borns entering or leaving the disability state or +dying at each age according to the incidence rates measured today on different cohorts. The +proportion of people disabled at each age in this simulated cohort will be much +lower that the proportions observed at each age in a cross-sectional survey. +This new prevalence curve introduced in a life table will give a more realistic +HE level than the Sullivan method which mostly reflects the history of health +conditions in a country.

+

Therefore, the main question is how can we measure incidence rates from +cross-longitudinal surveys? This is the goal of the IMaCH program. From your +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.

+

A cross-longitudinal survey consists of a first survey ("cross") where +individuals of different ages are interviewed about their health status or degree +of disability. At least a second wave of interviews ("longitudinal") should +measure each individual new health status. Health expectancies are computed from +the transitions observed between waves (interviews) and are computed for each degree of +severity of disability (number of health states). The more degrees of severity considered, the more +time is necessary to reach the Maximum Likelihood of the parameters involved in +the model. Considering only two states of disability (disabled and healthy) is +generally enough but the computer program works also with more health +states.

The simplest model for the transition probabilities is the multinomial logistic model where +pij is the probability to be observed in state j at the second +wave conditional to be observed in state i at the first wave. Therefore +a simple model is: log(pij/pii)= aij + bij*age+ cij*sex, where +'age' is age and 'sex' is a covariate. The advantage that this +computer program claims, is that if the delay between waves is not +identical for each individual, or if some individual missed an interview, the +information is not rounded or lost, but taken into account using an +interpolation or extrapolation. hPijx is the probability to be observed +in state i at age x+h conditional on the observed state i +at age x. The delay 'h' can be split into an exact number +(nh*stepm) of unobserved intermediate states. This elementary transition +(by month or quarter, trimester, semester or year) is modeled as the above multinomial +logistic. The hPx matrix is simply the matrix product of nh*stepm +elementary matrices and the contribution of each individual to the likelihood is +simply hPijx.

+

The program presented in this manual is a general program named +IMaCh (for Interpolated +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 +transition probabilities between an initial state and a final state. +From there, the computer program produces indicators such as the observed and +stationary prevalence, life expectancies and their variances both numerically and graphically. Our +transition model consists of absorbing and non-absorbing states assuming the +possibility of return across the non-absorbing states. The main advantage of +this package, compared to other programs for the analysis of transition data +(for example: Proc Catmod of SAS®) is that the whole individual +information is used even if an interview is missing, a state or a date is +unknown or when the delay between waves is not identical for each individual. +The program is dependent upon a set of parameters inputted by the user: selection of a sub-sample, +number of absorbing and non-absorbing states, number of waves to be taken in account , a tolerance level for the +maximization function, the periodicity of the transitions (we can compute +annual, quarterly or monthly transitions), covariates in the model. IMaCh works on +Windows or on Unix platform.

+
+ +

(1) Laditka S. B. and Wolf, D. (1998), New Methods for Analyzing +Active Life Expectancy. Journal of Aging and Health. Vol 10, No. 2.

+

(2) Lièvre +A., Brouard N. and Heathcote Ch. (2003) Estimating Health Expectancies from +Cross-longitudinal surveys. Mathematical Population Studies.- 10(4), +pp. 211-248 +


+ +

What kind of data is required?

+

The minimum data required for a transition model is the recording of a set of +individuals interviewed at a first date and interviewed once more. From the observations of an individual, we obtain a follow-up over +time of the occurrence of a specific event. In this documentation, the event is +related to health state, but the program can be applied to many +longitudinal studies with different contexts. To build the data file +as explained +in the next section, you must have the month and year of each interview and +the corresponding health state. In order to get age, date of birth (month +and year) are required (missing values are allowed for month). Date of death +(month and year) is an important information also required if the individual is +dead. Shorter steps (i.e. a month) will more closely take into account the +survival time after the last interview.

+
+ +

The data file

+

In this example, 8,000 people have been interviewed in a cross-longitudinal +survey of 4 waves (1984, 1986, 1988, 1990). Some people missed 1, 2 or 3 +interviews. Health states are healthy (1) and disabled (2). The survey is not a +real one but a simulation of the American Longitudinal Survey on Aging. The +disability state is defined as dependence in at least one of four ADLs (Activities +of daily living, like bathing, eating, walking). Therefore, even if the +individuals interviewed in the sample are virtual, the information in +this sample is close to reality for the United States. Sex is not recorded +is this sample. The LSOA survey is biased in the sense that people +living in an institution were not included in the first interview in +1984. Thus the prevalence of disability observed in 1984 is lower than +the true prevalence at old ages. However when people moved into an +institution, they were interviewed there in 1986, 1988 and 1990. Thus +the incidences of disabilities are not biased. Cross-sectional +prevalences of disability at old ages are thus artificially increasing in 1986, +1988 and 1990 because of a greater proportion of the sample +institutionalized. Our article (Lièvre A., Brouard N. and Heathcote +Ch. (2003)) shows the opposite: the period prevalence based on the +incidences is lower at old +ages than the adjusted cross-sectional prevalence illustrating that +there has been significant progress against disability.

+

Each line of the data set (named data1.txt in this first +example) is an individual record. Fields are separated by blanks:

+ +

 

+

If you do not wish to include information on weights or +covariates, you must fill the column with a number (e.g. 1) since all +fields must be present.

+
+ +

Your first example parameter file

+

#Imach version 0.97b, June 2004, INED-EUROREVES

+

This first line was a comment. Comments line start with a '#'.

+

First uncommented line

title=1st_example datafile=data1.txt lastobs=8600 firstpass=1 lastpass=4
+ +

 

+

Second uncommented +line

ftol=1.e-08 stepm=1 ncovcol=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0
+ +

Covariates

+

Intercept and age are automatically included in the model. Additional +covariates can be included with the command:

model=list of covariates
+ +

In this example, we have two covariates in the data file (fields 2 and 3). +The number of covariates included in the data file between the id and the date +of birth is ncovcol=2 (it was named ncov in version prior to 0.8). If you have 3 +covariates in the datafile (fields 2, 3 and 4), you will set ncovcol=3. Then you +can run the programme with a new parametrisation taking into account the third +covariate. For example, model=V1+V3 estimates a model with the +first and third covariates. More complicated models can be used, but this will +take more time to converge. With a simple model (no covariates), the programme +estimates 8 parameters. Adding covariates increases the number of parameters : +12 for model=V1, 16 for model=V1+V1*age and 20 +for model=V1+V2+V3.

+

Guess values for optimization +

+

You must write the initial guess values of the parameters for optimization. +The number of parameters, N depends on the number of absorbing states +and non-absorbing states and on the number of covariates in the model (ncovmodel).
N is +given by the formula N=(nlstate + +ndeath-1)*nlstate*ncovmodel .

Thus in +the simple case with 2 covariates in the model(the model is log (pij/pii) = aij + bij * age +where intercept and age are the two covariates), and 2 health states (1 for +disability-free and 2 for disability) and 1 absorbing state (3), you must enter +8 initials values, a12, b12, a13, b13, a21, b21, a23, b23. You can start with +zeros as in this example, but if you have a more precise set (for example from +an earlier run) you can enter it and it will speed up the convergence
Each of the four +lines starts with indices "ij": ij aij bij

+
# Guess values of aij and bij in log (pij/pii) = aij + bij * age
 12 -14.155633  0.110794 
 13  -7.925360  0.032091 
 21  -1.890135 -0.029473 
-23  -6.234642  0.022315 
-
- -

or, to simplify (in most of cases it converges but there is no -warranty!):

- -
-
12 0.0 0.0
+23  -6.234642  0.022315 
+

or, to simplify (in most of cases it converges but there is no warranty!): +

+
12 0.0 0.0
 13 0.0 0.0
 21 0.0 0.0
-23 0.0 0.0
-
- -

In order to speed up the convergence you can make a first run -with a large stepm i.e stepm=12 or 24 and then decrease the stepm -until stepm=1 month. If newstepm is the new shorter stepm and -stepm can be expressed as a multiple of newstepm, like newstepm=n -stepm, then the following approximation holds:

- -
aij(stepm) = aij(n . stepm) - ln(n)
-
- -

and

- -
bij(stepm) = bij(n . stepm) .
- -

For example if you already ran for a 6 months interval and -got:
-

- -
# Parameters
+23 0.0 0.0
+

In order to speed up the convergence you can make a first run with a large +stepm i.e stepm=12 or 24 and then decrease the stepm until stepm=1 month. If +newstepm is the new shorter stepm and stepm can be expressed as a multiple of +newstepm, like newstepm=n stepm, then the following approximation holds:

aij(stepm) = aij(n . stepm) - ln(n)
+
+

and

bij(stepm) = bij(n . stepm) .
+

For example if you already ran with stepm=6 (a 6 months interval) and got:

# Parameters
 12 -13.390179  0.126133 
 13  -7.493460  0.048069 
 21   0.575975 -0.041322 
 23  -4.748678  0.030626 
-
- -

If you now want to get the monthly estimates, you can guess -the aij by substracting ln(6)= 1,7917
-and running
-

- -
12 -15.18193847  0.126133 
+
+

Then you now want to get the monthly estimates, you can guess the aij by +subtracting ln(6)= 1.7917
and running using

12 -15.18193847  0.126133 
 13 -9.285219469  0.048069
 21 -1.215784469 -0.041322
 23 -6.540437469  0.030626
-
- -

and get
-

- -
12 -15.029768 0.124347 
+
+

and get

12 -15.029768 0.124347 
 13 -8.472981 0.036599 
 21 -1.472527 -0.038394 
 23 -6.553602 0.029856 
 
-which is closer to the results. The approximation is probably useful
+

which is closer to the results. The approximation is probably useful only for very small intervals and we don't have enough experience to -know if you will speed up the convergence or not. -

- -
         -ln(12)= -2.484
+know if you will speed up the convergence or not.

+
         -ln(12)= -2.484
  -ln(6/1)=-ln(6)= -1.791
  -ln(3/1)=-ln(3)= -1.0986
 -ln(12/6)=-ln(2)= -0.693
-
- -In version 0.9 and higher you can still have valuable results even if -your stepm parameter is bigger than a month. The idea is to run with -bigger stepm in order to have a quicker convergence at the price of a -small bias. Once you know which model you want to fit, you can put -stepm=1 and wait hours or days to get the convergence! - -To get unbiased results even with large stepm we introduce the idea of -pseudo likelihood by interpolating two exact likelihoods. Let us -detail this: -

-If the interval of d months between two waves is not a -mutliple of 'stepm', but is comprised between (n-1) stepm and -n stepm then both exact likelihoods are computed (the -contribution to the likelihood at n stepm requires one matrix -product more) (let us remember that we are modelling the probability -to be observed in a particular state after d months being -observed at a particular state at 0). The distance, (bh in -the program), from the month of interview to the rounded date of n -stepm is computed. It can be negative (interview occurs before -n stepm) or positive if the interview occurs after n -stepm (and before (n+1)stepm). -
-Then the final contribution to the total likelihood is a weighted -average of these two exact likelihoods at n stepm (out) and -at (n-1)stepm(savm). We did not want to compute the third -likelihood at (n+1)stepm because it is too costly in time, so -we used an extrapolation if bh is positive.
Formula of -inter/extrapolation may vary according to the value of parameter mle: -

-mle=1	  lli= log((1.+bbh)*out[s1][s2]- bbh*savm[s1][s2]); /* linear interpolation */
-
-mle=2	lli= (savm[s1][s2]>(double)1.e-8 ? \
+
In version 0.9 and higher you can still have valuable results even if your +stepm parameter is bigger than a month. The idea is to run with bigger stepm in +order to have a quicker convergence at the price of a small bias. Once you know +which model you want to fit, you can put stepm=1 and wait hours or days to get +the convergence! To get unbiased results even with large stepm we introduce the +idea of pseudo likelihood by interpolating two exact likelihoods. In +more detail: +

If the interval of d months between two waves is not a multiple of +'stepm', but is between (n-1) stepm and n stepm then +both exact likelihoods are computed (the contribution to the likelihood at n +stepm requires one matrix product more) (let us remember that we are +modelling the probability to be observed in a particular state after d +months being observed at a particular state at 0). The distance, (bh in +the program), from the month of interview to the rounded date of n +stepm is computed. It can be negative (interview occurs before n +stepm) or positive if the interview occurs after n stepm (and +before (n+1)stepm).
Then the final contribution to the total +likelihood is a weighted average of these two exact likelihoods at n +stepm (out) and at (n-1)stepm(savm). We did not want to compute +the third likelihood at (n+1)stepm because it is too costly in time, so +we used an extrapolation if bh is positive.
The formula +for the inter/extrapolation may vary according to the value of parameter mle:

mle=1	  lli= log((1.+bbh)*out[s1][s2]- bbh*savm[s1][s2]); /* linear interpolation */
+ 
+mle=2	lli= (savm[s1][s2]>(double)1.e-8 ? \
           log((1.+bbh)*out[s1][s2]- bbh*(savm[s1][s2])): \
           log((1.+bbh)*out[s1][s2])); /* linear interpolation */
-mle=3	lli= (savm[s1][s2]>1.e-8 ? \
+mle=3	lli= (savm[s1][s2]>1.e-8 ? \
           (1.+bbh)*log(out[s1][s2])- bbh*log(savm[s1][s2]): \
           log((1.+bbh)*out[s1][s2])); /* exponential inter-extrapolation */
 
 mle=4   lli=log(out[s[mw[mi][i]][i]][s[mw[mi+1][i]][i]]); /* No interpolation  */
         no need to save previous likelihood into memory.
-
-

-If the death occurs between first and second pass, and for example -more precisely between n stepm and (n+1)stepm the -contribution of this people to the likelihood is simply the difference -between the probability of dying before n stepm and the -probability of dying before (n+1)stepm. There was a bug in -version 0.8 and death was treated as any other state, i.e. as if it -was an observed death at second pass. This was not precise but -correct, but when information on the precise month of death came -(death occuring prior to second pass) we did not change the likelihood -accordingly. Thanks to Chris Jackson for correcting us. In earlier + +

If the death occurs between the first and second pass, and for example more +precisely between n stepm and (n+1)stepm the contribution of +these people to the likelihood is simply the difference between the probability +of dying before n stepm and the probability of dying before +(n+1)stepm. There was a bug in version 0.8 and death was treated as any +other state, i.e. as if it was an observed death at second pass. This was not +precise but correct, although when information on the precise month of +death came (death occuring prior to second pass) we did not change the +likelihood accordingly. We thank Chris Jackson for correcting it. In earlier versions (fortunately before first publication) the total mortality -was overestimated (people were dying too early) of about 10%. Version -0.95 and higher are correct. - -

Our suggested choice is mle=1 . If stepm=1 there is no difference -between various mle options (methods of interpolation). If stepm is -big, like 12 or 24 or 48 and mle=4 (no interpolation) the bias may be -very important if the mean duration between two waves is not a -multiple of stepm. See the appendix in our main publication concerning -the sine curve of biases. - - -

Guess values for computing variances

+was thus overestimated (people were dying too early) by about 10%. Version +0.95 and higher are correct. -

These values are output by the maximisation of the likelihood mle=1. These valuse can be used as an input of a -second run in order to get the various output data files (Health -expectancies, period prevalence etc.) and figures without rerunning -the long maximisation phase (mle=0).

- -

These 'scales' are small values needed for the computing of -numerical derivatives. These derivatives are used to compute the -hessian matrix of the parameters, that is the inverse of the -covariance matrix. They are often used for estimating variances and -confidence intervals. Each line consists in indices "ij" -followed by the initial scales (zero to simplify) associated with aij -and bij.

- - +

Covariance matrix of parameters

+

The covariance matrix is output if mle=1. But it can be +also be used as an input to get the various output data files (Health +expectancies, period prevalence etc.) and figures without rerunning +the maximisation phase (mle=0).
Each line starts with indices +"ijk" followed by the covariances between aij and bij:
+

   121 Var(a12) 
    122 Cov(b12,a12)  Var(b12) 
           ...
-   232 Cov(b23,a12)  Cov(b23,b12) ... Var (b23) 
- - +

Age range for calculation of stationary prevalences and +health expectancies

agemin=70 agemax=100 bage=50 fage=100
+

Once we obtained the estimated parameters, the program is able to calculate +period prevalence, transitions probabilities and life expectancies at any age. +Choice of the age range is useful for extrapolation. In this example, +the age of people interviewed varies from 69 to 102 and the model is +estimated using their exact ages. But if you are interested in the +age-specific period prevalence you can start the simulation at an +exact age like 70 and stop at 100. Then the program will draw at +least two curves describing the forecasted prevalences of two cohorts, +one for healthy people at age 70 and the second for disabled people at +the same initial age. And according to the mixing property +(ergodicity) and because of recovery, both prevalences will tend to be +identical at later ages. Thus if you want to compute the prevalence at +age 70, you should enter a lower agemin value. +

Setting bage=50 (begin age) and fage=100 (final age), let the program compute +life expectancy from age 'bage' to age 'fage'. As we use a model, we can +interessingly compute life expectancy on a wider age range than the age range +from the data. But the model can be rather wrong on much larger intervals. +Program is limited to around 120 for upper age!


+
+

Computing the cross-sectional prevalence

begin-prev-date=1/1/1984 end-prev-date=1/6/1988 estepm=1
+

Statements 'begin-prev-date' and 'end-prev-date' allow the user to +select the period in which the observed prevalences in each state. In +this example, the prevalences are calculated on data survey collected +between 1 January 1984 and 1 June 1988.

+ +

Population- or status-based health +expectancies

pop_based=0
+

The program computes status-based health expectancies, i.e health +expectancies which depend on the initial health state. If you are healthy, your +healthy life expectancy (e11) is higher than if you were disabled (e21, with e11 +> e21).
To compute a healthy life expectancy 'independent' of the initial +status we have to weight e11 and e21 according to the probability of +being in each state at initial age which correspond to the proportions +of people in each health state (cross-sectional prevalences). +

We could also compute e12 and e12 and get e.2 by weighting them according to +the observed cross-sectional prevalences at initial age. +

In a similar way we could compute the total life expectancy by summing e.1 +and e.2 .
The main difference between 'population based' and 'implied' or +'period' is in the weights used. 'Usually', cross-sectional prevalences of +disability are higher than period prevalences particularly at old ages. This is +true if the country is improving its health system by teaching people how to +prevent disability by promoting better screening, for example of people +needing cataract surgery. Then the proportion of disabled people at +age 90 will be lower than the current observed proportion. +

Thus a better Health Expectancy and even a better Life Expectancy value is +given by forecasting not only the current lower mortality at all ages but also a +lower incidence of disability and higher recovery.
Using the period +prevalences as weight instead of the cross-sectional prevalences we are +computing indices which are more specific to the current situations and +therefore more useful to predict improvements or regressions in the future as to +compare different policies in various countries. +

+

Prevalence forecasting (Experimental)

starting-proj-date=1/1/1989 final-proj-date=1/1/1992 mov_average=0 
+

Prevalence and population projections are only available if the interpolation +unit is a month, i.e. stepm=1 and if there are no covariate. The programme +estimates the prevalence in each state at a precise date expressed in +day/month/year. The programme computes one forecasted prevalence a year from a +starting date (1 January 1989 in this example) to a final date (1 January +1992). The statement mov_average allows computation of smoothed forecasted +prevalences with a five-age moving average centered at the mid-age of the +fiveyear-age period.

+

Population forecasting (Experimental)

+ + +
+ +

Running Imach with this +example

+

We assume that you have already typed your 1st_example parameter +file as explained above. To run +the program under Windows you should either:

+ +

The time to converge depends on the step unit used (1 month is more +precise but more cpu time consuming), on the number of cases, and on the number of +variables (covariates). +

The program outputs many files. Most of them are files which will be plotted +for better understanding.

To run under Linux is mostly the same. +

It is no more difficult to run IMaCh on a MacIntosh. +


+ +

Output of the program and graphs +

+

Once the optimization is finished (once the convergence is reached), many +tables and graphics are produced. +

The IMaCh program will create a subdirectory with the same name as your +parameter file (here mypar) where all the tables and figures will be +stored.
Important files like the log file and the output parameter file +(the latter contains the maximum likelihood estimates) are stored at +the main level not in this subdirectory. Files with extension .log and +.txt can be edited with a standard editor like wordpad or notepad or +even can be viewed with a browser like Internet Explorer or Mozilla. +

The main html file is also named with the same name biaspar.htm. You can +click on it by holding your shift key in order to open it in another window +(Windows). +

Our grapher is Gnuplot, an interactive plotting program (GPL) which can +also work in batch mode. A gnuplot reference manual is available here.
When the run is finished, and in +order that the window doesn't disappear, the user should enter a character like +q for quitting.
These characters are:

+The main gnuplot file is named +biaspar.gp and can be edited (right click) and run again. +

Gnuplot is easy and you can use it to make more complex graphs. Just click on +gnuplot and type plot sin(x) to see how easy it is. +

Results files

- cross-sectional prevalence in each state (and at first pass): biaspar/prbiaspar.txt
+

The first line is the title and displays each field of the file. First column +corresponds to age. Fields 2 and 6 are the proportion of individuals in states 1 +and 2 respectively as observed at first exam. Others fields are the numbers of +people in states 1, 2 or more. The number of columns increases if the number of +states is higher than 2.
The header of the file is

# Age Prev(1) N(1) N Age Prev(2) N(2) N
 70 1.00000 631 631 70 0.00000 0 631
 71 0.99681 625 627 71 0.00319 2 627 
-72 0.97125 1115 1148 72 0.02875 33 1148 
- -

It means that at age 70 (between 70 and 71), the prevalence in state 1 is 1.000 -and in state 2 is 0.00 . At age 71 the number of individuals in -state 1 is 625 and in state 2 is 2, hence the total number of -people aged 71 is 625+2=627.
-

- -
- Estimated parameters and -covariance matrix: rbiaspar.imach
- -

This file contains all the maximisation results:

- -
 -2 log likelihood= 21660.918613445392
+72 0.97125 1115 1148 72 0.02875 33 1148 
+

It means that at age 70 (between 70 and 71), the prevalence in state 1 is +1.000 and in state 2 is 0.00 . At age 71 the number of individuals in state 1 is +625 and in state 2 is 2, hence the total number of people aged 71 is 625+2=627. +

+
- Estimated parameters and covariance +matrix: rbiaspar.imach
+

This file contains all the maximisation results:

 -2 log likelihood= 21660.918613445392
  Estimated parameters: a12 = -12.290174 b12 = 0.092161 
                        a13 = -9.155590  b13 = 0.046627 
                        a21 = -2.629849  b21 = -0.022030 
@@ -946,126 +679,103 @@ covariance matrix: 
-
-

By substitution of these parameters in the regression model, -we obtain the elementary transition probabilities:

- -

- -
- Transition probabilities: -biaspar/pijrbiaspar.txt
- -

Here are the transitions probabilities Pij(x, x+nh). The second -column is the starting age x (from age 95 to 65), the third is age -(x+nh) and the others are the transition probabilities p11, p12, p13, -p21, p22, p23. The first column indicates the value of the covariate -(without any other variable than age it is equal to 1) For example, line 5 of the file -is:

- -
1 100 106 0.02655 0.17622 0.79722 0.01809 0.13678 0.84513 
- -

and this means:

- -
p11(100,106)=0.02655
+ 
+

By substitution of these parameters in the regression model, we obtain the +elementary transition probabilities:

+

+
- Transition probabilities: +biaspar/pijrbiaspar.txt
+

Here are the transitions probabilities Pij(x, x+nh). The second column is the +starting age x (from age 95 to 65), the third is age (x+nh) and the others are +the transition probabilities p11, p12, p13, p21, p22, p23. The first column +indicates the value of the covariate (without any other variable than age it is +equal to 1) For example, line 5 of the file is:

1 100 106 0.02655 0.17622 0.79722 0.01809 0.13678 0.84513 
+

and this means:

p11(100,106)=0.02655
 p12(100,106)=0.17622
 p13(100,106)=0.79722
 p21(100,106)=0.01809
 p22(100,106)=0.13678
-p22(100,106)=0.84513 
- -
- Period prevalence in each state: -biaspar/plrbiaspar.txt
- -
#Prevalence
+p22(100,106)=0.84513 
+
- Period +prevalence in each state: biaspar/plrbiaspar.txt
#Prevalence
 #Age 1-1 2-2
 
 #************ 
 70 0.90134 0.09866
 71 0.89177 0.10823 
 72 0.88139 0.11861 
-73 0.87015 0.12985 
- -

At age 70 the period prevalence is 0.90134 in state 1 and 0.09866 -in state 2. This period prevalence differs from the cross-sectional -prevalence. Here is the point. The cross-sectional prevalence at age -70 results from the incidence of disability, incidence of recovery and -mortality which occurred in the past of the cohort. Period prevalence -results from a simulation with current incidences of disability, -recovery and mortality estimated from this cross-longitudinal -survey. It is a good predictin of the prevalence in the -future if "nothing changes in the future". This is exactly -what demographers do with a period life table. Life expectancy is the -expected mean survival time if current mortality rates (age-specific incidences -of mortality) "remain constant" in the future.

- -
- Standard deviation of -period prevalence: biaspar/vplrbiaspar.txt
- -

The period prevalence has to be compared with the cross-sectional -prevalence. But both are statistical estimates and therefore -have confidence intervals. -
For the cross-sectional prevalence we generally need information on -the design of the surveys. It is usually not enough to consider the -number of people surveyed at a particular age and to estimate a -Bernouilli confidence interval based on the prevalence at that -age. But you can do it to have an idea of the randomness. At least you -can get a visual appreciation of the randomness by looking at the -fluctuation over ages. - -

For the period prevalence it is possible to estimate the -confidence interval from the Hessian matrix (see the publication for -details). We are supposing that the design of the survey will only -alter the weight of each individual. IMaCh is scaling the weights of -individuals-waves contributing to the likelihood by making the sum of -the weights equal to the sum of individuals-waves contributing: a -weighted survey doesn't increase or decrease the size of the survey, -it only give more weights to some individuals and thus less to the -others. - -

-cross-sectional and period -prevalence in state (2=disable) with confidence interval: -biaspar/vbiaspar21.png
- -

This graph exhibits the period prevalence in state (2) with the -confidence interval in red. The green curve is the observed prevalence -(or proportion of individuals in state (2)). Without discussing the -results (it is not the purpose here), we observe that the green curve -is rather below the period prevalence. It the data where not biased by -the non inclusion of people living in institutions we would have -concluded that the prevalence of disability will increase in the -future (see the main publication if you are interested in real data -and results which are opposite).

- -

- -
-Convergence to the -period prevalence of disability: biaspar/pbiaspar11.png
-
- -

This graph plots the conditional transition probabilities from -an initial state (1=healthy in red at the bottom, or 2=disable in -green on top) at age x to the final state 2=disable at -age x+h. Conditional means at the condition to be alive -at age x+h which is hP12x + hP22x. The -curves hP12x/(hP12x + hP22x) and hP22x/(hP12x -+ hP22x) converge with h, to the period -prevalence of disability. In order to get the period -prevalence at age 70 we should start the process at an earlier -age, i.e.50. If the disability state is defined by severe -disability criteria with only a few chance to recover, then the -incidence of recovery is low and the time to convergence is -probably longer. But we don't have experience yet.

- -
- Life expectancies by age -and initial health status with standard deviation: biaspar/erbiaspar.txt
- -
# Health expectancies 
+73 0.87015 0.12985 
+

At age 70 the period prevalence is 0.90134 in state 1 and 0.09866 in state 2. +This period prevalence differs from the cross-sectional prevalence and +we explaining. The cross-sectional prevalence at age 70 results from +the incidence of disability, incidence of recovery and mortality which +occurred in the past for the cohort. Period prevalence results from a +simulation with current incidences of disability, recovery and +mortality estimated from this cross-longitudinal survey. It is a good +prediction of the prevalence in the future if "nothing changes in the +future". This is exactly what demographers do with a period life +table. Life expectancy is the expected mean survival time if current +mortality rates (age-specific incidences of mortality) "remain +constant" in the future. +

+
- Standard deviation of period +prevalence: biaspar/vplrbiaspar.txt
+

The period prevalence has to be compared with the cross-sectional prevalence. +But both are statistical estimates and therefore have confidence intervals. +
For the cross-sectional prevalence we generally need information on the +design of the surveys. It is usually not enough to consider the number of people +surveyed at a particular age and to estimate a Bernouilli confidence interval +based on the prevalence at that age. But you can do it to have an idea of the +randomness. At least you can get a visual appreciation of the randomness by +looking at the fluctuation over ages. +

For the period prevalence it is possible to estimate the confidence interval +from the Hessian matrix (see the publication for details). We are supposing that +the design of the survey will only alter the weight of each individual. IMaCh +scales the weights of individuals-waves contributing to the likelihood by +making the sum of the weights equal to the sum of individuals-waves +contributing: a weighted survey doesn't increase or decrease the size of the +survey, it only give more weight to some individuals and thus less to the +others. +

-cross-sectional and period prevalence in state +(2=disable) with confidence interval: biaspar/vbiaspar21.png
+

This graph exhibits the period prevalence in state (2) with the confidence +interval in red. The green curve is the observed prevalence (or proportion of +individuals in state (2)). Without discussing the results (it is not the purpose +here), we observe that the green curve is somewhat below the period +prevalence. If the data were not biased by the non inclusion of people +living in institutions we would have concluded that the prevalence of +disability will increase in the future (see the main publication if +you are interested in real data and results which are opposite).

+

+
-Convergence to the period prevalence of +disability: biaspar/pbiaspar11.png
+

This graph plots the conditional transition probabilities from an initial +state (1=healthy in red at the bottom, or 2=disabled in green on the top) at age +x to the final state 2=disabled at age x+h + where conditional means conditional on being alive at age x+h which is +hP12x + hP22x. The curves hP12x/(hP12x + hP22x) +and hP22x/(hP12x + hP22x) converge with h, to the +period prevalence of disability. In order to get the period prevalence +at age 70 we should start the process at an earlier age, i.e.50. If the +disability state is defined by severe disability criteria with only a +small chance of recovering, then the incidence of recovery is low and the time to convergence is +probably longer. But we don't have experience of this yet.

+
- Life expectancies by age and initial health +status with standard deviation: biaspar/erbiaspar.txt
# Health expectancies 
 # Age 1-1 (SE) 1-2 (SE) 2-1 (SE) 2-2 (SE)
  70   11.0180 (0.1277)    3.1950 (0.3635)    4.6500 (0.0871)    4.4807 (0.2187)
  71   10.4786 (0.1184)    3.2093 (0.3212)    4.3384 (0.0875)    4.4820 (0.2076)
@@ -1079,304 +789,215 @@ href="biaspar/erbiaspar.txt">biaspar/
  79    6.7464 (0.0867)    3.3220 (0.1124)    2.3794 (0.1112)    4.4646 (0.1364)
  80    6.3538 (0.0868)    3.3354 (0.1014)    2.1949 (0.1168)    4.4587 (0.1331)
  81    5.9775 (0.0873)    3.3484 (0.0933)    2.0222 (0.1230)    4.4520 (0.1320)
-
- -
For example  70  11.0180 (0.1277) 3.1950 (0.3635) 4.6500 (0.0871)  4.4807 (0.2187)
+
For example  70  11.0180 (0.1277) 3.1950 (0.3635) 4.6500 (0.0871)  4.4807 (0.2187)
 means
-e11=11.0180 e12=3.1950 e21=4.6500 e22=4.4807 
- -
- -

For example, life expectancy of a healthy individual at age 70 -is 11.0 in the healthy state and 3.2 in the disability state -(total of 14.2 years). If he was disable at age 70, his life expectancy -will be shorter, 4.65 years in the healthy state and 4.5 in the -disability state (=9.15 years). The total life expectancy is a -weighted mean of both, 14.2 and 9.15. The weight is the proportion -of people disabled at age 70. In order to get a period index -(i.e. based only on incidences) we use the stable or -period prevalence at age 70 (i.e. computed from -incidences at earlier ages) instead of the cross-sectional prevalence -(observed for example at first medical exam) (see -below).

- -
- Variances of life -expectancies by age and initial health status: biaspar/vrbiaspar.txt
- -

For example, the covariances of life expectancies Cov(ei,ej) -at age 50 are (line 3)

- -
   Cov(e1,e1)=0.4776  Cov(e1,e2)=0.0488=Cov(e2,e1)  Cov(e2,e2)=0.0424
- -
-Variances of one-step -probabilities : biaspar/probrbiaspar.txt
- -

For example, at age 65

- -
   p11=9.960e-001 standard deviation of p11=2.359e-004
- -
- Health -expectancies -with standard errors in parentheses: biaspar/trbiaspar.txt
- -
#Total LEs with variances: e.. (std) e.1 (std) e.2 (std) 
- -
70 13.26 (0.22) 9.95 (0.20) 3.30 (0.14) 
- -

Thus, at age 70 the total life expectancy, e..=13.26 years is -the weighted mean of e1.=13.46 and e2.=11.35 by the period -prevalences at age 70 which are 0.90134 in state 1 and 0.09866 in -state 2 respectively (the sum is equal to one). e.1=9.95 is the -Disability-free life expectancy at age 70 (it is again a weighted -mean of e11 and e21). e.2=3.30 is also the life expectancy at age -70 to be spent in the disability state.

- -
-Total life expectancy by -age and health expectancies in states (1=healthy) and (2=disable): -biaspar/ebiaspar1.png
- -

This figure represents the health expectancies and the total -life expectancy with a confidence interval (dashed line).

- -
        
- -

Standard deviations (obtained from the information matrix of -the model) of these quantities are very useful. -Cross-longitudinal surveys are costly and do not involve huge -samples, generally a few thousands; therefore it is very -important to have an idea of the standard deviation of our -estimates. It has been a big challenge to compute the Health -Expectancy standard deviations. Don't be confuse: life expectancy -is, as any expected value, the mean of a distribution; but here -we are not computing the standard deviation of the distribution, -but the standard deviation of the estimate of the mean.

- -

Our health expectancies estimates vary according to the sample -size (and the standard deviations give confidence intervals of -the estimates) but also according to the model fitted. Let us -explain it in more details.

- -

Choosing a model means at least two kind of choices. At first we -have to decide the number of disability states. And at second we have to -design, within the logit model family, the model itself: variables, -covariables, confounding factors etc. to be included.

- -

More disability states we have, better is our demographical -approach of the disability process, but smaller are the number of -transitions between each state and higher is the noise in the -measurement. We do not have enough experiments of the various -models to summarize the advantages and disadvantages, but it is -important to say that even if we had huge and unbiased samples, -the total life expectancy computed from a cross-longitudinal -survey, varies with the number of states. If we define only two -states, alive or dead, we find the usual life expectancy where it -is assumed that at each age, people are at the same risk to die. -If we are differentiating the alive state into healthy and -disable, and as the mortality from the disability state is higher -than the mortality from the healthy state, we are introducing -heterogeneity in the risk of dying. The total mortality at each -age is the weighted mean of the mortality in each state by the -prevalence in each state. Therefore if the proportion of people -at each age and in each state is different from the period -equilibrium, there is no reason to find the same total mortality -at a particular age. Life expectancy, even if it is a very useful -tool, has a very strong hypothesis of homogeneity of the -population. Our main purpose is not to measure differential -mortality but to measure the expected time in a healthy or -disability state in order to maximise the former and minimize the -latter. But the differential in mortality complexifies the -measurement.

- -

Incidences of disability or recovery are not affected by the number -of states if these states are independent. But incidences estimates -are dependent on the specification of the model. More covariates we -added in the logit model better is the model, but some covariates are -not well measured, some are confounding factors like in any -statistical model. The procedure to "fit the best model' is -similar to logistic regression which itself is similar to regression -analysis. We haven't yet been sofar because we also have a severe -limitation which is the speed of the convergence. On a Pentium III, -500 MHz, even the simplest model, estimated by month on 8,000 people -may take 4 hours to converge. Also, the IMaCh program is not a -statistical package, and does not allow sophisticated design -variables. If you need sophisticated design variable you have to them -your self and and add them as ordinary variables. IMaCX allows up to 8 -variables. The current version of this program allows only to add -simple variables like age+sex or age+sex+ age*sex but will never be -general enough. But what is to remember, is that incidences or -probability of change from one state to another is affected by the -variables specified into the model.

- -

Also, the age range of the people interviewed is linked -the age range of the life expectancy which can be estimated by -extrapolation. If your sample ranges from age 70 to 95, you can -clearly estimate a life expectancy at age 70 and trust your -confidence interval because it is mostly based on your sample size, -but if you want to estimate the life expectancy at age 50, you -should rely in the design of your model. Fitting a logistic model on a age -range of 70 to 95 and estimating probabilties of transition out of -this age range, say at age 50, is very dangerous. At least you -should remember that the confidence interval given by the -standard deviation of the health expectancies, are under the -strong assumption that your model is the 'true model', which is -probably not the case outside the age range of your sample.

- -
- Copy of the parameter -file: orbiaspar.txt
- -

This copy of the parameter file can be useful to re-run the -program while saving the old output files.

- -
- Prevalence forecasting: -biaspar/frbiaspar.txt
- -

- -First, -we have estimated the observed prevalence between 1/1/1984 and -1/6/1988 (June, European syntax of dates). The mean date of all interviews (weighted average of the -interviews performed between 1/1/1984 and 1/6/1988) is estimated -to be 13/9/1985, as written on the top on the file. Then we -forecast the probability to be in each state.

- -

-For example on 1/1/1989 :

- -
# StartingAge FinalAge P.1 P.2 P.3
+e11=11.0180 e12=3.1950 e21=4.6500 e22=4.4807 
+

For example, life expectancy of a healthy individual at age 70 is 11.0 in the +healthy state and 3.2 in the disability state (total of 14.2 years). If he was +disabled at age 70, his life expectancy will be shorter, 4.65 years in the +healthy state and 4.5 in the disability state (=9.15 years). The total life +expectancy is a weighted mean of both, 14.2 and 9.15. The weight is the +proportion of people disabled at age 70. In order to get a period index (i.e. +based only on incidences) we use the stable +or period prevalence at age 70 (i.e. computed from incidences at earlier +ages) instead of the cross-sectional +prevalence (observed for example at first interview) (see +below).

+
- Variances of life expectancies by age and +initial health status: biaspar/vrbiaspar.txt
+

For example, the covariances of life expectancies Cov(ei,ej) at age 50 are +(line 3)

   Cov(e1,e1)=0.4776  Cov(e1,e2)=0.0488=Cov(e2,e1)  Cov(e2,e2)=0.0424
+
-Variances of one-step probabilities +: biaspar/probrbiaspar.txt
+

For example, at age 65

   p11=9.960e-001 standard deviation of p11=2.359e-004
+
- Health +expectancies with standard errors +in parentheses: biaspar/trbiaspar.txt
#Total LEs with variances: e.. (std) e.1 (std) e.2 (std) 
70 13.26 (0.22) 9.95 (0.20) 3.30 (0.14) 
+

Thus, at age 70 the total life expectancy, e..=13.26 years is the weighted +mean of e1.=13.46 and e2.=11.35 by the period prevalences at age 70 which are +0.90134 in state 1 and 0.09866 in state 2 respectively (the sum is equal to +one). e.1=9.95 is the Disability-free life expectancy at age 70 (it is again a +weighted mean of e11 and e21). e.2=3.30 is also the life expectancy at age 70 to +be spent in the disability state.

+
-Total life expectancy by age and health +expectancies in states (1=healthy) and (2=disable): biaspar/ebiaspar1.png
+

This figure represents the health expectancies and the total life expectancy +with a confidence interval (dashed line).

        
+

Standard deviations (obtained from the information matrix of the model) of +these quantities are very useful. Cross-longitudinal surveys are costly and do +not involve huge samples, generally a few thousands; therefore it is very +important to have an idea of the standard deviation of our estimates. It has +been a big challenge to compute the Health Expectancy standard deviations. Don't +be confused: life expectancy is, as any expected value, the mean of a +distribution; but here we are not computing the standard deviation of the +distribution, but the standard deviation of the estimate of the mean.

+

Our health expectancy estimates vary according to the sample size (and the +standard deviations give confidence intervals of the estimates) but also +according to the model fitted. We explain this in more detail.

+

Choosing a model means at least two kind of choices. First we have to +decide the number of disability states. And second we have to design, within +the logit model family, the model itself: variables, covariates, confounding +factors etc. to be included.

+

The more disability states we have, the better is our demographical +approximation of the disability process, but the smaller the number of +transitions between each state and the higher the noise in the +measurement. We have not experimented enough with the various models +to summarize the advantages and disadvantages, but it is important to +note that even if we had huge unbiased samples, the total life +expectancy computed from a cross-longitudinal survey would vary with +the number of states. If we define only two states, alive or dead, we +find the usual life expectancy where it is assumed that at each age, +people are at the same risk of dying. If we are differentiating the +alive state into healthy and disabled, and as mortality from the +disabled state is higher than mortality from the healthy state, we are +introducing heterogeneity in the risk of dying. The total mortality at +each age is the weighted mean of the mortality from each state by the +prevalence of each state. Therefore if the proportion of people at each age and +in each state is different from the period equilibrium, there is no reason to +find the same total mortality at a particular age. Life expectancy, even if it +is a very useful tool, has a very strong hypothesis of homogeneity of the +population. Our main purpose is not to measure differential mortality but to +measure the expected time in a healthy or disabled state in order to maximise +the former and minimize the latter. But the differential in mortality +complicates the measurement.

+

Incidences of disability or recovery are not affected by the number of states +if these states are independent. But incidence estimates are dependent on the +specification of the model. The more covariates we add in the logit +model the better +is the model, but some covariates are not well measured, some are confounding +factors like in any statistical model. The procedure to "fit the best model' is +similar to logistic regression which itself is similar to regression analysis. +We haven't yet been sofar because we also have a severe limitation which is the +speed of the convergence. On a Pentium III, 500 MHz, even the simplest model, +estimated by month on 8,000 people may take 4 hours to converge. Also, the IMaCh +program is not a statistical package, and does not allow sophisticated design +variables. If you need sophisticated design variable you have to them your self +and and add them as ordinary variables. IMaCh allows up to 8 variables. The +current version of this program allows only to add simple variables like age+sex +or age+sex+ age*sex but will never be general enough. But what is to remember, +is that incidences or probability of change from one state to another is +affected by the variables specified into the model.

+

Also, the age range of the people interviewed is linked the age range of the +life expectancy which can be estimated by extrapolation. If your sample ranges +from age 70 to 95, you can clearly estimate a life expectancy at age 70 and +trust your confidence interval because it is mostly based on your sample size, +but if you want to estimate the life expectancy at age 50, you should rely in +the design of your model. Fitting a logistic model on a age range of 70 to 95 +and estimating probabilties of transition out of this age range, say at age 50, +is very dangerous. At least you should remember that the confidence interval +given by the standard deviation of the health expectancies, are under the strong +assumption that your model is the 'true model', which is probably not the case +outside the age range of your sample.

+
- Copy of the parameter file: +orbiaspar.txt
+

This copy of the parameter file can be useful to re-run the program while +saving the old output files.

+
- Prevalence forecasting: biaspar/frbiaspar.txt
+

First, we have estimated the observed prevalence between 1/1/1984 and +1/6/1988 (June, European syntax of dates). The mean date of all interviews +(weighted average of the interviews performed between 1/1/1984 and 1/6/1988) is +estimated to be 13/9/1985, as written on the top on the file. Then we forecast +the probability to be in each state.

+

For example on 1/1/1989 :

# StartingAge FinalAge P.1 P.2 P.3
 # Forecasting at date 1/1/1989
-  73 0.807 0.078 0.115
- -

- -Since the minimum age is 70 on the 13/9/1985, the youngest forecasted -age is 73. This means that at age a person aged 70 at 13/9/1989 has a -probability to enter state1 of 0.807 at age 73 on 1/1/1989. -Similarly, the probability to be in state 2 is 0.078 and the -probability to die is 0.115. Then, on the 1/1/1989, the prevalence of -disability at age 73 is estimated to be 0.088.

- -
- Population forecasting: -biaspar/poprbiaspar.txt
- -
# Age P.1 P.2 P.3 [Population]
+  73 0.807 0.078 0.115
+

Since the minimum age is 70 on the 13/9/1985, the youngest forecasted age is +73. This means that at age a person aged 70 at 13/9/1989 has a probability to +enter state1 of 0.807 at age 73 on 1/1/1989. Similarly, the probability to be in +state 2 is 0.078 and the probability to die is 0.115. Then, on the 1/1/1989, the +prevalence of disability at age 73 is estimated to be 0.088.

+
- Population forecasting: biaspar/poprbiaspar.txt
# Age P.1 P.2 P.3 [Population]
 # Forecasting at date 1/1/1989 
 75 572685.22 83798.08 
 74 621296.51 79767.99 
-73 645857.70 69320.60 
- -
# Forecasting at date 1/1/19909 
+73 645857.70 69320.60 
# Forecasting at date 1/1/19909 
 76 442986.68 92721.14 120775.48
 75 487781.02 91367.97 121915.51
-74 512892.07 85003.47 117282.76 
- -

From the population file, we estimate the number of people in -each state. At age 73, 645857 persons are in state 1 and 69320 -are in state 2. One year latter, 512892 are still in state 1, -85003 are in state 2 and 117282 died before 1/1/1990.

- -
- -

Trying an example

- -

Since you know how to run the program, it is time to test it -on your own computer. Try for example on a parameter file named imachpar.imach which is a copy -of mypar.imach included -in the subdirectory of imach, mytry. -Edit 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 file mydata.txt -is a smaller file of 3,000 people but still with 4 waves.

- -

Right click on the .imach file and a window will popup with the -string 'Enter the parameter file name:'

- - - - - -
IMACH, Version 0.97b

Enter - the parameter file name: imachpar.imach

-
- -

Most of the data files or image files generated, will use the -'imachpar' string into their name. The running time is about 2-3 -minutes on a Pentium III. If the execution worked correctly, the -outputs files are created in the current directory, and should be -the same as the mypar files initially included in the directory mytry.

- -
+

Once the running is finished, the program requires a +character:

+ + + +
Type e to edit output files, g to graph again, c + to start again, and q for exiting:
In order to +have an idea of the time needed to reach convergence, IMaCh gives an estimation +if the convergence needs 10, 20 or 30 iterations. It might be useful. +

First you should enter e to edit the master +file mypar.htm.

+ +

This software have been partly granted by Euro-REVES, a concerted action from the +European Union. It will be copyrighted identically to a GNU software product, +i.e. program and software can be distributed freely for non commercial use. +Sources are not widely distributed today. You can get them by asking us with a +simple justification (name, email, institute) mailto:brouard@ined.fr and mailto:lievre@ined.fr .

+

Latest version (0.97b of June 2004) can be accessed at http://euroreves.ined.fr/imach