Version 0.7,
+February 2002
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
+
- -
Version -0.7, February 2002
- -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
-
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 +
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 +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 @@ -95,50 +511,52 @@ population. Life expectancy (LE) (or tot 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).
+of the author who first described it).Age-specific proportions of people disable are very difficult -to forecast because each proportion corresponds to historical -conditions of the cohort and it is the result of the historical -flows from entering disability and recovering in the past until -today. The age-specific intensities (or incidence rates) of -entering disability or recovering a good health, are reflecting -actual conditions and therefore can be used at each age to -forecast the future of this cohort. 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 (using the exemple of -an improvement) that the proportions observed at each age in a -cross-sectional survey. This new prevalence curve introduced in a -life table will give a much more actual and 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.
+
Age-specific proportions of people
+disable are very difficult to forecast because each proportion
+corresponds to historical conditions of the cohort and it is the
+result of the historical flows from entering disability and
+recovering in the past until today. The age-specific intensities
+(or incidence rates) of entering disability or recovering a good
+health, are reflecting actual conditions and therefore can be
+used at each age to forecast the future of this cohort. 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 (using the example of an improvement) that the proportions
+observed at each age in a cross-sectional survey. This new
+prevalence curve introduced in a life table will give a much more
+actual and 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
@@ -158,22 +576,21 @@ month or quarter trimester, semester or
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, +
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
+(For example: Proc Catmod of SAS(r)) 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
@@ -181,199 +598,334 @@ according to parameters: selection of a
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, quaterly or monthly
+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 +
(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.
+Aging and Health. Vol 10, No. 2.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 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.
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 is 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.
- -Each line of the data set (named data1.txt -in this first example) is an individual record which fields are:
- -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 is 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.
Each line of the data set (named data1.txt
+in this first example) is an individual record which fields are:
+
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.
+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.
This is a comment. Comments start with a '#'.
+This is a comment. Comments start with a '#'.
title=1st_example datafile=data1.txt lastobs=8600 firstpass=1 lastpass=4- -
title=1st_example datafile=data1.txt lastobs=8600 firstpass=1 lastpass=4
+
+- -
ftol=1.e-08 stepm=1 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0- -
ftol=1.e-08 stepm=1 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0
+
+Intercept and age are systematically included in the model. -Additional covariates can be included with the command
+model=list of covariates- -
Intercept
+and age are systematically included in the model. Additional
+covariates can be included with the command
model=list of covariates
+
+You must write the initial guess values of the parameters for -optimization. The number of parameters, N depends on the +
You
+must write the initial guess values of the parameters for
+optimisation. 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 +
@@ -388,325 +940,562 @@ start with zeros as in this example, but
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 -12 -14.155633 0.110794 -13 -7.925360 0.032091 -21 -1.890135 -0.029473 -23 -6.234642 0.022315-
or, to simplify:
- --- -12 0.0 0.0 -13 0.0 0.0 -21 0.0 0.0 -23 0.0 0.0-
This is an output if mle=1. But it can be -used as an input to get the various output data files (Health +
# 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:
12 0.0 0.0
+
+13 0.0 0.0
+
+21 0.0 0.0
+
+23 0.0 0.0
+
+This +is an output if mle=1. But it can be used as +an input to get the various output data files (Health expectancies, stationary prevalence etc.) and figures without -rerunning the rather long maximisation phase (mle=0).
+rerunning the rather long maximisation phase (mle=0).The scales are small values for the evaluation of numerical +
The +scales are small values for the evaluation of numerical derivatives. These derivatives are used to compute the hessian matrix of the parameters, that is the inverse of the covariance matrix, and the variances of health expectancies. Each line consists in indices "ij" followed by the initial scales -(zero to simplify) associated with aij and bij.
+(zero to simplify) associated with aij and bij.-- -# Scales (for hessian or gradient estimation) -12 0. 0. -13 0. 0. -21 0. 0. -23 0. 0.-
# Scales (for hessian or gradient estimation)
+
+12 0. 0.
+
+13 0. 0.
+
+21 0. 0.
+
+23 0. 0.
+
+This is an output if mle=1. But it can be -used as an input to get the various output data files (Health +
This +is an output if mle=1. But it can be used as +an input to get the various output data files (Health expectancies, stationary prevalence etc.) and figures without -rerunning the rather long maximisation phase (mle=0).
+rerunning the rather long 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)
-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)+
232 Cov(b23,a12) Cov(b23,b12) ... Var (b23)
+
+# Covariance matrix
+
+121 0.
+
+122 0. 0.
+
+131 0. 0. 0.
+
+132 0. 0. 0. 0.
+
+211 0. 0. 0. 0. 0.
+
+212 0. 0. 0. 0. 0. 0.
+
+231 0. 0. 0. 0. 0. 0. 0.
+
+232 0. 0. 0. 0. 0. 0. 0. 0.
+
+agemin=70 agemax=100 bage=50 fage=100
+
+Once
+we obtained the estimated parameters, the program is able to
+calculated stationary prevalence, transitions probabilities and
+life expectancies at any age. Choice of age range is useful for
+extrapolation. In our data file, ages varies from age 70 to 102.
+Setting bage=50 and fage=100, makes the program computing life
+expectancy from age bage to age fage. As we use a model, we can
+compute life expectancy on a wider age range than the age range
+from the data. But the model can be rather wrong on big
+intervals.
Similarly,
+it is possible to get extrapolated stationary prevalence by age
+ranging from agemin to agemax.
-- -# Covariance matrix -121 0. -122 0. 0. -131 0. 0. 0. -132 0. 0. 0. 0. -211 0. 0. 0. 0. 0. -212 0. 0. 0. 0. 0. 0. -231 0. 0. 0. 0. 0. 0. 0. -232 0. 0. 0. 0. 0. 0. 0. 0.-
begin-prev-date=1/1/1984 end-prev-date=1/6/1988
+
+Statements
+'begin-prev-date' and 'end-prev-date' allow to select the period
+in which we calculate 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.
pop_based=0
+
+The
+user has the possibility to choose between population-based or
+status-based health expectancies. If pop_based=0 then
+status-based health expectancies are computed and if pop_based=1,
+the programme computes population-based health expectancies.
+Health expectancies are weighted averages of health expectancies
+respective of the initial state. For a status-based index, the
+weights are the cross-sectional prevalences observed between two
+dates, as previously explained, whereas
+for a population-based index, the weights are the stationary
+prevalences.
starting-proj-date=1/1/1989 final-proj-date=1/1/1992 mov_average=0
+
+Prevalence
+and population projections are available only if the
+interpolation unit is a month, i.e. stepm=1. 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 of
+1989 in this example) to a final date (1 January 1992). The
+statement mov_average allows to compute smoothed forecasted
+prevalences with a five-age moving average centred at the mid-age
+of the five-age period.
agemin=70 agemax=100 bage=50 fage=100+
Once we obtained the estimated parameters, the program is able -to calculated stationary prevalence, transitions probabilities -and life expectancies at any age. Choice of age range is useful -for extrapolation. In our data file, ages varies from age 70 to -102. Setting bage=50 and fage=100, makes the program computing -life expectancy from age bage to age fage. As we use a model, we -can compute life expectancy on a wider age range than the age -range from the data. But the model can be rather wrong on big -intervals.
- -Similarly, it is possible to get extrapolated stationary -prevalence by age ranging from agemin to agemax.
- -popforecast=0 popfile=pyram.txt popfiledate=1/1/1989 last-popfiledate=1/1/1992
+
+This
+command is available if the interpolation unit is a month, i.e.
+stepm=1 and if popforecast=1. From a data file including age and
+number of persons alive at the precise date ‘popfiledate’,
+you can forecast the number of persons in each state until date
+‘last-popfiledate’. In this example, the popfile pyram.txt includes real
+data which are the Japanese population in 1989.
begin-prev-date=1/1/1984 end-prev-date=1/6/1988+
We
+assume that you entered your 1st_example
+parameter file as explained above. To
+run the program you should click on the imach.exe icon and enter
+the name of the parameter file which is for example C:\usr\imach\mle\biaspar.txt (you
+also can click on the biaspar.txt icon located in C:\usr\imach\mle and put it with the mouse on
+the imach window).
The
+time to converge depends on the step unit that you used (1 month
+is cpu consuming), on the number of cases, and on the number of
+variables.
The
+program outputs many files. Most of them are files which will be
+plotted for better understanding.
Statements 'begin-prev-date' and 'end-prev-date' allow to -select the period in which we calculate 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.
- -Once
+the optimization is finished, some graphics can be made with a
+grapher. We use Gnuplot which is an interactive plotting program
+copyrighted but freely distributed. A gnuplot reference manual is
+available here.
+When the running is finished, the user should enter a character
+for plotting and output editing.
These
+characters are:
The
+first line is the title and displays each field of the file. The
+first column is age. The fields 2 and 6 are the proportion of
+individuals in states 1 and 2 respectively as observed during the
+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
pop_based=0+
# Age Prev(1) N(1) N Age Prev(2) N(2) N
-The user has the possibility to choose between -population-based or status-based health expectancies. If -pop_based=0 then status-based health expectancies are computed -and if pop_based=1, the programme computes population-based -health expectancies. Health expectancies are weighted averages of -health expectancies respective of the initial state. For a -status-based index, the weights are the cross-sectional -prevalences observed between two dates, as previously -explained, whereas for a population-based index, the weights -are the stationary prevalences.
+70 1.00000 631 631 70 0.00000 0 631
-71 0.99681 625 627 71 0.00319 2 627
-starting-proj-date=1/1/1989 final-proj-date=1/1/1992 mov_average=0+
72 0.97125 1115 1148 72 0.02875 33 1148
-Prevalence and population projections are only available if -the interpolation unit is a month, i.e. stepm=1. 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 of -1989 in this example) to a final date (1 january 1992). The -statement mov_average allows to compute smoothed forecasted -prevalences with a five-age moving average centered at the -mid-age of the five-age period.
+It
+means that at age 70, 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.
This
+file contains all the maximisation results:
popforecast=0 popfile=pyram.txt popfiledate=1/1/1989 last-popfiledate=1/1/1992+
-2 log likelihood= 21660.918613445392
-This command is available if the interpolation unit is a -month, i.e. stepm=1 and if popforecast=1. From a data file
+ Estimated parameters: a12 = -12.290174 b12 = 0.092161
-Structure of the data file pyram.txt -: age numbers
+ a13 = -9.155590 b13 = 0.046627
-+
a21 = -2.629849 b21 = -0.022030
- a23 = -7.958519 b23 = 0.042614
- Covariance matrix: Var(a12) = 1.47453e-001
-We assume that you entered your 1st_example
-parameter file as explained above. To
-run the program you should click on the imach.exe icon and enter
-the name of the parameter file which is for example C:\usr\imach\mle\biaspar.txt
-(you also can click on the biaspar.txt icon located in
-C:\usr\imach\mle and put it with
-the mouse on the imach window).
-
The time to converge depends on the step unit that you used (1 -month is cpu consuming), on the number of cases, and on the -number of variables.
+ Var(b12) = 2.18676e-005
-The program outputs many files. Most of them are files which -will be plotted for better understanding.
+ Var(a13) = 2.09715e-001
- Var(b13) = 3.28937e-005
- Var(a21) = 9.19832e-001
-Once the optimization is finished, some graphics can be made
-with a grapher. We use Gnuplot which is an interactive plotting
-program copyrighted but freely distributed. A gnuplot reference
-manual is available here.
-When the running is finished, the user should enter a caracter
-for plotting and output editing.
These caracters are:
- - Var(b21) = 1.29229e-004
-The first line is the title and displays each field of the
-file. The first column is age. The fields 2 and 6 are the
-proportion of individuals in states 1 and 2 respectively as
-observed during the 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, 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.
-
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 - a23 = -7.958519 b23 = 0.042614 - Covariance matrix: Var(a12) = 1.47453e-001 - Var(b12) = 2.18676e-005 - Var(a13) = 2.09715e-001 - Var(b13) = 3.28937e-005 - Var(a21) = 9.19832e-001 - Var(b21) = 1.29229e-004 - Var(a23) = 4.48405e-001 - Var(b23) = 5.85631e-005 -- -
By substitution of these parameters in the regression model, -we obtain the elementary transition probabilities:
- -
Here are the transitions probabilities Pij(x, x+nh) where nh -is a multiple of 2 years. The first column is the starting age x -(from age 50 to 100), the second is age (x+nh) and the others are -the transition probabilities p11, p12, p13, p21, p22, p23. For -example, line 5 of the file is:
- -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- -
#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+
Var(a23) = 4.48405e-001+ +
Var(b23) = 5.85631e-005
+
+
+
+By
+substitution of these parameters in the regression model, we
+obtain the elementary transition probabilities:
Here
+are the transitions probabilities Pij(x, x+nh) where nh is a
+multiple of 2 years. The first column is the starting age x (from
+age 50 to 100), the second is age (x+nh) and the others are the
+transition probabilities p11, p12, p13, p21, p22, p23. For
+example, line 5 of the file is:
100 106 0.02655 0.17622 0.79722 0.01809 0.13678 0.84513
+
+and
+this means:
p11(100,106)=0.02655
-At age 70 the stationary prevalence is 0.90134 in state 1 and +
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
+
+#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 stationary prevalence is 0.90134 in state 1 and 0.09866 in state 2. This stationary prevalence differs from observed prevalence. Here is the point. The observed prevalence at age 70 results from the incidence of disability, incidence of @@ -718,40 +1507,52 @@ future if "nothing changes in the f exactly what demographers do with a Life table. Life expectancy is the expected mean time to survive if observed mortality rates (incidence of mortality) "remains constant" in the -future.
+future.The stationary prevalence has to be compared with the observed +
The +stationary prevalence has to be compared with the observed prevalence by age. But both are statistical estimates and subjected to stochastic errors due to the size of the sample, the design of the survey, and, for the stationary prevalence to the model used and fitted. It is possible to compute the standard -deviation of the stationary prevalence at each age.
- -This graph exhibits the stationary 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 stationary -prevalence. It suggests an increase of the disability prevalence -in the future.
- -
This graph plots the conditional transition probabilities from -an initial state (1=healthy in red at the bottom, or 2=disable in +
This
+graph exhibits the stationary 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 stationary prevalence.
+It suggests an increase of the disability prevalence in the
+future.
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 @@ -762,99 +1563,128 @@ prevalence at age 70 we should start the 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.
+probably longer. But we don't have experience yet.# Health expectancies
-# Health expectancies -# Age 1-1 1-2 2-1 2-2 -70 10.9226 3.0401 5.6488 6.2122 -71 10.4384 3.0461 5.2477 6.1599 -72 9.9667 3.0502 4.8663 6.1025 -73 9.5077 3.0524 4.5044 6.0401+
# Age 1-1 1-2 2-1 2-2
-For example 70 10.9226 3.0401 5.6488 6.2122 means: -e11=10.9226 e12=3.0401 e21=5.6488 e22=6.2122+
70 10.9226 3.0401 5.6488 6.2122
-+
71 10.4384 3.0461 5.2477 6.1599
-For example, life expectancy of a healthy individual at age 70 -is 10.92 in the healthy state and 3.04 in the disability state +
72 9.9667 3.0502 4.8663 6.1025
+
+73 9.5077 3.0524 4.5044 6.0401
+
+For example 70 10.9226 3.0401 5.6488 6.2122 means:
+
+e11=10.9226 e12=3.0401 e21=5.6488 e22=6.2122
+
++ +
For +example, life expectancy of a healthy individual at age 70 is +10.92 in the healthy state and 3.04 in the disability state (=13.96 years). If he was disable at age 70, his life expectancy will be shorter, 5.64 in the healthy state and 6.21 in the disability state (=11.85 years). The total life expectancy is a weighted mean of both, 13.96 and 11.85; weight is the proportion of people disabled at age 70. In order to get a pure period index -(i.e. based only on incidences) we use the computed or -stationary prevalence at age 70 (i.e. computed from -incidences at earlier ages) instead of the observed prevalence -(for example at first exam) (see -below).
- -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- -
#Total LEs with variances: e.. (std) e.1 (std) e.2 (std)- -
70 13.76 (0.22) 10.40 (0.20) 3.35 (0.14)- -
Thus, at age 70 the total life expectancy, e..=13.76years is
-the weighted mean of e1.=13.96 and e2.=11.85 by the stationary
+(i.e. based only on incidences) we use the computed or
+stationary prevalence at age 70 (i.e. computed from
+incidences at earlier ages) instead of the observed prevalence
+(for example at first exam) (see
+below).
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+ +
#Total LEs with variances: e.. (std) e.1 (std) e.2 (std)
+
+70 13.76 (0.22) 10.40 (0.20) 3.35 (0.14)
+
+Thus, +at age 70 the total life expectancy, e..=13.76years is the +weighted mean of e1.=13.96 and e2.=11.85 by the stationary prevalence 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=10.40 is the Disability-free life expectancy at age 70 (it is again a weighted mean of e11 and e21). e.2=3.35 is also the life expectancy at age -70 to be spent in the disability state.
+70 to be spent in the disability state.This figure represents the health expectancies and the total -life expectancy with the confident interval in dashed curve.
- -- -
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 estimate) but also according to the model fitted. Let us -explain it in more details.
- -Choosing a model means ar least two kind of choices. First we -have to decide the number of disability states. Second we have to -design, within the logit model family, the model: variables, -covariables, confonding 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 +
This
+figure represents the health expectancies and the total life
+expectancy with the confident interval in dashed curve.
+ +
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
+estimate) but also according to the model fitted. Let us explain
+it in more details.
Choosing
+a model means at least two kind of choices. First we have to
+decide the number of disability states. Second we have to design,
+within the logit model family, the model: 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 @@ -877,134 +1707,204 @@ population. Our main purpose is not to m 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.
+measurement.Incidences of disability or recovery are not affected by the -number of states if these states are independant. But incidences -estimates are dependant 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 program is not yet a statistical package, which permits -a simple writing of the variables and the model to take into -account in the maximisation. The actual 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 +of disability or recovery are not affected by the number of +states if these states are independant. But incidences estimates +are dependant 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 so far +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 program is not yet a statistical package, +which permits a simple writing of the variables and the model to +take into account in the maximisation. The actual 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.
+affected by the variables specified into the model.Also, the age range of the people interviewed has a link with -the age range of the life expectancy which can be estimated by +
Also, +the age range of the people interviewed has a link with 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 which is mostly based on your sample size, but if you want to estimate the life expectancy at age 50, you should rely in your model, but fitting a logistic model on a age -range of 70-95 and estimating probabilties of transition out of +range of 70-95 and estimating probabilities 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.
+probably not the case.This copy of the parameter file can be useful to re-run the -program while saving the old output files.
+This
+copy of the parameter file can be useful to re-run the program
+while saving the old output files.
On a d'abord estimé la date moyenne des interviaew. ie -13/9/1995. This file contains
+First,
+we have estimated the observed prevalence between 1/1/1984 and
+1/6/1988. The mean date of interview (weighed average of
+the interviews performed between1/1/1984 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.
Example, at date 1/1/1989 :
+Example,
+at date 1/1/1989 :
73 0.807 0.078 0.115
+# StartingAge FinalAge P.1 P.2 P.3
This means that at age 73, the probability for a person age 70 -at 13/9/1989 to be in state 1 is 0.807, in state 2 is 0.078 and -to die is 0.115 at 1/1/1989.
+# Forecasting at date 1/1/1989
73 0.807 0.078 0.115
# 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+
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.
# 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+
# Age P.1 P.2 P.3 [Population]
-# Forecasting at date 1/1/1989
-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.txt which is a copy of mypar.txt included in the -subdirectory of imach, mytry. -Edit it to change the name of the data file to ..\data\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.
+75 572685.22 83798.08
-Click on the imach.exe icon to open a window. Answer to the -question:'Enter the parameter file name:'
+74 621296.51 79767.99
-| IMACH, Version 0.7 Enter - the parameter file name: ..\mytry\imachpar.txt + | IMACH,
+ Version 0.7 Enter
+ the parameter file name: ..\mytry\imachpar.txt |
Most of the data files or image files generated, will use the +
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.
+the same as the mypar files initially included in the directory mytry.Output on the screen The output screen looks like this Log file -# +-· Output on the screen The output screen looks like this Log file-title=MLE datafile=..\data\mydata.txt lastobs=3000 firstpass=1 lastpass=3 -ftol=1.000000e-008 stepm=24 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0
Total number of individuals= 2965, Agemin = 70.00, Agemax= 100.92
+
-Warning, no any valid information for:126 line=126
-Warning, no any valid information for:2307 line=2307
-Delay (in months) between two waves Min=21 Max=51 Mean=24.495826
-These lines give some warnings on the data file and also some raw statistics on frequencies of transitions.
-Age 70 1.=230 loss[1]=3.5% 2.=16 loss[2]=12.5% 1.=222 prev[1]=94.1% 2.=14
- prev[2]=5.9% 1-1=8 11=200 12=7 13=15 2-1=2 21=6 22=7 23=1
-Age 102 1.=0 loss[1]=NaNQ% 2.=0 loss[2]=NaNQ% 1.=0 prev[1]=NaNQ% 2.=0
- #title=MLE datafile=..\data\mydata.txt lastobs=3000 firstpass=1 lastpass=3
+
+ftol=1.000000e-008 stepm=24 ncov=2 nlstate=2 ndeath=1 maxwav=4 mle=1 weight=0
+
+Total number of individuals= 2965, Agemin = 70.00, Agemax= 100.92
+
+
+
+Warning, no any valid information for:126 line=126
+
+Warning, no any valid information for:2307 line=2307
-+
Delay (in months) between two waves Min=21 Max=51 Mean=24.495826
-These lines give some warnings on the data file and also some raw statistics on frequencies of transitions.+ +
Age 70 1.=230 loss[1]=3.5% 2.=16 loss[2]=12.5% 1.=222 prev[1]=94.1% 2.=14
+
+ prev[2]=5.9% 1-1=8 11=200 12=7 13=15 2-1=2 21=6 22=7 23=1
+
+Age 102 1.=0 loss[1]=NaNQ% 2.=0 loss[2]=NaNQ% 1.=0 prev[1]=NaNQ% 2.=0
+
+Calculation of the hessian matrix. Wait... -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 - -Inverting the hessian to get the covariance matrix. Wait... - -#Hessian matrix# -3.344e+002 2.708e+004 -4.586e+001 -3.806e+003 -1.577e+000 -1.313e+002 3.914e-001 3.166e+001 -2.708e+004 2.204e+006 -3.805e+003 -3.174e+005 -1.303e+002 -1.091e+004 2.967e+001 2.399e+003 --4.586e+001 -3.805e+003 4.044e+002 3.197e+004 2.431e-002 1.995e+000 1.783e-001 1.486e+001 --3.806e+003 -3.174e+005 3.197e+004 2.541e+006 2.436e+000 2.051e+002 1.483e+001 1.244e+003 --1.577e+000 -1.303e+002 2.431e-002 2.436e+000 1.093e+002 8.979e+003 -3.402e+001 -2.843e+003 --1.313e+002 -1.091e+004 1.995e+000 2.051e+002 8.979e+003 7.420e+005 -2.842e+003 -2.388e+005 -3.914e-001 2.967e+001 1.783e-001 1.483e+001 -3.402e+001 -2.842e+003 1.494e+002 1.251e+004 -3.166e+001 2.399e+003 1.486e+001 1.244e+003 -2.843e+003 -2.388e+005 1.251e+004 1.053e+006 -# Scales -12 1.00000e-004 1.00000e-006 -13 1.00000e-004 1.00000e-006 -21 1.00000e-003 1.00000e-005 -23 1.00000e-004 1.00000e-005 -# Covariance - 1 5.90661e-001 - 2 -7.26732e-003 8.98810e-005 - 3 8.80177e-002 -1.12706e-003 5.15824e-001 - 4 -1.13082e-003 1.45267e-005 -6.50070e-003 8.23270e-005 - 5 9.31265e-003 -1.16106e-004 6.00210e-004 -8.04151e-006 1.75753e+000 - 6 -1.15664e-004 1.44850e-006 -7.79995e-006 1.04770e-007 -2.12929e-002 2.59422e-004 - 7 1.35103e-003 -1.75392e-005 -6.38237e-004 7.85424e-006 4.02601e-001 -4.86776e-003 1.32682e+000 - 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 -# agemin agemax for lifexpectancy, bage fage (if mle==0 ie no data nor Max likelihood). - - -agemin=70 agemax=100 bage=50 fage=100 -Computing prevalence limit: result on file 'plrmypar.txt' -Computing pij: result on file 'pijrmypar.txt' -Computing Health Expectancies: result on file 'ermypar.txt' -Computing Variance-covariance of DFLEs: file 'vrmypar.txt' -Computing Total LEs with variances: file 'trmypar.txt' -Computing Variance-covariance of Prevalence limit: file 'vplrmypar.txt' -End of Imach --
Once the running is finished, the program -requires a caracter:
+· Calculation of the hessian matrix. Wait...+ +
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
+
+
+
+Inverting the hessian to get the covariance matrix. Wait...
+
+-
| Type e to edit output files, c - to start again, and q for exiting: | +Type
+ e to edit output files, c to start again, and q for
+ exiting: |
First you should enter e to -edit the master file mypar.htm.
- -First
+you should enter e to edit the master file
+mypar.htm.
This software have been partly granted by Euro-REVES, a concerted +
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 (0.7 of February 2002) can be accessed at http://euroreves.ined.fr/imach
-
Latest
+version (0.7 of February 2002) can be accessed at http://euroreves.ined.fr/imach
