-
-ij aij bij +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 @@ -397,7 +397,7 @@ Each of the four lines starts with indic 23 -6.234642 0.022315
or, to simplify:
+or, to simplify (in most of cases it converges but there is no warranty!):
: This file contains all the maximisation results: -12 0.0 0.0 @@ -409,7 +409,7 @@ Each of the four lines starts with indicGuess values for computing variances
This is an output if mle=1. But it can be -used as an input to get the vairous output data files (Health +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).
@@ -440,7 +440,7 @@ consists in indices "ij" folloCovariance matrix of parameters
This is an output if mle=1. But it can be -used as an input to get the vairous output data files (Health +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).
@@ -475,23 +475,24 @@ covariances between aij and bij: -last -uncommented line
+Age range for calculation of stationary +prevalences and health expectancies
agemin=70 agemax=100 bage=50 fage=100Once we obtained the estimated parameters, the program is able to calculated stationary prevalence, transitions probabilities -and life expectancies at any age. Choice of age ranges is useful +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.
+102. It is possible to get extrapolated stationary +prevalence by age ranging from agemin to agemax. + -Similarly, it is possible to get extrapolated stationary -prevalence by age raning from agemin to agemax.
+Setting bage=50 (begin age) and fage=100 (final age), makes the program computing +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!
+
- agemin= Minimum age for calculation of the @@ -500,16 +501,120 @@ prevalence by age raning from agemin to stationary prevalence
- bage= Minimum age for calculation of the health expectancies
-- fage= Maximum ages for calculation of the health +
- fage= Maximum age for calculation of the health expectancies
Computing the observed prevalence
+ +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.
+ ++
+ +- begin-prev-date= Starting date + (day/month/year)
+- end-prev-date= Final date + (day/month/year)
+Population- or status-based health +expectancies
+ +pop_based=0+ +The program computes status-based health expectancies, i.e health +expectancies which depends on your 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 independant of the initial status +we have to weight e11 and e21 according to the probability to be in +each state at initial age or, with other word, according to the +proportion of people in each state.
+ +We prefer computing a 'pure' period healthy life expectancy based only +on the transtion forces. Then the weights are simply the stationnary +prevalences or 'implied' prevalences at the initial age.
+ +Some other people would like to use the cross-sectional prevalences +(the "Sullivan prevalences") observed at the initial age during a +period of time defined just above. + ++
+ + + +- popbased= 0 Health expectancies are computed + at each age from stationary prevalences 'expected' at this initial age.
+- popbased= 1 Health expectancies are computed + at each age from cross-sectional 'observed' prevalence at this + initial age. As all the population is not observed at the same exact date we + define a short period were the observed prevalence is computed.
+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 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.
+ ++
+ +- starting-proj-date= starting date + (day/month/year) of forecasting
+- final-proj-date= final date + (day/month/year) of forecasting
+- mov_average= smoothing with a five-age + moving average centered at the mid-age of the five-age + period. The command mov_average takes + value 1 if the prevalences are smoothed and 0 otherwise.
+Last uncommented line : Population +forecasting
+ +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.
+ ++
+- popforecast= + 0 Option for population forecasting. If + popforecast=1, the programme does the forecasting.
+- popfile= + name of the population file
+- popfiledate= + date of the population population
+- last-popfiledate= + date of the last population projection
+
Running Imach with this example
-We assume that you entered your 1st_example +
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
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. Imach outputs the -source of a gnuplot file, named 'graph.gp', which can be directly -input into gnuplot.
@@ -543,16 +647,8 @@ for plotting and output editing.
+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.@@ -578,9 +674,6 @@ The header of the file is 71 0.99681 625 627 71 0.00319 2 627 72 0.97125 1115 1148 72 0.02875 33 1148 -
- 'c' to start again the program from the beginning.
-- 'g' to made graphics. The output graphs are in GIF format - and you have no control over which is produced. If you - want to modify the graphics or make another one, you - should modify the parameters in the file graph.gp - located in imach\bin. A gnuplot reference manual is - available here. -
-- 'e' opens the index.htm file to edit the - output files and graphs.
+- 'e' opens the biaspar.htm + file to edit the output files and graphs.
- 'q' for exiting.
# Age Prev(1) N(1) N Age Prev(2) N(2) N - 70 0.95721 604 631 70 0.04279 27 631-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 @@ -592,22 +685,26 @@ covariance matrix
Number of iterations=47 - -2 log likelihood=46553.005854373667 - Estimated parameters: a12 = -12.691743 b12 = 0.095819 - a13 = -7.815392 b13 = 0.031851 - a21 = -1.809895 b21 = -0.030470 - a23 = -7.838248 b23 = 0.039490 - Covariance matrix: Var(a12) = 1.03611e-001 - Var(b12) = 1.51173e-005 - Var(a13) = 1.08952e-001 - Var(b13) = 1.68520e-005 - Var(a21) = 4.82801e-001 - Var(b21) = 6.86392e-005 - Var(a23) = 2.27587e-001 - Var(b23) = 3.04465e-005 ++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 --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:
+ ++
- Transition probabilities: pijrbiaspar.txt
@@ -617,30 +714,33 @@ is a multiple of 2 years. The first colu the transition probabilities p11, p12, p13, p21, p22, p23. For example, line 5 of the file is: -100 106 0.03286 0.23512 0.73202 0.02330 0.19210 0.78460+100 106 0.02655 0.17622 0.79722 0.01809 0.13678 0.84513and this means:
-p11(100,106)=0.03286 -p12(100,106)=0.23512 -p13(100,106)=0.73202 -p21(100,106)=0.02330 -p22(100,106)=0.19210 -p22(100,106)=0.78460+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- Stationary prevalence in each state: plrbiaspar.txt
-#Age 1-1 2-2 -70 0.92274 0.07726 -71 0.91420 0.08580 -72 0.90481 0.09519 -73 0.89453 0.10547+#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 0.92274 in state 1 and -0.07726 in state 2. This stationary prevalence differs from +
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 recovery and mortality which occurred in the past of the cohort. @@ -664,24 +764,24 @@ design of the survey, and, for the stati model used and fitted. It is possible to compute the standard deviation of the stationary prevalence at each age.
-Observed and stationary +
+vbiaspar21.gif --Observed and stationary prevalence in state (2=disable) with the confident interval: -vbiaspar2.gif
- -
-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.- -
Convergence to the -stationary prevalence of disability: pbiaspar1.gif
+
-
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.
+ ++ +
-Convergence to the +stationary prevalence of disability: pbiaspar11.gif
+
This graph plots the conditional transition probabilities from an initial state (1=healthy in red at the bottom, or 2=disable in @@ -703,23 +803,23 @@ href="erbiaspar.txt">erbiaspar.txt# Health expectancies # Age 1-1 1-2 2-1 2-2 -70 10.7297 2.7809 6.3440 5.9813 -71 10.3078 2.8233 5.9295 5.9959 -72 9.8927 2.8643 5.5305 6.0033 -73 9.4848 2.9036 5.1474 6.0035
For example 70 10.7297 2.7809 6.3440 5.9813 means: -e11=10.7297 e12=2.7809 e21=6.3440 e22=5.9813+
For example 70 10.4227 3.0402 5.6488 5.7123 means: +e11=10.4227 e12=3.0402 e21=5.6488 e22=5.7123-
+
For example, life expectancy of a healthy individual at age 70 -is 10.73 in the healthy state and 2.78 in the disability state -(=13.51 years). If he was disable at age 70, his life expectancy -will be shorter, 6.34 in the healthy state and 5.98 in the -disability state (=12.32 years). The total life expectancy is a -weighted mean of both, 13.51 and 12.32; weight is the proportion +is 10.42 in the healthy state and 3.04 in the disability state +(=13.46 years). If he was disable at age 70, his life expectancy +will be shorter, 5.64 in the healthy state and 5.71 in the +disability state (=11.35 years). The total life expectancy is a +weighted mean of both, 13.46 and 11.35; 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 @@ -736,7 +836,7 @@ href="vrbiaspar.txt">vrbiaspar.txtFor example, the covariances of life expectancies Cov(ei,ej) at age 50 are (line 3)
-Cov(e1,e1)=0.4667 Cov(e1,e2)=0.0605=Cov(e2,e1) Cov(e2,e2)=0.0183+
Cov(e1,e1)=0.4776 Cov(e1,e2)=0.0488=Cov(e2,e1) Cov(e2,e2)=0.0424
- Health
@@ -746,24 +846,24 @@ href="trbiaspar.txt">Total life expectancy by
+-Total life expectancy by
age and health expectancies in states (1=healthy) and (2=disable):
-ebiaspar.gif
+
ebiaspar1.gif
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. @@ -826,9 +926,7 @@ estimated by month on 8,000 people may t 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 without covariations, like age+sex but -without age+sex+ age*sex . This can be done from the source code -(you have to change three lines in the source code) but will +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.
@@ -853,27 +951,73 @@ file: - Prevalence forecasting: +frbiaspar.txt + +
