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lifetablegen

Generate life table series from calibrated mortality model

Syntax

[qx,lx,dx] = lifetablegen(x,a)
[qx,lx,dx] = lifetablegen(x,a,lifemodel)

Description

example

[qx,lx,dx] = lifetablegen(x,a) generates a life table series from a calibrated mortality model.

example

[qx,lx,dx] = lifetablegen(x,a,lifemodel) generates a life table series from a calibrated mortality model using the optional argument for lifemodel.

Examples

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Load the life table data file.

load us_lifetable_2009

Calibrate the life table from survival data using the default heligman-pollard parametric model.

a = lifetablefit(x, lx)
a = 

    0.0005    0.0006    0.0004
    0.0592    0.0819    0.0192
    0.1452    0.1626    0.1048
    0.0007    0.0011    0.0007
    6.2843    6.7634    1.1038
   24.1387   24.2898   53.1782
    0.0000    0.0000    0.0000
    1.0971    1.0987    1.1100

Generate a life table series from the calibrated mortality model.

qx = lifetablegen(x,a);
display(qx(1:20,:))
    0.0063    0.0069    0.0057
    0.0005    0.0006    0.0004
    0.0002    0.0003    0.0002
    0.0002    0.0002    0.0002
    0.0001    0.0001    0.0001
    0.0001    0.0001    0.0001
    0.0001    0.0001    0.0001
    0.0001    0.0001    0.0001
    0.0001    0.0001    0.0001
    0.0001    0.0001    0.0001
    0.0001    0.0001    0.0001
    0.0001    0.0001    0.0001
    0.0002    0.0002    0.0001
    0.0002    0.0002    0.0002
    0.0002    0.0003    0.0002
    0.0003    0.0004    0.0002
    0.0004    0.0005    0.0002
    0.0005    0.0006    0.0003
    0.0006    0.0008    0.0003
    0.0007    0.0009    0.0003

Plot the qx series and display the legend. The series qx is the conditional probability that a person at age x will die between age x and the next age in the series.

plot(x,log(qx))
legend(series)

Load the life table data file.

load us_lifetable_2009

Convert the life table series into life tables with forced termination.

[~, lx] = lifetableconv(x, qx, 'qx');

Calibrate the life table from survival data using the default heligman-pollard parametric model.

a = lifetablefit(x, lx)
a = 

    0.0005    0.0006    0.0004
    0.0592    0.0819    0.0192
    0.1452    0.1626    0.1048
    0.0007    0.0011    0.0007
    6.2855    6.7639    1.1034
   24.1383   24.2895   53.1966
    0.0000    0.0000    0.0000
    1.0971    1.0987    1.1100

Generate a life table series from the calibrated mortality model.

qx = lifetablegen((0:100), a)
qx = 

    0.0063    0.0069    0.0057
    0.0005    0.0006    0.0004
    0.0002    0.0003    0.0002
    0.0002    0.0002    0.0002
    0.0001    0.0001    0.0001
    0.0001    0.0001    0.0001
    0.0001    0.0001    0.0001
    0.0001    0.0001    0.0001
    0.0001    0.0001    0.0001
    0.0001    0.0001    0.0001

Plot the qx series and display the legend. The series qx is the conditional probability that a person at age x will die between age x and the next age in the series.

plot((0:100), log(qx));
legend(series, 'location', 'southeast');
title('Conditional Probability of Dying within One Year of Current Age');
xlabel('Age');
ylabel('Log Probability');

Input Arguments

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Increasing ages for raw data, specified as a N vector of nonnegative integer values. The ages must start at 0 (birth).

Data Types: double

Model parameters for num models, specified as a numparam-by-num matrix, where the number of parameters (numparam) depends on the model specified using the lifemodel argument.

Data Types: double

(Optional) Parametric mortality model type, specified as a character vector with one of the following values:

  • 'heligman-pollard' — Eight-parameter Heligman-Pollard model (version 1), specified in terms of the discrete hazard function:

    q(x)1q(x)=A(x+B)C+Dexp(E(logxF)2)+GHX

    for ages x0, with parameters A, B, C, D, E, F, G, H0.

  • 'heligman-pollard-2' — Eight-parameter Heligman-Pollard model (version 2), specified in terms of the discrete hazard function:

    q(x)1q(x)=A(x+B)C+Dexp(E(logxF)2)+GHX1+GHX

    for ages x0, with parameters A, B, C, D, E, F, G, H0.

  • 'heligman-pollard-3' — Eight-parameter Heligman-Pollard model (version 3), specified in terms of the discrete hazard function:

    q(x)=A(x+B)C+Dexp(E(logxF)2)+GHX

    for ages x0, with parameters A, B, C, D, E, F, G, H0.

  • 'gompertz' — Two-parameter Gompertz model, specified in terms of the continuous hazard function:

    h(x) = A exp(Bx)
    for ages x0, with parameters A, B0.

  • 'makeham' — Three-parameter Gompertz-Makeham model, specified in terms of the continuous hazard function:

    h(x) = A exp(Bx) + C
    for ages x0, with parametersA, B, C0.

  • 'siler' — Five-parameter Siler model, specified in terms of the continuous hazard function:

    h(x) = A exp(Bx) + C + D exp(-Ex)
    for ages x0, with parameters A, B, C, D, E0.

Data Types: char

Output Arguments

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Conditional probabilities of dying for N ages and num series, returned as an N-by-num matrix. The series qx is the conditional probability that a person at age x will die between age x and the next age in the series. For the last age, qx represents probabilities or counts for all ages after the last age.

The last row of the N-by-num output for qx is the values for all ages on or after the last age in x (due to Forced Termination). Therefore, the last row of qx contains 1 (100% probability of dying on or after the last age).

Survival counts for N ages and num series, returned as an N-by-num matrix. The series lx is the number of people alive at age x, given 100,000 alive at birth.

Decrement counts for N ages and num series, returned as an N-by-num matrix. The series dx is the number of people out of 100,000 alive at birth who die between age x and the next age in the series. For the last age, dx represent probabilities or counts for all ages after the last age.

The last row of the N-by-num output for dx are values for all ages on or after the last age in x (due to Forced Termination). Therefore, the last row of dx contains the remaining count of 100,000 people alive at birth who have not died by the last age.

More About

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Forced Termination

Most modern life tables have “forced” termination. Forced termination means that the last row of the life table applies for all persons with ages on or after the last age in the life table.

This sample illustrates forced termination.

In this case, the last row of the life table applies for all persons aged 100 or older. Specifically, qx probabilities are 1qx for ages less than 100 and, technically, qx for age 100.

Forced termination has terminal age values that apply to all ages after the terminal age so that lx is positive, qx is 1, and dx is positive. Ages after the terminal age are NaN values, although lx and dx can be 0 and qx can be 1 for input series. Forced termination is triggered by a naturally terminating series, the last age in a truncated series, or the first NaN value in a series.

References

[1] Arias, E. “United States Life Tables.” National Vital Statistics Reports, U.S. Department of Health and Human Services. Vol. 62, No. 7, 2009.

[2] Carriere, F. “Parametric Models for Life Tables.” Transactions of the Society of Actuaries. Vol. 44, 1992, pp. 77–99.

[3] Gompertz, B. “On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies.” Philosophical Transactions of the Royal Society. Vol. 115, 1825, pp. 513–582.

[4] Heligman, L. M. .A., and J. H. Pollard. “The Age Pattern of Mortality.” Journal of the Institute of Actuaries Vol. 107, Pt. 1, 1980, pp. 49–80.

[5] Makeham, W .M. “On the Law of Mortality and the Construction of Annuity Tables.” Journal of the Institute of Actuaries Vol. 8, 1860 . pp. 301–310.

[6] Siler, W. “A Competing-Risk Model for Animal Mortality.” Ecology Vol. 60, pp. 750–757, 1979.

[7] Siler, W. “Parameters of Mortality in Human Populations with Widely Varying Life Spans.” Statistics in Medicine Vol. 2, 1983, pp. 373–380.

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