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lifetablefit

Calibrate life table from survival data with parametric models

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Syntax

[a,elx] = lifetablefit(x,lx)
[a,elx] = lifetablefit(___,lifemodel,objtype,interpmethod,a0)

Description

example

[a,elx] = lifetablefit(x,lx)calibrates a life table,x, from survival data,lx, using parametric models.

example

[a,elx] = lifetablefit(___,lifemodel,objtype,interpmethod,a0)calibrates a life table,x, from survival data,lx, using parametric models using optional arguments forlifemodel,objtype,interpmethod, anda0.

Examples

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Open Live Script

Load the life table data file.

loadus_lifetable_2009

Calibrate the life table from survival data using the defaultheligman-pollardparametric model.

[a,elx] = lifetablefit(x,lx); display(a)
a =8×30.0005 0.0006 0.0004 0.0592 0.0819 0.0192 0.1452 0.1626 0.1048 0.0007 0.0011 0.0007 6.2848 6.7635 1.1038 24.1386 24.2897 53.1772 0.0000 0.0000 0.0000 1.0971 1.0987 1.1100
display(elx(1:20,:))
1.0e+05 * 1.0000 1.0000 1.0000 0.9937 0.9931 0.9943 0.9932 0.9926 0.9939 0.9930 0.9923 0.9936 0.9928 0.9921 0.9935 0.9926 0.9920 0.9933 0.9925 0.9919 0.9932 0.9924 0.9918 0.9931 0.9923 0.9917 0.9930 0.9922 0.9916 0.9929 0.9921 0.9914 0.9928 0.9920 0.9913 0.9927 0.9919 0.9912 0.9926 0.9917 0.9910 0.9924 0.9915 0.9908 0.9923 0.9913 0.9905 0.9921 0.9910 0.9901 0.9919 0.9906 0.9896 0.9917 0.9901 0.9890 0.9914 0.9895 0.9882 0.9912

Plot theqxseries and display the legend. The seriesqxis the conditional probability that a person at agexwill die between agexand the next age in the series

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

Figure contains an axes object. The axes object contains 3 objects of type line. These objects represent All, Male, Female.

Input Arguments

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Increasing ages for raw data, specified as aNvector for nonnegative integers.

Data Types:double

Collection ofnumdiscrete survival counts, specified as anN-by-num矩阵。输入lxseries is the number of people alive at agex, given 100,000 alive at birth. Values of0orNaNin the input tablelxare ignored.

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 ) 1 − q ( x ) = A ( x + B ) C + D exp ( − E ( log x F ) 2 ) + G H X

    for agesx≥0, with parametersA,B,C,D,E,F,G,H≥0.

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

    q ( x ) 1 − q ( x ) = A ( x + B ) C + D exp ( − E ( log x F ) 2 ) + G H X 1 + G H X

    for agesx≥0, with parametersA,B,C,D,E,F,G,H≥0.

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

    q ( x ) = A ( x + B ) C + D exp ( − E ( log x F ) 2 ) + G H X

    for agesx≥0, with parametersA,B,C,D,E,F,G,H≥0.

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

    h(x) = A exp(Bx)
    for agesx≥0, with parametersA,B≥0.

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

    h(x) = A exp(Bx) + C
    for agesx≥0, with parametersA,B,C≥0.

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

    h(x) = A exp(Bx) + C + D exp(-Ex)
    for agesx≥0, with parametersA,B,C,D,E≥0.

Data Types:char

(Optional) Objective for nonlinear least-squares estimation, specified as a character vector with the following values:

  • 'ratio'— Given raw dataqxand model estimates q ^ x forx=1, ... ,N, the first objective (which is the preferred objective) has the form

    Φ = ∑ x = 1 N ( 1 − q ^ x q x ) 2

  • 'logratio'— Given raw dataqxand model estimates q ^ x forx=1, ... ,N, the second objective has the form

    Φ = ∑ x = 1 N ( log ( q ^ x ) − log ( q x ) ) 2

Data Types:char

(Optional) Interpolation method to use for abridged life table inputs, specified as a character vector with the following values:

  • 'cubic'— Cubic interpolation that uses“pchip”method ininterp1.

  • 'linear'— Linear interpolation.

  • 'none'— No interpolation.

Note

If the ages inx不是连续年和插值设置to'none', then the estimates for the parameters are suitable only for the age vectorx.

If you use the parameter estimates to compute life table values for arbitrary years, interpolate using the default'cubic'method.

Interpolation with abridged life tables forms internal interpolated full life tables, which usually improves model fits.

Data Types:char

(Optional) Initial parameter estimate to be applied to all series, specified as anumparamvector. This vector must conform to the number of parameters in the model specified using thelifemodelargument.

Data Types:double

Output Arguments

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Parameter estimates for eachnumseries, returned as anumparam-by-num矩阵。

Estimated collection ofnumstandardized survivor series, returned as anN-by-num矩阵。Theelxoutput series is the number of people alive at agex, given 100,000 alive at birth. Values of0orNaNin the input tablelxare ignored.

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 ActuariesVol. 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 ActuariesVol. 8, 1860, pp. 301–310.

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

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

Version History

Introduced in R2015a

See Also

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