lifetablefit

Calibrate life table from survival data with parametric models

Syntax

[a,elx] = lifetablefit(x,lx)

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

Description

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

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

Examples

Calibrate Life Table from Survival Data Using a Heligman-Pollard Parametric Model

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,elx] = lifetablefit(x,lx);
display(a)

a = 8×3

    0.0005    0.0006    0.0004
    0.0592    0.0819    0.0192
    0.1452    0.1626    0.1048
    0.0007    0.0011    0.0007
    6.2852    6.7639    1.1038
   24.1386   24.2896   53.1786
    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 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)

Input Arguments

x — Increasing ages for raw data

vector of nonnegative integers

Increasing ages for raw data, specified as a N vector for nonnegative integers.

Data Types: double

lx — Collection of num discrete survival counts

matrix

Collection of num discrete survival counts, specified as an N-by-num matrix. The input lx series is the number of people alive at age x, given 100,000 alive at birth. Values of 0 or NaN in the input table lx are ignored.

Data Types: double

lifemodel — Parametric mortality model type

'heligman-pollard' (default) | character vector with values 'heligman-pollard', 'heligman-pollard-2''heligman-pollard-3', 'gompertz', 'makeham', 'siler'

(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:

  $$\frac{q(x)}{1-q(x)}=A^{{(x+B)}^C}+D\exp (-E{(\log \frac{x}{F})}^2)+G{H}^X$$

  for ages x ≥ 0, with parameters A, 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:

  $$\frac{q(x)}{1-q(x)}=A^{{(x+B)}^C}+D\exp (-E{(\log \frac{x}{F})}^2)+\frac{G{H}^X}{1+G{H}^X}$$

  for ages x ≥ 0, with parameters A, 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:

  $$q(x)=A^{{(x+B)}^C}+D\exp (-E{(\log \frac{x}{F})}^2)+G{H}^X$$

  for ages x ≥ 0, with parameters A, 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 ages x ≥ 0, with parameters A, B ≥ 0.

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

  h(x) = A exp(Bx) + C

  for ages x ≥ 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 ages x ≥ 0, with parameters A, B, C, D, E ≥ 0.

Data Types: char

objtype — Objective for nonlinear least-squares estimation

'ratio' (default) | character vector with values 'ratio' 'logratio'

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

• 'ratio' — Given raw data q_x and model estimates $${\widehat{q}}_x$$ for x = 1, ... , N, the first objective (which is the preferred objective) has the form

  $$\Phi ={{\displaystyle \sum _{x=1}^N\left(1-\frac{{\widehat{q}}_x}{q_x}\right)}}^2$$

• 'logratio' — Given raw data q_x and model estimates $${\widehat{q}}_x$$ for x = 1, ... , N, the second objective has the form

  $$\Phi ={{\displaystyle \sum _{x=1}^N\left(\log ({\widehat{q}}_x)-\log (q_x)\right)}}^2$$

Data Types: char

interpmethod — Interpolation method to use for abridged life table inputs

'cubic' (default) | character vector with values 'cubic', 'linear', 'none'

(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 in interp1.

• 'linear' — Linear interpolation.

• 'none' — No interpolation.

Note

If the ages in x are not consecutive years and interpolation is set to 'none', then the estimates for the parameters are suitable only for the age vector x.

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

a0 — Initial parameter estimate to be applied to all series

vector

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

Data Types: double

Output Arguments

a — Parameter estimates for each num series

matrix

Parameter estimates for each num series, returned as a numparam-by-num matrix.

elx — Estimated collection of num standardized survivor series

matrix

Estimated collection of num standardized survivor series, returned as an N-by-num matrix. The elx output series is the number of people alive at age x, given 100,000 alive at birth. Values of 0 or NaN in the input table lx are 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 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.

Version History

Introduced in R2015a