# lifetableconv

Convert life table series into life tables with forced termination

## Description

## Examples

### Convert Life Table Series into Life Tables with Forced Termination

Load the life table data file.

`load us_lifetable_2009`

Convert life table series into life tables with forced termination.

[qx,lx,dx] = lifetableconv(x,lx); display(qx(1:20,:))

0.0064 0.0070 0.0057 0.0004 0.0004 0.0004 0.0003 0.0003 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0001 0.0001 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 0.0002 0.0002 0.0002 0.0003 0.0004 0.0002 0.0004 0.0005 0.0002 0.0005 0.0006 0.0003 0.0005 0.0007 0.0003 0.0006 0.0009 0.0004 0.0007 0.0010 0.0004

display(lx(1:20,:))

1.0e+05 * 1.0000 1.0000 1.0000 0.9936 0.9930 0.9943 0.9932 0.9926 0.9939 0.9930 0.9923 0.9937 0.9927 0.9920 0.9935 0.9926 0.9919 0.9933 0.9924 0.9917 0.9932 0.9923 0.9916 0.9931 0.9922 0.9914 0.9930 0.9921 0.9913 0.9929 0.9920 0.9912 0.9928 0.9919 0.9911 0.9927 0.9918 0.9910 0.9926 0.9917 0.9909 0.9925 0.9915 0.9907 0.9923 0.9912 0.9903 0.9921 0.9908 0.9898 0.9919 0.9904 0.9892 0.9916 0.9899 0.9885 0.9913 0.9892 0.9876 0.9909

display(dx(1:20,:))

637.2266 698.8750 572.6328 40.4062 43.9297 36.7188 27.1875 30.0938 24.1406 20.7656 23.0781 18.3359 15.9141 17.2109 14.5625 14.8672 16.3125 13.3516 13.3672 14.7891 11.8750 12.1328 13.3828 10.8203 10.8125 11.6094 9.9844 9.4609 9.5781 9.3438 8.6172 8.1328 9.1172 9.2656 8.8359 9.7188 12.5938 13.5078 11.6328 19.1016 22.9844 15.0234 27.6719 35.5938 19.3516 36.6328 48.5703 24.0547 45.0156 60.7109 28.4844 53.1406 72.8906 32.2812 60.8984 85.1172 35.2578 68.3438 97.2266 37.6875

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)

### Convert the Life Table dx Series After Fitting and Generating the Life Table Series

Load the life table data file.

`load us_lifetable_2009`

Calibrate life table from survival data with the default Heligman-Pollard parametric model.

a = lifetablefit(x,lx)

`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
6.2843 6.7637 1.1038
24.1387 24.2895 53.1783
0.0000 0.0000 0.0000
1.0971 1.0987 1.1100

Generate life table series from the calibrated mortality model.

qx = lifetablegen((0:120),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

Convert life table series into life tables with forced termination.

```
[~,~,dx] = lifetableconv((0:120),qx,'qx');
display(dx(1:20,:))
```

630.9924 686.9527 571.6098 48.7928 55.1029 40.9870 24.8018 26.3778 23.6167 17.0832 17.5878 17.0317 13.6181 13.8188 13.6143 11.8662 12.0076 11.6314 10.9782 11.1573 10.4905 10.5996 10.8604 9.9488 10.5757 10.9396 9.8952 10.8789 11.3612 10.2718 11.6084 12.2508 11.0418 12.9921 13.9271 12.1763 15.3478 16.8835 13.6482 18.9936 21.6793 15.4300 24.1390 28.7665 17.4943 30.8004 38.3215 19.8134 38.7713 50.1489 22.3603 47.6532 63.6909 25.1099 56.9300 78.1270 28.0386 66.0576 92.5258 31.1259

Plot the `dx`

series and display the legend. The series `dx`

is the number of people who die out of 100,000 alive at birth between age *x* and the next age in the series.

plot((0:119),dx(1:end-1,:)); legend(series, 'location', 'northwest'); title('\bfLife Table Yearly Decrements'); xlabel('Age'); ylabel('Number Dying within a Given Year');

## Input Arguments

`x0`

— Increasing ages for raw data

vector of nonnegative integer values

Increasing ages for raw data, specified as nonnegative integer values in
an `N0`

vector.

The vector of ages *x* must contain nonnegative integer
values. If the input series is the discrete survival function
`lx`

, then the starting age need only be nonnegative.
Otherwise, the starting age must be `0`

.

**Data Types: **`double`

`lx0`

— Collection of `num`

standardized survivor series

matrix

Collection of `num`

standardized survivor series,
specified as an `N0`

-by-`num`

matrix. The
input `lx0`

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
`lx0`

are ignored.

**Data Types: **`double`

`y0`

— Collection of `num`

life table series to be converted

matrix

Collection of `num`

life table series to be converted,
specified as an `N0`

-by-`num`

matrix. The
default `y0`

series is `lx0`

.

**Data Types: **`double`

`y0type`

— Type of mortality series for input `y0`

with default `'lx'`

`'lx'`

(default) | character vector with values `'qx'`

,
`'lx'`

, `'dx'`

(Optional) Type of mortality series for input `y0`

,
specified as a character vector with one of the following values:

`'qx'`

— Input is a table of discrete hazards (`qx`

).`'lx'`

— Input is a table of discrete survival counts (`lx`

).`'dx'`

— Input is a table of discrete decrements (`dx`

).

Whereas the output series have forced termination, the input
series (`y0`

) can have one of several types of termination:

Natural termination runs out to the last person so that

`lx`

goes to`0`

,`qx`

goes to`1`

, and`dx`

goes to`0`

. For more information, see Natural Termination.Truncated termination stops at a terminal age so that

`lx`

is positive,`qx`

is less than`1`

, and`dx`

is positive. Any ages after the terminal age are`NaN`

values. For more information, see Truncated Termination.

**Data Types: **`char`

## Output Arguments

`qx`

— Discrete hazard function

matrix

Discrete hazard function, returned as an
`N0`

-by-`num`

matrix with forced
termination. For more information, see Forced Termination.

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.

`lx`

— Discrete survival function

matrix

Discrete survival function, returned as an
`N0`

-by-`num`

matrix with forced
termination. For more information, see Forced Termination.

The series `lx`

is the number of people alive at age
*x*, given 100,000 alive at birth.

`dx`

— Discrete decrements function

matrix

Discrete decrements function, returned as an
`N0`

-by-`num`

matrix with forced
termination. For more information, see Forced Termination.

The series `dx`

is the number of people who die out of
100,000 alive at birth, between age *x* and the next age in
the series.

## More About

### 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, *q _{x}*
probabilities are

_{1}

*q*

_{x}for ages less than 100 and, technically,

_{∞}

*q*

_{x}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.

### Natural Termination

Before 1970, life tables were often published with data that included all ages for which persons associated with a given series were still alive. Tables in this form have "natural" termination. In natural termination, the last row of the life table for each series counts the deaths or probabilities of deaths of the last remaining person at the corresponding age. Tables in this form can be problematic due to the granularity of the data and the fact that groups of series can terminate at distinct ages. Natural termination is illustrated in the following sample of the last few years of a life table.

This form for life tables poses a number of issues that go beyond the obvious
statistical issues. First, the
*l*_{x} table on the
left terminates at ages 108, 109, 109, and 113 for the four series in the table.
Technically, the numbers after these ages are `0`

, but can also be
`NaN`

values because no person is alive after these terminating
ages. Second, the probabilities
*q*_{x} on the right
terminate with fluctuating probabilities that go from `0`

to
`1`

in some cases. In this case, however, all probabilities are
_{1}*q*_{x}
probabilities (unlike the forced termination probabilities). You can argue that the
probabilities after the ages of termination can be `1`

(anyone
alive at this age is expected to die in the next year), `0`

(the
age lies outside the support of the probability distribution), or
`NaN`

values.

### Truncated Termination

Truncated termination occurs with truncation of life tables at an
arbitrary age. For example, from 1970–1990, United States life tables truncated at
age 85. This format is problematic because life table probabilities must either
terminate with probability `1`

(forced termination) or discard data
that exceeds the terminating age. This sample of the last few years of a life table
illustrates truncated termination. The raw data for this table is the
*l*_{x} series. The
*q*_{x} series is
derived from this series.

This life table format poses problems for termination because, for example, over
27% of the population for the fourth
*l*_{x} series is
still alive at age 85. To claim that the probability of dying for all ages after age
85 is 100% might be true but is uninformative. Notwithstanding the statistical
issues, however, tables in this form are completed by forced termination.

## 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.

## Version History

**Introduced in R2015a**

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