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Convert life table series into life tables with forced termination

```
[qx,lx,dx]
= lifetableconv(x0,lx0)
```

```
[qx,lx,dx]
= lifetableconv(x0,y0,y0type)
```

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)

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
6.2840 6.7639 1.1040
24.1387 24.2894 53.1672
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,:))
```

630.9953 686.9417 571.6095 48.7927 55.1040 40.9872 24.8017 26.3779 23.6167 17.0832 17.5877 17.0317 13.6181 13.8187 13.6143 11.8661 12.0075 11.6314 10.9781 11.1572 10.4905 10.5995 10.8604 9.9488 10.5757 10.9395 9.8951 10.8789 11.3612 10.2717 11.6084 12.2508 11.0417 12.9922 13.9271 12.1763 15.3480 16.8834 13.6482 18.9940 21.6791 15.4301 24.1395 28.7663 17.4945 30.8009 38.3212 19.8136 38.7717 50.1487 22.3607 47.6536 63.6907 25.1104 56.9302 78.1270 28.0392 66.0576 92.5260 31.1265

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');

`x0`

— Increasing ages for raw datavector 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 seriesmatrix

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 convertedmatrix

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`

`qx`

— Discrete hazard functionmatrix

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 functionmatrix

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 functionmatrix

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.

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

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.

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 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.

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

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