Period Life Table from Age-Specific Mortality Rates

Builds a standard period life table from age-specific mortality rates (nMx). Works with both abridged tables (age groups 0, 1, 5, 10, ..., 85+) and complete single-year tables (0, 1, 2, ..., 100+); the last age group is always treated as open-ended.

Usage

life_table(
  age,
  mx,
  sex = c("total", "male", "female"),
  ax = NULL,
  radix = 1e+05
)

Arguments

age: Numeric vector of age-group lower bounds, strictly increasing, e.g. c(0, 1, seq(5, 85, by = 5)) for a standard abridged table. The last group is open-ended.
mx: Numeric vector of age-specific mortality rates (deaths per person-year), the same length as age. Must be non-negative, with no missing values.
sex: Character. "total" (default), "male", or "female". Only used for the default infant and child ax (see Details).
ax: Optional numeric vector overriding the default person-years assumptions, the same length as age. NA elements fall back to the defaults.
radix: Numeric. The starting cohort size l0. Default 100000; use 1 for survivorship proportions.

Value

A tibble with one row per age group and columns:

age: Age-group lower bound (as supplied).
n: Width of the age interval; Inf for the open interval.
mx: Age-specific mortality rate (as supplied).
ax: Average person-years lived in the interval by those dying in it.
qx: Probability of dying in the interval; 1 in the open interval.
lx: Survivors at exact age x out of radix.
dx: Deaths in the interval.
Lx: Person-years lived in the interval.
Tx: Person-years lived above exact age x.
ex: Remaining life expectancy at exact age x. NA where lx has reached 0.

Details

The conversion from the mortality rate mx to the probability of dying qx uses the standard relation $${}_nq_x = \frac{n \, {}_nm_x}{1 + (n - {}_na_x) \, {}_nm_x},$$ where ${}_na_x$ is the average number of person-years lived in the interval by those dying in it. Values of qx are capped at 1 (with a warning, since capping signals implausibly high rates). In the open interval, qx = 1 and Lx = lx / mx.

The ax assumption. By default:

age 0 (when the first group is age 0 with width 1): the Coale-Demeny West formulas keyed on m0 (Preston, Heuveline and Guillot 2001, Table 3.3), by sex. For sex = "total", the male and female values are averaged.
ages 1-4 (when the second group is ages 1-4): the corresponding Coale-Demeny West formula.
all other closed intervals: n / 2 (the midpoint assumption).
open interval: 1 / mx (the life expectancy implied by a constant rate).

Pass your own ax vector to override all of this, e.g. to match a published table exactly.

Life expectancy at any tabulated age is read off the ex column: ex[1] is life expectancy at birth when the table starts at age 0. The table may also start above age 0 (e.g. age = c(60, 65, ..., 85)) to compute remaining life expectancy conditional on survival to the first age.

References

Preston SH, Heuveline P, Guillot M (2001). Demography: Measuring and Modeling Population Processes. Blackwell, Oxford. Chapter 3.

Coale AJ, Demeny P, Vaughan B (1983). Regional Model Life Tables and Stable Populations. 2nd ed. Academic Press, New York.

Examples

# Abridged life table for a typical middle-income mortality schedule
age <- c(0, 1, seq(5, 85, by = 5))
mx  <- c(0.0200, 0.0010, 0.0004, 0.0003, 0.0005, 0.0007, 0.0009,
         0.0012, 0.0016, 0.0022, 0.0032, 0.0048, 0.0075, 0.0120,
         0.0190, 0.0310, 0.0520, 0.0860, 0.1500)
lt <- life_table(age, mx)
lt
#> # A tibble: 19 × 10
#>      age     n     mx    ax      qx      lx     dx      Lx       Tx    ex
#>    <dbl> <dbl>  <dbl> <dbl>   <dbl>   <dbl>  <dbl>   <dbl>    <dbl> <dbl>
#>  1     0     1 0.02   0.104 0.0196  100000   1965.  98239. 7563915. 75.6 
#>  2     1     4 0.001  1.54  0.00399  98035.   391. 391180. 7465676. 76.2 
#>  3     5     5 0.0004 2.5   0.00200  97644.   195. 487732. 7074496. 72.5 
#>  4    10     5 0.0003 2.5   0.00150  97449.   146. 486880. 6586764. 67.6 
#>  5    15     5 0.0005 2.5   0.00250  97303.   243. 485907. 6099884. 62.7 
#>  6    20     5 0.0007 2.5   0.00349  97060.   339. 484452. 5613977. 57.8 
#>  7    25     5 0.0009 2.5   0.00449  96721.   434. 482518. 5129525. 53.0 
#>  8    30     5 0.0012 2.5   0.00598  96287.   576. 479993. 4647007. 48.3 
#>  9    35     5 0.0016 2.5   0.00797  95711.   763. 476646. 4167014. 43.5 
#> 10    40     5 0.0022 2.5   0.0109   94948.  1039. 472143. 3690368. 38.9 
#> 11    45     5 0.0032 2.5   0.0159   93909.  1491. 465819. 3218225. 34.3 
#> 12    50     5 0.0048 2.5   0.0237   92419.  2192. 456614. 2752406. 29.8 
#> 13    55     5 0.0075 2.5   0.0368   90227.  3321. 442831. 2295792. 25.4 
#> 14    60     5 0.012  2.5   0.0583   86906.  5062. 421872. 1852961. 21.3 
#> 15    65     5 0.019  2.5   0.0907   81843.  7423. 390659. 1431089. 17.5 
#> 16    70     5 0.031  2.5   0.144    74421. 10706. 345339. 1040430. 14.0 
#> 17    75     5 0.052  2.5   0.230    63715. 14660. 281925.  695091. 10.9 
#> 18    80     5 0.086  2.5   0.354    49055. 17361. 201872.  413165.  8.42
#> 19    85   Inf 0.15   6.67  1        31694. 31694. 211293.  211293.  6.67

# Life expectancy at birth and at age 60
lt$ex[1]
#> [1] 75.63915
lt$ex[lt$age == 60]
#> [1] 21.32154

# Sex-specific infant ax (affects e0 slightly)
life_table(age, mx, sex = "female")$ex[1]
#> [1] 75.63891

# Survivorship proportions instead of a 100,000 radix
life_table(age, mx, radix = 1)$lx
#>  [1] 1.0000000 0.9803522 0.9764404 0.9744894 0.9730288 0.9705993 0.9672081
#>  [8] 0.9628654 0.9571055 0.9494792 0.9390920 0.9241858 0.9022684 0.8690560
#> [15] 0.8184314 0.7442061 0.6371509 0.4905498 0.3169396