Derive Life Tables that Match Life Expectancies, using a Brass Logit Model

Turn life expectancies at birth into full life tables, using the Brass logit model. The method is simple and is designed for simulations or for settings with little or no data on age-specific mortality rates. In settings where data on age-specific mortality is available, other methods might be more appropriate.

Usage

ex_to_lifetab_brass(
  target,
  standard,
  infant = c("constant", "linear", "CD", "AK"),
  child = c("constant", "linear", "CD"),
  closed = c("constant", "linear"),
  open = "constant",
  radix = 1e+05,
  suffix = NULL
)

Arguments

target: A data frame containing a variable called "ex", and possibly other varibles. See Details.
standard: A data frame containing variables called age and lx, and possibly others. See Details.
infant, child, closed, open: Methods used to calculate life expectancy. See lifetab() for details.
radix: Initial population for the lx column in the derived life table(s). Default is 100000.
suffix: Optional suffix added to life table columns.

Value

A tibble.

Method

The method implemented by ex_to_lifetab_brass() is based on the observation that, if populations A and B are demographically similar, then, in many cases,

$$\text{logit}(l_x^{\text{B}}) \approx \alpha + \beta \text{logit}(l_x^{\text{A}})$$

where $l_x$ is the "survivorship probability" quantity from a life table. When populations are similar, $beta$ is often close to 1.

Given (i) target life expectancy, (ii) a set of $l_x^{\text{A}}$), (referred to as a "standard"), and (iii) a value for $\beta$, ex_to_lifetab_brass() finds a value for $\alpha$ that yields a set of $l_x^{\text{B}}$) with the required life expectancy.

The `target` argument

target is a data frame specifying life expectancies for each population being modelled, and, optionally, inputs to the calculations, and 'by' variables.

target contains the following variables:

A variable called "ex" giving life expectancy at birth.
Optionally, a variable called "beta" with values for $\beta$. Can be an ordinary numeric vector or an rvec. If target does not include a "beta" variable, then ex_to_lifetab_brass() sets $\beta$ to 1.
A variable called "sex". The "sex" variable must be supplied if the infant argument to ex_to_lifetab_brass() is "CD" or "AK", or if the child argument is "CD".
Optionally, 'by' variables. Typical examples are time, region, and model variant.

The `standard` argument

standard is a data frame specifying the $l_x$ to be used with each life expectancy in target, and, optionally, values for the average age person-years lived by people who die in each group, $_na_x$. Values in standard are age-specific.

standard contains the following variables:

A variable called "age", with labels that can be parsed by reformat_age().
A variable called "lx". Cannot be an rvec.
Additional variables used to match rows in standard to rows in target.

References

Brass W, Coale AJ. 1968. “Methods of analysis and estimation,” in Brass, W, Coale AJ, Demeny P, Heisel DF, et al. (eds). The Demography of Tropical Africa. Princeton NJ: Princeton University Press, pp. 88–139.

Moultrie TA, Timæus IM. 2013. Introduction to Model Life Tables. In Moultrie T, Dorrington R, Hill A, Hill K, Timæus I, Zaba B. (eds). Tools for Demographic Estimation. Paris: International Union for the Scientific Study of Population. online version.

Examples

## create new life tables based on level-1
## 'West' model life tables, but with lower
## life expectancy

library(dplyr, warn.conflicts = FALSE)

target <- data.frame(sex = c("Female", "Male"), 
                     ex = c(17.5, 15.6))

standard <- west_lifetab |>
    filter(level == 1) |>
    select(sex, age, lx)
    
ex_to_lifetab_brass(target = target,
                    standard = standard,
                    infant = "CD",
                    child = "CD")
#> # A tibble: 42 × 7
#>    sex    age       qx      lx     dx      Lx    ex
#>    <chr>  <fct>  <dbl>   <dbl>  <dbl>   <dbl> <dbl>
#>  1 Female 0     0.412  100000  41204.  73218.  17.5
#>  2 Female 1-4   0.286   58796. 16790. 190877.  28.5
#>  3 Female 5-9   0.0793  42006.  3331. 201590.  35.4
#>  4 Female 10-14 0.0617  38675.  2387. 187346.  33.2
#>  5 Female 15-19 0.0793  36288.  2879. 174145.  30.2
#>  6 Female 20-24 0.0979  33409.  3270. 158729.  27.6
#>  7 Female 25-29 0.109   30139.  3273. 142356.  25.4
#>  8 Female 30-34 0.122   26866.  3264. 125994.  23.1
#>  9 Female 35-39 0.131   23602.  3103. 110070.  21.0
#> 10 Female 40-44 0.138   20499.  2838.  95223.  18.8
#> # ℹ 32 more rows

## target is an rvec
library(rvec, warn.conflicts = FALSE)
target_rvec <- data.frame(sex = c("Female", "Male"), 
                          ex = rnorm_rvec(n = 2,
                                          mean = c(17.5, 15.6),
                                          n_draw = 1000))
ex_to_lifetab_brass(target = target_rvec,
                    standard = standard)
#> # A tibble: 42 × 7
#>    sex    age                     qx                   lx                   dx
#>    <chr>  <fct>         <rdbl<1000>>         <rdbl<1000>>         <rdbl<1000>>
#>  1 Female 0        0.42 (0.38, 0.46) 1e+05 (1e+05, 1e+05) 41526 (37788, 45580)
#>  2 Female 1-4      0.29 (0.27, 0.31) 58474 (54420, 62212) 16777 (16660, 16791)
#>  3 Female 5-9    0.08 (0.075, 0.085) 41683 (37735, 45524)    3322 (3194, 3408)
#>  4 Female 10-14 0.062 (0.058, 0.066) 38360 (34541, 42117)    2379 (2266, 2462)
#>  5 Female 15-19  0.08 (0.075, 0.084) 35981 (32275, 39655)    2868 (2709, 2993)
#>  6 Female 20-24   0.098 (0.094, 0.1) 33114 (29567, 36662)    3254 (3044, 3430)
#>  7 Female 25-29     0.11 (0.1, 0.11) 29859 (26523, 33232)    3254 (3012, 3465)
#>  8 Female 30-34    0.12 (0.12, 0.13) 26606 (23510, 29767)    3243 (2971, 3490)
#>  9 Female 35-39    0.13 (0.13, 0.14) 23363 (20539, 26277)    3080 (2794, 3349)
#> 10 Female 40-44    0.14 (0.13, 0.14) 20283 (17746, 22928)    2815 (2530, 3091)
#> # ℹ 32 more rows
#> # ℹ 2 more variables: Lx <rdbl<1000>>, ex <rdbl<1000>>