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Create a Gompertz mortality with an age and a period slope

Source: R/mortality_gompertz.R
gompertz_mortality.Rd

Creates a mortality whose \(\log\mu\) is linear in age and in time:

$$\log\mu_{xt} = \texttt{log\_mu\_0} + \texttt{slope\_x}\,(x - \texttt{x\_0}) + \texttt{slope\_t}\,(t - \texttt{t\_0})$$

Both slopes are additive, so a period improvement is a negative slope_t. That is the usual case: mortality rises with age and falls with time.

This is the natural choice of test mortality for the \(Z\) scaling of a log-likelihood, where only slope_x materially matters, and it also serves as a reference mortality or as a simple model in its own right.

Every parameter has a default, so gompertz_mortality() on its own gives a broadly reasonable pensioner basis:

$$\log\mu_{xt} = -3.8 + 0.1\,(x - 75) - 0.01\,(t - 2020)$$

Usage

gompertz_mortality(
  log_mu_0 = -3.8,
  slope_x = 0.1,
  x_0 = 75,
  slope_t = -0.01,
  t_0 = 2020,
  name = NULL
)

Arguments

log_mu_0

The value of \(\log\mu\) at age x_0 and time t_0.

slope_x

The slope of \(\log\mu\) in age, per year. Positive for mortality that rises with age.

x_0

The age at which \(\log\mu\) takes the value log_mu_0, as a durationy or a number of years.

slope_t

The slope of \(\log\mu\) in time, per year. Typically negative, i.e. mortality improving.

t_0

The time at which \(\log\mu\) takes the value log_mu_0, as a datey or a year.

name

An optional name for this mortality, read back by name() and printed wherever the mortality appears.

Value

A mortality.

Details

Internally the law is held in its cohort form, which is the shape that integrates cheaply. Writing \(b\) for the date of birth and \(b_0 = \texttt{t\_0} - \texttt{x\_0}\) for the cohort origin the two offsets imply,

$$\log\mu_{bt} = \texttt{log\_mu\_0} - \texttt{slope\_x}\,(b - b_0) + (\texttt{slope\_x} + \texttt{slope\_t})\,(t - \texttt{t\_0})$$

which is identical to the age-period form above because \(x = t - b\). Birth is fixed for an individual, so the first two terms are a per-individual constant that lifts out of the integral, leaving a single term linear in \(t\) whose slope every individual shares.

Examples

# The default basis.
log_mu(gompertz_mortality(), list(birth = datey::datey(1945)), datey::datey(2020))
#> [1] -3.8

m <- gompertz_mortality(
  log_mu_0 = -4.6, slope_x = 0.10, x_0 = 70,
  slope_t = -0.02, t_0 = 2020
)
# log mu at age 70 in 2020, for someone born in 1950:
log_mu(m, list(birth = datey::datey(1950)), datey::datey(2020))
#> [1] -4.6