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Mortality Rate Changing

Mortality rates can be expressed in different types, such as central death rates, 1-year death probabilities and force of mortality. Different mortality rate completion methods require different types of mortality rates so this module includes a helper function to change between the types of mortality rates.

This module uses the following notation for an individual aged \(x\) in year \(y\) for each simulation \(i\):

  • \(m_{x, y}^{(i)}\): central death rate
  • \(q_{x, y}^{(i)}\): 1-year death probability
  • \(\mu_{x, y}^{(i)}\): force of mortality

Implementation assumes a constant force of mortality for fractional ages, resulting in the following conversions:

  • \(q_{x, y}^{(i)} = 1 - e^{-\mu_{x, y}^{(i)}}\)
  • \(\mu_{x, y}^{(i)} = m_{x, y}^{(i)}\)
  • \(m_{x, y}^{(i)} = -\log(1 - q_{x, y}^{(i)})\)

Mortality Rate Changing

rate2rate(rates, from, to)

   Parameters:

     rates : matrix/array

       mortality rates with age rows, cohort/year columns

       (and simulation number 3rd dimension)

     from : character

       input type of mortality rates: 'central' for central death rates,

       'prob' for 1 year death probabilities, 'force' for force of mortality

     to : character

       desired output type of mortality rates: 'central' for central death rates,

       'prob' for 1 year death probabilities, 'force' for force of mortality

   Returns:

     matrix/array of converted mortality rates with age rows, cohort/year columns

     (and simulation number 3rd dimension)

   Usage:

# consider the male mortality rates from the data file 'mortality_AUS_data'

central_rates <- mortality_AUS_data$rate$male

# convert to 1-yr death probabilities

death_prob <- rate2rate(central_rates, from = "central", to = "prob")