Package 'MortalityLaws' reference manual

Title:	Parametric Mortality Models, Life Tables and HMD
Description:	Fit the most popular human mortality 'laws', and construct full and abridge life tables given various input indices. A mortality law is a parametric function that describes the dying-out process of individuals in a population during a significant portion of their life spans. For a comprehensive review of the most important mortality laws see Tabeau (2001) <doi:10.1007/0-306-47562-6_1>. Practical functions for downloading data from various human mortality databases are provided as well.
Authors:	Marius D. Pascariu [aut, cre, cph] , Vladimir Canudas-Romo [ctb]
Maintainer:	Marius D. Pascariu <[email protected]>
License:	MIT + file LICENSE
Version:	2.1.0
Built:	2024-11-10 03:14:38 UTC
Source:	https://github.com/mpascariu/mortalitylaws

MortalityLaws Test Data

Description

Dataset containing altered death rates (mx), death counts (Dx) and exposures (Ex) for the female population living in England & Wales in four different years: 1850, 1900, 1950 and 2010. The data-set is provided for testing purposes only. Download the actual data free of charge from https://www.mortality.org. Once a username and a password are created on the website the function ReadHMD can be used for downloading.

Usage

ahmd
ahmd

Format

An object of class list of length 3.

Source

Human Mortality Database

Examples

head(ahmd$mx)
head(ahmd$mx)

Check Data Availability in HMD

Description

The function returns information about available data in the Human Mortality Database, HMD (period life tables etc.), with the range of years covered by the life tables.

Usage

availableHMD(link = "https://www.mortality.org/Data/DataAvailability")
availableHMD(link = "https://www.mortality.org/Data/DataAvailability")

Arguments

link

URL to the HMD available data. Default: "https://www.mortality.org/Data/DataAvailability"

Value

A tibble.

Author(s)

Marius D. Pascariu

Examples

availableHMD()

availableHMD()

Check Available Mortality Laws

Description

The function returns information about the parametric models that can be called and fitted in the MortalityLaw function. For a comprehensive review of the most important mortality laws, Tabeau (2001) is a good starting point.

Usage

availableLaws(law = NULL)
availableLaws(law = NULL)

Arguments

law

Optional. Default: NULL. One can extract details about a certain model by specifying its codename.

Value

The output is of the "availableLaws" class with the components:

`table`	Table with mortality models and codes to be used in `MortalityLaw`
`legend`	Table with details about the section of the mortality curve

Author(s)

Marius D. Pascariu

References

Gompertz, B. (1825). On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies. Philosophical Transactions of the Royal Society of London, 115, 513-583.
Makeham, W. (1860). On the Law of Mortality and Construction of Annuity Tables. The Assurance Magazine and Journal of the Institute of Actuaries, 8(6), 301-310. doi:10.1017/S204616580000126X
Thiele, T. (1871). On a Mathematical Formula to express the Rate of Mortality throughout the whole of Life, tested by a Series of Observations made use of by the Danish Life Insurance Company of 1871. Journal of the Institute of Actuaries and Assurance Magazine, 16(5), 313-329. doi:10.1017/S2046167400043688
Oppermann, L. H. F. (1870). On the graduation of life tables, with special application to the rate of mortality in infancy and childhood. The Insurance Record Minutes from a meeting in the Institute of Actuaries, 42.
Wittstein, T. and D. Bumsted. (1883). The Mathematical Law of Mortality. Journal of the Institute of Actuaries and Assurance Magazine, 24(3), 153-173.
Steffensen, J. (1930). Infantile mortality from an actuarial point of view. Skandinavisk Aktuarietidskrift 13, 272-286. doi:10.1080/03461238.1930.10416902
Perks, W. (1932). On Some Experiments in the Graduation of Mortality Statistics. Journal of the Institute of Actuaries, 63(1), 12-57. doi:10.1017/S0020268100046680
Harper, F. S. (1936). An actuarial study of infant mortality. Scandinavian Actuarial Journal 1936 (3-4), 234-270. doi:10.1080/03461238.1936.10405113
Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of applied mechanics 103, 293-297.
Beard, R. E. (1971). Some aspects of theories of mortality, cause of death analysis, forecasting and stochastic processes. Biological aspects of demography 999, 57-68.
Vaupel, J., Manton, K.G., and Stallard, E. (1979). The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography 16(3): 439-454. doi:10.2307/2061224
Siler, W. (1979), A Competing-Risk Model for Animal Mortality. Ecology, 60: 750-757. doi:10.2307/1936612
Heligman, L., & Pollard, J. (1980). The age pattern of mortality. Journal of the Institute of Actuaries, 107(1), 49-80. doi:10.1017/S0020268100040257
Rogers A and Planck F (1983). MODEL: A General Program for Estimating Parametrized Model Schedules of Fertility, Mortality, Migration, and Marital and Labor Force Status Transitions. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-83-102
Martinelle S. (1987). A generalized Perks formula for old-age mortality. Stockholm, Sweden, Statistiska Centralbyran, 1987. 55 p. (R&D Report, Research-Methods-Development, U/STM No. 38)
Carriere J.F. (1992). Parametric models for life tables. Transactions of the Society of Actuaries. Vol.44
Kostaki A. (1992). A nine-parameter version of the Heligman-Pollard formula. Mathematical Population Studies. Vol. 3 277-288. doi:10.1080/08898489209525346
Thatcher AR, Kannisto V and Vaupel JW (1998). The force of mortality at ages 80 to 120. Odense Monographs on Population Aging Vol. 5, Odense University Press, 1998. 104, 20 p. Odense, Denmark
Tabeau E. (2001). A Review of Demographic Forecasting Models for Mortality. In: Tabeau E., van den Berg Jeths A., Heathcote C. (eds) Forecasting Mortality in Developed Countries. European Studies of Population, vol 9. Springer, Dordrecht. doi:10.1007/0-306-47562-6_1
Finkelstein M. (2012) Discussing the Strehler-Mildvan model of mortality Demographic Research, Vol. 26(9), 191-206. doi:10.4054/DemRes.2012.26.9

Examples

availableLaws()
availableLaws()

Check Available Loss Function

Description

The function returns information about the implemented loss function used by the optimization procedure in the MortalityLaw function.

Usage

availableLF()
availableLF()

Value

A list of class availableLF with the components:

`table`	Table with loss functions and codes to be used in `MortalityLaw`.
`legend`	Table with details about the abbreviation used.

Author(s)

Marius D. Pascariu

Examples

availableLF()
availableLF()

Convert Life Table Indicators

Description

Easy conversion between the life table indicators. This function is based on the LifeTable function and methods behind it.

Usage

convertFx(x, data, from, to, ...)
convertFx(x, data, from, to, ...)

Arguments

`x`	Vector of ages at the beginning of the age interval.
`data`	Vector or data.frame/matrix containing the mortality indicators.
`from`	Specify the life table indicator in the input `data`. Character. Options: `mx, qx, dx, lx`.
`to`	What indicator would you like to obtain? Character. Options: `mx, qx, dx, lx, Lx, Tx, ex`.
`...`	Further arguments to be passed to the `LifeTable` function with impact on the results to be produced.

Value

A matrix or array containing life table indicators.

Author(s)

Marius D. Pascariu

Examples

# Data ---
x  <- 0:110
mx <- ahmd$mx

# mx to qx
qx <- convertFx(x, data = mx, from = "mx", to = "qx")
# mx to dx
dx <- convertFx(x, data = mx, from = "mx", to = "dx")
# mx to lx
lx <- convertFx(x, data = mx, from = "mx", to = "lx")


# There are 28 possible combinations --------------------------------
# Let generate all of them.
from <- c("mx", "qx", "dx", "lx")
to   <- c("mx", "qx", "dx", "lx", "Lx", "Tx", "ex")
K    <- expand.grid(from = from, to = to) # all possible cases/combinations

for (i in 1:nrow(K)) {
  In  <- as.character(K[i, "from"])
  Out <- as.character(K[i, "to"])
  N <- paste0(Out, "_from_", In)
  cat(i, " Create", N, "\n")
  # Create the 28 sets of results
  assign(N, convertFx(x = x, data = get(In), from = In, to = Out))
}
# Data ---
x  <- 0:110
mx <- ahmd$mx

# mx to qx
qx <- convertFx(x, data = mx, from = "mx", to = "qx")
# mx to dx
dx <- convertFx(x, data = mx, from = "mx", to = "dx")
# mx to lx
lx <- convertFx(x, data = mx, from = "mx", to = "lx")


# There are 28 possible combinations --------------------------------
# Let generate all of them.
from <- c("mx", "qx", "dx", "lx")
to   <- c("mx", "qx", "dx", "lx", "Lx", "Tx", "ex")
K    <- expand.grid(from = from, to = to) # all possible cases/combinations

for (i in 1:nrow(K)) {
  In  <- as.character(K[i, "from"])
  Out <- as.character(K[i, "to"])
  N <- paste0(Out, "_from_", In)
  cat(i, " Create", N, "\n")
  # Create the 28 sets of results
  assign(N, convertFx(x = x, data = get(In), from = In, to = Out))
}

Compute Life Tables from Parameters of a Mortality Law

Description

Compute Life Tables from Parameters of a Mortality Law

Usage

LawTable(x, par, law, sex = NULL, lx0 = 1e+05, ax = NULL)
LawTable(x, par, law, sex = NULL, lx0 = 1e+05, ax = NULL)

Arguments

`x`	Vector of ages at the beginning of the age interval.
`par`	The parameters of the mortality model.
`law`	The name of the mortality law/model to be used. e.g. `gompertz`, `makeham`, ... To investigate all the possible options, see `availableLaws` function.
`sex`	Sex of the population considered here. Default: `NULL`. This argument affects the first two values in the life table ax column. If sex is specified the values are computed based on the Coale-Demeny method and are slightly different for males than for females. Options: `NULL, male, female, total`.
`lx0`	Radix. Default: 100 000.
`ax`	Numeric scalar. Subject-time alive in age-interval for those who die in the same interval. If `NULL` this will be estimated. A common assumption is `ax = 0.5`, i.e. the deaths occur in the middle of the interval. Default: `NULL`.

Details

The "life table" is also called "mortality table" or "actuarial table". This shows, for each age, what the probability is that a person of that age will die before his or her next birthday, the expectation of life across different age ranges or the survivorship of people from a certain population.

Value

The output is of the "LifeTable" class with the components:

`lt`	Computed life table;
`call`	`Call` in which all of the specified arguments are specified by their full names;
`process_date`	Time stamp.

Author(s)

Marius D. Pascariu

Examples

# Example 1 --- Makeham --- 4 tables ----------
x1 = 45:100
L1 = "makeham"
C1 = matrix(c(0.00717, 0.07789, 0.00363,
              0.01018, 0.07229, 0.00001,
              0.00298, 0.09585, 0.00002,
              0.00067, 0.11572, 0.00078),
            nrow = 4, dimnames = list(1:4, c("A", "B", "C")))

LawTable(x = 45:100, par = C1, law = L1)

# WARNING!!!

# It is important to know how the coefficients have been estimated. If the
# fitting of the model was done over the [x, x+) age-range, the LawTable
# function can be used to create a life table only for age x onward.

# What can go wrong?

# ** Example 1B - is OK.
LawTable(x = 45:100, par = c(0.00717, 0.07789, 0.00363), law = L1)

# ** Example 1C - Not OK, because the life expectancy at age 25 is
# equal with life expectancy at age 45 in the previous example.
LawTable(x = 25:100, par = c(0.00717, 0.07789, 0.00363), law = L1)

# Why is this happening?

# If we have a model that covers only a part of the human mortality curve
# (e.g. adult mortality), in fitting the x vector is scaled down, meaning
# age (x) becomes (x - min(x) + 1). And, the coefficients are estimated on
# a scaled x in ordered to obtain meaningful estimates. Otherwise the
# optimization process might not converge.

# What can we do about it?

# a). Know which mortality laws are rescaling the x vector in the fitting
# process. If these models are fitted with the MortalityLaw() function, you
# can find out like so:
A <- availableLaws()$table
A[, c("CODE", "SCALE_X")]

# b). If you are using one of the models that are applying scaling,
# be aware over what age-range the coefficients have been estimated. If they
# have been estimated using, say, ages 50 to 80, you can use the
# LawTable() to build a life tables from age 50 onwards.


# Example 2 --- Heligman-Pollard -- 1 table ----
x2 = 0:110
L2 = "HP"
C2 = c(0.00223, 0.01461, 0.12292, 0.00091,
       2.75201, 29.01877, 0.00002, 1.11411)

LawTable(x = x2, par = C2, law = L2)

# Because "HP" is not scaling down the x vector, the output is not affected
# by the problem described above.

# Check
LawTable(x = 3:110, par = C2, law = L2)
# Note the e3 = 70.31 in both tables
# Example 1 --- Makeham --- 4 tables ----------
x1 = 45:100
L1 = "makeham"
C1 = matrix(c(0.00717, 0.07789, 0.00363,
              0.01018, 0.07229, 0.00001,
              0.00298, 0.09585, 0.00002,
              0.00067, 0.11572, 0.00078),
            nrow = 4, dimnames = list(1:4, c("A", "B", "C")))

LawTable(x = 45:100, par = C1, law = L1)

# WARNING!!!

# It is important to know how the coefficients have been estimated. If the
# fitting of the model was done over the [x, x+) age-range, the LawTable
# function can be used to create a life table only for age x onward.

# What can go wrong?

# ** Example 1B - is OK.
LawTable(x = 45:100, par = c(0.00717, 0.07789, 0.00363), law = L1)

# ** Example 1C - Not OK, because the life expectancy at age 25 is
# equal with life expectancy at age 45 in the previous example.
LawTable(x = 25:100, par = c(0.00717, 0.07789, 0.00363), law = L1)

# Why is this happening?

# If we have a model that covers only a part of the human mortality curve
# (e.g. adult mortality), in fitting the x vector is scaled down, meaning
# age (x) becomes (x - min(x) + 1). And, the coefficients are estimated on
# a scaled x in ordered to obtain meaningful estimates. Otherwise the
# optimization process might not converge.

# What can we do about it?

# a). Know which mortality laws are rescaling the x vector in the fitting
# process. If these models are fitted with the MortalityLaw() function, you
# can find out like so:
A <- availableLaws()$table
A[, c("CODE", "SCALE_X")]

# b). If you are using one of the models that are applying scaling,
# be aware over what age-range the coefficients have been estimated. If they
# have been estimated using, say, ages 50 to 80, you can use the
# LawTable() to build a life tables from age 50 onwards.


# Example 2 --- Heligman-Pollard -- 1 table ----
x2 = 0:110
L2 = "HP"
C2 = c(0.00223, 0.01461, 0.12292, 0.00091,
       2.75201, 29.01877, 0.00002, 1.11411)

LawTable(x = x2, par = C2, law = L2)

# Because "HP" is not scaling down the x vector, the output is not affected
# by the problem described above.

# Check
LawTable(x = 3:110, par = C2, law = L2)
# Note the e3 = 70.31 in both tables

Compute Life Tables from Mortality Data

Description

Construct either a full or abridged life table with various input choices like: death counts and mid-interval population estimates (Dx, Ex) or age-specific death rates (mx) or death probabilities (qx) or survivorship curve (lx) or a distribution of deaths (dx). If one of these options is specified, the other can be ignored. The input data can be an object of class: numerical vector, matrix or data.frame.

Usage

LifeTable(x, Dx = NULL, Ex = NULL,
             mx = NULL,
             qx = NULL,
             lx = NULL,
             dx = NULL,
             sex = NULL,
             lx0 = 1e5,
             ax  = NULL)
LifeTable(x, Dx = NULL, Ex = NULL,
             mx = NULL,
             qx = NULL,
             lx = NULL,
             dx = NULL,
             sex = NULL,
             lx0 = 1e5,
             ax  = NULL)

Arguments

`x`	Vector of ages at the beginning of the age interval.
`Dx`	Object containing death counts. An element of the `Dx` object represents the number of deaths during the year to persons aged x to x+n.
`Ex`	Exposure in the period. `Ex` can be approximated by the mid-year population aged x to x+n.
`mx`	Life table death rate in age interval [x, x+n).
`qx`	Probability of dying in age interval [x, x+n).
`lx`	Probability of survival up until exact age x (if l(0) = 1), or the number of survivors at exact age x, assuming l(0) > 1.
`dx`	Deaths by life-table population in the age interval [x, x+n).
`sex`	Sex of the population considered here. Default: `NULL`. This argument affects the first two values in the life table ax column. If sex is specified the values are computed based on the Coale-Demeny method and are slightly different for males than for females. Options: `NULL, male, female, total`.
`lx0`	Radix. Default: 100 000.
`ax`	Numeric scalar. Subject-time alive in age-interval for those who die in the same interval. If `NULL` this will be estimated. A common assumption is `ax = 0.5`, i.e. the deaths occur in the middle of the interval. Default: `NULL`.

Details

Value

The output is of the "LifeTable" class with the components:

`lt`	Computed life table;
`call`	`Call` in which all of the specified arguments are specified by their full names;
`process_date`	Time stamp.

Author(s)

Marius D. Pascariu

Examples

# Example 1 --- Full life tables with different inputs ---

y  <- 1900
x  <- as.numeric(rownames(ahmd$mx))
Dx <- ahmd$Dx[, paste(y)]
Ex <- ahmd$Ex[, paste(y)]

LT1 <- LifeTable(x, Dx = Dx, Ex = Ex)
LT2 <- LifeTable(x, mx = LT1$lt$mx)
LT3 <- LifeTable(x, qx = LT1$lt$qx)
LT4 <- LifeTable(x, lx = LT1$lt$lx)
LT5 <- LifeTable(x, dx = LT1$lt$dx)

LT1
LT5
ls(LT5)

# Example 2 --- Compute multiple life tables at once ---

LTs = LifeTable(x, mx = ahmd$mx)
LTs
# A warning is printed if the input contains missing values.
# Some of the missing values can be handled by the function.

# Example 3 --- Abridged life table ------------

x  <- c(0, 1, seq(5, 110, by = 5))
mx <- c(.053, .005, .001, .0012, .0018, .002, .003, .004,
        .004, .005, .006, .0093, .0129, .019, .031, .049,
        .084, .129, .180, .2354, .3085, .390, .478, .551)
LT6 <- LifeTable(x, mx = mx, sex = "female")
LT6

# Example 4 --- Abridged life table w using my own 'ax' ------------
# In this examples we are using the ages (x) and death rates (mx) from
# example 3. Note that 'ax' must have the same length as the 'x' vector
# otherwise an error message will be returned.

my_ax <- c(0.1, 1.5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
           2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1)

LT7 <- LifeTable(x = x, mx = mx, ax = my_ax)

# Example 1 --- Full life tables with different inputs ---

y  <- 1900
x  <- as.numeric(rownames(ahmd$mx))
Dx <- ahmd$Dx[, paste(y)]
Ex <- ahmd$Ex[, paste(y)]

LT1 <- LifeTable(x, Dx = Dx, Ex = Ex)
LT2 <- LifeTable(x, mx = LT1$lt$mx)
LT3 <- LifeTable(x, qx = LT1$lt$qx)
LT4 <- LifeTable(x, lx = LT1$lt$lx)
LT5 <- LifeTable(x, dx = LT1$lt$dx)

LT1
LT5
ls(LT5)

# Example 2 --- Compute multiple life tables at once ---

LTs = LifeTable(x, mx = ahmd$mx)
LTs
# A warning is printed if the input contains missing values.
# Some of the missing values can be handled by the function.

# Example 3 --- Abridged life table ------------

x  <- c(0, 1, seq(5, 110, by = 5))
mx <- c(.053, .005, .001, .0012, .0018, .002, .003, .004,
        .004, .005, .006, .0093, .0129, .019, .031, .049,
        .084, .129, .180, .2354, .3085, .390, .478, .551)
LT6 <- LifeTable(x, mx = mx, sex = "female")
LT6

# Example 4 --- Abridged life table w using my own 'ax' ------------
# In this examples we are using the ages (x) and death rates (mx) from
# example 3. Note that 'ax' must have the same length as the 'x' vector
# otherwise an error message will be returned.

my_ax <- c(0.1, 1.5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
           2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1)

LT7 <- LifeTable(x = x, mx = mx, ax = my_ax)

Fit Mortality Laws

Description

Fit parametric mortality models given a set of input data which can be represented by death counts and mid-interval population estimates (Dx, Ex) or age-specific death rates (mx) or death probabilities (qx). Using the argument law one can specify the model to be fitted. So far more than 27 parametric models have been implemented; check the availableLaws function to learn about the available options. The models can be fitted under the maximum likelihood methodology or by selecting a loss function to be optimised. See the implemented loss function by running the availableLF function.

Usage

MortalityLaw(x, Dx = NULL, Ex = NULL, mx = NULL, qx = NULL,
                law = NULL,
                opt.method = "LF2",
                parS = NULL,
                fit.this.x = x,
                custom.law = NULL,
                show = FALSE, ...)
MortalityLaw(x, Dx = NULL, Ex = NULL, mx = NULL, qx = NULL,
                law = NULL,
                opt.method = "LF2",
                parS = NULL,
                fit.this.x = x,
                custom.law = NULL,
                show = FALSE, ...)

Arguments

`x`	Vector of ages at the beginning of the age interval.
`Dx`	Object containing death counts. An element of the `Dx` object represents the number of deaths during the year to persons aged x to x+n.
`Ex`	Exposure in the period. `Ex` can be approximated by the mid-year population aged x to x+n.
`mx`	Life table death rate in age interval [x, x+n).
`qx`	Probability of dying in age interval [x, x+n).
`law`	The name of the mortality law/model to be used. e.g. `gompertz`, `makeham`, ... To investigate all the possible options, see `availableLaws` function.
`opt.method`	How would you like to find the parameters? Specify the function to be optimize. Available options: the Poisson likelihood function `poissonL`; the Binomial likelihood function -`binomialL`; and 6 other loss functions. For more details, check the `availableLF` function.
`parS`	Starting parameters used in the optimization process (optional).
`fit.this.x`	Select the ages to be considered in model fitting. By default `fit.this.x = x`. One may want to exclude from the fitting procedure, say, the advanced ages where the data is sparse.
`custom.law`	Allows you to fit a model that is not defined in the package. Accepts as input a function.
`show`	Choose whether to display a progress bar during the fitting process. Logical. Default: `FALSE`.
`...`	Arguments to be passed to or from other methods.

Details

Depending on the complexity of the model, one of following optimization strategies is employed:

Nelder-Mead method: approximates a local optimum of a problem with n variables when the objective function varies smoothly and is unimodal. For details see optim;
PORT routines: provides unconstrained optimization and optimization subject to box constraints for complicated functions. For details check nlminb;
Levenberg-Marquardt algorithm: damped least-squares method. For details check nls.lm.

Value

The output is of the "MortalityLaw" class with the components:

`input`	List with arguments provided in input. Saved for convenience.
`info`	Brief information about the model.
`coefficients`	Estimated coefficients.
`fitted.values`	Fitted values of the selected model.
`residuals`	Deviance residuals.
`goodness.of.fit`	List containing goodness of fit measures like AIC, BIC and log-Likelihood.
`opt.diagnosis`	Resultant optimization object useful for checking the convergence etc.

Author(s)

Marius D. Pascariu

Examples

# Example 1: --------------------------
# Fit Makeham Model for Year of 1950.

x  <- 45:75
Dx <- ahmd$Dx[paste(x), "1950"]
Ex <- ahmd$Ex[paste(x), "1950"]

M1 <- MortalityLaw(x = x, Dx = Dx, Ex = Ex, law = 'makeham')

M1
ls(M1)
coef(M1)
summary(M1)
fitted(M1)
predict(M1, x = 45:95)
plot(M1)


# Example 2: --------------------------
# We can fit the same model using a different data format
# and a different optimization method.
x  <- 45:75
mx <- ahmd$mx[paste(x), ]
M2 <- MortalityLaw(x = x, mx = mx, law = 'makeham', opt.method = 'LF1')
M2
fitted(M2)
predict(M2, x = 55:90)

# Example 3: --------------------------
# Now let's fit a mortality law that is not defined
# in the package, say a reparameterized Gompertz in
# terms of modal age at death
# hx = b*exp(b*(x-m)) (here b and m are the parameters to be estimated)

# A function with 'x' and 'par' as input has to be defined, which returns
# at least an object called 'hx' (hazard rate).
my_gompertz <- function(x, par = c(b = 0.13, M = 45)){
  hx  <- with(as.list(par), b*exp(b*(x - M)) )
  return(as.list(environment()))
}

M3 <- MortalityLaw(x = x, Dx = Dx, Ex = Ex, custom.law = my_gompertz)
summary(M3)
# predict M3 for different ages
predict(M3, x = 85:130)


# Example 4: --------------------------
# Fit Heligman-Pollard model for a single
# year in the dataset between age 0 and 100 and build a life table.

x  <- 0:100
mx <- ahmd$mx[paste(x), "1950"] # select data
M4 <- MortalityLaw(x = x, mx = mx, law = 'HP', opt.method = 'LF2')
M4
plot(M4)

LifeTable(x = x, qx = fitted(M4))
# Example 1: --------------------------
# Fit Makeham Model for Year of 1950.

x  <- 45:75
Dx <- ahmd$Dx[paste(x), "1950"]
Ex <- ahmd$Ex[paste(x), "1950"]

M1 <- MortalityLaw(x = x, Dx = Dx, Ex = Ex, law = 'makeham')

M1
ls(M1)
coef(M1)
summary(M1)
fitted(M1)
predict(M1, x = 45:95)
plot(M1)


# Example 2: --------------------------
# We can fit the same model using a different data format
# and a different optimization method.
x  <- 45:75
mx <- ahmd$mx[paste(x), ]
M2 <- MortalityLaw(x = x, mx = mx, law = 'makeham', opt.method = 'LF1')
M2
fitted(M2)
predict(M2, x = 55:90)

# Example 3: --------------------------
# Now let's fit a mortality law that is not defined
# in the package, say a reparameterized Gompertz in
# terms of modal age at death
# hx = b*exp(b*(x-m)) (here b and m are the parameters to be estimated)

# A function with 'x' and 'par' as input has to be defined, which returns
# at least an object called 'hx' (hazard rate).
my_gompertz <- function(x, par = c(b = 0.13, M = 45)){
  hx  <- with(as.list(par), b*exp(b*(x - M)) )
  return(as.list(environment()))
}

M3 <- MortalityLaw(x = x, Dx = Dx, Ex = Ex, custom.law = my_gompertz)
summary(M3)
# predict M3 for different ages
predict(M3, x = 85:130)


# Example 4: --------------------------
# Fit Heligman-Pollard model for a single
# year in the dataset between age 0 and 100 and build a life table.

x  <- 0:100
mx <- ahmd$mx[paste(x), "1950"] # select data
M4 <- MortalityLaw(x = x, mx = mx, law = 'HP', opt.method = 'LF2')
M4
plot(M4)

LifeTable(x = x, qx = fitted(M4))

MortalityLaws: Parametric Mortality Models, Life Tables and HMD

Description

Fit the most popular human mortality 'laws', and construct full and abridge life tables given various input indices. A mortality law is a parametric function that describes the dying-out process of individuals in a population during a significant portion of their life spans. For a comprehensive review of the most important mortality laws see Tabeau (2001) doi:10.1007/0-306-47562-6_1. Practical functions for downloading data from various human mortality databases are provided as well.

Details

To learn more about the package, start with the vignettes: browseVignettes(package = "MortalityLaws")

Author(s)

Maintainer: Marius D. Pascariu [email protected] (ORCID) [copyright holder]

Other contributors:

Vladimir Canudas-Romo [contributor]

Plot Function for MortalityLaw

Description

Plot Function for MortalityLaw

Usage

## S3 method for class 'MortalityLaw'
plot(x, ...)
## S3 method for class 'MortalityLaw'
plot(x, ...)

Arguments

`x`	An object of class MortalityLaw
`...`	Arguments to be passed to methods, such as graphical parameters (see `par`).

Value

generate a plot

Author(s)

Marius D. Pascariu

Examples

# See complete example in MortalityLaw help page
# See complete example in MortalityLaw help page

Predict function for MortalityLaw

Description

Predict function for MortalityLaw

Usage

## S3 method for class 'MortalityLaw'
predict(object, x, ...)
## S3 method for class 'MortalityLaw'
predict(object, x, ...)

Arguments

`object`	An object of class `"MortalityLaw"`
`x`	Vector of ages to be considered in prediction
`...`	Additional arguments affecting the predictions produced.

Value

A vector of predicted hazard rates

Author(s)

Marius D. Pascariu

Examples

# Extrapolate old-age mortality with the Kannisto model
# Fit ages 80-94 and extrapolate up to 120.

Mx <- ahmd$mx[paste(80:94), "1950"]
M1 <- MortalityLaw(x = 80:94, mx  = Mx, law = 'kannisto')
fitted(M1)
predict(M1, x = 80:120)

# See more examples in MortalityLaw function help page.
# Extrapolate old-age mortality with the Kannisto model
# Fit ages 80-94 and extrapolate up to 120.

Mx <- ahmd$mx[paste(80:94), "1950"]
M1 <- MortalityLaw(x = 80:94, mx  = Mx, law = 'kannisto')
fitted(M1)
predict(M1, x = 80:120)

# See more examples in MortalityLaw function help page.

Download the Australian Human Mortality Database (AHMD)

Description

Download detailed mortality and population data for different provinces and territories in Australia, in a single object from the Australian Human Mortality Database.

Usage

ReadAHMD(what, regions = NULL, interval = "1x1", save = FALSE, show = TRUE)
ReadAHMD(what, regions = NULL, interval = "1x1", save = FALSE, show = TRUE)

Arguments

`what`	What type of data are you looking for? The following options might be available for some or all the countries and regions: `"births"` – birth records; `"Dx_lexis"` – deaths by Lexis triangles; `"Ex_lexis"` – exposure-to-risk by Lexis triangles; `"population"` – population size; `"Dx"` – death counts; `"Ex"` – exposure-to-risk; `"mx"` – central death-rates; `"LT_f"` – period life tables for females; `"LT_m"` – period life tables for males; `"LT_t"` – period life tables both sexes combined; `"e0"` – period life expectancy at birth; `"Exc"` – cohort exposures; `"mxc"` – cohort death-rates; `"LT_fc"` – cohort life tables for females; `"LT_mc"` – cohort life tables for males; `"LT_tc"` – cohort life tables both sexes combined; `"e0c"` – cohort life expectancy at birth;
`regions`	Specify the region specific data you want to download by adding the AHMD region code/s. Options: `"ACT"` – Australian Capital Territory; `"NSW"` – New South Wales; `"NT"` – Northern Territory; `"QLD"` – Queensland; `"SA"` – South Australia; `"TAS"` – Tasmania; `"VIC"` – Victoria; `"WA"` – Western Australia; `NULL` – if `NULL` data for all the regions are downloaded.
`interval`	Datasets are given in various age and time formats based on which the records are agregated. Interval options: `"1x1"` – by age and year; `"1x5"` – by age and 5-year time interval; `"1x10"` – by age and 10-year time interval; `"5x1"` – by 5-year age group and year; `"5x5"` – by 5-year age group and 5-year time interval; `"5x10"` –by 5-year age group and 10-year time interval.
`save`	Do you want to save a copy of the dataset on your local machine? Logical. Default: `FALSE`.
`show`	Choose whether to display a progress bar. Logical. Default: `TRUE`.

Details

(Description taken from the AHMD website).

The Australian Human Mortality Database (AHMD) was created to provide detailed Australian mortality and population data to researchers, students, journalists, policy analysts, and others interested in the history of human longevity. The project is an achievement of the Mortality, Ageing & Health research team in the ANU School of Demography under the supervision of Associate Professor Vladimir Canudas-Romo, in collaboration with demographers at the Max Plank Institute for Demographic Research (Rostock, Germany) and the Department of Demography, University of California at Berkeley.

The AHMD is a "satellite" of the Human Mortality Database (HMD), an international database which currently holds detailed data for multiple countries or regions. Consequently, the AHMD's underlying methodology corresponds to the one used for the HMD.

The AHMD gathers all required data (deaths counts, births counts, population size, exposure-to-risk, death rates) to compute life tables for Australia, its states and its territories. One of the great advantages of the database is to include data that is validated and corrected, when required, and rendered comparable, if possible, for the period ranging from 1971 thru 2016. For comparison purposes, various life tables published by governmental organizations are also available for download in PDF format.

Value

A ReadAHMD object that contains:

`input`	List with the input values;
`data`	Data downloaded from AHMD;
`download.date`	Time stamp;
`years`	Numerical vector with the years covered in the data;
`ages`	Numerical vector with ages covered in the data.

Author(s)

Marius D. Pascariu

Examples


# Download demographic data for Australian Capital Territory and
# Tasmania regions in 5x1 format

# Death counts. We don't want to export data outside R.
AHMD_Dx <- ReadAHMD(what = "Dx",
                    regions = c('ACT', 'TAS'),
                    interval  = "5x1",
                    save = FALSE)
AHMD_Dx

# Download life tables for female population in all the states and export data.
LTF <- ReadAHMD(what = "LT_f", interval  = "5x1", save = FALSE)
LTF

# Download demographic data for Australian Capital Territory and
# Tasmania regions in 5x1 format

# Death counts. We don't want to export data outside R.
AHMD_Dx <- ReadAHMD(what = "Dx",
                    regions = c('ACT', 'TAS'),
                    interval  = "5x1",
                    save = FALSE)
AHMD_Dx

# Download life tables for female population in all the states and export data.
LTF <- ReadAHMD(what = "LT_f", interval  = "5x1", save = FALSE)
LTF

Download the Canadian Human Mortality Database (CHMD)

Description

Download detailed mortality and population data for different provinces and territories in Canada, in a single object from the Canadian Human Mortality Database.

Usage

ReadCHMD(what, regions = NULL, interval = "1x1", save = FALSE, show = TRUE)
ReadCHMD(what, regions = NULL, interval = "1x1", save = FALSE, show = TRUE)

Arguments

`what`	What type of data are you looking for? The following options are available: `"births"` – birth records; `"Dx_lexis"` – deaths by Lexis triangles; `"population"` – population size; `"Dx"` – death counts; `"Ex"` – exposure-to-risk; `"mx"` – central death-rates; `"LT_f"` – period life tables for females; `"LT_m"` – period life tables for males; `"LT_t"` – period life tables both sexes combined; `"e0"` – period life expectancy at birth;
`regions`	Specify the region specific data you want to download by adding the CHMD region code/s. Options: `"CAN"` – Canada - Sum of Canadian provinces and territories; `"NFL"` – Newfoundland & Labrador; `"PEI"` – Prince Edward Island; `"NSC"` – Nova Scotia; `"NBR"` – New Brunswick; `"QUE"` – Quebec; `"ONT"` – Ontario; `"MAN"` – Manitoba; `"SAS"` – Saskatchewan; `"ALB"` – Alberta; `"BCO"` – British Columbia; `"NWT"` – Northwest Territories & Nunavut; `"YUK"` – Yukon; `NULL` – if `NULL` data for all the regions are downloaded.
`interval`	Datasets are given in various age and time formats based on which the records are agregated. Interval options: `"1x1"` – by age and year; `"1x5"` – by age and 5-year time interval; `"1x10"` – by age and 10-year time interval; `"5x1"` – by 5-year age group and year; `"5x5"` – by 5-year age group and 5-year time interval; `"5x10"` –by 5-year age group and 10-year time interval.
`save`	Do you want to save a copy of the dataset on your local machine? Logical. Default: `FALSE`.
`show`	Choose whether to display a progress bar. Logical. Default: `TRUE`.

Details

(Description taken from the CHMD website).

The Canadian Human Mortality Database (CHMD) was created to provide detailed Canadian mortality and population data to researchers, students, journalists, policy analysts, and others interested in the history of human longevity. The project is an achievement of the Mortality and Longevity research team at the Department of Demography, Universite de Montreal, under the supervision of Professor Robert Bourbeau, in collaboration with demographers at the Max Plank Institute for Demographic Research (Rostock, Germany) and the Department of Demography, University of California at Berkeley. Nadine Ouellette, researcher at the Institut national d'etudes demographiques in Paris and member of the Mortality and Longevity research team at the Universite de Montreal, is in charge of computing all CHMD life tables and updating the CHMD web site.

The CHMD is a "satellite" of the Human Mortality Database (HMD), an international database which currently holds detailed data for multiple countries or regions. Consequently, the CHMD's underlying methodology corresponds to the one used for the HMD.

The CHMD gathers all required data (deaths counts, births counts, population size, exposure-to-risk, death rates) to compute life tables for Canada, its provinces and its territories. One of the great advantages of the database is to include data that is validated and corrected, when required, and rendered comparable, if possible, for the period ranging from 1921 thru 2011. For comparison purposes, various life tables published by governmental organizations are also available for download in PDF format.

Value

A ReadCHMD object that contains:

`input`	List with the input values;
`data`	Data downloaded from CHMD;
`download.date`	Time stamp;
`years`	Numerical vector with the years covered in the data;
`ages`	Numerical vector with ages covered in the data.

Author(s)

Marius D. Pascariu

Examples


# Download demographic data for Quebec and Saskatchewan regions in 1x1 format

# Death counts. We don't want to export data outside R.
CHMD_Dx <- ReadCHMD(what = "Dx",
                    regions = c('QUE', 'SAS'),
                    interval  = "1x1",
                    save = FALSE)

# Download life tables for female population. To export data use save = TRUE.
LTF <- ReadCHMD(what = "LT_f",
                regions = c('QUE', 'SAS'),
                interval  = "1x1",
                save = FALSE)

# Download demographic data for Quebec and Saskatchewan regions in 1x1 format

# Death counts. We don't want to export data outside R.
CHMD_Dx <- ReadCHMD(what = "Dx",
                    regions = c('QUE', 'SAS'),
                    interval  = "1x1",
                    save = FALSE)

# Download life tables for female population. To export data use save = TRUE.
LTF <- ReadCHMD(what = "LT_f",
                regions = c('QUE', 'SAS'),
                interval  = "1x1",
                save = FALSE)

Download The Human Mortality Database (HMD)

Description

Download detailed mortality and population data for different countries and regions in a single object from the Human Mortality Database.

Usage

ReadHMD(
  what,
  countries = NULL,
  interval = "1x1",
  username,
  password,
  save = FALSE,
  show = TRUE
)
ReadHMD(
  what,
  countries = NULL,
  interval = "1x1",
  username,
  password,
  save = FALSE,
  show = TRUE
)

Arguments

`what`	What type of data are you looking for? The following options might be available for some or all the countries and regions: `"births"` – birth records; `"Dx_lexis"` – deaths by Lexis triangles; `"Ex_lexis"` – exposure-to-risk by Lexis triangles; `"population"` – population size; `"Dx"` – death counts; `"Ex"` – exposure-to-risk; `"mx"` – central death-rates; `"LT_f"` – period life tables for females; `"LT_m"` – period life tables for males; `"LT_t"` – period life tables both sexes combined; `"e0"` – period life expectancy at birth; `"Exc"` – cohort exposures; `"mxc"` – cohort death-rates; `"LT_fc"` – cohort life tables for females; `"LT_mc"` – cohort life tables for males; `"LT_tc"` – cohort life tables both sexes combined; `"e0c"` – cohort life expectancy at birth;
`countries`	Specify the country data you want to download by adding the HMD country code/s. Options: `"AUS" "AUT", "BEL", "BGR", "BLR", "CAN", "CHL", "HRV", "HKG", "CHE", "CZE", "DEUTNP", "DEUTE", "DEUTW", "DNK", "ESP", "EST", "FIN", "FRATNP","FRACNP", "GRC", "HUN", "IRL", "ISL" "ISR", "ITA", "JPN", "KOR", "LTU", "LUX", "LVA", "NLD", "NOR", "NZL_NP", "NZL_MA" "NZL_NM", "POL", "PRT" "RUS", "SVK", "SVN", "SWE", "TWN", "UKR", "GBR_NP","GBRTENW","GBRCENW","GBR_SCO", "GBR_NIR","USA"`. If `NULL` data for all the countries are downloaded at once;
`interval`	Datasets are given in various age and time formats based on which the records are agregated. Interval options: `"1x1"` – by age and year; `"1x5"` – by age and 5-year time interval; `"1x10"` – by age and 10-year time interval; `"5x1"` – by 5-year age group and year; `"5x5"` – by 5-year age group and 5-year time interval; `"5x10"` –by 5-year age group and 10-year time interval.
`username`	Your HMD username. If you don't have one you can sign up for free on the Human Mortality Database website.
`password`	Your HMD password.
`save`	Do you want to save a copy of the dataset on your local machine? Logical. Default: `FALSE`.
`show`	Choose whether to display a progress bar. Logical. Default: `TRUE`.

Details

The Human Mortality Database (HMD) was created to provide detailed mortality and population data to researchers, students, journalists, policy analysts, and others interested in the history of human longevity. The project began as an outgrowth of earlier projects in the Department of Demography at the University of California, Berkeley, USA, and at the Max Planck Institute for Demographic Research in Rostock, Germany (see history). It is the work of two teams of researchers in the USA and Germany (see research teams), with the help of financial backers and scientific collaborators from around the world (see acknowledgements). The Center on the Economics and Development of Aging (CEDA) French Institute for Demographic Studies (INED) has also supported the further development of the database in recent years.

Value

A ReadHMD object that contains:

`input`	List with the input values (except the password).
`data`	Data downloaded from HMD.
`download.date`	Time stamp.
`years`	Numerical vector with the years covered in the data.
`ages`	Numerical vector with ages covered in the data.

Author(s)

Marius D. Pascariu

Examples

## Not run: 


# Download demographic data for 3 countries in 1x1 format
age_int  <- 1  # age interval: 1,5
year_int <- 1  # year interval: 1,5,10
interval <- paste0(age_int, "x", year_int)  # --> 1x1
# And the 3 countries: Sweden Denmark and USA. We have to use the HMD codes
cntr  <- c('SWE', 'DNK', 'USA')

# Download death counts. We don't want to export data outside R.
HMD_Dx <- ReadHMD(what = "Dx",
                  countries = cntr,
                  interval  = interval,
                  username  = "[email protected]",
                  password  = "password",
                  save = FALSE)
HMD_Dx

# Download life tables for female population and export data.
LTF <- ReadHMD(what = "LT_f",
               countries = cntr,
               interval  = interval,
               username  = "[email protected]",
               password  = "password",
               save = TRUE)
LTF

## End(Not run)
## Not run: 


# Download demographic data for 3 countries in 1x1 format
age_int  <- 1  # age interval: 1,5
year_int <- 1  # year interval: 1,5,10
interval <- paste0(age_int, "x", year_int)  # --> 1x1
# And the 3 countries: Sweden Denmark and USA. We have to use the HMD codes
cntr  <- c('SWE', 'DNK', 'USA')

# Download death counts. We don't want to export data outside R.
HMD_Dx <- ReadHMD(what = "Dx",
                  countries = cntr,
                  interval  = interval,
                  username  = "[email protected]",
                  password  = "password",
                  save = FALSE)
HMD_Dx

# Download life tables for female population and export data.
LTF <- ReadHMD(what = "LT_f",
               countries = cntr,
               interval  = interval,
               username  = "[email protected]",
               password  = "password",
               save = TRUE)
LTF

## End(Not run)

Download the Japanese Mortality Database (JMD)

Description

Download detailed mortality and population data of the 47 prefectures in Japan, in a single object. The source of data is the Japanese Mortality Database.

Usage

ReadJMD(what, regions = NULL, interval = "1x1", save = FALSE, show = TRUE)
ReadJMD(what, regions = NULL, interval = "1x1", save = FALSE, show = TRUE)

Arguments

`what`	What type of data are you looking for? The following options might be available for some or all the countries and regions: `"births"` – birth records; `"Dx_lexis"` – deaths by Lexis triangles; `"Ex_lexis"` – exposure-to-risk by Lexis triangles; `"population"` – population size; `"Dx"` – death counts; `"Ex"` – exposure-to-risk; `"mx"` – central death-rates; `"LT_f"` – period life tables for females; `"LT_m"` – period life tables for males; `"LT_t"` – period life tables both sexes combined; `"e0"` – period life expectancy at birth; `"Exc"` – cohort exposures; `"mxc"` – cohort death-rates; `"LT_fc"` – cohort life tables for females; `"LT_mc"` – cohort life tables for males; `"LT_tc"` – cohort life tables both sexes combined; `"e0c"` – cohort life expectancy at birth;
`regions`	Specify the region specific data you want to download by adding the JMD region code/s. Options: `"Japan", "Hokkaido", "Aomori", "Iwate", "Miyagi","Akita", "Yamagata", "Fukushima", "Ibaraki", "Tochigi", "Gunma", "Saitama", "Chiba", "Tokyo", "Kanagawa", "Niigata", "Toyama", "Ishikawa", "Fukui", "Yamanashi", "Nagano", "Gifu", "Shizuoka","Aichi", "Mie", "Shiga", "Kyoto", "Osaka", "Hyogo", "Nara", "Wakayama", "Tottori", "Shimane", "Okayama", "Hiroshima", "Yamaguchi", "Tokushima", "Kagawa", "Ehime", "Kochi", "Fukuoka", "Saga", "Nagasaki", "Kumamoto", "Oita", "Miyazaki", "Kagoshima", "Okinawa"`. If `NULL` data for all the regions are downloaded at once.
`interval`	Datasets are given in various age and time formats based on which the records are agregated. Interval options: `"1x1"` – by age and year; `"1x5"` – by age and 5-year time interval; `"1x10"` – by age and 10-year time interval; `"5x1"` – by 5-year age group and year; `"5x5"` – by 5-year age group and 5-year time interval; `"5x10"` –by 5-year age group and 10-year time interval.
`save`	Do you want to save a copy of the dataset on your local machine? Logical. Default: `FALSE`.
`show`	Choose whether to display a progress bar. Logical. Default: `TRUE`.

Details

(Description taken from the JMD website).

The Japanese Mortality Database is a comprehensively-reorganized mortality database that is optimized for mortality research and consistent with the Human Mortality Database. This database is provided as a part of the research project "Demographic research on the causes and the socio-economic consequence of longetivity extension in Japan" (2011-2013), "Demographic research on longevity extension, population aging, and their effects on the social security and socio-economic structures in Japan" (2014-2016), and "Comprehensive research from a demographic viewpoint on the longevity revolution" (2017-2019) at the National Institute of Population and Social Security Research.

The Japanese Mortality Database is designed to provide the life tables to all the people who are interested in Japanese mortality including domestic and foreign mortality researchers for the purpose of mortality research. Especially because we have structured it to conform with the HMD, our database is suitable for international comparison, we put emphasis on the compatibility with the HMD more than our country's particular characteristics. Therefore, the life tables by JMD do not necessarily exhibit the same values as ones by the official life tables prepared and released by the Statistics and Information Department, Minister's Secretariat, Ministry of Health, Labor and Welfare according to the different base population or the methods for estimating the tables. When doing things other than mortality research, if life table that statistically displays our country's mortality situation is necessary, please use the official life table that has been prepared by the Statistics and Information Department, Minister's Secretariat, Ministry of Health, Labor and Welfare.

At the present time, we offer the data for All Japan and by prefecture. The project team is studying the methodology for estimating life tables along with data preparation. Therefore, the data may be updated when a new methodology is adopted. Please refer to "Methods" for further information.

Value

A ReadJMD object that contains:

`input`	List with the input values;
`data`	Data downloaded from JMD;
`download.date`	Time stamp;
`years`	Numerical vector with the years covered in the data;
`ages`	Numerical vector with ages covered in the data.

Author(s)

Marius D. Pascariu

Examples


# Download demographic data for Fukushima and Tokyo regions in 1x1 format

# Death counts. We don't want to export data outside R.
JMD_Dx <- ReadJMD(what = "Dx",
                  regions = c('Fukushima', 'Tokyo'),
                  interval  = "1x1",
                  save = FALSE)
JMD_Dx

# Download life tables for female population in all the states and export data.
LTF <- ReadJMD(what = "LT_f", interval  = "5x5", save = FALSE)
LTF

# Download demographic data for Fukushima and Tokyo regions in 1x1 format

# Death counts. We don't want to export data outside R.
JMD_Dx <- ReadJMD(what = "Dx",
                  regions = c('Fukushima', 'Tokyo'),
                  interval  = "1x1",
                  save = FALSE)
JMD_Dx

# Download life tables for female population in all the states and export data.
LTF <- ReadJMD(what = "LT_f", interval  = "5x5", save = FALSE)
LTF

Package 'MortalityLaws'

Help Index

MortalityLaws Test Data

Description

Usage

Format

Source

See Also

Examples

Check Data Availability in HMD

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Check Available Mortality Laws

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Check Available Loss Function

Description

Usage

Value

Author(s)

See Also

Examples

Convert Life Table Indicators

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Compute Life Tables from Parameters of a Mortality Law

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

Compute Life Tables from Mortality Data

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

Fit Mortality Laws

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

MortalityLaws: Parametric Mortality Models, Life Tables and HMD

Description

Details

Author(s)

See Also

Plot Function for MortalityLaw

Description

Usage

Arguments

Value

Author(s)

See Also