MortCast: Estimation and Projection of Age-Specific Mortality Rates

Description

Age-specific mortality rates are estimated and projected using the Kannisto, Lee-Carter and related methods as described in Sevcikova et al. (2016) doi:10.1007/978-3-319-26603-9_15.

Details

The package implements methodology described in Sevcikova et al. (2016) that is related to estimating and predicting age-specific mortality rates. The main functions are:

• cokannisto: Extrapolates given mortality rates into higher ages using the Coherent Kannisto method. The original Kannisto method (with sex-independent extrapolation) is avalable in the function kannisto.

• lileecarter.estimate: Estimates the coherent Lee-Carter parameters for male and female mortality rates (Li and Lee 2005), i.e. sex-independent parameters a_x and k_t, and the coherent parameter b_x. In addition, it computes the ultimate b^u_x for rotation (Li et al. 2013). The underlying sex-independent estimation is implemented in the function leecarter.estimate.

• mortcast: Using estimated coherent Lee-Carter parameters and given future sex-specific life expectancies, it projects age-specific mortality rates, while (by default) rotating the b_x parameter as described in Li et al. (2013).

Functions contained in the package can be used to apply Algorithm 2 in Sevcikova et al. (2016) as shown in the Example below. It can be used for both, 5-year and 1-year age groups.

Other methods for forecasting mortality rates are available:

• pmd: pattern of mortality decline

• mlt: model life tables

• logquad: log-quadratic mortality model

• mortcast.blend: combining two different methods

A life table can be constructed using the life.table function.

Author(s)

Hana Sevcikova, Nan Li and Patrick Gerland

References

Li, N. and Lee, R. D. (2005). Coherent mortality forecasts for a group of populations: An extension of the Lee-Carter method. Demography, 42, 575-594.

Li, N., Lee, R. D. and Gerland, P. (2013). Extending the Lee-Carter method to model the rotation of age patterns of mortality decline for long-term projections. Demography, 50, 2037-2051.

Sevcikova H., Li N., Kantorova V., Gerland P., Raftery A.E. (2016). Age-Specific Mortality and Fertility Rates for Probabilistic Population Projections. In: Schoen R. (eds) Dynamic Demographic Analysis. The Springer Series on Demographic Methods and Population Analysis, vol 39. Springer, Cham. Earlier version.

Examples

# This example applies Algorithm 2 in Sevcikova et al. (2016) 
# on data from WPP2017 for China
#
data(mxM, mxF, e0Fproj, e0Mproj, package = "wpp2017")
country <- "China"

# extract observed mortality rates for male and female
mxm <- subset(mxM, name == country)[,4:16]
mxf <- subset(mxF, name == country)[,4:16]
rownames(mxm) <- rownames(mxf) <- c(0,1, seq(5, 100, by=5))

# Step 1: extrapolate from 100+ to 130+ using Coherent Kannisto
mx130 <- cokannisto(mxm, mxf)
 
# Steps 2-5: estimate coherent Lee-Carter parameters
# (here ax is computed from the last observed period 
# and smoothened over ages)
lc.est <- lileecarter.estimate(mx130$male, mx130$female, 
            ax.index = ncol(mx130$male), ax.smooth = TRUE)
        
# Steps 6-9: project future mortality rates based on future 
# life expectancies from WPP2017
e0f <- as.numeric(subset(e0Fproj, name == country)[-(1:2)])
e0m <- as.numeric(subset(e0Mproj, name == country)[-(1:2)])
names(e0f) <- names(e0m) <- colnames(e0Fproj)[-(1:2)]
pred <- mortcast(e0m, e0f, lc.est)

# plot projection for the first and last future time period
plot(pred$female$mx[,"2015-2020"], type="l", log="y", 
    ylim=range(pred$female$mx, pred$male$mx), xaxt="n",
    ylab="mx", xlab="Age", main=country, col="red")
axis(1, at=1:nrow(pred$female$mx),
    labels=rownames(pred$female$mx))
lines(pred$male$mx[,"2015-2020"], col="blue")
lines(pred$female$mx[,"2095-2100"], col="red", lty=2)
lines(pred$male$mx[,"2095-2100"], col="blue", lty=2)
legend("topleft", legend=c("male 2015-2020", "female 2015-2020",
    "male 2095-2100", "female 2095-2100"), bty="n",
    col=rep(c("blue", "red"),2), lty=c(1,1,2,2))

Coefficients for the Log-Quadratic Mortality Model

Description

Data object containing a table of coefficients to be used in the Log-Quadratic Model as implemented in the logquad function.

Usage

data(LQcoef)

Format

Data frame containing columns “sex”, “age”, “ax”, “bx”, “cx”, “vx”. Rows correspond to sex and age groups.

Source

https://u.demog.berkeley.edu/~jrw/LogQuad/

References

Wilmoth, J., Zureick, S., Canudas-Romo, V., Inoue, M., Sawyer, C. (2012). A Flexible Two-Dimensional Mortality Model for Use in Indirect Estimation. Population studies, 66(1), 1-28. doi:10.1080/00324728.2011.611411

See Also

logquad

Examples

data(LQcoef)
head(LQcoef)

Model Life Tables Lookup

Description

Lookup tables containing values for various model life tables, including Coale-Demeny and UN life tables.

Usage

data(MLTlookup)
data(MLT1Ylookup)

Format

Data frame with the following columns:

type

Type of the model life table. Available options are “CD_East”, “CD_North”, “CD_South”, “CD_West”, “UN_Chilean”, “UN_Far_Eastern”, “UN_General”, “UN_Latin_American”, “UN_South_Asian”. For the CD types, see Coale et al. (1983). For the UN types, see the link in References below.

sex

Code for distinguishing sexes. 1 is for male, 2 is for female.

age

Starting age of an age group. In MLTlookup these are 0, 1, 5, 10, ... 130. The MLT1Ylookup table contains 1-year ages ranging from 0 to 130.

e0

Level of life expectancy, starting at 20 and going by steps of 2.5 up to 115.

mx

Mortality rates.

lx, Lx, sx

Other life table columns.

Source

An updated version of these datasets were provided by Sara Hertog, United Nations Population Division, in October 2021 (package version >= 2.6-0). For previous version of the tables, install MortCast 2.5-0: 'devtools::install_github("PPgp/MortCast@v2.5-0")'

References

Coale, A., P. Demeny, and B. Vaughn. 1983. Regional model life tables and stable populations. 2nd ed. New York: Academic Press.

https://www.un.org/development/desa/pd/data/extended-model-life-tables

See Also

mlt

Examples

data(MLTlookup)
str(MLTlookup)
# CD West life table for male at e0 of 80
subset(MLTlookup, type == "CD_West" & sex == 1 & e0 == 80)

Coefficients for Sex Ratio Adjustments in the PMD Method

Description

Data object containing a table of coefficients to be used to adjust the sex ratio in the coherent Pattern Mortality Decline method as implemented in the copmd function. To invoke the adjustment, argument sexratio.adjust should be set to TRUE.

Usage

data(PMDadjcoef)

Format

Data frame containing columns “age”, “intercept”, “lmxf”, “e0f”, “e0f2”, and “gap”. Rows correspond to age groups. The values are estimates of the following regression

\log_{10} mx^M = \beta_0 + \beta_1\log_{10} mx^F + \beta_2 e_0^F + \beta_3 (e_0^F)^2 + \beta_4(e_0^F - e_0^M)

The order of the columns starting with intercept corresponds to the order of the coefficients in the above equation.

Source

The coefficients were estimated and provided by Danan Gu, UN Population Division.

References

Gu, D., Pelletier, F. and Sawyer, C. (2017). Projecting Age-sex-specific Mortality: A Comparison of the Modified Lee-Carter and Pattern of Mortality Decline Methods, UN Population Division, Technical Paper No. 6. New York: United Nations. https://population.un.org/wpp/Publications/Files/WPP2017_TechnicalPaperNo6.pdf

See Also

copmd

Examples

data(PMDadjcoef)
PMDadjcoef

Pattern Mortality Decline Lookup Tables

Description

Data object containing two tables with \rho coefficients for the Pattern Mortality Decline method as implemented in the pmd function.

Usage

data(PMDrho)

Format

Using data(PMDrho) loads two objects into memory: RhoFemales and RhoMales. They both are data frames with 22 rows corresponding to age groups, and 17 columns corresponding to different levels of life expectancy in 5-years intervals (from 50 to 135). The names of the columns reflect the middle of the respective interval.

References

Andreev, K. Gu, D., Gerland, P. (2013). Age Patterns of Mortality Improvement by Level of Life Expectancy at Birth with Applications to Mortality Projections. Paper presented at the Annual Meeting of the Population Association of America, New Orleans, LA. https://paa2013.populationassociation.org/papers/132554.

Gu, D., Pelletier, F. and Sawyer, C. (2017). Projecting Age-sex-specific Mortality: A Comparison of the Modified Lee-Carter and Pattern of Mortality Decline Methods, UN Population Division, Technical Paper No. 6. New York: United Nations. https://population.un.org/wpp/Publications/Files/WPP2017_TechnicalPaperNo6.pdf

See Also

pmd

Examples

data(PMDrho)
head(RhoFemales)
head(RhoMales)

# plot a few male patterns
e0lev <- colnames(RhoMales)[c(1, 5, 9, 13, 17)]
plot(RhoMales[, e0lev[1]], type="l", log="y", ylim=range(RhoMales[,e0lev]),
    ylab="male rho", xlab="Age")
for(i in 2:length(e0lev)) lines(RhoMales[,e0lev[i]], lty = i)
legend("bottomleft", legend = e0lev, lty = 1:length(e0lev), bty= "n")

Coherent Kannisto Method

Description

Extrapolate given mortality rates into higher ages using the Coherent Kannisto method as described in Sevcikova et al. (2016).

Usage

cokannisto(
  mxM,
  mxF,
  est.ages = seq(80, 95, by = 5),
  proj.ages = seq(100, 130, by = 5)
)

Arguments

Details

The function first estimates the coherent Kannisto parameters by passing mortality rates for age groups est.ages into the cokannisto.estimate function. The estimated parameters are then passed to the projection function kannisto.predict to extrapolate into ages proj.ages. Lastly, the input mortality objects are extended by results for the extrapolated ages. If proj.ages contains age groups that are included in mxM and mxF, values for those age groups are overwritten.

Value

A list of two vectors or matrices (for male and female) containing the input motality objects extended by the extrapolated age groups.

References

Sevcikova H., Li N., Kantorova V., Gerland P., Raftery A.E. (2016). Age-Specific Mortality and Fertility Rates for Probabilistic Population Projections. In: Schoen R. (eds) Dynamic Demographic Analysis. The Springer Series on Demographic Methods and Population Analysis, vol 39. Springer, Cham

See Also

cokannisto.estimate, kannisto.predict

Examples

data(mxM, mxF, package = "wpp2017")
country <- "South Africa"
mxm <- subset(mxM, name == country)[,-(1:3)]
mxf <- subset(mxF, name == country)[,-(1:3)]
rownames(mxm) <- rownames(mxf) <- c(0,1, seq(5, 100, by=5))
mxnew <- cokannisto(mxm, mxf)
ages <- as.integer(rownames(mxnew$male))
plot(ages, mxnew$male[,"2095-2100"], type="l", log="y", 
    xlab="age", ylab="mx", col="blue", main=country)
lines(ages, mxnew$female[,"2095-2100"], col="red")
lines(ages, mxnew$male[,"2010-2015"], lty=2, col="blue")
lines(ages, mxnew$female[,"2010-2015"], lty=2, col="red")
legend("bottomright", legend=c("male 2010-2015", "female 2010-2015",
    "male 2095-2100", "female 2095-2100"), bty="n",
    col=rep(c("blue", "red"),2), lty=c(2,2,1,1))

Coherent Kannisto Estimation

Description

Estimate the coherent Kannisto parameters as described in Sevcikova et al. (2016).

Usage

cokannisto.estimate(mxM, mxF, ages, fitted = TRUE)

Arguments

Details

Given the Kannisto equation logit(m_x) = \log(c) + dx, the Coherent Kannisto method estimates the d parameter jointly for male and female data, in order to prevent mortality crossovers in higher ages.

Value

List of two lists, one for male and one for female. Each of the two lists contains the following components:

coefficients:

named vector with the coherent Kannisto coefficients c and d. The d values are the same in both lists.

fitted.values:

the fitted values (not included if fitted is FALSE)

residuals:

input rates minus the fitted values (not included if fitted is FALSE)

References

Sevcikova H., Li N., Kantorova V., Gerland P., Raftery A.E. (2016). Age-Specific Mortality and Fertility Rates for Probabilistic Population Projections. In: Schoen R. (eds) Dynamic Demographic Analysis. The Springer Series on Demographic Methods and Population Analysis, vol 39. Springer, Cham

See Also

cokannisto, kannisto.predict, kannisto

Examples

data(mxM, mxF, package = "wpp2017")
country <- "Brazil"
mxm <- subset(mxM, name == country)[,"2010-2015"]
mxf <- subset(mxF, name == country)[,"2010-2015"]
cokannisto.estimate(mxm[18:21], mxf[18:21], ages = 18:21)

Kannisto Method

Description

Extrapolate given mortality rates using the original Kannisto method.

Usage

kannisto(mx, est.ages = seq(80, 95, by = 5), proj.ages = seq(100, 130, by = 5))

Arguments

Details

The function first estimates the original Kannisto parameters by passing mortality rates for age groups est.ages into the kannisto.estimate function. The estimated parameters are then passed to the projection function kannisto.predict to extrapolate into ages proj.ages. Lastly, the input mortality object is extended by results for the extrapolated ages. If proj.ages contains age groups that are included in mx, values for those age groups are overwritten.

Value

A vector or matrix containing the input mortality object mx extended by the extrapolated age groups.

References

Thatcher, A. R., Kannisto, V. and Vaupel, J. W. (1998). The Force of Mortality at Ages 80 to 120, volume 5 of Odense Monographs on Population Aging Series. Odense, Denmark: Odense University Press.

See Also

kannisto.estimate, kannisto.predict, cokannisto

Examples

data(mxM, package = "wpp2017")
mx <- subset(mxM, name == "Burkina Faso")[,-(1:3)]
rownames(mx) <- c(0,1, seq(5, 100, by=5))
mxnew <- kannisto(mx)
ages <- as.integer(rownames(mxnew))
plot(ages, mxnew[,"2095-2100"], type="l", log="y", 
    xlab="age", ylab="mx", col="red")
lines(ages, mxnew[,"2010-2015"])

# Kannisto for 1-year age groups
# derive toy 1-year mx using model life tables at e0 of 70
mx1y <- mlt(70, sex = "male", nx = 1)
# Pretend we only observed mx for ages 0:100. 
# Use 90-99 for estimation and extend mx from 100 to 140
mx1ynew <- kannisto(mx1y[1:100, , drop = FALSE], est.ages = 90:99, proj.ages = 100:140)
# Plot the new mx for old ages
plot(80:140, mx1ynew[81:141], type = "l", xlab="age", ylab="mx", col="red")
# Check how it compares to the original mx that was not used in the estimation
lines(100:130, mx1y[101:nrow(mx1y)])

Kannisto Estimation

Description

Estimate the Kannisto parameters (Thatcher et al. 1998).

Usage

kannisto.estimate(mx, ages)

Arguments

Details

Given the Kannisto equation logit(m_x) = \log(c) + dx, the function estimates the c and d parameters using values of ages as the covariate x.

Value

List with the following components:

coefficients:

named vector with Kannisto coefficients c and d.

fitted.values:

the fitted values

residuals:

input rates minus the fitted values

References

Thatcher, A. R., Kannisto, V. and Vaupel, J. W. (1998). The Force of Mortality at Ages 80 to 120, volume 5 of Odense Monographs on Population Aging Series. Odense, Denmark: Odense University Press.

See Also

kannisto.predict, kannisto, cokannisto.estimate

Examples

data(mxM, package = "wpp2017")
mx <- subset(mxM, name == "Canada")[,"2010-2015"]
kannisto.estimate(mx[18:21], ages = 18:21)

Kannisto Prediction

Description

Given estimated Kannisto parameters (coherent or original), it predicts mortality rates for given ages.

Usage

kannisto.predict(pars, ages)

Arguments

Details

Given parameters c and d in pars, the function uses the Kannisto equation logit(m_x) = \log(c) + dx, to predict mortality rates for age groups x given by ages.

Value

Vector of predicted mortality rates.

References

Thatcher, A. R., Kannisto, V. and Vaupel, J. W. (1998). The Force of Mortality at Ages 80 to 120, volume 5 of Odense Monographs on Population Aging Series. Odense, Denmark: Odense University Press.

See Also

cokannisto, kannisto.estimate, cokannisto.estimate

Examples

data(mxM, mxF, package = "wpp2017")
mxm <- subset(mxM, name == "Germany")[,"2010-2015"]
ages <- c(0, 1, seq(5, 130, by=5))

# using original Kannisto parameters
pars <- kannisto.estimate(mxm[18:21], ages = ages[18:21])
mxm.pred <- kannisto.predict(pars$coefficients, ages = ages[22:28])
plot(ages, c(mxm[1:21], mxm.pred), type="l", log="y", 
    xlab="age", ylab="mx")
    
# Coherent Kannisto
mxf <- subset(mxF, name == "Germany")[,"2010-2015"]
copars <- cokannisto.estimate(
   mxm[18:21], mxf[18:21], ages = ages[18:21])
cmxm.pred <- kannisto.predict(copars[["male"]]$coefficients, ages = ages[22:28])
cmxf.pred <- kannisto.predict(copars[["female"]]$coefficients, ages = ages[22:28])
plot(ages, c(mxm[1:21], cmxm.pred), type="l", log="y", 
    xlab="age", ylab="mx", col="blue")
lines(ages, c(mxf[1:21], cmxf.pred), col="red")

Lee-Carter Estimation

Description

Estimate Lee-Carter parameters (Lee and Carter 1992).

Usage

leecarter.estimate(
  mx,
  ax.index = NULL,
  ax.smooth = FALSE,
  ax.smooth.df = NULL,
  bx.postprocess = TRUE,
  nx = 5
)

Arguments

Details

The function estimates parameters of \log(m_x(t)) = a_x + b_x k(t) + \epsilon_x(t) (Lee and Carter 1992). The argument ax.index determines which time periods to use to estimate the a_x parameter, while ax.smooth controls if the resulting a_x should be smoothened over ages (see Sevcikova et al. 2016 for details).

Value

List with elements ax, bx and kt corresponding to the estimated parameters.

References

Lee, R. D. and Carter, L. (1992). Modeling and forecasting the time series of US mortality. Journal of the American Statistical Association, 87, 659-671.

Sevcikova H., Li N., Kantorova V., Gerland P., Raftery A.E. (2016). Age-Specific Mortality and Fertility Rates for Probabilistic Population Projections. In: Schoen R. (eds) Dynamic Demographic Analysis. The Springer Series on Demographic Methods and Population Analysis, vol 39. Springer, Cham

See Also

mortcast, lileecarter.estimate

Examples

data(mxM, package = "wpp2017")
mx <- subset(mxM, name == "Netherlands")[,4:16]
rownames(mx) <- c(0,1, seq(5, 100, by=5))
lc.ax.avg <- leecarter.estimate(mx)
lc.ax.last <- leecarter.estimate(mx, ax.index=ncol(mx))
plot(lc.ax.avg$ax, type="l")
lines(lc.ax.last$ax, col="blue")

Life Table Function

Description

Function for obtaining life table quantities from mortality rates.

Usage

life.table(
  mx,
  sex = c("male", "female", "total"),
  abridged = TRUE,
  a0rule = c("ak", "cd"),
  radix = 1,
  open.age = 130
)

Arguments

Details

Computes a life table corresponding to given mortality rates for either 5- or 1-year age groups. The implementation follows Preston et al. (2001).

Value

Data frame with rows corresponding to age groups and the following columns:

age

Starting year of the age group.

mx

Age-specific mortality rates as passed into the mx argument.

qx

Probability of dying between ages x and x+n.

lx

Number of survivors at age x.

dx

Number of deaths between ages x and x+n.

Lx

Person-years lived between ages x and x+n.

sx

Survival rate from age x to x+n. Note that in an abridged life table, sx always refers to 5-year intervals. Here, sx in the first row is the survival from births to the second age group, sx in the second row is the survival from age 0-4 to age 5-9, third row has the survival from 5-9 to 10-14 etc.

Tx

Person-years lived after age x.

ex

Life expectancy at age x.

ax

Average person-years lived in the interval by those dying in the interval. For young ages, it follows Preston et al. (2001), Table 3.3 on page 48. If a0rule is "ak" (default) the Andreev-Kingkade method is used for a0. For compatibility with computations done at the UN, we set ax for ages 5 and 10 in the abridged version to 2.5. For an unabridged life table, ax is set to 0.5 for all but first and last age groups.

References

Preston, S.H., Heuveline, P., Guillot, M. (2001). Demography: Measuring and Modeling Population Processes. Oxford: Blackwell Publishers Ltd.

Andreev, E.M. and Kingkade, W.W. (2015). Average age at death in infancy and infant mortality level: Reconsidering the Coale-Demeny formulas at current levels of low mortality. Demographic Research, 33(13), p.363-390. DOI: 10.4054/DemRes.2015.33.13

Examples

data(mxF, e0Fproj, package = "wpp2017")
# get female mortality of Mexico for the current year
country <- "Mexico"
mxf <- subset(mxF, name == country)[,"2010-2015"]
life.table(mxf, sex = "female")

Coherent Lee-Carter Estimation

Description

Estimate coherent Lee-Carter parameters (Li and Lee 2005).

Usage

lileecarter.estimate(mxM, mxF, nx = 5, ...)

Arguments

Details

The coherent Lee-Carter parameters for male and female mortality rates share the same b_x which is the average of the age-specific b_x parameters.

The function in addition computes the ultimate b^u_x as defined in Li et al. (2013) based on the coherent b_x.

Value

List containing elements bx (coherent b_x parameter), ultimate.bx (ultimate b^u_x parameter), ages (age groups), nx (age group interval), and lists female and male, each with the Lee-Carter parameters.

References

Li, N. and Lee, R. D. (2005). Coherent mortality forecasts for a group of populations: An extension of the Lee-Carter method. Demography, 42, 575-594.

Li, N., Lee, R. D. and Gerland, P. (2013). Extending the Lee-Carter method to model the rotation of age patterns of mortality decline for long-term projections. Demography, 50, 2037-2051.

Examples

data(mxM, mxF, package = "wpp2017")
country <- "Germany"
mxm <- subset(mxM, name == country)[,4:16]
mxf <- subset(mxF, name == country)[,4:16]
rownames(mxm) <- rownames(mxf) <- c(0,1, seq(5, 100, by=5))
lc <- lileecarter.estimate(mxm, mxf)
plot(lc$bx, type="l")
lines(lc$ultimate.bx, lty=2)

Log-Quadratic Mortality Model

Description

Predict age-specific mortality rates using the Log-Quadratic Mortality Model (Wilmoth et al. 2012).

Usage

logquad(
  e0,
  sex = c("male", "female", "total"),
  my.coefs = NULL,
  q5ranges = c(1e-04, 0.9),
  k = 0,
  keep.lt = FALSE,
  ...
)

logquadj(e0m, e0f, ...)

Arguments

Details

The LogQuad method in this implementation projects mortality rates using the equation

\log(m_x) = a_x + b_x h + c_x h^2 + v_x k

where a_x, b_x, c_x and v_x are age-specific coefficients, h = \log( 5q0 ) (i.e. reflects child mortality), and k should be chosen to match 45q15 (adult mortality) or set to 0 (default). The coefficients can be passed as inputs, or taken from the package default dataset LQcoef which are taken from https://u.demog.berkeley.edu/~jrw/LogQuad/.

For the given inputs and values of life expectancy e0, the function finds values of h that best match e0, using life tables and the bisection method. It returns the corresponding mortality schedule for each value of e0.

Function logquad is for one sex, while logquadj can be used for both sexes.

Value

Function logquad returns a list with the following elements: a matrix mx with the predicted mortality rates. If keep.lt is TRUE, it also contains matrices sr (survival rates), and life table quantities Lx and lx. Function logquadj returns a list of objects, one for each sex.

References

Wilmoth, J., Zureick, S., Canudas-Romo, V., Inoue, M., Sawyer, C. (2012). A Flexible Two-Dimensional Mortality Model for Use in Indirect Estimation. Population studies, 66(1), 1-28. doi:10.1080/00324728.2011.611411

See Also

LQcoef, mortcast.blend, mortcast, pmd, mlt

Examples

data(e0Mproj, package = "wpp2017")
country <- "Brazil"
# get target e0
e0m <- as.numeric(subset(e0Mproj, name == country)[-(1:2)])
# project into future
pred <- logquad(e0m, sex = "male")
# plot first projection in black and the remaining ones in heat colors 
plot(pred$mx[,1], type = "l", log = "y", ylim = range(pred$mx),
    ylab = "male mx", xlab = "Age", main = country)
for(i in 2:ncol(pred$mx)) lines(pred$mx[,i], 
    col = heat.colors(20)[i])
    

Model Life Tables Mortality Patterns

Description

Predict age-specific mortality rates using Coale-Demeny and UN model life tables.

Usage

mlt(e0, sex = c("male", "female"), type = "CD_West", nx = 5, ...)

mltj(e0m, e0f, ..., nx = 5)

Arguments

Details

Given a level of life expectancy (e0), sex and a type of model life table, the function extracts the corresponding mortality pattern from MLTlookup (for abridged LT) or MLT1Ylookup (for 1-year LT), while interpolating between neighboring e0 groups. Function mlt is for one sex, while mltj can be used for both sexes.

Value

Function mlt returns a matrix with the predicted mortality rates. Columns correspond to the values in the e0 vector and rows correspond to age groups. Function mltj returns a list of such matrices, one for each sex.

References

https://www.un.org/development/desa/pd/data/extended-model-life-tables

Coale, A., P. Demeny, and B. Vaughn. 1983. Regional model life tables and stable populations. 2nd ed. New York: Academic Press.

See Also

mortcast, mortcast.blend, pmd, MLTlookup

Examples

data(e0Fproj, package = "wpp2017")
country <- "Uganda"
# get target e0
e0f <- subset(e0Fproj, name == country)[-(1:2)]
# project into future using life table Cole-Demeny North
mx <- mlt(e0f, sex = "female", type = "CD_North")
# plot first projection in black and the remaining ones in grey 
plot(mx[,1], type = "l", log = "y", ylim = range(mx),
    ylab = "female mx", xlab = "Age", 
    main = paste(country, "5-year age groups"))
for(i in 2:ncol(mx)) lines(mx[,i], col = "grey")

# MLT for 1-year age groups
mx1y <- mlt(e0f, sex = "female", type = "CD_North", nx = 1)
plot(mx1y[,1], type = "l", log = "y", ylim = range(mx1y),
    ylab = "female mx", xlab = "Age", 
    main = paste(country, "1-year age groups"))
for(i in 2:ncol(mx1y)) lines(mx1y[,i], col = "grey")
    

Coherent Rotated Lee-Carter Prediction

Description

Predict age-specific mortality rates using the coherent rotated Lee-Carter method.

Usage

mortcast(
  e0m,
  e0f,
  lc.pars,
  rotate = TRUE,
  keep.lt = FALSE,
  constrain.all.ages = FALSE,
  ...
)

Arguments

Details

This function implements Steps 6-9 of Algorithm 2 in Sevcikova et al. (2016). It uses the abridged or unabridged life table function to find the level of mortality that coresponds to the given life expectancy. Thus, it can be used for both, mortality for 5- or 1-year age groups.

Value

List with elements female and male, each of which contains a matrix mx with the predicted mortality rates. If keep.lt is TRUE, it also contains matrices sr (survival rates), and life table quantities Lx and lx.

References

Li, N., Lee, R. D. and Gerland, P. (2013). Extending the Lee-Carter method to model the rotation of age patterns of mortality decline for long-term projections. Demography, 50, 2037-2051.

Sevcikova H., Li N., Kantorova V., Gerland P., Raftery A.E. (2016). Age-Specific Mortality and Fertility Rates for Probabilistic Population Projections. In: Schoen R. (eds) Dynamic Demographic Analysis. The Springer Series on Demographic Methods and Population Analysis, vol 39. Springer, Cham

See Also

rotate.leecarter, leecarter.estimate, lileecarter.estimate, mortcast.blend

Examples

# estimate parameters from historical mortality data (5-year age groups)
data(mxM, mxF, e0Fproj, e0Mproj, package = "wpp2017")
country <- "Brazil"
mxm <- subset(mxM, name == country)[,4:16]
mxf <- subset(mxF, name == country)[,4:16]
rownames(mxm) <- rownames(mxf) <- c(0,1, seq(5, 100, by=5))
lc <- lileecarter.estimate(mxm, mxf)

# project into future for given levels of life expectancy
e0f <- as.numeric(subset(e0Fproj, name == country)[-(1:2)])
e0m <- as.numeric(subset(e0Mproj, name == country)[-(1:2)])
pred <- mortcast(e0m, e0f, lc)

# plot first projection in black and the remaining ones in grey 
plot(lc$ages, pred$female$mx[,1], type="b", log="y", ylim=range(pred$female$mx),
    ylab="female mx", xlab="Age", main=paste(country, "(5-year age groups)"), cex=0.5)
for(i in 2:ncol(pred$female$mx)) lines(lc$ages, pred$female$mx[,i], col="grey")

# similarly for 1-year age groups
# derive toy 1-year mx using model life tables at given level of e0
mxm1y <- mlt(seq(65, 71, length = 4), sex = "male", nx = 1)
mxf1y <- mlt(seq(73, 78, length = 4), sex = "female", nx = 1)

# estimate parameters
lc1y <- lileecarter.estimate(mxm1y, mxf1y, nx = 1)
 
# project into the future 
pred1y <- mortcast(e0m, e0f, lc1y)

# plot first projection in black and the remaining ones in grey 
plot(lc1y$ages, pred1y$female$mx[,1], type="b", log="y", ylim=range(pred1y$female$mx),
    ylab="female mx", xlab="Age", main="1-year age groups", cex=0.5)
for(i in 2:ncol(pred1y$female$mx)) lines(lc1y$ages, pred1y$female$mx[,i], col="grey")

Mortality Prediction by Method Blending

Description

Predict age-specific mortality rates using a blend of two different methods (Coherent Lee-Carter, Coherent Pattern Mortality Decline, Log-Quadratic model, or Model Life Tables). Weights can be applied to fine-tune the blending mix.

Usage

mortcast.blend(
  e0m,
  e0f,
  meth1 = "lc",
  meth2 = "mlt",
  weights = c(1, 0.5),
  nx = 5,
  apply.kannisto = TRUE,
  min.age.groups = if (nx == 5) 28 else 131,
  match.e0 = TRUE,
  keep.lt = FALSE,
  meth1.args = NULL,
  meth2.args = NULL,
  kannisto.args = NULL,
  ...
)

Arguments

Details

The function allows to combine two different methods using given weights. The weights can change over time - by default they are interpolated from the starting weight to the end weight. As the blended mortality rates do not necessarily match the target life expectancy, scaling is applied to improve the match, controlled by the match.e0 argument. The projection is done for both sexes, so that coherent methods can be applied.

Value

List with elements female and male, each of which contains a matrix mx with the predicted mortality rates. If the result has been scaled (match.e0 is TRUE), the element mx.rawblend contains the mx before scaling. Also in such a case, if keep.lt is TRUE, it also contains matrices sr (survival rates), and life table quantities Lx and lx. In addition, the return object contains elements meth1res and meth2res which contain the results of the functions corresponding to the two methods. Elements meth1 and meth2 contain the names of the methods. A vector weights contains the final (possibly interpolated) weights.

See Also

mortcast, copmd, mltj, logquad, cokannisto

Examples

data(mxM, mxF, e0Fproj, e0Mproj, package = "wpp2017")
country <- "Brazil"
# estimate parameters from historical mortality data
mxm <- subset(mxM, name == country)[,4:16]
mxf <- subset(mxF, name == country)[,4:16]
rownames(mxm) <- rownames(mxf) <- c(0,1, seq(5, 100, by=5))
lcest <- lileecarter.estimate(mxm, mxf)
# project into future
e0f <- subset(e0Fproj, name == country)[-(1:2)]
e0m <- subset(e0Mproj, name == country)[-(1:2)]

# Blend LC and MLT
pred1 <- mortcast.blend(e0m, e0f, meth1 = "lc", meth2 = "mlt",
    meth1.args = list(lc.pars = lcest),
    meth2.args = list(type = "CD_North"),
    weights = c(1,0.25))
    
# Blend PMD and MLT
pred2 <- mortcast.blend(e0m, e0f, meth1 = "pmd", meth2 = "mlt",
    meth1.args = list(mxm0 = mxm[, "2010-2015"],
                      mxf0 = mxf[, "2010-2015"]))
                      
# plot projection by time
plotmx <- function(pred, iage, main) 
    with(pred, {
        # blended projections 
        plot(female$mx[iage,], type="l", 
            ylim=range(meth1res$female$mx[iage,], 
                       meth2res$female$mx[iage,]),
            ylab="female mx", xlab="Time", main=main, col = "red")
        lines(meth1res$female$mx[iage,], lty = 2)
        lines(meth2res$female$mx[iage,], lty = 3)
        legend("topright", legend=c("blend", meth1, meth2),
               lty = 1:3, col = c("red", "black", "black"), bty = "n")
    })
age.group <- 3 # 5-9 years old
par(mfrow=c(1,2))
plotmx(pred1, age.group, "LC-MLT (age 5-9)")
plotmx(pred2, age.group, "PMD-MLT (age 5-9)")

# Blend LC and MLT for 1-year age groups
#########################################
# First interpolate e0 to get 1-year life expectancies (for first five years)
e0m1y <- approx(as.double(e0m[,1:2]), n = 5)$y
e0f1y <- approx(as.double(e0f[,1:2]), n = 5)$y
# derive toy mx in order to get some LC parameters
mxm1y <- mlt(seq(70, 72, length = 4), sex = "male", nx = 1)
mxf1y <- mlt(seq(78, 79, length = 4), sex = "female", nx = 1)
lcest1y <- lileecarter.estimate(mxm1y, mxf1y, nx = 1)

# projections
pred3 <- mortcast.blend(e0m1y, e0f1y, meth1 = "lc", meth2 = "mlt",
    weights = c(1,0.25), min.age.groups = 131, nx = 1, 
    meth1.args = list(lc.pars = lcest1y),
    kannisto.args = list(est.ages = 90:99, proj.ages = 100:130))
    
# plot results
par(mfrow=c(1,1))
plot(0:130, pred3$female$mx[,5], log = "y", type = "l", col = "red")
lines(0:130, pred3$male$mx[,5], col = "blue")

Pattern of Mortality Decline Prediction

Description

Predict age-specific mortality rates using the Pattern of mortality decline (PMD) method (Andreev et al. 2013).

Usage

pmd(
  e0,
  mx0,
  sex = c("male", "female"),
  nx = 5,
  interp.rho = FALSE,
  kranges = c(0, 25),
  keep.lt = FALSE,
  keep.rho = FALSE,
  ...
)

modpmd(
  e0,
  mx0,
  sex = c("male", "female"),
  nx = 5,
  interp.rho = FALSE,
  kranges = c(0, 25),
  ax.index = NULL,
  ax.smooth = FALSE,
  ax.smooth.df = NULL,
  keep.lt = FALSE,
  keep.rho = FALSE,
  ...
)

copmd(
  e0m,
  e0f,
  mxm0,
  mxf0,
  nx = 5,
  interp.rho = FALSE,
  keep.rho = FALSE,
  use.modpmd = FALSE,
  ...
)

Arguments

Details

These functions implements the PMD method introduced in Andreev et al. (2013) and its modifications. It assumes that the future decline in age-specific mortality will follow a certain pattern with the increase in life expectancy at birth (e0):

log[mx(t)] = log[mx(t-1)] - k(t) \rho_x(t)

Here, \rho_x(t) is the age-specific pattern of mortality decline between t-1 and t. Such patterns for each sex and various levels of e0 are stored in the dataset PMDrho. The pmd function can be instructed to interpolate between neighboring levels of e0 by setting the argument interp.rho to TRUE. The k parameter is estimated to match the e0 level using the bisection method.

Function pmd evaluates the method for a single sex, while copmd does it coherently for both sexes. In the latter case, the same \rho_x (namely the average over sex-specific \rho_x) is used for both, male and female.

Function modpmd implements a modified version of pmd where the initial log[mx(t_0)] is replaced by an a_x estimated as in leecarter.estimate, i.e. using possibly multiple years of historical mx and optionally smoothed. Arguments ax.index, ax.smooth and ax.smooth.df determine the estimation years and parameters of the smoothing.

Value

Function pmd and modpmd return a list with the following elements: a matrix mx with the predicted mortality rates. If keep.lt is TRUE, it also contains matrices sr (survival rates), and life table quantities Lx and lx. If keep.rho is TRUE, it contains a matrix rho where columns correspond to the values in the e0 vector and rows correspond to age groups.

Function copmd returns a list with one element for each sex (male and female) where each of them is a list as described above. In addition if keep.rho is TRUE, element rho.sex gives the sex-dependent (i.e. not averaged) \rho_x coefficient.

References

Andreev, K., Gu, D., Gerland, P. (2013). Age Patterns of Mortality Improvement by Level of Life Expectancy at Birth with Applications to Mortality Projections. Paper presented at the Annual Meeting of the Population Association of America, New Orleans, LA. https://paa2013.populationassociation.org/papers/132554.

Gu, D., Pelletier, F., Sawyer, C. (2017). Projecting Age-sex-specific Mortality: A Comparison of the Modified Lee-Carter and Pattern of Mortality Decline Methods, UN Population Division, Technical Paper No. 6. New York: United Nations. https://population.un.org/wpp/Publications/Files/WPP2017_TechnicalPaperNo6.pdf

See Also

mortcast, mortcast.blend, PMDrho

Examples

data(mxF, e0Fproj, package = "wpp2017")
country <- "Hungary"
# get initial mortality for the current year
mxf <- subset(mxF, name == country)[,"2010-2015"]
names(mxf) <- c(0,1, seq(5, 100, by=5))
# get target e0
e0f <- subset(e0Fproj, name == country)[-(1:2)]
# project into future
pred <- pmd(e0f, mxf, sex = "female")
# plot first projection in black and the remaining ones in grey 
plot(pred$mx[,1], type = "l", log = "y", ylim = range(pred$mx),
    ylab = "female mx", xlab = "Age", main = country)
for(i in 2:ncol(pred$mx)) lines(pred$mx[,i], col = "grey")

Rotated Lee-Carter

Description

Rotate the Lee-Carter parameter b_x over time to reach an ultimate b^u_x, as described in Li et al. (2013).

Usage

rotate.leecarter(bx, ultimate.bx, e0, e0l = 80, e0u = 102, p = 0.5)

ultimate.bx(bx)

Arguments

Value

Function rotate.leecarter returns a matrix of rotated B_x(t) where rows correspond to age groups and columns correspond to time periods (given by the vector e0).

Function ultimate.bx returns a vector of the ultimate b^u_x.

References

Li, N., Lee, R. D. and Gerland, P. (2013). Extending the Lee-Carter method to model the rotation of age patterns of mortality decline for long-term projections. Demography, 50, 2037-2051.

Examples

data(mxF, mxM, e0Fproj, e0Mproj, package = "wpp2017")
country <- "Japan"
mxm <- subset(mxM, name == country)[,4:16]
mxf <- subset(mxF, name == country)[,4:16]
e0f <- as.numeric(subset(e0Fproj, name == country)[-(1:2)])
e0m <- as.numeric(subset(e0Mproj, name == country)[-(1:2)])
rownames(mxm) <- rownames(mxf) <- c(0,1, seq(5, 100, by=5))
lc <- lileecarter.estimate(mxm, mxf)
rotlc <- rotate.leecarter(lc$bx, lc$ultimate.bx, (e0f + e0m)/2)
plot(lc$bx, type="l")
lines(lc$ultimate.bx, col="red")
for(i in 1:ncol(rotlc)) lines(rotlc[,i], col="grey")