Package ‘MortCast’

May 29, 2020

Type Package
Title Estimation and Projection of Age-Specific Mortality Rates
Version 2.3-0
Date 2020-05-28
Author Hana Sevcikova, Nan Li and Patrick Gerland
Maintainer Hana Sevcikova <hanas@uw.edu>
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>.
License GPL (>= 2)
Depends R (>= 3.5.0), wpp2017
RoxygenNote 7.0.2
LazyLoad True
LazyData True
NeedsCompilation yes
Repository CRAN
Date/Publication 2020-05-29 05:10:17 UTC

R topics documented:
MortCast-package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
cokannisto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
cokannisto.estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
kannisto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
kannisto.estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
kannisto.predict . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
leecarter.estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
life.table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
lileecarter.estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
logquad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
LQcoef . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1
2 MortCast-package

mlt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
MLTlookup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
mortcast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
mortcast.blend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
pmd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
PMDadjcoef . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
PMDrho . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
rotate.leecarter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Index 30

MortCast-package 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 esti-
mating 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 ax and kt , and the coherent
parameter bx . In addition, it computes the ultimate bux 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 bx param-
eter 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.

MortCast-package 3

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))
4 cokannisto

cokannisto 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

mxM A vector or matrix of male mortality rates. If it is a matrix, rows correspond

to age groups with rownames identifying the corresponding age as integers. By
default five-years age groups are assigned to rows if rownames are not given.
mxF A vector or matrix of female mortality rates. Its length or dimension should be
the same mxM.
est.ages A vector of integers identifying the ages to be used for estimation. It should be
a subset of rownames of mxM.
proj.ages A vector of integers identifying the age groups for which mortality rates are to
be projected.

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 ex-
tended by the extrapolated age groups.
cokannisto.estimate 5

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))

cokannisto.estimate 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
mxM A vector of male mortality rates.
mxF A vector of female mortality rates.
ages A vector of ages corresponding to mxM and mxF.
fitted Logical. If TRUE the fitted values and residuals are returned.
6 kannisto

Details
Given the Kannisto equation logit(mx ) = 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 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))
kannisto 7

Arguments

mx A vector or matrix of mortality rates. If it is a matrix, rows correspond to age

groups with rownames identifying the corresponding age as integers. By default
five-years age groups are assigned to rows if rownames are not given.
est.ages A vector of integers identifying the ages to be used for estimation. It should be
a subset of rownames of mx.
proj.ages A vector of integers identifying the age groups for which mortality rates are to
be projected.

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"])
8 kannisto.estimate

kannisto.estimate Kannisto Estimation

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

Usage
kannisto.estimate(mx, ages)

Arguments
mx A vector of mortality rates.
ages A vector of ages corresponding to mx.

Details
Given the Kannisto equation logit(mx ) = 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.predict 9

kannisto.predict Kannisto Prediction

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

Usage
kannisto.predict(pars, ages)

Arguments
pars A named vector with Kanisto coefficients c and d (e.g. result of kannisto.estimate
or cokannisto.estimate).
ages A vector of ages to make prediction for.

Details
Given parameters c and d in pars, the function uses the Kannisto equation logit(mx ) = 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")
10 leecarter.estimate

# 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")

leecarter.estimate Lee-Carter Estimation

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

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

Arguments
mx A matrix of age-specific mortality rates where rows correspond to age groups
and columns correspond to time periods. Rownames define the starting ages of
the age groups.
ax.index A vector of column indices of mx to be used to estimate the ax parameter. By
default all time periods are used.
ax.smooth Logical allowing to smooth the ax over ages.
bx.postprocess Logical determining if numerical anomalies in bx should be dealt with.
nx Size of age groups. By default ages are determined by rownames of mx. This
argument is only used if mx has no rownames. If nx is 5, the age groups are
interpreted as 0, 1, 5, 10, . . . . For nx equals 1, the age groups are interpreted as
0, 1, 2, 3, . . . .

Details
The function estimates parameters of log(mx (t)) = ax + bx k(t) + x (t) (Lee and Carter 1992).
The argument ax.index determines which time periods to use to estimate the ax parameter, while
ax.smooth controls if the resulting ax should be smoothened over ages (see Sevcikova et al. 2016
for details).
life.table 11

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 Life Table Function

Description

Function for obtaining life table quantities from mortality rates.

Usage

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

Arguments
mx Vector of age-specific mortality rates nmx. If abridged is TRUE (default), the
elements correspond to 1m0, 4m1, 5m5, 5m10, . . . . If abridged is FALSE, they
correspond to 1m0, 1m1, 1m2, 1m3, . . . .
sex Which sex the mortality rates correspond to.
abridged Is it an abridged life table (TRUE, default) or not (FALSE). In the former case, the
mx vector is interpreted as corresponding to age groups 0, 1-4, 5-9, 10-14, . . . .
If FALSE, the mx vector is interpreted as corresponding to one-year age groups,
i.e. 0, 1, 2, 3, . . . .
radix Base of the life table.
open.age Open age group. If smaller than the last age group of mx, the life table is trun-
cated. It does not have any effect if larger than the last age group.

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), including the choice of ax (see Table 3.3 on page 48).

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. 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.
lileecarter.estimate 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")

lileecarter.estimate Coherent Lee-Carter Estimation

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

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

Arguments
mxM A matrix of male age-specific mortality rates where rows correspond to age
groups and columns correspond to time periods. For 5-year age groups, the first
and second rows corresponds to 0-1 and 1-5 age groups, respectively. Row-
names define the starting ages of the respective groups.
mxF A matrix of female mortality rates of the same shape as mxM.
nx Size of age groups. Should be either 5 or 1.
... Additional arguments passed to leecarter.estimate.

Details
The coherent Lee-Carter parameters for male and female mortality rates share the same bx which is
the average of the age-specific bx parameters.
The function in addition computes the ultimate bux as defined in Li et al. (2013) based on the
coherent bx .

Value
List containing elements bx (coherent bx parameter), ultimate.bx (ultimate bux parameter), ages
(age groups), nx (age group interval), and lists female and male, each with the Lee-Carter parame-
ters.

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.
14 logquad

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)

logquad 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
e0 Vector of target life expectancies.
sex Which sex does the give e0 corresponds to.
my.coefs Data frame with columns “sex”, “age”, “ax”, “bx”, “cx”, “vx”. The “sex” col-
umn should contain values “female”, “male” and/or “total”. The “age” column
must be sorted so that it assures that rows correspond to ages in increasing or-
der. Any NAs are internally converted to zeros. If not given, the dataset LQcoef
is used.
q5ranges A vector of size two, giving the min and max of 5q0 used in the bisection
method.
k Value of the k parameter.
keep.lt Logical. If TRUE additional life table columns are kept in the resulting object.
e0m A time series of target male life expectancy.
e0f A time series of target female life expectancy.
... Additional arguments passed to the underlying function.
logquad 15

Details

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

log(mx ) = ax + bx h + cx h2 + vx k

where ax , bx , cx and vx 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
http://www.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 mortal-
ity 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])
16 LQcoef

LQcoef 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

http://www.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)
mlt 17

mlt 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, ...)

Arguments
e0 A time series of target life expectancy.
sex Either "male" or "female".
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”.
nx Size of age groups. Should be either 5 or 1.
e0m A time series of target male life expectancy.
e0f A time series of target female life expectancy.
... Additional arguments passed to the underlying function.

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.
18 MLTlookup

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")

MLTlookup 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.
mortcast 19

e0 Level of life expectancy, starting at 20 and going by steps of 2.5 up to 130.

mx Mortality rates.
lx, Lx, sx Other life table columns.

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)

mortcast 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
e0m A time series of future male life expectancy.
e0f A time series of future female life expectancy.
20 mortcast

lc.pars A list of coherent Lee-Carter parameters with elements bx, ultimate.bx, ages,
nx, female and male as returned by lileecarter.estimate. The female and
male objects are again lists that should contain a vector ax and optionally a
matrix axt if the ax parameter needs to be defined as time dependent. In such
a case, rows are age groups and columns are time periods corresponding to the
length of the e0f and e0m vectors.
rotate If TRUE the rotation method of bx is used as described in Li et al. (2013).
keep.lt Logical. If TRUE additional life table columns are kept in the resulting object.
constrain.all.ages
By default the method constrains the male mortality to be above female mortality
for old ages if the male life expectancy is below the female life expectancy.
Setting this argument to TRUE causes this constraint to be applied to all ages.

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

mortcast.blend 21

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

# interpolate to get toy 1-year mx for estimation
interp <- function(x)
approx(c(0,1, seq(5, 100, by=5)), x, xout = seq(0, 100), method = "linear")$y
mxm1y <- apply(mxm, 2, interp)
mxf1y <- apply(mxf, 2, interp)
rownames(mxm1y) <- rownames(mxf1y) <- seq(0, 100)

# estimate parameters
lc1y <- lileecarter.estimate(mxm1y, mxf1y, nx = 1)

# project into 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$mx),
ylab="female mx", xlab="Age", main=paste(country, "(1-year age groups)"), cex=0.5)
for(i in 2:ncol(pred1y$female$mx)) lines(lc1y$ages, pred1y$female$mx[,i], col="grey")

mortcast.blend 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, 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),
apply.kannisto = TRUE,
min.age.groups = 28,
meth1.args = NULL,
22 mortcast.blend

meth2.args = NULL,
kannisto.args = NULL
)

Arguments

e0m A time series of future male life expectancy.

e0f A time series of future female life expectancy.
meth1 Character string giving the name of the first method to blend. It is one of “lc”,
“pmd”, “mlt” or “logquad”, corresponding to Coherent Lee-Carter (function
mortcast), Pattern Mortality Decline (function copmd), Log-Quadratic model
(function logquadj), and Model Life Tables (function mltj), respectively.
meth2 Character string giving the name of the second method to blend. One of the
same choices as meth1.
weights Numeric vector with values between 0 and 1 giving the weight of meth1. If
it is a single value, the same weight is applied for all time periods. If it is a
vector of size two, it is assumed these are weights for the first and the last time
period. Remaining weights will be interpolated. Note that meth2 is weighted by
1 -weights.
apply.kannisto Logical. If TRUE and if any of the methods results in less than min.age.groups
age categories, the coherent Kannisto method (cokannisto) is applied to extend
the age groups into old ages.
min.age.groups Minimum number of age groups. Triggers the application of Kannisto, see
above.
meth1.args List of arguments passed to the function that corresponds to meth1.
meth2.args List of arguments passed to the function that corresponds to meth2.
kannisto.args List of arguments passed to the cokannisto function if Kannisto is applied.

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. The projec-
tion 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. In addition, it 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

pmd 23

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)")

pmd Pattern of Mortality Decline Prediction

Description

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

Usage

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

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

Arguments

e0 A vector of target life expectancy, one element for each predicted time point.
mx0 A vector with starting age-specific mortality rates.
sex Either "male" or "female".
nx Size of age groups. Should be either 5 or 1.
interp.rho Logical controlling if the ρ coefficients should be interpolated (TRUE) or if the
raw (binned) version should be used (FALSE), as stored in the dataset PMDrho.
kranges A vector of size two, giving the min and max of the k parameter which is esti-
mated to match the target e0 using the bisection method.
keep.lt Logical. If TRUE additional life table columns are kept in the resulting object.
keep.rho Logical. If TRUE the ρ coefficients are included in the resulting object.
e0m A time series of target male life expectancy.
e0f A time series of target female life expectancy.
mxm0 A vector with starting age-specific male mortality rates.
mxf0 A vector with starting age-specific female mortality rates.
... Additional arguments passed to the underlying function. For copmd, in addi-
tion to kranges and keep.lt, it can be sexratio.adjust which is a logical
controlling if a sex-ratio adjustment should be applied to prevent crossovers
between male and female mx. In such a case it uses coefficients from the
PMDadjcoef dataset. However, if the argument adjust.with.mxf is set to TRUE
(in addition to sexratio.adjust), the adjustment is done using the female mor-
tality rates as the lower constraint for male mortality rates. If the argument
adjust.sr.if.needed is set to TRUE, a sex-ratio adjustment is performed dy-
namically, using the sex ratio in the previous time point. In such a case, an
adjustment in time t is applied only if there was a drop of sex ratio below one at
time t-1.
pmd 25

Details
These functions implements the PMD method introduced in Andreev et al. (2013). 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)ρx (t)

Here, ρx (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 ρx (namely the average over sex-specific ρx ) is used for both, male and
female.

Value
Function pmd 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. If keep.rho is TRUE, it contains a matrix rho where columns correpond 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) ρx 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 An-
nual Meeting of the Population Association of America, New Orleans, LA. http://paa2013.
princeton.edu/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)]
26 PMDadjcoef

# 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")

PMDadjcoef 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 corre-
spond to age groups. The values are estimates of the following regression

log10 mxM = β0 + β1 log10 mxF + β2 eF F 2 F M

0 + β3 (e0 ) + β4 (e0 − e0 )

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 Compari-
son of the Modified Lee-Carter and Pattern of Mortality Decline Methods, UN Population Divi-
sion, Technical Paper No. 6. New York: United Nations. https://population.un.org/wpp/
Publications/Files/WPP2017_TechnicalPaperNo6.pdf

See Also

copmd
PMDrho 27

Examples
data(PMDadjcoef)
PMDadjcoef

PMDrho Pattern Mortality Decline Lookup Tables

Description
Data object containing two tables with ρ 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 An-
nual Meeting of the Population Association of America, New Orleans, LA. http://paa2013.
princeton.edu/papers/132554.
Gu, D., Pelletier, F. and Sawyer, C. (2017). Projecting Age-sex-specific Mortality: A Compari-
son of the Modified Lee-Carter and Pattern of Mortality Decline Methods, UN Population Divi-
sion, 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)]
28 rotate.leecarter

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")

rotate.leecarter Rotated Lee-Carter

Description

Rotate the Lee-Carter parameter bx over time to reach an ultimate bux , 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

bx A vector of the Lee-Carter bx parameter (from e.g. lileecarter.estimate or

leecarter.estimate).
ultimate.bx A vector of the ultimate bux parameter as defined in Li, Lee, Gerland (2013)
(obtained using lileecarter.estimate or ultimate.bx).
e0 A time series of life expectancies.
e0l Level of life expectancy at which the rotation starts.
e0u Level of life expectancy at which the rotation finishes.
p Exponent of the smooth function.

Value

Function rotate.leecarter returns a matrix of rotated Bx (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 bux .

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.
rotate.leecarter 29

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")
Index

∗Topic datasets ultimate.bx, 28

LQcoef, 16 ultimate.bx (rotate.leecarter), 28
MLTlookup, 18
PMDadjcoef, 26
PMDrho, 27

cokannisto, 2, 4, 6, 7, 9, 22
cokannisto.estimate, 4, 5, 5, 8, 9
copmd, 22, 26
copmd (pmd), 23

kannisto, 2, 6, 6, 8
kannisto.estimate, 7, 8, 9
kannisto.predict, 4–8, 9

leecarter.estimate, 2, 10, 13, 20, 28

life.table, 2, 11
lileecarter.estimate, 2, 11, 13, 20, 28
logquad, 2, 14, 16, 22
logquadj, 22
logquadj (logquad), 14
LQcoef, 14, 15, 16

mlt, 2, 15, 17, 19

MLT1Ylookup, 17
MLT1Ylookup (MLTlookup), 18
mltj, 22
mltj (mlt), 17
MLTlookup, 17, 18, 18
MortCast (MortCast-package), 2
mortcast, 2, 11, 15, 18, 19, 22, 25
MortCast-package, 2
mortcast.blend, 2, 15, 18, 20, 21, 25

pmd, 2, 15, 18, 23, 27

PMDadjcoef, 24, 26
PMDrho, 24, 25, 27

RhoFemales (PMDrho), 27
RhoMales (PMDrho), 27
rotate.leecarter, 20, 28

30