Package 'LEdecomp'

Description

Abridge a single-year mx schedule to 0,1,5,... using lifetable quantities

Usage

abridge_mx(mx, age, sex = "t", closeout = TRUE)

Arguments

Value

numeric vector of abridged mx at ages c(0, 1, 5, 10, ...)

Description

Lx is defined as the integration of lx in the interval ⁠[x,x+n)⁠, where n is the width of the interval. There are many approximations for this. Here we use HMD Method Protocol equation 78. You can think of Lx as lifetable exposure, or person-years lived in each age interval.

Usage

ald_to_Lx(ax, lx, dx, nx)

Arguments

Value

numeric vector of Lx values

References

Wilmoth JR, Andreev K, Jdanov D, Glei DART, Boe C, Bubenheim M, Philipov D, Shkolnikov V, Vachon P, Winant C, M B (2021). “Methods protocol for the human mortality database. Version 6.” University of California, Berkeley, and Max Planck Institute for Demographic Research, Rostock. URL: http://mortality.org [version 26/01/2021].

Description

Implements the age-interval component formula (telescoping form) from Andreev (1982): eps[x1,x2) = l_x1*(e2_x1 - e1_x1) - l_x2*(e2_x2 - e1_x2), where l_ and e_ without prime are from a chosen baseline life table.

Usage

andreev(
  mx1,
  mx2,
  age = 0:(length(mx1) - 1),
  nx = rep(1, length(mx1)),
  sex1 = "t",
  sex2 = sex1,
  closeout = TRUE
)

Arguments

Details

setting closeout to TRUE will result in value of 1/mx for the final age group, of ax and a value of 1 for the closeout of qx.

Value

cc numeric vector with one element per age group, and which sums to the total difference in life expectancy between population 1 and 2.

vector of age-specific contributions to e0 gap

References

Andreev EM, others (1982). “Metod komponent v analize prodoljitelnosty zjizni.[The method of components in the analysis of length of life].” Vestnik Statistiki, 9(3), 42–47.

Examples

a   <- .001
b   <- .07
x   <- 0:100
mx1 <- a * exp(x * b)
mx2 <- a/2 * exp(x * b)
cc <- andreev(mx1, mx2, age = x)

e01 <- mx_to_e0(mx1, age = x)
e02 <- mx_to_e0(mx2, age = x)
(delta <- e02 - e01)
sum(cc)

Description

This approach conducts a classic Andreev (1982) decomposition in both directions, averaging the (sign-adjusted) result, i.e. a_avg = (andreev(mx1,mx2, ...) - andreev(mx2, mx1, ...)) / 2. #@note The final age group's contribution from the reversed decomposition is halved before averaging. This empirical adjustment ensures symmetry and numeric stability, though the theoretical basis requires further exploration.

Usage

andreev_sym(
  mx1,
  mx2,
  age = 0:(length(mx1) - 1),
  nx = rep(1, length(mx1)),
  sex1 = "t",
  sex2 = sex1,
  closeout = TRUE
)

Arguments

Value

numeric vector of contributions summing to the gap in life expectancy implied by mx1 and mx2.

See Also

andreev

Examples

a <- .001
b <- .07
x <- 0:100
mx1 <- a * exp(x * b)
mx2 <- a/2 * exp(x * b)
d <- andreev_sym(mx1, mx2, age = x)

e01 <- mx_to_e0(mx1,age=x)
e02 <- mx_to_e0(mx2,age=x)
(Delta <- e02 - e01)
sum(d)

d12 <- andreev(mx1, mx2, age = x)
d21 <- andreev(mx2, mx1, age = x) # direction opposite

plot(x, d, type= 'l')
  lines(x, d12, col = "red")
  lines(x, -d21, col = "blue")

Description

Following the notation given in Preston et al (2000), Arriaga's decomposition method can written as:

$_{n}\Delta_{x} = \frac{l_x^1}{l_0^1}\cdot \left( \frac{_{n}L_{x}^{2}}{l_{x}^{2}} - \frac{_{n}L_{x}^{1}}{l_{x}^{1}}\right) + \frac{T^{2}_{x+n}}{l_{0}^{1}} \cdot \left( \frac{l_{x}^{1}}{l_{x}^{2}} - \frac{l_{x+n}^{1}}{l_{x+n}^{2}} \right)$

Usage

arriaga(
  mx1,
  mx2,
  age = 0:(length(mx1) - 1),
  nx = rep(1, length(mx1)),
  sex1 = "t",
  sex2 = sex1,
  closeout = TRUE
)

Arguments

Details

A little-known property of this decomposition method is that it is directional, in the sense that we are comparing a movement of mx1 to mx2, and this is not exactly symmetrical with a comparison of mx2 with mx1. Note also, if decomposing in reverse from the usual, you may need a slight adjustment to the closeout value in order to match sums properly (see examples for a demonstration).

setting closeout to TRUE will result in value of 1/mx for the final age group, of ax and a value of 1 for the closeout of qx.

Value

cc numeric vector with one element per age group, and which sums to the total difference in life expectancy between population 1 and 2.

References

Arriaga EE (1984). “Measuring and explaining the change in life expectancies.” Demography, 21, 83–96. Preston S, Heuveline P, Guillot M (2000). Demography: measuring and modeling population processes. Wiley-Blackwell.

Examples

a <- .001
b <- .07
x <- 0:100
mx1 <- a * exp(x * b)
mx2 <- a/2 * exp(x * b)
cc <- arriaga(mx1, mx2, age = x)
e01 <- mx_to_e0(mx1, age = x)
e02 <- mx_to_e0(mx2, age = x)
(delta <- e02 - e01)
sum(cc)


 plot(x, cc)

# asymmetrical with a decomposition in the opposite direction
cc2 <- -arriaga(mx1 = mx2, mx2 = mx1, age = x)
plot(x, cc)
lines(x,cc2)
# also we lose some precision?
sum(cc);sum(cc2)
# found it!
delta-sum(cc2); cc2[length(cc2)] / 2

# But this is no problem if closeout = FALSE
-arriaga(mx1 = mx2, mx2 = mx1, age = x,closeout=FALSE) |> sum()
arriaga(mx1 = mx1, mx2 = mx2, age = x,closeout=FALSE) |> sum()

Description

This approach conducts a classic Arriaga decomposition in both directions, averaging the (sign-adjusted) result, i.e. a_avg = (arriaga(mx1,mx2, ...) - arriaga(mx2, mx1, ...)) / 2. #@note The final age group's contribution from the reversed decomposition is halved before averaging. This empirical adjustment ensures symmetry and numeric stability, though the theoretical basis requires further exploration. Note this method is identical to Arriaga III as extended by Ponnapalli (2005). It is also equivalent to the symmetrical variants of Andreev and Lopez-Ruzicka, as well as chandrasekaran.

Usage

arriaga_sym(
  mx1,
  mx2,
  age = 0:(length(mx1) - 1),
  nx = rep(1, length(mx1)),
  sex1 = "t",
  sex2 = sex1,
  closeout = TRUE
)

Arguments

Value

numeric vector of contributions summing to the gap in life expectancy implied by mx1 and mx2.

References

Arriaga EE (1984). “Measuring and explaining the change in life expectancies.” Demography, 21, 83–96. Ponnapalli KM (2005). “A comparison of different methods for decomposition of changes in expectation of life at birth and differentials in life expectancy at birth.” Demographic Research, 12, 141–172.

See Also

arriaga

Examples

a <- .001
b <- .07
x <- 0:100
mx1 <- a * exp(x * b)
mx2 <- a/2 * exp(x * b)
d <- arriaga_sym(mx1, mx2, age = x)

e01 <- mx_to_e0(mx1,age=x)
e02 <- mx_to_e0(mx2,age=x)
(Delta <- e02 - e01)
sum(d)

d12 <- arriaga(mx1, mx2, age = x)
d21 <- arriaga(mx2, mx1, age = x) # direction opposite

plot(x, d, type= 'l')
  lines(x, d12, col = "red")
  lines(x, -d21, col = "blue")

Description

Returns a table of all implemented methods, their function name, and category.

Usage

available_methods(category = NULL)

Arguments

Value

A data frame of available decomposition methods.

Examples

available_methods()

Description

Following the notation given in Ponnapalli (2005), and the decomposition method can written as:

$_{n}\Delta_{x} = \frac{(e_x^2 - e_x^1)(l_x^2 + l_x^1)}{2} - \frac{(e_{x+n}^2 - e_{x+n}^1)(l_{x+n}^2 + l_{x+n}^1)}{2} - \frac{_{n}L_{x}^{1}}{l_{x}^{1}} + \frac{T_{x+n}^{2}}{l_{0}^{1}} \left( \frac{l_{x}^{1}}{l_{x}^{2}} - \frac{l_{x+n}^{1}}{l_{x+n}^{2}} \right)$

where $_{n}\Delta_{x}$ is the contribution of rate differences in age $x$ to the difference in life expectancy implied by mx1 and mx2. This formula can be averaged between 'effect interaction deferred' and 'effect interaction forwarded' from the Ponnapalli (2005).

Usage

chandrasekaran_II(
  mx1,
  mx2,
  age = (1:length(mx1)) - 1,
  nx = rep(1, length(mx1)),
  sex1 = "t",
  sex2 = sex1,
  closeout = TRUE
)

Arguments

Details

setting closeout to TRUE will result in value of 1/mx for the final age group, of ax and a value of 1 for the closeout of qx. This function gives numerically identical results to arriaga_sym(), lopez_ruzicka_sym(), and chandrasekaran_III().

Value

cc numeric vector with one element per age group, and which sums to the total difference in life expectancy between population 1 and 2.

References

Arriaga EE (1984). “Measuring and explaining the change in life expectancies.” Demography, 21, 83–96. Preston S, Heuveline P, Guillot M (2000). Demography: measuring and modeling population processes. Wiley-Blackwell. Ponnapalli KM (2005). “A comparison of different methods for decomposition of changes in expectation of life at birth and differentials in life expectancy at birth.” Demographic Research, 12, 141–172.

Examples

a <- .001
b <- .07
x <- 0:100
mx1 <- a * exp(x * b)
mx2 <- a/2 * exp(x * b)
cc <- chandrasekaran_II(mx1, mx2, age = x)
e01 <- mx_to_e0(mx1, age = x)
e02 <- mx_to_e0(mx2, age = x)
(delta <- e02 - e01)
sum(cc)


 plot(x, cc)

Description

Implements the Chandrasekaran III decomposition as described in Ponnapalli (2005), which combines multiple directional effects into a symmetric average. The method constructs a decomposition of the difference in life expectancy into four parts: the main effect, the operative effect, their average (exclusive effect), and a non-linear interaction term. These are calculated based on life table values. Let $e_x^i$ denote remaining life expectancy at age $x$ for population $i$, and $l_x^i$ the number of survivors to age $x$. Then:

• Main effect:

  $\frac{l_x^1}{l_x^2} \left[ l_x^2 (e_x^2 - e_x^1) - l_{x+n}^2 (e_{x+n}^2 - e_{x+n}^1) \right]$

• Operative effect:

  $\frac{l_x^2}{l_x^1} \left[ l_x^1 (e_x^2 - e_x^1) - l_{x+n}^1 (e_{x+n}^2 - e_{x+n}^1) \right]$

• Exclusive effect:

  $\frac{\text{Main effect} + \text{Operative effect}}{2}$

• Interaction effect:

  $(e_{x+n}^2 - e_{x+n}^1) \cdot \frac{1}{2} \left[ \frac{l_x^1 \cdot l_{x+n}^2}{l_x^2} + \frac{l_x^2 \cdot l_{x+n}^1}{l_x^1} - (l_{x+n}^1 + l_{x+n}^2) \right]$

The final contribution by age group is the sum of exclusive and interaction effects.

Usage

chandrasekaran_III(
  mx1,
  mx2,
  age = (1:length(mx1)) - 1,
  nx = rep(1, length(mx1)),
  sex1 = "t",
  sex2 = sex1,
  closeout = TRUE
)

Arguments

Details

This decomposition gives numerically identical results to arriaga_sym(), lopez_ruzicka_sym(), and chandrasekaran_II(), despite conceptual differences in their derivation. Included here for methodological completeness.

Value

Numeric vector of contributions by age group that sum to the total difference in life expectancy between the two mortality schedules.

References

Ponnapalli KM (2005). “A comparison of different methods for decomposition of changes in expectation of life at birth and differentials in life expectancy at birth.” Demographic Research, 12, 141–172.

See Also

chandrasekaran_II, arriaga_sym, lopez_ruzicka_sym

Examples

a <- 0.001
b <- 0.07
x <- 0:100
mx1 <- a * exp(x * b)
mx2 <- a/2 * exp(x * b)
cc <- chandrasekaran_III(mx1, mx2, age = x)
e01 <- mx_to_e0(mx1, age = x)
e02 <- mx_to_e0(mx2, age = x)
(delta <- e02 - e01)
sum(cc)

plot(x, cc, type = "l")

Description

A variety of exact or asymptotically exact life expectancy decomposition methods are implemented. Also, several life-expectancy decomposition sensitivity methods are implemented to answer how each age will change with an increase/decrease in life expectancy. See the package README and references for details.

Usage

LEdecomp(
  mx1,
  mx2,
  age = NULL,
  nx = NULL,
  n_causes = NULL,
  cause_names = NULL,
  sex1 = "t",
  sex2 = sex1,
  method = c("lifetable", "arriaga", "arriaga_sym", "sen_arriaga", "sen_arriaga_sym",
    "sen_arriaga_inst", "sen_arriaga_inst2", "sen_arriaga_sym_inst",
    "sen_arriaga_sym_inst2", "andreev", "andreev_sym", "sen_andreev", "sen_andreev_sym",
    "sen_andreev_inst", "sen_andreev_inst2", "sen_andreev_sym_inst",
    "sen_andreev_sym_inst2", "chandrasekaran_ii", "sen_chandrasekaran_ii",
    "sen_chandrasekaran_ii_inst", "sen_chandrasekaran_ii_inst2", "chandrasekaran_iii",
    "sen_chandrasekaran_iii", "sen_chandrasekaran_iii_inst", 
    
    "sen_chandrasekaran_iii_inst2", "lopez_ruzicka", "lopez_ruzicka_sym",
    "sen_lopez_ruzicka", "sen_lopez_ruzicka_sym", "sen_lopez_ruzicka_inst",
    "sen_lopez_ruzicka_inst2", "sen_lopez_ruzicka_sym_inst",
    "sen_lopez_ruzicka_sym_inst2", "horiuchi", "stepwise", "numerical"),
  closeout = TRUE,
  opt = TRUE,
  tol = 1e-10,
  Num_Intervals = 20,
  symmetrical = TRUE,
  direction = "both",
  perturb = 1e-06,
  ...
)

Arguments

Value

An object of class "LEdecomp":

• mx1, mx2, age, sex1, sex2, method, closeout, opt, tol, Num_Intervals, symmetrical, direction, perturb

• sens: vector/matrix of sensitivities (same dimensions as inputs)

• LE1, LE2: life expectancy for mx1 and mx2

• LEdecomp: vector/matrix of contributions (same shape as inputs)

References

Arriaga EE (1984). “Measuring and explaining the change in life expectancies.” Demography, 21, 83–96. Chandrasekaran C (1986). “Assessing the effect of mortality change in an age group on the expectation of life at birth.” Janasamkhya, 4(1), 1–9. Preston S, Heuveline P, Guillot M (2000). Demography: measuring and modeling population processes. Wiley-Blackwell. Ponnapalli KM (2005). “A comparison of different methods for decomposition of changes in expectation of life at birth and differentials in life expectancy at birth.” Demographic Research, 12, 141–172. Horiuchi S, Wilmoth JR, Pletcher SD (2008). “A decomposition method based on a model of continuous change.” Demography, 45(4), 785–801. Andreev EM, others (1982). “Metod komponent v analize prodoljitelnosty zjizni.[The method of components in the analysis of length of life].” Vestnik Statistiki, 9(3), 42–47. Andreev EM, Shkolnikov VM, Begun AZ (2002). “Algorithm for decomposition of differences between aggregate demographic measures and its application to life expectancies, healthy life expectancies, parity-progression ratios and total fertility rates.” Demographic research, 7, 499–522.

See Also

sen_e0_mx_lt(), arriaga(), arriaga_sym(), sen_arriaga(), sen_arriaga_sym()

Examples

## Simple reproducible setup
set.seed(123)
a <- 0.001
b <- 0.07

## 1) Vector (single cause), single-year ages
age <- 0:50
mx1 <- a * exp(age * b)
mx2 <- (a / 2) * exp(age * b)

res_vec <- LEdecomp(
  mx1 = mx1, mx2 = mx2,
  age = age, nx = rep(1, length(age)),
  sex1 = "t", method = "sen_arriaga", opt = TRUE
)
round(sum(res_vec$LEdecomp), 4)

## 2) Matrix (multiple causes): rows = age, cols = causes
##    Build 3 causes with random positive weights per age
k <- 3
w1 <- matrix(runif(length(age) * k, 0.9, 1.1), nrow = length(age)); w1 <- w1 / rowSums(w1)
w2 <- matrix(runif(length(age) * k, 0.9, 1.1), nrow = length(age)); w2 <- w2 / rowSums(w2)
mx1_mat <- (mx1) * w1
mx2_mat <- (mx2) * w2
colnames(mx1_mat) <- colnames(mx2_mat) <- paste0("c", 1:k)
rownames(mx1_mat) <- rownames(mx2_mat) <- as.character(age)

res_mat <- LEdecomp(
  mx1 = mx1_mat, mx2 = mx2_mat,
  age = age, nx = rep(1, length(age)),
  sex1 = "t", method = "sen_arriaga", opt = TRUE
)
## Check: row-summed cause contributions equal all-cause result
res_all <- LEdecomp(
  mx1 = mx1, mx2 = mx2,
  age = age, nx = rep(1, length(age)),
  sex1 = "t", method = "sen_arriaga", opt = TRUE
)
all.equal(rowSums(res_mat$LEdecomp), res_all$LEdecomp, tolerance = 1e-7)

## 3) Data frame (wide): same as matrix but with an 'age' column
df1 <- data.frame(age = age, mx1_mat, check.names = FALSE)
df2 <- data.frame(age = age, mx2_mat, check.names = FALSE)
res_df <- LEdecomp(
  mx1 = df1, mx2 = df2,
  age = NULL, nx = rep(1, length(age)),
  sex1 = "t", method = "sen_arriaga", opt = TRUE
)
all.equal(res_df$LEdecomp, res_mat$LEdecomp, tolerance = 1e-8)

## 4) Stacked vector (long/concatenated): all ages for cause 1, then cause 2, etc.
##    If 'age' is repeated per cause, LEdecomp infers n_causes and collapses age.
mx1_stack <- as.vector(mx1_mat)  # column-major flattening
mx2_stack <- as.vector(mx2_mat)
age_rep   <- rep(age, k)         # typical tidy pipeline: age repeated per cause

res_stack <- LEdecomp(
  mx1 = mx1_stack, mx2 = mx2_stack,
  age = age_rep, nx = NULL,
  sex1 = "t", method = "sen_arriaga", opt = TRUE
)
## Output is a stacked vector matching the matrix baseline when flattened
all.equal(res_stack$LEdecomp, c(res_mat$LEdecomp), tolerance = 1e-8)

## 5) Abridged ages (0,1,5,10,...,110): inference when labels are missing
age_ab <- c(0L, 1L, seq.int(5L, 110L, by = 5L))
nx_ab  <- c(diff(age_ab), tail(diff(age_ab), 1L))
mx1_ab <- a * exp(age_ab * b)
mx2_ab <- (a / 2) * exp(age_ab * b)

## Explicit abridged example
res_ab_explicit <- LEdecomp(
  mx1 = mx1_ab, mx2 = mx2_ab,
  age = age_ab, nx = nx_ab,
  sex1 = "t", method = "sen_arriaga", opt = TRUE
)

## Unlabeled abridged vector of the same length: age and nx inferred
res_ab_infer <- LEdecomp(
  mx1 = mx1_ab, mx2 = mx2_ab,
  age = NULL, nx = NULL,
  sex1 = "t", method = "sen_arriaga", opt = TRUE
)
all.equal(res_ab_infer$age, as.numeric(age_ab))
all.equal(res_ab_infer$LEdecomp, res_ab_explicit$LEdecomp, tolerance = 1e-8)

## 6) Rownames override age when they look like ages
##    Here we give the wrong 'age' but set rownames to "0","1",...,"50".
wrong_age <- age + 10
mx1_rn <- mx1_mat; mx2_rn <- mx2_mat
rownames(mx1_rn) <- rownames(mx2_rn) <- as.character(age)
res_rn <- suppressWarnings(LEdecomp(
  mx1 = mx1_rn, mx2 = mx2_rn,
  age = wrong_age, nx = rep(1, length(age)),
  sex1 = "t", method = "sen_arriaga", opt = TRUE
))
all.equal(res_rn$age, as.numeric(age))

## 7) List available methods
available_methods()

Description

Here we combine HMD Method Protocol equations 79 and 80. We calculate all the remaining years left to live at each age, then condition this on survival to each age.

Usage

lL_to_ex(lx, Lx)

Arguments

Value

numeric vector of remaining life expectancy ex

References

Wilmoth JR, Andreev K, Jdanov D, Glei DART, Boe C, Bubenheim M, Philipov D, Shkolnikov V, Vachon P, Winant C, M B (2021). “Methods protocol for the human mortality database. Version 6.” University of California, Berkeley, and Max Planck Institute for Demographic Research, Rostock. URL: http://mortality.org [version 26/01/2021].

Description

Implements the decomposition of life expectancy proposed by Lopez and Ruzicka, as described in Ponnapalli (2005) (there labelled Lopez-Ruzicka II). This method expresses the difference in life expectancy between two mortality schedules in terms of an exclusive effect and an interaction effect, using life table quantities.

Let $e_x^i$ denote remaining life expectancy at age $x$ for population $i$, and $l_x^i$ the number of survivors to age $x$. Then:

• Exclusive effect:

  $\frac{l_x^1}{l_x^2} \left[ l_x^2 (e_x^2 - e_x^1) - l_{x+n}^2 (e_{x+n}^2 - e_{x+n}^1) \right]$

• Interaction effect:

  $(e_{x+n}^2 - e_{x+n}^1) \cdot \left[ \frac{l_x^1 \cdot l_{x+n}^2}{l_x^2} - l_{x+n}^1 \right]$

The total contribution to life expectancy difference in age group $x$ is the sum of the exclusive and interaction effects.

Usage

lopez_ruzicka(
  mx1,
  mx2,
  age = (1:length(mx1)) - 1,
  nx = rep(1, length(mx1)),
  sex1 = "t",
  sex2 = sex1,
  closeout = TRUE
)

Arguments

Details

This method produces numerically identical results to arriaga().

Value

Numeric vector of contributions by age group that sum to the total difference in life expectancy between the two mortality schedules.

References

Ponnapalli KM (2005). “A comparison of different methods for decomposition of changes in expectation of life at birth and differentials in life expectancy at birth.” Demographic Research, 12, 141–172.

See Also

arriaga, chandrasekaran_III, lopez_ruzicka_sym

Examples

a <- 0.001
b <- 0.07
x <- 0:100
mx1 <- a * exp(x * b)
mx2 <- a/2 * exp(x * b)
cc <- lopez_ruzicka(mx1, mx2, age = x)
sum(cc)

Description

Implements a symmetric version of the Lopez-Ruzicka decomposition by averaging the results from the forward and reverse directions. That is, lopez_ruzicka_sym(mx1, mx2) returns

$\frac{1}{2} \left( \text{lopez\_ruzicka}(mx1, mx2) - \text{lopez\_ruzicka}(mx2, mx1) \right)$

This symmetric adjustment ensures that the decomposition is directionally neutral. This procedure is equivalent to what Ponnapalli (2005) calls Lopez-Ruzicka III. It is also equivalent to the symmetrical variants of Andreev and Arriaga, as well as chandrasekaran.

Usage

lopez_ruzicka_sym(
  mx1,
  mx2,
  age = (1:length(mx1)) - 1,
  nx = rep(1, length(mx1)),
  sex1 = "t",
  sex2 = sex1,
  closeout = TRUE
)

Arguments

Details

This symmetric version gives numerically identical results to arriaga_sym(), chandrasekaran_II(), and chandrasekaran_III().

Value

Numeric vector of contributions by age group that sum to the total difference in life expectancy between the two mortality schedules.

References

Ponnapalli KM (2005). “A comparison of different methods for decomposition of changes in expectation of life at birth and differentials in life expectancy at birth.” Demographic Research, 12, 141–172. Lopez AD, Ruzicka LT (1977). “The Differential Mortality of the Sexes in Australia.” In McGlashan ND (ed.), Studies in Australian Mortality, volume 4 of Environmental Studies Occasional Paper, 89–94. University of Tasmania, Hobart, Tasmania. Occasional Paper No.\ 4.

See Also

lopez_ruzicka, arriaga_sym, chandrasekaran_II, chandrasekaran_III

Examples

a <- 0.001
b <- 0.07
x <- 0:100
mx1 <- a * exp(x * b)
mx2 <- a/2 * exp(x * b)
d <- lopez_ruzicka_sym(mx1, mx2, age = x)

# compare to arriaga_sym()
d2 <- arriaga_sym(mx1, mx2, age = x)
all.equal(d, d2)

Description

Minus the decumulation of the survival curve gives the death distribution. Or the element-wise product of lx and the conditional death probabilities qx gives the same thing.

Usage

lx_to_dx(lx)

Arguments

Value

numeric vector of dx values

Description

We assume mid-interval ax except for age 0 and potentially the open age group. ax is defined as the average years lived in each age interval by those that die within the interval, and it is used to increase the precision of lifetable estimates. We allow ourselves the midpoint rule for single ages because it has little leverage. If we were working with abridged ages then we would need to use a more sophisticated method.

Usage

mx_to_ax(
  mx,
  nx = rep(1, length(mx)),
  age = 0:(length(mx) - 1),
  sex = "t",
  closeout = TRUE
)

Arguments

Details

For the case of Total sex, we estimate the male and female $a(0)$ using the Andreev-Kingkade rule of thumb, and then average them. We assume a value of 1/2 for all other ages, unless closeout = TRUE, in which case we close with 1/mx for the final value.

Value

numeric vector of ax values, the same length as mx

References

Andreev EM, Kingkade WW (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, 363–390.

Description

We follow the full chain of standard lifetable column calculations to translate mx to ex, then select the first element of ex. If min(age) > 0, then we return remaining life expectancy at the lowest given age.

Usage

mx_to_e0(mx, age, sex = "t", nx = rep(1, length(age)), closeout = TRUE)

Arguments

Value

numeric scalar of e0

Description

We follow the full chain of standard lifetable column calculations to translate mx to ex.

Usage

mx_to_ex(mx, age, nx = rep(1, length(age)), sex = "t", closeout = TRUE)

Arguments

Value

numeric vector of ex, the same length as mx

Description

qx gives conditional death probabilities, in this case forced to be consistent with a set of mx and ax values per HMD Method Protocol eq 71.

Usage

mx_to_qx(mx, ax, nx = rep(1, length(mx)), closeout = TRUE)

Arguments

Value

numeric vector of qx, the same length as mx

Description

Plot contributions (or sensitivities) to a life expectancy difference by age or by age and cause using ggplot2. This is just for a quick default plot method.

Usage

## S3 method for class 'LEdecomp'
plot(
  x,
  what = c("LEdecomp", "sens"),
  geom = c("auto", "line", "bar"),
  col = NULL,
  lwd = 1.2,
  xlab = "Age",
  ylab = NULL,
  main = NULL,
  legend = TRUE,
  legend_pos = "right",
  abridged_midpoints = FALSE,
  layout = c("overlay", "facet"),
  ncol = NULL,
  ...
)

Arguments

Details

By default, if what = "LEdecomp" we plot using bars (geom = "bar"), but you can override this. For bar plots, recall it's the area, not the height that the eye reads; for this reason, if your data is in non-single ages, we divide out the interval width, so that the implied uniform graduation to single ages still sums to the gap. If what = "sens" note we plot on a single-age scale even if the data are in abridged ages.

Value

Invisibly returns the ggplot object (after printing).

Examples

data("US_data", package = "LEdecomp")
allc <- subset(US_data, year == 2010)

# Make Female vs Male all-cause schedules, Age 0:100
ac_w <- reshape(allc[, c("sex","age","mx")],
                timevar = "sex", idvar = "age", direction = "wide")
names(ac_w) <- sub("^mx\\.", "", names(ac_w))
ac_w <- ac_w[order(ac_w$age), ]

dec_ac <- LEdecomp(
  mx1 = ac_w$Male,
  mx2 = ac_w$Female,
  age = 0:100,
  method = "sen_arriaga"
)

# Simple single-line plot

plot(dec_ac, main = "All-cause Arriaga, 2010 Female vs Male")

## End(Not run)
## Example 2: Cause of death, one year, Female vs Male
data("US_data_CoD", package = "LEdecomp")

codf <- subset(US_data_CoD, year == 2010 & sex == "Female")
codm <- subset(US_data_CoD, year == 2010 & sex == "Male")

dec_cod <- LEdecomp(
  mx1 = codf$mxc,
  mx2 = codm$mxc,
  age = 0:100,
  n_causes = length(unique(codf$cause)),
  cause_names = unique(codf$cause_id),
  method = "sen_arriaga"
)

# Overlay of all causes

plot(dec_cod, layout = "overlay", main = "Arriaga CoD, 2010 Female vs Male", legend.pos = "top")

# Facet by cause (3 columns)
plot(dec_cod, layout = "facet", ncol = 3, main = "Arriaga by cause (faceted)")


## Example 3: How to add an all-cause total line yourself (overlay)

p <- plot(dec_cod, layout = "overlay", main = "Overlay with manual Total")
y_mat <- if (is.matrix(dec_cod$LEdecomp)) dec_cod$LEdecomp else
  matrix(dec_cod$LEdecomp, nrow = length(dec_cod$age))
total <- rowSums(y_mat)
p + ggplot2::geom_line(
  data = data.frame(age = dec_cod$age, total = total),
  mapping = ggplot2::aes(x = .data$age, y = .data$total),
  inherit.aes = FALSE, color = "black", linewidth = 1.1)

Description

The survival curve is calculated as the cumulative product of the conditional survival probabilities, which are the complement of conditional death probabilities, qx, except we take care to start with a clean 1. This function no radix option. lx with a radix of 1 can be interpreted as the probability of surviving from birth to age x.

Usage

qx_to_lx(qx, radix = 1)

Arguments

Value

numeric vector of lx values

Description

Why write x |> rev() |> cumsum() |> rev() when you can just write rcumsum(x)?

Usage

rcumsum(x)

Arguments

Value

numeric vector the same length as x

Description

The sensitivity of life expectancy to a perturbation in mortality rates can be derived by dividing the Andreev decomposition result $\Delta$ by the difference mx2-mx1.

$s_{x} = \frac{\Delta}{_{n}M^{2}_x - _{n}M^{1}_x}$

Usage

sen_andreev(
  mx1,
  mx2,
  age = 0:(length(mx1) - 1),
  nx = rep(1, length(mx1)),
  sex1 = "t",
  sex2 = sex1,
  closeout = TRUE
)

Arguments

Value

s numeric vector with one element per age group, and which gives the sensitivity values for each age.

References

Andreev EM, others (1982). “Metod komponent v analize prodoljitelnosty zjizni.[The method of components in the analysis of length of life].” Vestnik Statistiki, 9(3), 42–47.

See Also

andreev

Examples

a <- .001
b <- .07
x <- 0:100
mx1 <- a * exp(x * b)
mx2 <- a/2 * exp(x * b)
cc <- andreev(mx1, mx2, age = x)
# examples can come from above too
s <- sen_andreev(mx1, mx2, age = x)

plot(x, s)

cc_check <- s * (mx2 - mx1)

plot(x,cc)
lines(x,cc_check)

Description

This implementation gives a good approximation of the sensitivity of life expectancy to perturbations in mortality rates (central death rates). Since the Andreev approach requires two versions of mortality rates mx1, mx2, we create these by slightly perturbing mx up and down. Then we calculate the Andreev sensitivity in each direction and take the average. Specifically, we create mx1 and mx2 as

$m_{x}^{1} = m_x \cdot \left(\frac{1}{1 - h}\right)$

$m_{x}^{2} = m_x \cdot \left(1 - h\right)$

where h is small value given by the argument perturb.

Usage

sen_andreev_instantaneous(
  mx,
  age = 0:(length(mx1) - 1),
  sex = "t",
  nx = rep(1, length(mx)),
  perturb = 1e-06,
  closeout = TRUE
)

Arguments

Details

A minor correction might be needed for the final age group for the case of the reverse-direction Andreev sensitivity. Note also for values of perturb (h) that are less than 1e-7 we might lose stability in results.

Value

numeric vector of sensitivity of life expectancy to perturbations in mx.

Examples

a   <- .001
b   <- .07
x   <- 0:100
mx1 <- a * exp(x * b)
mx2 <- a/2 * exp(x * b)
mx  <- (mx1 + mx2) / 2
s     <- sen_andreev_instantaneous(mx, age = x)
s1    <- sen_andreev_instantaneous(mx1, age = x)
s2    <- sen_andreev_instantaneous(mx2, age = x)
s1_d  <- sen_andreev(mx1, mx2, age = x)
s2_d  <- sen_andreev(mx2, mx1, age = x)
delta <- mx2 - mx1
# dots give our point estimate of sensitivity at the average of the rates,
# which is different from the

plot(x,s*delta, ylim = c(0,.3))
lines(x,s1*delta,col = "red")
lines(x,s2*delta,col = "blue")
# the sensitivity of the average is different
# from the average of the sensitivities!
lines(x, ((s1+s2)) / 2 * delta)
# and these are different from the directional sensitivities
# covering the whole space from mx1 to mx2:
lines(x, s1_d*delta, col = "red", lty =2)
lines(x, s2_d*delta, col = "blue", lty =2)

Description

This is a second approach for estimating the sensitivity for a single set of rates. Here, rather than directly expanding and contracting rates to convert mx into mx1 and mx2 we instead shift the logged mortality rates up and down by the factor perturb = h. Specifically:

$m_{x}^{1} = e^{\ln\left(m_x\right) + h}$

$m_{x}^{2} = e^{\ln\left(m_x\right) - h}$

Usage

sen_andreev_instantaneous2(
  mx,
  age = 0:(length(mx1) - 1),
  sex = "t",
  nx = rep(1, length(mx)),
  perturb = 1e-06,
  closeout = TRUE
)

Arguments

Value

numeric vector of sensitivity of life expectancy to perturbations in mx

See Also

sen_andreev_instantaneous

Examples

a   <- .001
b   <- .07
x   <- 0:100
mx <- a * exp(x * b)
# the multiplicative perturbation:
s1 <- sen_andreev_instantaneous(mx)
s2 <- sen_andreev_instantaneous2(mx)
plot(x,
     s1 - s2,
     pch = 16,
     cex=.5,
     main = "very similar")

Description

This approach conducts a classic Andreev (1982) decomposition in both directions, averaging the (sign-adjusted) result, i.e. a_avg = (andreev(mx1,mx2, ...) - andreev(mx2, mx1, ...)) / 2, then approximates the sensitivity by dividing out the rate differences, i.e. s = a_avg / (mx2 - mx1). A resulting decomposition will be exact because the two andreev directions are exact, but this method might be vulnerable to 0s in the denominator.

Usage

sen_andreev_sym(
  mx1,
  mx2,
  age = 0:(length(mx1) - 1),
  nx = rep(1, length(mx1)),
  sex1 = "t",
  sex2 = sex1,
  closeout = TRUE
)

Arguments

Value

numeric vector of life expectancy sensitivity to perturbations in mx evaluated at the average of mx1 and mx2.

See Also

arriaga

Examples

a <- .001
b <- .07
x <- 0:100
mx1 <- a * exp(x * b)
mx2 <- a/2 * exp(x * b)
s <- sen_andreev_sym(mx1, mx2, age = x)

e01 <- mx_to_e0(mx1,age=x)
e02 <- mx_to_e0(mx2,age=x)
(Delta <- e02 - e01)
deltas <- mx2- mx1
sum(deltas * s)
mx_avg <- (mx1+mx2) / 2

mx_avg <- (mx1 + mx2) / 2
plot(x, s, type = 'l')
lines(x, sen_andreev_instantaneous(mx_avg, age=x),col = "blue")

Description

Estimates the sensitivity of life expectancy to small changes in age-specific mortality rates using the symmetrical Andreev decomposition. This is done by applying a small multiplicative perturbation to the input mortality rates and using the symmetrical sensitivity function sen_andreev_sym().

Specifically, the function constructs:

$m_{x}^{1} = m_x \cdot \left(\frac{1}{1 - h}\right)$

$m_{x}^{2} = m_x \cdot (1 - h)$

and applies sen_andreev_sym(mx1, mx2, ...) to the result.

Usage

sen_andreev_sym_instantaneous(
  mx,
  age = 0:(length(mx1) - 1),
  sex = "t",
  nx = rep(1, length(mx)),
  perturb = 1e-06,
  closeout = TRUE
)

Arguments

Details

This function yields an instantaneous approximation to the derivative of life expectancy with respect to mortality, evaluated at the input schedule. Because sen_andreev_sym() is itself symmetrical, only the "forward" perturbation is required.

Value

numeric vector of life expectancy sensitivity to perturbations in mx.

See Also

sen_andreev_sym, sen_andreev_sym_instantaneous2, sen_lopez_ruzicka_instantaneous

Examples

a <- 0.001
b <- 0.07
x <- 0:100
mx <- a * exp(x * b)
s <- sen_andreev_sym_instantaneous(mx, age = x)

plot(x, s, type = "l")

Description

This is a second approach for estimating the sensitivity for a single set of rates. Here, rather than directly expanding and contracting rates to convert mx into mx1 and mx2 we instead shift the logged mortality rates up and down by the factor perturb = h. Specifically:

$m_{x}^{1} = e^{\ln\left(m_x\right) + h}$

$m_{x}^{2} = e^{\ln\left(m_x\right) - h}$

Usage

sen_andreev_sym_instantaneous2(
  mx,
  age = 0:(length(mx1) - 1),
  sex = "t",
  nx = rep(1, length(mx)),
  perturb = 1e-06,
  closeout = TRUE
)

Arguments

Value

numeric vector of life expectancy sensitivity to perturbations in mx.

See Also

sen_andreev_instantaneous

Examples

a <- 0.001
b <- 0.07
x <- 0:100
mx <- a * exp(x * b)
s <- sen_andreev_sym_instantaneous2(mx, age = x)

plot(x, s, type = "l")

Description

The sensitivity of life expectancy to a perturbation in mortality rates can be derived by dividing the Arriaga decomposition result $\Delta$ by the difference mx2-mx1.

$s_{x} = \frac{\Delta}{_{n}M^{2}_x - _{n}M^{1}_x}$

Usage

sen_arriaga(
  mx1,
  mx2,
  age = 0:(length(mx1) - 1),
  nx = rep(1, length(mx1)),
  sex1 = "t",
  sex2 = sex1,
  closeout = TRUE
)

Arguments

Value

s numeric vector with one element per age group, and which gives the sensitivity values for each age.

References

Arriaga EE (1984). “Measuring and explaining the change in life expectancies.” Demography, 21, 83–96. Preston S, Heuveline P, Guillot M (2000). Demography: measuring and modeling population processes. Wiley-Blackwell.

See Also

arriaga

Examples

a <- .001
b <- .07
x <- 0:100
mx1 <- a * exp(x * b)
mx2 <- a/2 * exp(x * b)
cc <- arriaga(mx1, mx2, age = x)
# examples can come from above too
s <- sen_arriaga(mx1, mx2, age = x)

plot(x, s)

cc_check <- s * (mx2 - mx1)

plot(x,cc)
lines(x,cc_check)

Description

This implementation gives a good approximation of the sensitivity of life expectancy to perturbations in mortality rates (central death rates). Since the Arriaga approach requires two versions of mortality rates mx1, mx2, we create these by slightly perturbing mx up and down. Then we calculate the Arriaga sensitivity in each direction and take the average. Specifically, we create mx1 and mx2 as

$m_{x}^{1} = m_x \cdot \left(\frac{1}{1 - h}\right)$

$m_{x}^{2} = m_x \cdot \left(1 - h\right)$

where h is small value given by the argument perturb.

Usage

sen_arriaga_instantaneous(
  mx,
  age = 0:(length(mx1) - 1),
  sex = "t",
  nx = rep(1, length(mx)),
  perturb = 1e-06,
  closeout = TRUE
)

Arguments

Details

A minor correction might be needed for the final age group for the case of the reverse-direction Arriaga sensitivity. Note also for values of perturb (h) that are less than 1e-7 we might lose stability in results.

Value

numeric vector of sensitivity of life expectancy to perturbations in mx.

Examples

a   <- .001
b   <- .07
x   <- 0:100
mx1 <- a * exp(x * b)
mx2 <- a/2 * exp(x * b)
mx  <- (mx1 + mx2) / 2
s     <- sen_arriaga_instantaneous(mx, age = x)
s1    <- sen_arriaga_instantaneous(mx1, age = x)
s2    <- sen_arriaga_instantaneous(mx2, age = x)
s1_d  <- sen_arriaga(mx1, mx2, age = x)
s2_d  <- sen_arriaga(mx2, mx1, age = x)
delta <- mx2 - mx1
# dots give our point estimate of sensitivity at the average of the rates,
# which is different from the

plot(x,s*delta, ylim = c(0,.3))
lines(x,s1*delta,col = "red")
lines(x,s2*delta,col = "blue")
# the sensitivity of the average is different
# from the average of the sensitivities!
lines(x, ((s1+s2)) / 2 * delta)
# and these are different from the directional sensitivities
# covering the whole space from mx1 to mx2:
lines(x, s1_d*delta, col = "red", lty =2)
lines(x, s2_d*delta, col = "blue", lty =2)

Description

This is a second approach for estimating the sensitivity for a single set of rates. Here, rather than directly expanding and contracting rates to convert mx into mx1 and mx2 we instead shift the logged mortality rates up and down by the factor perturb = h. Specifically:

$m_{x}^{1} = e^{\ln\left(m_x\right) + h}$

$m_{x}^{2} = e^{\ln\left(m_x\right) - h}$

Usage

sen_arriaga_instantaneous2(
  mx,
  age = 0:(length(mx1) - 1),
  sex = "t",
  nx = rep(1, length(mx)),
  perturb = 1e-06,
  closeout = TRUE
)

Arguments

Value

numeric vector of sensitivity of life expectancy to perturbations in mx

See Also

sen_arriaga_instantaneous

Examples

a   <- .001
b   <- .07
x   <- 0:100
mx <- a * exp(x * b)
# the multiplicative perturbation:
s1 <- sen_arriaga_instantaneous(mx)
s2 <- sen_arriaga_instantaneous2(mx)
plot(x,
     s1 - s2,
     pch = 16,
     cex=.5,
     main = "very similar")

Description

This approach conducts a classic Arriaga decomposition in both directions, averaging the (sign-adjusted) result, i.e. a_avg = (arriaga(mx1,mx2, ...) - arriaga(mx2, mx1, ...)) / 2, then approximates the sensitivity by dividing out the rate differences, i.e. s = a_avg / (mx2 - mx1). A resulting decomposition will be exact because the two arriaga directions are exact, but this method might be vulnerable to 0s in the denominator.

Usage

sen_arriaga_sym(
  mx1,
  mx2,
  age = 0:(length(mx1) - 1),
  nx = rep(1, length(mx1)),
  sex1 = "t",
  sex2 = sex1,
  closeout = TRUE
)

Arguments

Value

numeric vector of life expectancy sensitivity to perturbations in mx evaluated at the average of mx1 and mx2.

See Also

arriaga

Examples

a <- .001
b <- .07
x <- 0:100
mx1 <- a * exp(x * b)
mx2 <- a/2 * exp(x * b)
s <- sen_arriaga_sym(mx1, mx2, age = x)

e01 <- mx_to_e0(mx1,age=x)
e02 <- mx_to_e0(mx2,age=x)
(Delta <- e02 - e01)
deltas <- mx2- mx1
sum(deltas * s)
mx_avg <- (mx1+mx2) / 2

mx_avg <- (mx1 + mx2) / 2
plot(x, s, type = 'l')
lines(x, sen_arriaga_instantaneous(mx_avg, age=x),col = "blue")

Description

Estimates the sensitivity of life expectancy to small changes in age-specific mortality rates using the symmetrical Arriaga decomposition. This is done by applying a small multiplicative perturbation to the input mortality rates and using the symmetrical sensitivity function sen_arriaga_sym().

Specifically, the function constructs:

$m_{x}^{1} = m_x \cdot \left(\frac{1}{1 - h}\right)$

$m_{x}^{2} = m_x \cdot (1 - h)$

and applies sen_arriaga_sym(mx1, mx2, ...) to the result.

Usage

sen_arriaga_sym_instantaneous(
  mx,
  age = 0:(length(mx1) - 1),
  sex = "t",
  nx = rep(1, length(mx)),
  perturb = 1e-06,
  closeout = TRUE
)

Arguments

Details

This function yields an instantaneous approximation to the derivative of life expectancy with respect to mortality, evaluated at the input schedule. Because sen_arriaga_sym() is itself symmetrical, only the "forward" perturbation is required.

Value

numeric vector of life expectancy sensitivity to perturbations in mx.

See Also

sen_arriaga_sym, sen_arriaga_sym_instantaneous2, sen_lopez_ruzicka_instantaneous

Examples

a <- 0.001
b <- 0.07
x <- 0:100
mx <- a * exp(x * b)
s <- sen_arriaga_sym_instantaneous(mx, age = x)

plot(x, s, type = "l")

Description

This is a second approach for estimating the sensitivity for a single set of rates. Here, rather than directly expanding and contracting rates to convert mx into mx1 and mx2 we instead shift the logged mortality rates up and down by the factor perturb = h. Specifically:

$m_{x}^{1} = e^{\ln\left(m_x\right) + h}$

$m_{x}^{2} = e^{\ln\left(m_x\right) - h}$

Usage

sen_arriaga_sym_instantaneous2(
  mx,
  age = 0:(length(mx1) - 1),
  sex = "t",
  nx = rep(1, length(mx)),
  perturb = 1e-06,
  closeout = TRUE
)

Arguments

Value

numeric vector of life expectancy sensitivity to perturbations in mx.

See Also

sen_arriaga_instantaneous

Examples

a <- 0.001
b <- 0.07
x <- 0:100
mx <- a * exp(x * b)
s <- sen_arriaga_sym_instantaneous2(mx, age = x)

plot(x, s, type = "l")

Description

Computes the sensitivity of life expectancy to changes in age-specific mortality rates using the Chandrasekaran II decomposition approach described by Ponnapalli (2005). The sensitivity is obtained by dividing the age-specific contributions (from chandrasekaran_II()) by the differences in mortality rates (mx2 - mx1). This yields a pointwise estimate of the derivative of life expectancy with respect to each age-specific mortality rate evaluated at an imagined midpoint between the first a second set of mortality rates.

Usage

sen_chandrasekaran_II(
  mx1,
  mx2,
  age = (1:length(mx1)) - 1,
  nx = rep(1, length(mx1)),
  sex1 = "t",
  sex2 = sex1,
  closeout = TRUE
)

Arguments

Details

This give numerically identical results to sen_arriaga_sym(), sen_lopez_ruzicka_sym(), and sen_chandrasekaran_III().

Value

A numeric vector of sensitivity values by age group.

numeric vector of sensitivity of life expectancy to perturbations in mx between mx1 and mx2.

References

Ponnapalli KM (2005). “A comparison of different methods for decomposition of changes in expectation of life at birth and differentials in life expectancy at birth.” Demographic Research, 12, 141–172.

See Also

chandrasekaran_II, sen_arriaga, sen_arriaga_sym

Examples

a <- 0.001
b <- 0.07
x <- 0:100
mx1 <- a * exp(x * b)
mx2 <- a/2 * exp(x * b)
s <- sen_chandrasekaran_II(mx1, mx2, age = x)

# Check that multiplying sensitivity by rate difference approximates the decomposition
cc_check <- s * (mx2 - mx1)
cc <- chandrasekaran_II(mx1, mx2, age = x)

plot(x, cc, type = "l")
lines(x, cc_check, col = "red", lty = 2)

Description

Estimates the sensitivity of life expectancy to small changes in mortality rates using the Chandrasekaran II decomposition. This is done by perturbing the input mortality rates up and down by a small factor and calculating the directional sensitivity.

Specifically, the function constructs:

$m_{x}^{1} = m_x \cdot \left(\frac{1}{1 - h}\right)$

$m_{x}^{2} = m_x \cdot (1 - h)$

and applies sen_chandrasekaran_II(mx1, mx2, ...) to the result.

Usage

sen_chandrasekaran_II_instantaneous(
  mx,
  age = (1:length(mx1)) - 1,
  nx = rep(1, length(mx)),
  sex = "t",
  perturb = 1e-06,
  closeout = TRUE
)

Arguments

Details

This approach gives a reasonable approximation of the derivative of life expectancy with respect to each age-specific mortality rate. It gives numerically identical results to sen_arriaga_sym_instantaneous(), sen_lopez_ruzicka_instantaneous(), and sen_chandrasekaran_III_instantaneous().

Value

numeric vector of sensitivity of life expectancy to perturbations in mx

See Also

sen_chandrasekaran_II, sen_chandrasekaran_II_instantaneous2, sen_arriaga_sym_instantaneous

Examples

a <- 0.001
b <- 0.07
x <- 0:100
mx <- a * exp(x * b)
s <- sen_chandrasekaran_II_instantaneous(mx, age = x)

plot(x, s, type = "l")

Description

Estimates the sensitivity of life expectancy to small changes in mortality rates using the Chandrasekaran II decomposition. This variant perturbs the mortality rates in log space, creating two versions of mx by adding and subtracting a small constant to log(mx), then exponentiating.

Specifically:

$m_{x}^{1} = \exp\left( \ln m_x + h \right)$

$m_{x}^{2} = \exp\left( \ln m_x - h \right)$

and applies sen_chandrasekaran_II(mx1, mx2, ...) to the result.

Usage

sen_chandrasekaran_II_instantaneous2(
  mx,
  age = (1:length(mx1)) - 1,
  nx = rep(1, length(mx)),
  sex = "t",
  perturb = 1e-06,
  closeout = TRUE
)

Arguments

Details

This approach provides a log-linear perturbation of the mortality schedule and can be used to estimate the derivative of life expectancy with respect to logged mortality rates. It gives numerically identical results to sen_arriaga_sym_instantaneous2(), sen_lopez_ruzicka_instantaneous2(), and sen_chandrasekaran_III_instantaneous2().

Value

numeric vector of sensitivity of life expectancy to perturbations in mx.

See Also

sen_chandrasekaran_II_instantaneous, sen_arriaga_sym_instantaneous2, sen_lopez_ruzicka_instantaneous2

Examples

a <- 0.001
b <- 0.07
x <- 0:100
mx <- a * exp(x * b)
s <- sen_chandrasekaran_II_instantaneous2(mx, age = x)

plot(x, s, type = "l")

Description

Computes the implied sensitivity of life expectancy to changes in age-specific mortality rates using the Chandrasekaran III decomposition approach described by Ponnapalli (2005). The sensitivity is obtained by dividing the age-specific contributions (from chandrasekaran_III()) by the differences in mortality rates (mx2 - mx1). This yields a pointwise estimate of the derivative of life expectancy with respect to each age-specific mortality rate evaluated at an imagined midpoint between the first a second set of mortality rates.

Usage

sen_chandrasekaran_III(
  mx1,
  mx2,
  age = (1:length(mx1)) - 1,
  nx = rep(1, length(mx1)),
  sex1 = "t",
  sex2 = sex1,
  closeout = TRUE
)

Arguments

Details

This gives numerically identical results to sen_arriaga_sym(), sen_lopez_ruzicka_sym(), and sen_chandrasekaran_II().

Value

A numeric vector of sensitivity values by age group.

References

Ponnapalli KM (2005). “A comparison of different methods for decomposition of changes in expectation of life at birth and differentials in life expectancy at birth.” Demographic Research, 12, 141–172.

See Also

chandrasekaran_III, sen_chandrasekaran_II, sen_arriaga_sym, sen_lopez_ruzicka_sym

Examples

a <- 0.001
b <- 0.07
x <- 0:100
mx1 <- a * exp(x * b)
mx2 <- a/2 * exp(x * b)
s <- sen_chandrasekaran_III(mx1, mx2, age = x)

# Check that multiplying sensitivity by rate difference approximates the decomposition
cc_check <- s * (mx2 - mx1)
cc <- chandrasekaran_III(mx1, mx2, age = x)

plot(x, cc, type = "l")
lines(x, cc_check, col = "red", lty = 2)

Description

Estimates the sensitivity of life expectancy to small changes in mortality rates using the Chandrasekaran III decomposition. This is done by perturbing the input mortality rates up and down by a small factor and computing directional sensitivity from the result.

Specifically:

$m_x^{1} = m_x \cdot \left( \frac{1}{1 - h} \right)$

$m_x^{2} = m_x \cdot (1 - h)$

and applies sen_chandrasekaran_III(mx1, mx2, ...) to the result.

Usage

sen_chandrasekaran_III_instantaneous(
  mx,
  age = (1:length(mx1)) - 1,
  nx = rep(1, length(mx1)),
  sex = "t",
  perturb = 1e-06,
  closeout = TRUE
)

Arguments

Details

This approach provides an approximation of the derivative of life expectancy with respect to each age-specific mortality rate, evaluated near the input mx. It gives numerically identical results to sen_arriaga_sym_instantaneous(), sen_lopez_ruzicka_instantaneous(), and sen_chandrasekaran_II_instantaneous().

Value

numeric vector of sensitivity of life expectancy to perturbations in mx.

See Also

sen_chandrasekaran_III, sen_chandrasekaran_III_instantaneous2, sen_arriaga_sym_instantaneous, sen_lopez_ruzicka_instantaneous

Examples

a <- 0.001
b <- 0.07
x <- 0:100
mx <- a * exp(x * b)
s <- sen_chandrasekaran_III_instantaneous(mx, age = x)

plot(x, s, type = "l")

Description

Estimates the sensitivity of life expectancy to small changes in mortality rates using the Chandrasekaran III decomposition and log-transformed perturbations. The method perturbs mx up and down in log space and averages the directional sensitivities to approximate the derivative.

Specifically:

$m_x^{1} = \exp(\ln m_x + h)$

$m_x^{2} = \exp(\ln m_x - h)$

and applies sen_chandrasekaran_III(mx1, mx2, ...) and sen_chandrasekaran_III(mx2, mx1, ...), returning their average.

Usage

sen_chandrasekaran_III_instantaneous2(
  mx,
  age = (1:length(mx1)) - 1,
  nx = rep(1, length(mx1)),
  sex = "t",
  perturb = 1e-06,
  closeout = TRUE
)

Arguments

Details

This version uses symmetric log-space perturbations. It gives numerically identical results to sen_arriaga_sym_instantaneous2(), sen_lopez_ruzicka_instantaneous2(), and sen_chandrasekaran_II_instantaneous2().

Value

numeric vector of sensitivity of life expectancy to perturbations in mx.

See Also

sen_chandrasekaran_III_instantaneous, sen_arriaga_sym_instantaneous2, sen_lopez_ruzicka_instantaneous2, sen_chandrasekaran_II_instantaneous2

Examples

a <- 0.001
b <- 0.07
x <- 0:100
mx <- a * exp(x * b)
s <- sen_chandrasekaran_III_instantaneous2(mx, age = x)

plot(x, s, type = "l")

Description

This function tries to get the direct discrete life expectancy sensitivity to $m(x)$, in continuous math it's $-l(x)e(x)$, we just need to find the best approx with a discrete lifetable. This direct lifetable-based calculation requires a few approximations to get a usable value whenever we're working with discrete data. In continuous notation, we know that the sensitivity $s(x)$

$s(x) = -l(x)e(x)$

but it is not obvious what to use from a discrete lifetable. In this implementation, we use $L(x)$ and an $a(x)$-weighted average of successive $e(x)$ values, specifically, we calculate:

$s_x = -L_x \cdot \left( e_x \cdot \left( 1 - a_x \right) + e_{x+1} \cdot a_x\right)$

This seems to be a very good approximation for ages >0, but we still have a small, but unaccounted-for discrepancy in age 0, at least when comparing with also-imperfect numerical derivatives.

Usage

sen_e0_mx_lt(
  mx,
  age = 0:(length(mx) - 1),
  nx = rep(1, length(mx)),
  sex = "t",
  closeout = TRUE
)

Arguments

Value

numeric vector of sensitivity of life expectancy to perturbations in mx.

Examples

x <- 0:100
mx <- 0.001 * exp(x * 0.07)
sl <-  sen_e0_mx_lt(mx,age=x,sex='t',closeout=TRUE)
sn <- numDeriv::grad(mx_to_e0, mx, age=x, sex = 't', closeout=TRUE)

plot(x,sl)
lines(x,sn)

# examine residuals:
sl - sn
# Note discrepancies in ages >0 are due to numerical precision only

plot(x, sl - sn, main = "still uncertain what accounts for the age 0 discrepancy")

Description

Computes the sensitivity of life expectancy to changes in age-specific mortality rates using the Lopez-Ruzicka decomposition approach. The sensitivity is calculated by dividing the age-specific contributions (from lopez_ruzicka()) by the differences in mortality rates (mx2 - mx1). This gives a pointwise estimate of the derivative of life expectancy with respect to each age-specific mortality rate, evaluated at an imagined midpoint between the two input rate schedules.

Usage

sen_lopez_ruzicka(
  mx1,
  mx2,
  age = (1:length(mx1)) - 1,
  nx = rep(1, length(mx1)),
  sex1 = "t",
  sex2 = sex1,
  closeout = TRUE
)

Arguments

Details

This method gives numerically identical results to sen_arriaga().

Value

A numeric vector of sensitivity values by age group.

References

Ponnapalli KM (2005). “A comparison of different methods for decomposition of changes in expectation of life at birth and differentials in life expectancy at birth.” Demographic Research, 12, 141–172.

See Also

lopez_ruzicka, sen_arriaga, sen_chandrasekaran_III

Examples

a <- 0.001
b <- 0.07
x <- 0:100
mx1 <- a * exp(x * b)
mx2 <- a / 2 * exp(x * b)
s <- sen_lopez_ruzicka(mx1, mx2, age = x)

# Check that multiplying sensitivity by rate difference reproduces the decomposition
cc_check <- s * (mx2 - mx1)
cc <- lopez_ruzicka(mx1, mx2, age = x)

plot(x, cc, type = "l")
lines(x, cc_check, col = "red", lty = 2)

Description

Estimates the sensitivity of life expectancy to small changes in mortality rates using the Lopez-Ruzicka decomposition. This is done by perturbing the input mortality rates up and down by a small factor and averaging the resulting directional sensitivities to approximate a symmetric derivative.

Specifically:

$m_x^{1} = m_x \cdot \left( \frac{1}{1 - h} \right)$

$m_x^{2} = m_x \cdot (1 - h)$

and applies sen_lopez_ruzicka(mx1, mx2, ...) and sen_lopez_ruzicka(mx2, mx1, ...), returning their average.

Usage

sen_lopez_ruzicka_instantaneous(
  mx,
  age = (1:length(mx1)) - 1,
  nx = rep(1, length(mx1)),
  sex = "t",
  perturb = 1e-06,
  closeout = TRUE
)

Arguments

Details

This method gives numerically identical results to sen_arriaga_sym_instantaneous(), sen_chandrasekaran_II_instantaneous(), and sen_chandrasekaran_III_instantaneous().

Value

numeric vector of sensitivity of life expectancy to perturbations in mx.

See Also

sen_lopez_ruzicka, sen_lopez_ruzicka_instantaneous2, sen_arriaga_sym_instantaneous, sen_chandrasekaran_II_instantaneous

Examples

a <- 0.001
b <- 0.07
x <- 0:100
mx <- a * exp(x * b)
s <- sen_lopez_ruzicka_instantaneous(mx, age = x)

plot(x, s, type = "l")

Description

Estimates the sensitivity of life expectancy to small changes in mortality rates using the Lopez-Ruzicka decomposition and log-space perturbation. This is done by shifting the log of the input mortality rates up and down by a small constant, then exponentiating, and computing the average directional sensitivity.

Specifically:

$m_x^{1} = \exp(\ln m_x + h)$

$m_x^{2} = \exp(\ln m_x - h)$

and applies sen_lopez_ruzicka(mx1, mx2, ...) and sen_lopez_ruzicka(mx2, mx1, ...), returning their average.

Usage

sen_lopez_ruzicka_instantaneous2(
  mx,
  age = (1:length(mx1)) - 1,
  nx = rep(1, length(mx1)),
  sex = "t",
  perturb = 1e-06,
  closeout = TRUE
)

Arguments

Details

This approach gives numerically identical results to sen_arriaga_sym_instantaneous2(), sen_chandrasekaran_II_instantaneous2(), and sen_chandrasekaran_III_instantaneous2().

Value

numeric vector of sensitivity of life expectancy to perturbations in mx.

See Also

sen_lopez_ruzicka_instantaneous, sen_arriaga_sym_instantaneous2, sen_chandrasekaran_III_instantaneous2

Examples

a <- 0.001
b <- 0.07
x <- 0:100
mx <- a * exp(x * b)
s <- sen_lopez_ruzicka_instantaneous2(mx, age = x)

plot(x, s, type = "l")