In jschoeley/pash: Pace-Shape Analysis of Life-tables

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Notation

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$X$ Continuous random variable "age at death" $x$ Lower bound of an age group $\theta$ Starting age of life-table (usually $\theta = 0$) $\omega$ Lower bound of last life-table age group (discrete setting) or highest observed value of $X$ (continuous setting) $n_x$ Width of age group $x$ as $[x, x+n_x)$ $f(x) = P(X=x)$ Probability density function of ages at death $\ell(x) = \ell_x = P(X > x)$ Probability of surviving up until $x$ ${}nd_x = \int{x}^{x+n}f(x)\,\text{d}x$ Expected number of deaths in the life-table population over age interval $[x, x+n)$ $\mu(x)$ Force of mortality at age $x$ (hazard of death) ${}_nm_x$ Expected mortality rate over age interval $[x, x+n_x)$ $e(x)=e_x=\int_x^\omega\ell(x)\,\text{d}x$ Remaining life-expectancy at age $x$ ${}_na_x$ Avg. time spent in $[x, x+n)$ when dying there ${}_nA_x = \frac{{}_na_x} {n_x}$ Avg. fraction of interval alive when dying there Let $k \in \mathbb{R}$, then
$\lfloor k \rfloor$ Integer part of $k$. $\lceil k \rceil$ $\lfloor k \rfloor + 1$ $\text{frac}(k)$ $k - \lfloor k \rfloor$

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Shape measures in pash

Life-table entropy, entropy, $\text{H}$

Life-table entropy, also known as Keyfitz's entropy [4], is the average remaining life-expectancy lost upon death ($e^\dagger$ [5]) divided by the total life-expectancy.

Continuous

$$ \text{H} = \frac{e^\dagger} {e(\theta)} = \frac{\int\limits_{x=\theta}^{\omega} e(x)f(x)\,\text{d}x} {e(\theta)} $$

Discrete

$$ \widehat{\text{H}} = \frac {\sum_\limits{x=\theta}^{\omega} {}nd_x \left[{}_nA_x e{x+n_x} + (1-{}nA_x) e_x \right]} {e\theta}, $$ with

$$ {}nd\omega \left[{}nA\omega e_{\omega+n_\omega} + (1-{}nA\omega) e_x \right] = e_\omega $$

for the final age group.

Harmonizing Transformation

$$ \text{H}^* = 1-\text{H} $$

pash implementation

# Average Life-Expectancy in Age x
EDaggerx <- function(nax, nx, ex) {
  nAx = nax/nx
  edx = (nAx*c(ex[-1L], 0) + (1-nAx)*ex)
  edx[length(edx)] = ex[length(ex)]
  return(edx)
}

# Total Life Years Lost due to Death
EDagger <- function(nax, nx, ndx, ex) {
  edx = EDaggerx(nax, nx, ex)
  ed  = sum(ndx*edx)
  return(ed)
}

# Life Table Entropy
LifetableEntropy <- function(nax, nx, ndx, ex, harmonized) {
  ed = EDagger(nax, nx, ndx, ex)
  H  = ed/ex[1L]
  if (!isTRUE(harmonized)) {S = H}
  if (isTRUE(harmonized)) {S = 1-H}
  return(S)
}

Life-table coefficient of variation, cv, $\text{CV}$

The coefficient of variation of the age at death is the standard deviation in the age at death divided by the total life-expectancy.

Continuous

$$ \text{CV} = \frac {\sqrt{\text{Var}(X)}} {\text{E}(X)} =\frac {\sqrt{\int_{x=\theta}^{\omega} f(x)(x-e_0)^2\,\text{d}x}}{e_\theta}. $$

Discrete

$$ \widehat{\text{CV}} = \frac {\sqrt{\sum_\limits{x=\theta}^\omega {}nd_x(\overline{x}-e_0)^2}} {e\theta} = \frac {\sqrt{\sum_\limits{x=\theta}^\omega {}nd_x(x+{}_na_x-e_0)^2}} {e\theta}. $$

Harmonizing Transformation

$$ \text{CV}^* = 1-\text{CV} $$

pash implementation

# Life Table Variance
LifetableVar <- function(x, ndx, nax, ex) {
  var = sum(ndx*(x+nax-ex[1L])^2)
  return(var)
}

# Life Table Coefficient of Variation
LifetableCV <- function(x, ndx, nax, ex, harmonized) {
  var = LifetableVar(x, ndx, nax, ex)
  CV  = sqrt(var)/ex[1L]
  if (!isTRUE(harmonized)) {S = CV}
  if (isTRUE(harmonized)) {S = 1-CV}
  return(S)
}

Life-table Gini coefficient, gini, $\text{G}$

The Gini coefficient of the age-at-death distribution represents the mean absolute difference in the ages at death between any two individuals of a population.

We base our discretization of the Gini coefficient on the formulation by [3] because it allows for a straightforward inclusion of ${}_na_x$ and doesn't require special considerations in case of an open-ended age group. An alternative formulation in terms of the life-table survivorship curve proposed by [2] and discretized by [8, 9] is in widespread use in demography.

Continuous

$$ \text{G} = \frac{1}{2e_\theta} \int\limits_{x=\theta}^{\omega}\int\limits_{y=\theta}^{\omega} |x-y|\cdot f(x)f(y)\,\text{d}x\,\text{d}y $$

Discrete

Let $i, j$ be indices of life-table age groups $1\ldots k$. Let $\overline{x}_i = x_i + a_i$ be the average age at death for those who die in age group $i$. Then

$$ \widehat{\text{G}} = \frac {1} {2e_\theta} \sum\limits_{i,j=1}^{k}d_i \cdot d_j \cdot | \overline{x}_i - \overline{x}_j | $$

This expression can easily evaluated using the matrix facilities available in R. We therefore reformulate it in matrix algebra notation. Let $\mathbf{D}_{k\times k}=\mathbf{d} \otimes \mathbf{d}$ and let $\overline{\mathbf{X}} = | \mathbf{e}\overline{\mathbf{x}}^\top - \overline{\mathbf{x}}\mathbf{e}^\top|$, then

$$ \begin{aligned} \widehat{\text{G}} &= \frac {1} {2e_\theta} \sum\limits_{i,j=1}^{k} \mathbf{D} \circ \overline{\mathbf{X}} \ &= \frac {\mathbf{e}^\top\left[\mathbf{D} \circ \overline{\mathbf{X}}\right]\mathbf{e}} {2e_\theta} \end{aligned} $$

Harmonizing Transformation

$$ \text{G}^* = 1-2\text{G} $$

pash implementation

# Life Table Gini-Coefficient
LifetableGini <- function (x, nax, ndx, ex, harmonized) {
  e = rep(1, length(x))
  D = outer(ndx, ndx)
  x_ = x+nax
  X_ = abs(e%*%t(x_) - x_%*%t(e))
  G = sum(D*X_)/(2*ex[1L])
  if (!isTRUE(harmonized)) {S = G}
  if (isTRUE(harmonized)) {S = 1-2*G}
  return(S)
}

Mortality ratio, mr, $\text{MR}$

The ratio between mortality at age 0 and mortality at the age of total life-expectancy. First suggested by [1], further developed by [7].

Continuous

$$ \text{MR} = \frac{\mu(\theta)}{\mu(e_\theta)} $$

Discrete

$$ \widehat{\text{MR}} = \frac {{}nm\theta} {\widehat{\mu}{e\theta}} $$

where, using linear interpolation of the ${}_nm_x$ data, we have

$$ \widehat{\mu}{e\theta} = \frac {1} {n_{\lfloor e_\theta \rfloor}} \left[ {}nm{\lfloor e_\theta \rfloor} (n_{\lfloor e_\theta \rfloor} - \text{frac}~e_\theta) + {}nm{\lceil e_\theta \rceil} \text{frac}~e_\theta \right] $$

Harmonizing Transformation

$$ \text{MR}^* = 1-\text{MR} $$

pash implementation

# Mortality Ratio
MortalityRatio <- function(x, nx, nmx, ex, harmonized){
  m0   = nmx[1L]
  m_e0 = approx(x = x, y = nmx, xout = ex[1L])[["y"]]
  MR   = m0/m_e0
  if (!isTRUE(harmonized)) {S = MR}
  if (isTRUE(harmonized)) {S = 1 - MR}
  return(S)
}

Life expectancy ratio, ler, $\text{LER}$

The ratio between remaining life expectancy at the age of total life expectancy and total life expectancy. Suggested by [7].

Continuous

$$ \text{LER} = \frac{e(e_\theta)}{e(\theta)} $$

Discrete

$$ \widehat{\text{LER}} = 1 - \frac{\widehat{e}{e\theta}}{e_\theta} $$

where, using linear interpolation of the $e(x)$ function, we have

$$ \widehat{e}{e\theta} = \frac{1}{n_{\lfloor e_\theta \rfloor}} \left[ e_{\lfloor e_\theta \rfloor} (n_{\lfloor e_\theta \rfloor} - \text{frac}~e_\theta) + e_{\lceil e_\theta \rceil} \text{frac}~e_\theta \right]. $$

Harmonizing Transformation

$$ \text{LER}^* = 1-\text{LER} $$

pash implementation

# Life Expectancy Ratio
LER <- function(x, nx, ex, harmonized){
  e_e0 = approx(x = x, y = ex, xout = ex[1L])[["y"]]
  ler = e_e0/ex[1L]
  if (!isTRUE(harmonized)) {S = ler}
  if (isTRUE(harmonized)) {S = 1-ler}
  return(S)
}

Average change in force of mortality wrt. $\ell_x$, acfm, $\text{ACFM}$

A rescaled variant of the average lifetime change of the hazard of death divided by average mortality. Suggested by [7].

Continuous

$$ \text{ACFM} = \int\limits_{x=\theta}^{\omega} \frac{\mu(x) - \mu(\theta)}{\overline{\mu}} f(x)\,\text{d}x $$

Discrete

$$ \widehat{\text{ACFM}} = e_\theta \sum_{x=\theta}^{\omega} \left[ ({}nm_x - {}_nm\theta) {}_nd_x \right] $$

Harmonizing Transformation

$$ \text{ACFM}^* = 1-e^{-\text{ACFM}} $$

pash implementation

# Average of Change in Force of Mortality with respect to lx
ACFM <- function(nmx, ndx, ex, harmonized){
  acfm_x = (nmx - nmx[1L]) * ndx
  D = ex[1L] * sum(acfm_x)
  if (!isTRUE(harmonized)) {S = D}
  if (isTRUE(harmonized)) {S = 1-exp(-D)}
  return(S)
}

Probability to survive up to mean age at death, psmad, $\ell_{e_\theta}$

The probability to survive up to the mean age at death, suggested by [3].

Continuous

$$ \ell_{e_\theta} = \text{P}(X>e_\theta) $$

Discrete

Using linear interpolation of the $\ell(x)$ function we get

$$ \widehat{\ell}{e\theta} = \frac{1}{n_{\lfloor e_\theta \rfloor}} \left[ \ell_{\lfloor e_\theta \rfloor} (n_{\lfloor e_\theta \rfloor} - \text{frac}~e_\theta) + \ell_{\lceil e_\theta \rceil} \text{frac}~e_\theta \right]. $$

Harmonizing Transformation

$$ \ell_{e_\theta}^\ast = 1 + \log \ell_{e_\theta} $$

pash implementation

# Probability to Survive up to the Mean Age at Death
PSMAD <- function(x, nx, lx, ex, harmonized){
  l_e0  = approx(x = x, y = lx, xout = ex[1L])[["y"]]
  if (!isTRUE(harmonized)) {S = l_e0}
  if (isTRUE(harmonized)) {S = 1 + log(l_e0)}
  return(S)
}

References

[1]: Baudisch, A. (2011). The pace and shape of ageing. Methods in Ecology and Evolution, 2(4), 375-382. https://doi.org/10.1111/j.2041-210X.2010.00087.x

[2] Hanada, K. (1983). A Formula of Gini's Concentration Ratio and its Applications to Life Tables. Journal of the Japan Statistical Society, 13(2), 95-98. Retrieved from https://www.jstage.jst.go.jp/article/jjss1970/13/2/13_2_95/_pdf

[3]: Kendall, M. G. (1945). The Advanced Theory of Statistics. Volume I (2nd ed.). London: Charles Griffin.

[4]: Keyfitz, N., & Caswell, H. (2005). Applied Mathematical Demography (3rd ed.). New York, NY: Springer.

[5]: Vaupel, J. W., & Romo, V. C. (2003). Decomposing change in life expectancy: a bouquet of formulas in honor of Nathan Keyfitz’s 90th birthday. Demography, 40(2), 201-216.

[6]: Wrycza, T., & Baudisch, A. (2014). The pace of aging: Intrinsic time scales in demography. Demographic Research, 30(1), 1571-1590. https://doi.org/10.4054/DemRes.2014.30.57

[7]: Wrycza, T. F., Missov, T. I., & Baudisch, A. (2015). Quantifying the Shape of Aging. PLOS ONE, 10(3), 1-18. https://doi.org/10.1371/journal.pone.0119163

[8] Shkolnikov, V. M., & Andreev, E. M. (2010). Spreadsheet for calculation of life-table dispersion measures. Retrieved from http://www.demogr.mpg.de/papers/technicalreports/tr-2010-001.pdf

[9]: Shkolnikov, V., Andreev, E., & Begun, A. Z. (2003). Gini coefficient as a life table function. Demographic Research, 8(17), 305-358. https://doi.org/10.4054/DemRes.2003.8.11

jschoeley/pash documentation built on May 20, 2019, 2:07 a.m.