Perturbation Analysis of Indices of Lifespan Variability

Abstract

A number of indices exist to calculate lifespan variation, each with different underlying properties. Here, we present new formulae for the response of seven of these indices to changes in the underlying mortality schedule (life disparity, Gini coefficient, standard deviation, variance, Theil’s index, mean logarithmic deviation, and interquartile range). We derive each of these indices from an absorbing Markov chain formulation of the life table, and use matrix calculus to obtain the sensitivity and the elasticity (i.e., the proportional sensitivity) to changes in age-specific mortality. Using empirical French and Russian male data, we compare the underlying sensitivities to mortality change under different mortality regimes to determine the conditions under which the indices might differ in their conclusions about the magnitude of lifespan variation. Finally, we demonstrate how the sensitivities can be used to decompose temporal changes in the indices into contributions of age-specific mortality changes. The result is an easily computable method for calculating the properties of this important class of longevity indices.

Appendices

Appendix: Sensitivity Results and Derivations

This appendix provides details of the derivations. We begin with a summary of some basic matrix calculus techniques, and then present the derivations of the sensitivities of the indices of lifespan variation.

Matrix Calculus Preliminaries

The indices of lifespan variation in Table 3 are functions of scalars, vectors, and matrices. Matrix calculus permits differentiation of all three. The derivative of a scalar y with respect to a scalar x is the derivative $\frac {dy}{dx}$ familiar from basic calculus. The derivative of a n × 1 vector y with respect to a scalar x is the n × 1 vector

$$ \frac{d\mathbf{y}}{dx}=\left(\begin{aligned}{c} \displaystyle \frac{dy_{1}}{dx}\\ \vdots \\ \displaystyle \frac{dy_{n}}{dx}\ \end{aligned}\right). $$

(18)

The derivative of a scalar y with respect to a m × 1 vector x is the 1 × m gradient vector

$$ \frac{dy}{d\mathbf{x}^{\top}}=\left(\frac{\partial y}{dx_{1}} \cdots \frac{\partial y}{dx_{m}}\right). $$

(19)

The derivative of an n × 1 vector y with respect to a m × 1 vector x is the n × m Jacobian matrix, whose (i, j) entry is the derivative of y _i with respect to x _j :

$$ \frac{d\mathbf{y}}{d\mathbf{x}^{\top}}=\left(\frac{dy_{i}}{dx_{j}}\right). $$

(20)

The derivatives of matrices are computed by transforming the matrices into column vectors using the $\text {vec}$ operator and applying the rules for vector differentiation. Thus, the derivative of the m × n matrix Y with respect to the p × q matrix X is the mn × pq matrix

$$ \frac{d\mathbf{Y}}{d\mathbf{X}}=\frac{d\text{vec}\mathbf{Y}}{d\text{vec}^{\top}\mathbf{X}}. $$

(21)

For notational simplicity, we denote $(d\text {vec}\mathbf {X})^{\top }$ as $d\text {vec}^{\top }\mathbf {X}$.

These definitions imply the chain rule for matrix calculus; if Y is a function of X, and X is a function of Z, then

$$ \frac{d\text{vec}\textbf{Y}}{d\text{vec}^{\top}\textbf{Z}}=\frac{d\text{vec}\textbf{Y}}{d\text{vec}^{\top}\textbf{X}}\frac{d\text{vec}\textbf{X}}{d\text{vec}^{\top}\textbf{Z}}. $$

(22)

Matrix derivatives are constructed by forming differentials, where the differential of a matrix (or vector) is the matrix (or vector) of differentials of the elements; that is,

$$ d\mathbf{X}=(dx_{ij}). $$

(23)

If, for some matrix Q, it can be shown that

$$ d\mathbf{y}=\mathbf{Q}d\mathbf{x}, $$

(24)

then according to the “first identification theorem” of Magnus and Neudecker (1985),

$$ \frac{d\mathbf{y}}{d\mathbf{x}^{\top}}=\mathbf{Q}. $$

(25)

We will frequently obtain expressions of the form shown in Eq. (25) using a theorem originated by Roth (1934): if $\mathbf {Y}=\mathbf {ABC}$, then

$$ \text{vec}\mathbf{Y} = (\mathbf{C}^{\top} \otimes \mathbf{A})\text{vec}\mathbf{B}. $$

(26)

We will also simplify expressions involving Kronecker products using

$$ (\mathbf{A} \otimes \mathbf{B})(\mathbf{C} \otimes \mathbf{D}) = \mathbf{AC} \otimes \mathbf{BD} $$

(27)

whenever AC and BD are defined.

More details on matrix calculus can be found in Magnus and Neudecker (1988). A good mathematical introduction can be found in Abadir and Magnus (2005), and demographic discussions appear in Caswell (2007, 2008, 2010).

Sensitivities of the Indices of Lifespan Variation

Preliminaries

Differentiating the various indices made use of the following sensitivities. The vector of life expectancies as a function of age is given by

$$ \boldsymbol{\upeta}^{\top} = \mathbf{e}^{\top} \mathbf{N}. $$

(28)

The derivative of this vector with respect to mortality is (Caswell 2006, 2009)

$$ \frac{d\boldsymbol{\upeta}}{d\boldsymbol{\uptheta}^{\top}} = \left(\mathbf{I}\otimes\mathbf{e}^{\top}\right)\left(\mathbf{N}^{\top}\otimes\mathbf{N}\right)\frac{d\text{vec}\mathbf{U}}{d\boldsymbol{\uptheta}^{\top}}. $$

(29)

The life expectancy at birth is given by

$$ \upeta_{1} = \boldsymbol{\upeta}^{\top} \mathbf{e}_{1}, $$

(30)

and thus its derivative is

$$ \frac{d{\upeta_{1}}}{d\boldsymbol{\uptheta}^{\top}} = \left(\mathbf{e}_{1}^{\top} \otimes \mathbf{e}^{\top}\right) \frac{d\text{vec} \mathbf{N}}{d\boldsymbol{\uptheta}^{\top}} $$

(31)

$$ = \left(\mathbf{e}_{1}^{\top}\mathbf{N}^{\top} \otimes \mathbf{e}^{\top} \mathbf{N}\right) \frac{d\text{vec} \mathbf{U}}{d\boldsymbol{\uptheta}^{\top}}. $$

(32)

The distribution of age at death is given by the vector

$$ \mathbf{f} = \mathbf{MNe}_{1}. $$

(33)

Its derivative is given by (Caswell 2010),

$$ {\frac{d\mathbf{f}}{d\boldsymbol{\uptheta}^{\top}}} = {\left(\mathbf{e}_{1}^{\top}\mathbf{N}^{\top}\otimes\mathbf{I}\right)} \frac{d\text{vec}{\mathbf{M}}}{d\boldsymbol{\uptheta}^{\top}} + \left(\mathbf{e}_{1}^{\top}\mathbf{N}^{\top}\otimes\mathbf{B}\right)\frac{d\text{vec}{\mathbf{U}}}{d\boldsymbol{\uptheta}^{\top}}. $$

(34)

The derivatives of U and M depend on the structure of the life cycle; in the age-classified case under consideration here, M contains the probabilities of death q _i on the diagonal, and U contains the probabilities of survival 1−q _i on the subdiagonal.

Life Disparity, $\upeta ^{\dagger }$

The disparity can be written as

$$ \upeta^{\dagger} = \mathbf{f}^{\top} \boldsymbol{\upeta}. $$

(35)

As shown in Caswell (2010, 2011),

$$ d\upeta^{\top} = \left(d\mathbf{f}^{\top}\right)\boldsymbol{\upeta} + \mathbf{f}^{\top} \left(d\boldsymbol{\upeta}\right), $$

(36)

and thus

$$ \frac{d\upeta^{\dagger}}{d\boldsymbol{\uptheta}^{\top}} = \boldsymbol{\upeta}^{\top} \frac{d\mathbf{f}}{d\boldsymbol{\uptheta}^{\top}} + \mathbf{f}^{\top} \frac{d\boldsymbol{\upeta}}{d\boldsymbol{\uptheta}^{\top}}, $$

(37)

where $d\mathbf {f}/ d\boldsymbol {\uptheta }^{\top }$ is given by (34) and $d\boldsymbol {\upeta } / d \boldsymbol {\uptheta }^{\top }$ is given by (29).

Gini Coefficient

In matrix form, the Gini coefficient is given by

$$ G=1-\frac{1}{\upeta_{1}} \mathbf{e}^{\top}\left[\boldsymbol{\ell} \circ \boldsymbol{\ell} \right] , $$

(38)

where the survivorship vector is

$$ \boldsymbol{\ell} = \mathbf{e}-\mathbf{Cf} . $$

(39)

Differentiating (38), noting that $\upeta _{1}$ is a scalar, gives

$$ d G = \frac{1}{\upeta_{1}^{2}} \mathbf{e}^{\top} \left(\boldsymbol{\ell} \circ \boldsymbol{\ell} \right) d \upeta_{1} - \frac{2}{\upeta_{1}} \mathbf{e}^{\top} \left[ \boldsymbol{\ell} \circ (d \boldsymbol{\ell}) \right] . $$

(40)

We apply the vec operator to both sides of (40) and obtain

$$ d G = \frac{1}{\upeta_{1}^{2}} \mathbf{e}^{\top} \left(\boldsymbol{\ell} \circ \boldsymbol{\ell} \right) d \upeta_{1} - \frac{2}{\upeta_{1}} \ell^{\top} d \boldsymbol{\ell} . $$

(41)

Differentiating (39) gives $d \boldsymbol {\ell } = - \mathbf {C} d \mathbf {f}$; substituting this into (41) and using the chain rule gives

$$ \frac{dG}{d\boldsymbol{\uptheta}^{\top}} = \frac{1}{\upeta_{1}^{2}} \mathbf{e}^{\top} \left(\boldsymbol{\ell} \circ \boldsymbol{\ell} \right) \frac{d\upeta_{1}}{d\boldsymbol{\uptheta}^{\top}} + \frac{2}{\upeta_{1}} \ell^{\top} \mathbf{C} \frac{d\mathbf{f}}{d\boldsymbol{\uptheta}^{\top}}, $$

(42)

where $d \upeta _1/ d \boldsymbol {\uptheta }^{\top }$ is given by (32)

Mean Logarithmic Deviation

The mean logarithmic deviation in matrix notation is

$$ MLD = \mathbf{f}^{\top} \left[ \mathbf{e} \log \upeta_{1} - \log \mathbf{x} \right] , $$

(43)

where the logarithm is applied elementwise. Differentiating (43) gives

$$ dMLD = \left( d \mathbf{f}^{\top} \right) \left[ \mathbf{e} \log \upeta_{1} - \log \mathbf{x} \right] + \mathbf{f}^{\top} \mathbf{e} \left( d \log \upeta_{1} \right) . $$

(44)

However, $\mathbf {f}^{\top } \mathbf {e} =1$ because f is a probability distribution. Using this fact and also noting that $d \log \upeta _{1} = (1/\upeta _1) d \upeta _{1}$, we obtain

$$ \frac{dMLD}{d\boldsymbol{\uptheta}^{\top}} = \left[ \mathbf{e}^{\top} \log \upeta_{1} - \log \mathbf{x}^{\top} \right] \frac{d\mathbf{f}}{d\boldsymbol{\uptheta}^{\top}} + \frac{1}{\upeta_{1}} \frac{d\upeta_{1}}{d\boldsymbol{\uptheta}^{\top}}, $$

(45)

where $d \upeta _1/ d \boldsymbol {\uptheta }^{\top }$ is given by (32) and $d \mathbf {f}/ d\boldsymbol {\uptheta }^{\top }$ is given by (34).

Theil’s Index

The expression for Theil’s index in matrix notation is

$$ T = \mathbf{f}^{\top} \left[ \frac{\mathbf{x}}{\upeta_{1}} \circ \log \frac{\mathbf{x}}{\upeta1} \right] , $$

(46)

where the logarithm is applied elementwise. Differentiating (46) term by term yields

$$ {}dT \,=\, {} \left( d \mathbf{f}^{\top} \right) {} \left[ \frac{\mathbf{x}}{\upeta_{1}} \circ \log \frac{\mathbf{x}}{\upeta1} \right] + \mathbf{f}^{\top} {} \left[ d {} \left( \frac{\mathbf{x}}{\upeta_{1}} \right) \circ \log \frac{\mathbf{x}}{\upeta_{1}} \right] + \mathbf{f}^{\top} {} \left[ \frac{\mathbf{x}}{\upeta_{1}} \circ d {} \left( \log \frac{\mathbf{x}}{\upeta_{1}} \right) \right] {} . $$

(47)

However,

$$ d \left( \frac{\mathbf{x}}{\upeta_{1}} \right) = -\frac{\mathbf{x}}{\upeta_{1}^{2}} d \upeta_{1} $$

(48)

$$\begin{array}{rll} d \left( \log \frac{\mathbf{x}}{\upeta_{1}} \right) &=& d \left( \log \mathbf{x} - \mathbf{e} \log \upeta_{1} \right) \\ &=& -\frac{\mathbf{e}}{\upeta_{1}} d \upeta_1 . \end{array} $$

(49)

Substituting (48) and (49) into (47), and transposing the first term, gives

$$ dT = \left( \frac{\mathbf{x}^{\top}}{\upeta_{1}} \circ \log \frac{\mathbf{x}^{\top}}{\upeta_{1}} \right) d \mathbf{f} - \mathbf{f}^{\top} \left[ \left( \frac{\mathbf{x}}{\upeta_{1}^{2}} \circ \log \frac{\mathbf{x}}{\upeta_{1}} \right)+ \left( \frac{\mathbf{x}}{\upeta_{1}} \circ \frac{\mathbf{e}}{\upeta_{1}} \right) \right] d \upeta_1 . $$

(50)

Simplifying Eq. (50) and expressing the result in terms of a parameter vector θ gives

$$ \frac{dT}{d\boldsymbol{\uptheta}^{\top}} = \left( \frac{\mathbf{x}^{\top}}{\upeta_{1}} \circ \log \frac{\mathbf{x}^{\top}}{\upeta_{1}} \right) \frac{d\mathbf{f}}{d\boldsymbol{\uptheta}^{\top}} - \left( \frac{T}{\upeta_{1}} + \frac{\mathbf{f}^{\top} \mathbf{x}}{\upeta_{1}^{2}} \right) \frac{d\upeta_{1}}{d\boldsymbol{\uptheta}^{\top}}, $$

(51)

where $d \upeta _1/ d \boldsymbol {\uptheta }^{\top }$ is given by (32) and $d \mathbf {f}/ d\boldsymbol {\uptheta }^{\top }$ is given by (34).

The Variance and Standard Deviation of Longevity

The variance in longevity, conditional on survival to age class i, is given by the vector v, which satisfies

$$ \mathbf{v}^{\top} = \mathbf{e}^{\top} \mathbf{N} \left( 2 \mathbf{N} - \mathbf{I} \right) - \boldsymbol{\upeta}^{\top} \circ \boldsymbol{\upeta}^{\top} . $$

(52)

Caswell (2006, 2009, 2010) shows that

$$ \frac{d\mathbf{v}}{d\boldsymbol{\uptheta}^{\top}} = \left[2 \left( \mathbf{N}^{\top} \otimes \mathbf{e}^{\top} \right) + 2 \left( \mathbf{I} \otimes \mathbf{e}^{\top} \mathbf{N} \right) - \left( \mathbf{I} \otimes \mathbf{e}^{\top} \right) \right] \frac{d\text{vec}\mathbf{N}}{d\boldsymbol{\uptheta}^{\top}} - 2\text{diag}(\boldsymbol{\upeta} ) \frac{d\boldsymbol{\upeta}}{d\boldsymbol{\uptheta}^{\top}}, $$

(53)

where $d \boldsymbol {\upeta } / d \boldsymbol {\uptheta }^{\top }$ is given by (29) and

$$ \frac{d\text{vec} \mathbf{N}}{d\boldsymbol{\uptheta}^{\top}} = \left( \mathbf{N}^{\top} \otimes \mathbf{N} \right) \frac{d\text{vec} \mathbf{U}}{d\boldsymbol{\uptheta}^{\top}}. $$

(54)

The standard deviation of longevity is given by the vector

$$ \mathbf{s} = \sqrt{\mathbf{v}}, $$

(55)

where the square root is taken elementwise and its sensitivity, as derived in Caswell (2010), is

$$ \frac{d\mathbf{s}}{d\boldsymbol{\uptheta}^{\top}} = \frac{1}{2}\text{diag}\left(\mathbf{s}\right)^{-1}\frac{d\mathbf{v}}{d\boldsymbol{\uptheta}^{\top}}. $$

(56)

The Interquartile Range

The interquartile range is defined implicitly in terms of the distribution of ages at death. Let $f\left ( x \right )$be a probability density function and $F\!\!\left(x\right)=\displaystyle\int_{-\infty }^{x}f\!\!\left(s\right)ds$ be the cumulative distribution. The qth quantile is the value $\hat {x}$ satisfying

$$ F\left(\hat{x}\right)=q. $$

(57)

Let $F\left (\hat {x}_{1}\right )=q_{1}$ and $F\left (\hat {x}_{2}\right )=q_{2}$, assuming that q ₂ > q ₁. The interquantile range is

$$ R\left ( q_{1},q_{2} \right )=\hat{x_{2}}-\hat{x_{1}}. $$

(58)

The special case of the interquartile range refers to R(0.25,0.75).

Now we choose a set of probabilities of interest

$$ \textbf{q} =\left(\begin{aligned}{c} q_{1}\\ \vdots \\ q_{h} \end{aligned}\right), $$

(59)

and let $\hat {\mathbf {x}}$ be the vector of quantiles that satisfy

$$ F\left [ \boldsymbol{\uptheta},\hat{\mathbf{x}}\left (\boldsymbol{\uptheta}\right ) \right ]=\mathbf{q}, $$

(60)

where the distribution $f\left ( \cdot \right )$ depends on a parameter vector θ of dimension p × 1.

Next we differentiate Eq. (60) as follows:

$$ \frac{\partial F}{\partial \boldsymbol{\uptheta}^{\top}}d \boldsymbol{\uptheta}+\frac{\partial F}{\partial \hat{\mathbf{x}}^{\top}}d\hat{\mathbf{x}}=0, $$

(61)

and solve for $d\hat {\mathbf {x}}$, to obtain

$$ d\hat{\mathbf{x}}=-\left (\frac{\partial F}{\partial \hat{\mathbf{x}}^{\top}} \right )^{-1}\left (\frac{\partial F}{\partial \boldsymbol{\uptheta}^{\top}} \right )d \boldsymbol{\uptheta}. $$

(62)

The first identification theorem implies that

$$ \frac{d\hat{\mathbf{x}}}{d\boldsymbol{\uptheta}^{\top}}=-\left (\frac{\partial F}{\partial \hat{\mathbf{x}}^{\top}} \right)^{-1}\left ( \frac{\partial F}{\partial \boldsymbol{\uptheta}^{\top}} \right) . $$

(63)

The first term on the right side of of Eq. (63) is

$$ \left(\frac{\partial F}{\partial \hat{\mathbf{x}}^{\top}} \right)^{-1}=\left( \begin{aligned} \frac{1}{f\left(\hat{x}_{1}\right)} && 0\\ & \ddots & \\ 0 & & \frac{1}{f\left(\hat{x}_{h}\right)} \end{aligned}\right), $$

(64)

while the second term is

$$ \left (\frac{\partial F}{\partial \uptheta^{\top}} \right )=\left( \begin{aligned} \frac{\partial F\left(\hat{x}_{1}\right)}{\partial \uptheta_{1}} & \cdots & \frac{\partial F\left(\hat{x}_{1}\right)}{\partial \uptheta_{p}}\\ \vdots & & \vdots \\ \frac{\partial F\left(\hat{x}_{h}\right)}{\partial \uptheta_{1}} & \cdots & \frac{\partial F\left(\hat{x}_{h}\right)}{\partial \uptheta_{p}} \end{aligned}\right). $$

(65)

The product of Eqs. (64) and (65), following Eq. (63), gives

$$ \left (\frac{d\hat{x}}{d\uptheta^{\top}} \right )=-\left( \begin{aligned}f \frac{1}{f\left(\hat{x}_{1}\right)}\frac{\partial F\left(\hat{x}_{1}\right)}{\partial \uptheta_{1}} & \cdots & \frac{1}{f\left(\hat{x}_{1}\right)}\frac{\partial F\left(\hat{x}_{1}\right)}{\partial \uptheta_{p}}\\ \vdots & & \vdots \\ \frac{1}{f\left(\hat{x}_{h}\right)}\frac{\partial F\left(\hat{x}_{h}\right)}{\partial \uptheta_{1}} & \cdots & \frac{1}{f\left(\hat{x}_{h}\right)}\frac{\partial F\left(\hat{x}_{h}\right)}{\partial \uptheta_{p}} \end{aligned}\right). $$

(66)

The sensitivity of the interquantile range is the difference between row j and row i of (66):

$$ \frac{dR_{\left(i,j\right)}}{d\boldsymbol{\uptheta}^{\top}}= \frac{d\hat{x}_{j}}{d\boldsymbol{\uptheta}^{\top}}-\frac{d\hat{x}_{i}}{d\boldsymbol{\uptheta}^{\top}}. $$

(67)

When $\mathbf {f}(x)$is a discrete distribution, the quantiles will have to be interpolated. This is what we did to find the sensitivity of the IQR with quartiles $\hat {x}_{3}$ and $\hat {x}_{1}$.

Appendix Figures Depicting the Sensitivity and Elasticity of Indices Calculated From Age 10

Fig. 6

The sensitivity of each index conditional on survival to age 10 with respect to mortality change at different ages. The sensitivities were standardized to the value of each index, i.e. $y_{10}\frac {dy}{d\uptheta }$, to make them comparable. Note the difference in scale between the top and bottom panels, plotted separately to more clearly delineate behavior of the indices at early and later ages. French males, period life table data from the Human Mortality Database

Fig. 7

The proportional change in the index calculated conditional upon survival to age 10 from a 1 % change in mortality at each age on the x-axis. French males, period life table data from the Human Mortality Database