Why Lifespans Are More Variable Among Blacks Than Among Whites in the United States

Abstract

Lifespans are both shorter and more variable for blacks than for whites in the United States. Because their lifespans are more variable, there is greater inequality in length of life—and thus greater uncertainty about the future—among blacks. This study is the first to decompose the black-white difference in lifespan variability in America. Are lifespans more variable for blacks because they are more likely to die of causes that disproportionately strike the young and middle-aged, or because age at death varies more for blacks than for whites among those who succumb to the same cause? We find that it is primarily the latter. For almost all causes of death, age at death is more variable for blacks than it is for whites, especially among women. Although some youthful causes of death, such as homicide and HIV/AIDS, contribute to the black-white disparity in variance, those contributions are largely offset by the higher rates of suicide and drug poisoning deaths for whites. As a result, differences in the causes of death for blacks and whites account, on net, for only about one-eighth of the difference in lifespan variance.

Appendix: Derivation of Equations for Spread, Allocation, and Timing Components

As noted in the text, the ANOVA equation for the total variance in age at death, by cause, is

$$ {\upsigma}^2={\displaystyle \sum_{c=1}^C{\displaystyle \sum_{i=1}^{N_c}\frac{{\left({X}_{ic}-\overline{X}\right)}^2}{N}}}={\displaystyle \sum_{c=1}^C{\displaystyle \sum_{i=1}^{N_c}\frac{x_{ic}^2}{N}}}+{\displaystyle \sum_{c=1}^C{\displaystyle \sum_{i=1}^{N_c}\frac{{\overline{x}}_c^2}{N}}}, $$

(A1)

where N is the total number of deaths and the c = 1, 2, . . . , C causes of death are mutually exclusive and exhaustive. The first term in Eq. (A1) is the within-category component of the variance, and the second term is the between-category component of the variance.

Among those who are victims of a given cause, the variance in age at death is

$$ {\upsigma}_c^2={\displaystyle \sum_{i=1}^{N_c}\frac{x_{ic}^2}{N_c}.} $$

(A2)

Thus:

$$ {\displaystyle \sum_{i=1}^{N_c}{x}_{ic}^2={N}_c{\upsigma}_c^2}. $$

(A3)

Substituting Eq. (A3) into the ANOVA within-component in Eq. (A1), we see that the ANOVA within-cause component is the incidence-weighted sum of the within-cause variances, σ ² _c :

$$ {\displaystyle \sum_{c=1}^C{\displaystyle \sum_{i=1}^{N_c}\frac{x_{ic}^2}{N}}}={\displaystyle \sum_{c=1}^C\frac{N_c{\upsigma}_c^2}{N}}={\displaystyle \sum_{c=1}^C{p}_c{\upsigma}_c^2}, $$

(A4)

where p _c = N _c / N, the proportion of deaths due to cause c. The between-variance component in ANOVA is likewise an incidence-weighted sum of dispersion:

$$ {\displaystyle \sum_{c=1}^C{\displaystyle \sum_{i=1}^{N_c}\frac{{\overline{x}}_c^2}{N}}}={\displaystyle \sum_{c=1}^C\frac{N_c{\overline{x}}_c^2}{N}}={\displaystyle \sum_{c=1}^C{p}_c{\overline{x}}_c^2}. $$

(A5)

The difference in the variances for two populations is the difference in the ANOVA within- and between-variance components for the two populations. It follows that the difference in the overall variances of lifespans of two populations B and W, broken down by cause, is the sum of the within-component difference and the between-component difference in the two populations:

$$ {\upsigma}_B^2-{\upsigma}_W^2={\displaystyle \sum_{c=1}^C{p}_{cB}{\upsigma}_{cB}^2}-{\displaystyle \sum_{c=1}^C{p}_{cW}{\upsigma}_{cW}^2}+{\displaystyle \sum_{c=1}^C{p}_{cB}{\overline{x}}_{cB}^2}-{\displaystyle \sum_{c=1}^C{p}_{cW}{\overline{x}}_{cW}^2}, $$

(A6)

where ∑ _c p _cB = ∑ _c p _cW = 1.0. Because positive differences are more intuitive to interpret than negative differences, we designate the population with the greater variance as B, so σ ² _B − σ ² _W > 0.

Algebraic manipulation of Eq. (A6) separates the spread component, which captures the part of the total disparity in variance that is due to cause-specific differences in the variances of the two populations; the allocation component, which accounts for the contribution of differences in cause-specific incidence; and the timing component, which isolates the effect of differences between the two populations in the variability of the mean age at death across causes. Taking the population with the smaller variance as the reference, and summing over all C causes, the spread, allocation, timing, and joint components are as given in the text (also in Nau and Firebaugh 2012).