Artículo de revista: Permanyer, Iñaki; Sasson, Isaac; Villavicencio, Francisco (2023) «Group- and Individual-Based Approaches to Health Inequality: Towards an Integration». Journal of the Royal Statistical Society Series A: Statistics in Society, febrer, qnac001. (ISSN 1467-985X) https://doi.org/10.1093/jrsssa/qnac001. Journal of the Royal Statistical Society Series A: Statistics in Society, 2023, 0, 1–24 https://doi.org/10.1093/jrsssa/qnac001 Original Article Group- and individual-based approaches Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023 to health inequality: towards an integration Iñaki Permanyer1,2 , Isaac Sasson3 and Francisco Villavicencio4,5,6 1 Centre for Demographic Studies (CED), Universitat Autònoma de Barcelona, Bellaterra, Spain 2 ICREA, Passeig Lluís Companys 23, Barcelona, Spain 3 Department of Sociology and Anthropology, Tel Aviv University, Tel Aviv, Israel 4 Department of Economic, Financial and Actuarial Mathematics, University of Barcelona, Barcelona, Spain 5 Interdisciplinary Center on Population Dynamics, University of Southern Denmark, Odense, Denmark 6 Department of International Health, Johns Hopkins Bloomberg School of Public Health, Baltimore, MD, USA Address for correspondence: Iñaki Permanyer, Carrer de Ca n’Altayó, Edifici E2, Universitat Autònoma de Barcelona, 08193 Bellaterra/Barcelona, Spain. Email: ipermanyer@ced.uab.es Abstract When assessing health inequalities, should one compare health outcomes across predetermined groups (e.g., race, ethnicity, socioeconomic status), or across individuals? Group-based approaches comparing group- specific means do not account for intra-group heterogeneity. Yet, traditional approaches based on additive decompositions splitting total inequality in its within- and between-group components fail to elucidate the groups’ relative performance. Here, we develop a third approach based on pairwise comparisons to evaluatenot only the variability that might exist across individuals within and between groups, but also the relative performance of the different groups vis-a-vis each other—thus integrating both perspectives into a coherent framework. Keywords: inequality decomposition, life table, lifespan inequality, mortality differentials, social groups Introduction The study of health inequalities has become a prominent issue in global research and policy agen- das since the early 2000s. Average health attainments invariably mask the heterogeneity in the underlying distribution of health, and there is a widespread agreement on the need to go beyond the ‘study of means’ when studying countries’ overall performance. When assessing health in- equalities, the choice of the basic unit of analysis is essential: Should one compare health outcomes across predetermined groups (classified, for instance, by race, ethnicity, religion, or socioeconomic status), or across individuals? Around the early 2000s, there was a brief but intense debate on whether health differences should be based on comparisons among groups or individuals. The pro- ponents of the group-based approach claimed that individual-based approaches to inequality ‘re- move equity and human rights from the public health monitoring agenda’ because of their exclusive focus on individuals—rather than on the groups to which they belong (Braveman et al., 2001:678). Ignoring such characteristics, they argued, one would be unable to identify whether some groups were disadvantaged vis-a-vis the others (Braveman et al., 2000, 2001). On the contrary, supporters of the individual-based approach argued that focusing exclusively on between-group differences and ignoring the variability existing within groups would miss an important part of the story. Instead, they suggested investigating the inter-individual variations Received: October 26, 2021. Accepted: October 28, 2022 © (RSS) Royal Statistical Society 2023. This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial License (https://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact journals.permissions@oup.com 2 J R Stat Soc Series A: Statistics in Society, 2023, Vol. 0, No. 0 in health outcomes first and then trying to determine how much of such variability would be ex- plained by social class, race, or any other grouping one might be interested in (Gakidou et al., 2000; Murray, 2001). Almost two decades later, no consensus seems to have been reached and current approaches fail to provide answers that are satisfactory to both sides simultaneously. On the one hand, there are countless studies comparing mean health outcomes (say, life expectancy) across groups that do not Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023 incorporate the heterogeneity that might exist within those groups into the analysis. On the other hand, the individual-based approach has gained considerable traction during the last years (Edwards & Tuljapurkar, 2005; Gakidou et al., 2000; Gakidou & King, 2002; Murray, 2001; van Raalte et al., 2018), but still fails to integrate the demands/requirements of the group-based approach (namely: to account for the performance of the different groups vis-a-vis each other). State-of-the-art methods relying on additively decomposable measures (like the variance or the Theil index) inform users on whether within-group inequality is larger or smaller than between-group inequality (Gakidou & King, 2002; Permanyer et al., 2018; Permanyer & Scholl, 2019; Seaman et al., 2019; van Raalte et al., 2012), but tell us nothing about groups’ rela- tive performance (e.g., ‘Are the rich performing much better than the poor?’). In this paper, we develop a new approach to decompose a well-known class of individual-based health inequality measures that, at the same time, inform users about the groups’ relative perform- ances—thus integrating the requirements of both sides of the debate into a coherent framework. We focus on a very specific health outcome: length of life. Being alive is a precondition for any so- cial phenomenon we might be interested in, so inequality in length of life is ‘the most fundamental of all inequalities’ (van Raalte et al., 2018:1002). Longevity is a crude but extremely useful, non- intrusive, and easy-to-measure health outcome that is collected worldwide on a regular basis. While the study of average longevity (i.e., life expectancy) has attracted considerable attention in demography and other social sciences for a long time, in recent years we have witnessed an up- surge of interest to look beyond the means and study the levels, trends and determinants of lifespan inequality (see, among others, Aburto et al., 2020; Colchero et al., 2016; Edwards, 2011; Edwards & Tuljapurkar, 2005; Engelman et al., 2010; Gillespie et al., 2014, Nau & Firebaugh, 2012; Sasson, 2016; Seligman et al., 2016; Smits & Monden, 2009; van Raalte et al., 2014; van Raalte et al., 2018; van Raalte & Caswell, 2013; Vaupel et al., 2011; Wilmoth & Horiuchi, 1999). To illustrate the usefulness of our approach, we apply it to racial and ethnic inequalities in length-of-life distributions in the United States. The elimination of racial health disparities is a prominent and pressing issue in the United States (Wrigley-Field, 2020), and it has been one of the main goals in its public health policies. Black–White disparities in mortality are well docu- mented in the literature, and so are their underlying causes of death (Firebaugh et al., 2014a; Harper et al., 2012; Hummer & Chinn, 2011). However, with few exceptions (e.g., Firebaugh et al., 2014b; Gillespie et al., 2014), past studies have focused primarily on differences in life ex- pectancy between these two populations. With our new methodological approach, we capture dis- tributional differences in lifespans among various population groups and quantify precisely by how much an average person from one group (White) outlives a person from another group (Black) and vice versa. We apply this method to Black and White lifespan distributions from 1970 to 2018. Information about Hispanics is also incorporated in the analysis from 2006 on- wards. Even though the illustrations and empirical results presented in this paper are all based on the length of life, this new framework can be adapted to the study of other health outcome var- iables that are measured on a ratio scale, like length of healthy life, frailty, height, weight, blood pressure, or grip strength. Background and motivation There are several approaches to assess the extent of inequality in distributions of individual-based health-related outcomes in populations that are partitioned across G social groups. A very popular one is to generate group-specific averages μ1, μ2, …, μG to compare the relative performance of the different groups vis-a-vis each other. In this way, we can easily assess whether (and to what extent) group A is performing, on average, better or worse than group B. Unfortunately, such approach ignores the variability that might exist within the different groups—which, depending on the health outcome we are working with, can be potentially large. Permanyer et al. 3 An alternative to the group-based perspective is to assess the influence that the population par- tition has on the overall levels of inter-individual inequality. In this regard, the literature on income inequality provides useful examples of indicators that weigh the variation occurring within groups vis-a-vis the variation across groups (Silber, 1999). The so-called ‘additively decomposable in- equality measures’ can be decomposed as Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023 I = Ib + Iw (1) where I measures overall inter-individual inequality, Ib is the inequality that would be observed if all individuals in each group (g) attained the same value (μg), and Iw is the average level of within-group inequality. Thus, additively decomposable inequality indices can be nicely broken down into two clearly interpretable components: (a) the between-group component (which is ob- tained after suppressing within-group variation by assuming all members in each group take a common value: the group mean) and (b) the within-group component (which is a weighted average of the inequality within each group). As shown by Shorrocks (1980), the class of inequality meas- ures that are additively decomposable is quite restricted, though. For the case of relative inequality, it only comprises the one-parameter family of generalized entropy measures, such as the Theil in- dex and the mean log deviation.1 For absolute inequality measures, the variance is the only one that is additively decomposable. Most of these indicators have been used to study lifespan inequal- ity (see, for instance, Edwards, 2011; Permanyer et al., 2018; Smits & Monden, 2009; van Raalte et al., 2012). In Appendix A, we explicitly show the additive decomposition formulae of these in- dicators when a population is partitioned in several groups. While the additive decomposition approach can tell us whether differences between groups are more or less prominent than differences within groups, it does not provide any information about the relative performance of the different groups vis-a-vis each other—which is typically among the most pressing issues one is interested in when inspecting inequalities. Indeed, the notion of ‘in- equality’ is inextricably linked to the idea that some groups are advantaged while others are dis- advantaged. Usually, this kind of ‘relative performance’ information is inferred from the values and relative position of the group-specific means (i.e., the corresponding life expectancies)—which has been traditionally linked to the group-based approach. Yet, inferring groups’ relative perform- ance based solely on their corresponding life expectancies disregards the heterogeneity that might exist within those groups, thus missing potentially relevant information. This is illustrated in the two hypothetical scenarios comparing the age-at-death distributions for populations A and B shown in Figure 1. In the first scenario, there is almost no variability in the corresponding lifespan distributions, so ranking the two populations based on their corresponding life expectancies at birth (denoted by eA B 0 and e0 , respectively) is not particularly misleading. In the second scenario, A B both e0 and e0 are kept unchanged, but there is much greater variability in population A. Now, while individuals in population B, on average, tend to live longer lives than those in A, a non- negligible set of individuals in population A outlive by several years all individuals from popula- tion B. And yet, the between-group component of standard additively decomposable inequality measures like the Theil index or the variance would give the same result in both scenarios. In such circumstances, assessing the relative performance of the two populations vis-a-vis each other solely based on the values of eA B 0 and e0 misses an important part of the story. In this paper, we suggest a new approach to decompose overall inequality when the population we are studying is partitioned across socially relevant groups. The basic ideas our new method builds upon are simple, and differences to existing inequality measures pertain mainly to the com- putation of the between-group component. Specifically, rather than assuming that individuals’ at- tainments can be meaningfully represented by the average attainment of their group, the approach we adopt requires that the level of inequality in a given society be based on making all possible pairwise comparisons among individuals. In this way, a natural decomposition ensues in which overall inequality can be broken down into two clearly interpretable parts: the standard within-group component (consisting of a weighted average of inequalities within groups) and a new between-group component containing the average distances among all pairs of groups. In 1 While the Atkinson index of inequality (Atkinson 1970) is not additively decomposable, it is a non-linear and monotonic transformation of the class of Generalized Entropy measures (see Shorrocks 1980). 4 J R Stat Soc Series A: Statistics in Society, 2023, Vol. 0, No. 0 Scenario 1 Scenario 2 eA o eB o eA o eB o Distribution of deaths Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023 dA x dB x dB x dA x Age Figure 1. Two scenarios with two hypothetical age-at-death distributions dxA and dxB from two populations A and B. In both scenarios, the corresponding means of the distributions (e0A and e0B , respectively) are the same. addition, such distances can be further decomposed into two additional components: the part of the distance involving gaps in favour of one group and the part involving gaps in the opposite dir- ection. Thus, we are able to assess not only ‘how far apart’ any two groups are, but also their cor- responding relative position, that is, whether one group is more or less disadvantaged with respect to the other, and vice versa. This kind of information cannot be inferred from the comparison of group-based life expectancies alone. There is a long tradition of defining inequality measures based on all possible pairwise compar- isons. The well-known Gini coefficient and other conceptually related measures (which have often been used in lifespan inequality analysis, e.g., Aburto et al., 2022; Shkolnikov et al., 2003; Smits & Monden, 2009) are constructed exactly in that way. The novelty here is not on the inequality index itself but in the proposed decomposition and the insights it can generate for a better understanding of health inequality across and within social groups.2 Unlike current existing approaches, this new framework can explicitly assess how different groups perform vis-a-vis each other—thus integrat- ing comparisons across individuals and social groups. A new approach to decompose lifespan variation We present a new decomposition approach that, most notably, applies to the absolute and relative Gini coefficients. The Gini coefficient was originally proposed to measure income and wealth in- equality, and is one of the most popular indices in the social sciences. As thoroughly discussed by Yitzhaki & Schechtman (2013), there are several equivalent ways to define the Gini coefficient, which might depend on the context and the application. In this paper, we define it using life table notation. Life tables have been extensively used by demographers, biologists, epidemiologists, and actuarial scientists to describe the mortality and survival experience of a population (Preston et al., 2001). The literature on the measurement of lifespan inequality is based on the study of the age-at-death distributions described by such life tables, rather than by crude death counts by age, because the latter are affected by the age structure of the population(s) one is working with (e.g., crude death counts might be higher in low-mortality populations owing to the larger shares of individuals surviving to older ages, where the risk of mortality is inevitably higher). These prob- lems are sidestepped with the use of life tables, which render comparable populations with differ- ent age structures. 2 As discussed in further detail in Section 6, the Gini index is not additively decomposable as described in (1). However, it is amenable to other decomposition approaches, like the one proposed in this paper. Permanyer et al. 5 In life tables, ℓ0 denotes the initial population at age 0, which is usually set to 1 or 100,000 and referred to as the radix. Each life table has associated an age-at-death distribution for all the re- ported ages, commonly denoted by d = {d0 , · · · , dω }, where  dx is the number of individuals who die at age x and ω is the maximum possible age.3 Note that ωx=0 dx = ℓ0 , so d can be thought as the density function of the age-at-death distribution. For additional details on life tables, see for instance Preston et al., (2001). Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023 Using this life table notation, the absolute Gini coefficient can be defined as 1  ω  ω Δ(d): = da db |a − b| . (2) 2ℓ20 a=0 b=0 This measure of inequality considers all possible pairs of ages and calculates the average similarity/ dissimilarity among them—obtaining what is oftentimes referred to as the ‘average inter- individual difference’ (or AID). A related measure is the relative Gini coefficient of the life table, given by 1  ω  ω Δ(d) G(d) : = 2 da db |a − b| = , (3) 2ℓ0 e0 a=0 b=0 e0 where e0 is the life expectancy at birth. These two measures both have a clear interpretation: The absolute Gini coefficient measures half the expected age-at-death difference between two random- ly chosen individuals, whereas the relative Gini coefficient is defined as the ratio between the ab- solute Gini and the life expectancy associated to d (that is, it puts the AID in relation to the mean of the age-at-death distribution). In the remaining of this section, we present decompositions of Δ(d), which are later extended to G(d) in Appendix B. Decompositions: the two-group case Assume a population that is partitioned in two groups, for instance females (F) and males (M), with respective population sizes nF and nM, such that the total population is n = nF + nM. Given the group- specific age-at-death distributions dF = {dF0 , · · · , dFω } and dM = {dM 0 , · · · , dω }, it is easy to show M (see details in Appendix C) that the overall lifespan inequality can be broken down as Δ(d) = S F IW F + S M IW M + S FM IBFM , (4) where F IW : =Δ(dF ) M IW : =Δ(dM ) 1  ω  ω IBFM : = dF dM · |a − b| 2ℓ0 a=0 b=0 a b 2 n 2 F SF : = n n 2 M SM : = n nF nM S FM : =2 2 n 3 In this paper, we show results for ages ranging from 0 to ω, the minimum and maximum possible ages, respectively. The decomposition formulas we propose do not change if those bounds are changed to a and b, respectively (with 0 ≤ a < b ≤ ω). 6 J R Stat Soc Series A: Statistics in Society, 2023, Vol. 0, No. 0 Equation (4) has an intuitive interpretation: It decomposes overall inequality in a within-group F (S F IW + S M IW M ) and a between-group (S FM IBFM ) component. The within-group component is the weighted sum of the inequality within each of the groups (i.e., women and men), and the between-group takes into account the age-at-death differences between women and men. More spe- F M cifically, IW and IW are the pairwise comparisons as defined in (4) within each group, whereas IBFM measures half of the expected difference in years of life lived between a randomly chosen woman and Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023 a randomly chosen man. The latter is a simple way of measuring how ‘close’ or ‘far apart’ the two age-at-death distributions are.4 The coefficients S F , S M and S FM are the population weights of each component, satisfying S F + S M + S FM = 1. Decompositions: the multiple-group case Assume now that the population we are studying is partitioned in G ≥ 2 groups. For each popu- g lation subgroup g we have an age-at-death distribution dg = {d0 , · · · , dgω }. The population size of each group is denoted as ng. Analogously to the two-group case, Δ(d) can be decomposed as  G g  G  g−1 gh Δ(d) = ΔW (d) + ΔB (d) = S g IW + S gh IB , (5) g=1 g=2 h=1 where g IW : =Δ(dg ) gh 1  ω  ω IB : = dg dh · |a − b| 2ℓ0 a=0 b=0 a b 2 n 2 g Sg : = n ng nh S gh : =2 n2 Equation (5) breaks down total inequality (as measured by the absolute Gini) in a within-group (ΔW(d)) and a between-group (ΔB(d)) component. Like before, the within-group component is a g population-weighted average of the terms IW , which measure the extent of lifespan inequality within each of the groups. The between-group component is a population-weighted average of gh the terms IB , which measure half the expected age-at-death difference between two randomly chosen individuals, one from group g and another from group h. Once again, these terms should be interpreted as functions measuring the distances that exist among pairs of length-of-life distributions. Breaking down inequality among groups gh The distance function across group pairs IB in the between-group component of (5) can be further decomposed into two additional sub-components. Typically, the age-at-death differences |a − b| included in these terms sometimes go in favour of one group (say, g), and sometimes in favour of the other (h). Putting together the age-at-death differences to the advantage of one group on one side and the differences to the advantage of the other group on the other, we naturally obtain the following decomposition: gh IB = Ahg + Agh , (6) 4 There are other and even simpler ways to define distances between length-of-life distributions. For instance, to measure the distance between the corresponding life expectancies at birth: |eF0 − eM 0 |. Yet, none of them would fit into the decomposition shown in (4). Permanyer et al. 7 where   1  ω g  ω h Ahg = 2 d (b − a) · db (7) 2ℓ0 a=0 a b=a and Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023   1  ω ω Agh = 2 dh (a − b) · da . g (8) 2ℓ0 b=0 b a=b The term Ahg contains all age-at-death differences in favour of group h, that is: the cases when the randomly chosen individual from group h has a longer lifespan than the randomly chosen individ- ual from group g. When the randomly chosen individual in group g outlives the one from h, their contribution to Ahg is 0. The opposite happens with Agh. Thus, Ahg measures the ‘average advan- tage in length of life for those individuals of group h outliving those of g’. This natural decompos- ition quantifies not only whether two groups are ‘close’ or ‘far apart’ from each other, but also the extent to which one of them is systematically more advantaged than the other. In other words, it speaks about the groups’ relative position. To illustrate how the new Ahg, Agh terms behave, in Figure 2 we show some hypothetical exam- ples. In panel A, one has two age-at-death distributions that are identical (i.e., dg = dh). When this gh happens, then Ahg = Agh = IB /2, and none of the two groups can be considered to be disadvantaged with respect to the other. In real-world examples, though, age-at-death distributions are not identi- cal, and some of them might represent a better state of affairs than others. In panels B and C, we show examples where the two age-at-death distributions (dg and dh) partially overlap, but the individuals of group h tend to live longer than those of g. In these scenarios, 0 < Agh < Ahg, with higher values of Ahg indicating a better state of affairs for group h vis-a-vis group g. In the limit, when the two age-at-death distributions do not overlap (say, all individuals in group h live longer than all individ- gh uals in group g), then Ahg = I B and Agh = 0. This has been illustrated in panel D. Importantly, the Ahg and Agh terms are sensitive to the differences in longevity among all individual pairs from groups g and h. In panel D, the age-at-death distributions are further apart, so the value of Ahg is higher than the one observed in panel C (where the two distributions are closer to each other). Plugging the new terms of (7) and (8) into (5) yields  G g  G  g−1 Δ(d) = S g IW + S gh (Agh + Ahg ). (9) g=1 g=2 h=1 This is one of the key inequality decompositions introduced in this paper. On the one hand, it con- g tains terms describing the extent of inequality within each group (S g IW ). On the other hand, it con- tains terms describing not only the distances across groups, but also their relative performance vis-a-vis each other (S gh (Agh + Ahg )). Equation (9) nicely shows how total inter-individual vari- ability (Δ(d)) can be broken down into clearly interpretable pieces explaining not only how hetero- geneous the different subgroups are (the within-group component), but also how these groups fare against each other (the between-group component). Using matrix notation In some cases, it is useful to represent all the terms in (9) using a more compact matrix notation. For that purpose, we define the distance matrix ⎛ 1 ⎞ IW IB12 ··· IB1G ⎜ .. .. ⎟ ⎜ I21 I2 . . ⎟ ⎜ B W ⎟ MD : =⎜ ⎟, (10) ⎜ .. .. .. (G−1)G ⎟ ⎝ . . . I ⎠ B IBG1 ··· IBG(G−1) G IW 8 J R Stat Soc Series A: Statistics in Society, 2023, Vol. 0, No. 0 (a) gh IB = 3.4 (b) gh IB = 5.50 Agh = 1.7 Agh = 0.25 Ahg = 1.7 Ahg = 5.25 Pgh = 0.5 Pgh = 0.12 Phg = 0.5 Phg = 0.88 Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023 dg = dh dg dh Distribution of deaths (c) gh IB = 10.02 (d) gh IB = 25 Agh = 0.01 Agh = 0 Ahg = 10.01 Ahg = 25 Pgh = 0.01 Pgh = 0 Phg = 0.99 Phg = 1 dg dh dg dh Age Fig. 2. Four scenarios (a–d) with two hypothetical age-at-death distributions dxg and dxh from two population subgroups g and h. In each scenario, we report the values of IBgh , Agh , Ahg , Pgh , Phg (see (6)–(8), (13), (14)). g which has the within-group comparison values IW in the diagonal, and the between-group com- gh parison values IB off the diagonal. A generic member of the distance matrix in row g and column h can thus be interpreted as ‘half the expected difference in length-of-life between a randomly chos- en individual from group g and a randomly chosen individual from group h’. Similarly, we define the advantage matrix ⎛ 1 ⎞ IW /2 A12 ··· A1G ⎜ .. .. ⎟ ⎜ ⎟ ⎜ A21 IW 2 /2 . . ⎟ MA : =⎜ ⎟ (11) ⎜ .. .. .. ⎟ ⎝ . . . A(G−1)G ⎠ AG1 ··· AG(G−1) G IW /2 containing (half) the within-group inequality values in the diagonal, and the ‘advantage terms’ Agh off the diagonal. A generic member of the advantage matrix in row g and column h can thus be interpreted as ‘the average advantage in length of life for those individuals from group g with re- spect to those from group h’. gh From (6), we have that IB = Ahg + Agh for all pairs (g, h), which yields MD = MA + M′A , where M′A is the transposed matrix of MA . While both MD and MA are G × G matrices with posi- tive entries, the former is symmetrical, but the latter is not. Moreover, it is possible to retrieve the Permanyer et al. 9 distance matrix MD from the advantage matrix MA , but not vice versa. Using this notation, the decomposition in (9) can be re-written more compactly as Δ(d) = p′ × MD × p = p′ × (MA + M′A ) × p, (12) where p = (p1, …, pG) is the (column) vector of population shares across groups (i.e., pg = ng/n), Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023 and p′ denotes its transposed. Further extensions In this section, we present an extension of the decomposition approach introduced in (5) and (9) with potentially many empirical applications. We show how the ‘pairwise comparison approach’ introduced in this paper can be used to measure the probability that individuals from a given group outlive individuals from other groups. Over- and under-performance probability measures The between-group component of both the absolute or relative Gini coefficients is based on the expected age-at-death differences across individuals belonging to two different groups taken at random. The pairwise comparisons-based approach can also be used to measure a similar yet fun- damentally different question, like assessing the probability that individuals belonging to one group live longer than the individuals of another group. That is, rather than looking at ‘how many additional years (on average) are individuals from one group expected to live with respect to individuals from the other group’, we can simply ask how likely is it that individuals from one group outlive those from the other group. To answer this question, for any pair of groups g, h ∈ {1, · · · , G}, we define the probability that a randomly chosen individual from group g lives longer than a randomly chosen individual from group h as   1ω−1  ω Pgh := 2 dh dga , (13) ℓ0 b=0 b a=b+1 and the probability that the individual in group h outlives the one in group g as   1ω−1 g  ω Phg := 2 d dha . (14) ℓ0 b=0 b a=b+1 Therefore, whenever Pgh > Phg, individuals from group g are expected to live longer than those from group h. Note that (13) and (14) are formally equivalent to the terms Agh and Ahg defined in (7) and (8)). The only difference is that here the age-at-death gap (a − b or b − a) has been replaced by a value of 1 whenever one age is bigger than the other. Thus, the terms included in (13) and (14) are only sensitive to the relative position of longevity outcomes from groups g and h, but not to the distance that might exist among them. To illustrate this point, consider the panels C and D from Figure 2. Since in both panels one has that virtually all individuals in group h outlive those of group g, then Phg ≈ 1 and Pgh ≈ 0 in both cases, even if the gaps in longevity outcomes increases when moving from C to D—a change that is captured by the terms Agh and Ahg. Interestingly, the Pgh and Phg terms coincide with the outsurvival probability indicator φ intro- duced by Vaupel et al. (2021),—which is presented in a continuous setting and restricted to two groups only. Since there is the (relatively small) possibility of ties, the probabilities Pgh and Phg do not add up to one. For this to happen, we must add the probability of ties to Pgh and Phg, say 1 ω Pgh + Phg + dh dg = 1. (15) ℓ20 a=0 a a The last term is typically very small in relative terms, so Pgh + Phg ≈ 1. Based on these definitions, 10 J R Stat Soc Series A: Statistics in Society, 2023, Vol. 0, No. 0 and mimicking the advantage matrix MA form (11), we can define the performance probability matrix ⎛ ⎞ P11 P12 ··· P1G ⎜ .. .. ⎟ ⎜ P21 P22 . . ⎟ MP : =⎜ ⎜ . ⎟, ⎟ .. .. Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023 ⎝ .. . . P ⎠ (G−1)G PG1 ··· PG(G−1) PGG where Pgh and Phg are as in (13) and (14), and Pgg is the probability that a randomly chosen indi- vidual from group g lives longer than a randomly chosen individual from the same group. Adapting (15) to the case where g = h, yields   1 1 ω g 2 1 Pgg = 1− 2 (d ) ≈ , 2 ℓ0 a=0 a 2 provided that the summatory inside the parenthesis is relatively small. Like MA , MP is a G × G asymmetrical matrix. Its elements indicate the probabilities that the different groups in which the population is partitioned outperform one another. Application: Length-of-life inequality in the United States, 1970–2018 In this section, we illustrate the usefulness of our approach by looking at the evolution of lifespan inequality in the United States and its decomposition across racial groups since 1970, and by Hispanic origin from 2006 onward.5 First, we focus our attention on the US Black and White population regardless of ethnicity, treating women and men separately. This means that the over- all population in these cases consists only of US Blacks and Whites, thus ignoring what happens to other racial groups. Next, we widen the focus to incorporate ethnicity into the analysis, comparing Hispanics (of any race) with non-Hispanic Whites and non-Hispanic Blacks, so that the overall population consists of three groups. We carried out all our analyses using the open-source statis- tical software R (version 4.1.1) (R Core Team, 2021). Life expectancy among Black Americans is several years lower compared with their White coun- terparts. This gap, however, has varied widely over time—peaking at 7.1 years in 1989 and de- creasing steadily since to 3.5 years in 2017 (Arias & Xu, 2019). The Black mortality disadvantage diminished during this time due to reduction in mortality levels (HIV, unintentional injuries) or delayed disease onset (heart disease, cancer, stroke) in this group (Firebaugh et al., 2014a; Harper et al., 2012). The lifespans of Black Americans are not only shorter on average, but also more varied compared with Whites. The Black disadvantage in lifespan variability is largely attributed to preventable causes of death including heart disease, homicide, HIV/AIDS, and diabetes (Firebaugh et al., 2014b). It is import- ant to note, however, that Blacks do not exhibit higher mortality from all causes of death. The opioid epidemic, for example, led to far greater loss of life years among Whites (Sasson & Hayward, 2019). Firearm and COVID-19 deaths, on the other hand, are significantly more preva- lent among Blacks than Whites (Sasson & Hayward, 2019; Wrigley-Field, 2020). It is, therefore, imperative that we understand the evolution of racial inequality in age-at-death distributions, and the intricate ways in which members of each group outlive members of the other. Ethnicity is yet another important source of heterogeneity in US mortality, in addition to race. Hispanics in particular are a growing population group with unique mortality patterns, which will 5 In distinguishing between race and ethnicity, we follow Directive 15 of the Office of Management and Budget (OMB) concerning Race and Ethnic Standards for Federal Statistics and Administrative Reporting. According to the dir- ective, US census and vital statistics data are reported using separate questions regarding race (Black, White, American Indian or Alaskan Native, Asian or Pacific Islander) and ethnicity (Hispanic origin/Not of Hispanic origin). While the two classifications are not mutually exclusive, US life tables since 2006 are reported for Hispanics as a whole yet for non-Hispanics by race. Thus, our three-group analysis is divided into non-Hispanic Blacks, non-Hispanics Whites, and Hispanics of all races. For the two-group comparison we use a longer time-series distinguishing between Blacks and Whites, irrespective of Hispanic origin, since 1970. Permanyer et al. 11 undoubtedly play a role in shaping US mortality in the future. Estimates of life expectancy by Hispanic origin are relatively recent, starting in 2006 (Arias, 2010). In spite of their lower socio- economic status, on average, Hispanics consistently exhibit higher life expectancy and lower life- span variability than non-Hispanic Whites (Lariscy et al., 2015, 2016). Debates about the Hispanic mortality advantage are ongoing, but it appears to reflect a combination of selection mechanisms, both inward and outward, as well as favourable health behaviours such as lower smoking prevalence (Fenelon, 2013; Hummer et al., 2007; Palloni & Arias, 2004).6 Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023 Both race and ethnicity are important determinants of mortality in the United States and will con- tinue to be so in the coming years. We illustrate our method, and what new insights can be gained from it, using these two case studies. First, using US decennial (1970–2000) and annual (2001–2017) life tables (NCHS, 2021), we decompose Black–White mortality inequalities by gender, and show how they have evolved over time (Figures 3, 4 and 5). Second, to extend our method to three groups we add to our analysis Hispanic origin. From 2006 to 2018, we explore the probabilities that non-Hispanic Blacks, non-Hispanic Whites, and Hispanic Americans outlive each other (Figure 6). While all analyses are based on official US life tables, there have been important changes in racial and ethnic categorization over the years. Since 2000, the US Census permits mixed race identification whereas the National Vital Statistics System does not. In order to obtain consistent estimates of popu- lation shares throughout the study period, we relied on the NCHS bridged-population estimates from 2000 onward (US DHHS, 2020). Life tables have also changed slightly over the years. The US decen- nial life tables are given in single years of age all the way through 110, whereas the annual life tables from 2001 and onward end at age 100. In order to make them comparable we extended the latter to age 109+ using P-splines, a common method for smoothing mortality rates (Camarda, 2012). Assessing longevity inequalities in the United States Figure 3 shows the age-at-death distribution for Blacks and Whites by gender in the United States in 1970 and 2017. The vertical lines indicate the corresponding life expectancies at birth. In all cases, life expectancies are higher for Whites than for Blacks, but the gap has shrunk over time, a result that coheres with Harper et al. (2014), and Firebaugh et al. (2014a),. Beyond means, a vis- ual inspection of the shape of the age-at-death distributions suggests that: (a) lifespan inequality has decreased over time; (b) lifespan inequality among Whites is lower than that of Blacks; and (c) lifespan inequality among men is higher than among women. These observations are confirmed in Figure 4, which shows the trends of the Theil and the Gini indices of lifespan inequality since 1970 for the US Black and White population, and for women and men separately. Most importantly for the purposes of this paper, Figure 4 also depicts the de- composition of these indices in their basic constituents. The Theil index is representative of the current approaches to assess the influence of population par- titions on overall inequality levels (e.g., Gakidou & King, 2002; Permanyer et al., 2018; Permanyer & Scholl, 2019; Seaman et al., 2019; van Raalte et al., 2012). The values of the overall Theil index for US females decline from 0.041 in 1970 to 0.024 in 2017. For US males, it goes down from 0.056 in 1970 to 0.033 in 2017. In both cases, the overall Theil index has been declining from 1970 until 2000 and remained relatively stable from 2000 onwards. What about the additive decomposition of the Theil index in its within-group and between-group components (T = Tw + Tb; see Appendix A)? As shown in the graphs, the between-group component is extremely small, and only explains around 1% of total inequality in lifespans among the US Black and White population. The right panels of Figure 4 show the results corresponding to the absolute Gini coefficient and its different decompositions. The values of the overall Gini index (i.e., the Gini index applied to the whole US Black and White population) follow a very similar path when compared to those of the Theil index. While the levels and scale are different, the trends roughly go in the same direction: we also observe a decline in inequalities from 1970 to 2000, followed by a period of relative 6 The healthy immigrant effect is a well-established phenomenon, in the Unite States and elsewhere, in which immi- grants are healthier on average than the receiving population because they are selected based on their health status, socio- economic status, human capital, or character (Kennedy et al., 2015). By contrast, ‘salmon bias’ is a form of outward selection in which less healthy immigrants return to their country of origin as their health deteriorates, eventually dying abroad (Abraido-Lanza et al., 1999). US mortality data are census-unlinked, meaning that that numerator (death counts) and denominator (person-years) are estimated from different sources. Thus, returning migrants might be included in the at-risk population but their deaths might be omitted from the numerator if they had died outside the country, thus result- ing in downward bias of mortality rates for that group. 12 J R Stat Soc Series A: Statistics in Society, 2023, Vol. 0, No. 0 Females 1970 Females 2017 4% 3% Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023 2% 1% Distribution of deaths 0 Males 1970 Males 2017 4% 3% 2% 1% 0 0 10 20 30 40 50 60 70 80 90 100 110 0 10 20 30 40 50 60 70 80 90 100 110 Age (years) Blacks Whites Life expectancy at birth Fig. 3. Age-at-death distribution for Black and White females and males in the United States, 1970 (left panels) and 2017 (right panels). Source: NCHS (2021). stability until 2017. For females, the overall Gini coefficient goes down from 9.148 in 1970 to 7.965 in 2017, while for males it goes down from 10.050 to 9.087. Interestingly, these levels of overall lifespan inequality can be broken down into clearly interpretable components. Take the value of 10.050 observed for males in 1970. That year, the population share for Whites was p1 = 0.891, whereas Blacks represented p2 = 0.109. Following (12), one has that ′ 10.050 = p′ × M̃D × p = p′ × (M̃A + M̃A ) × p      9.685 11.416 0.891 = 0.891 0.109 = 11.416 12.081 0.109 ⎛⎛ ⎞ ⎛ ⎞⎞ 9.685 9.685    ⎜⎜ 2 7.695 3.721 ⎟ ⎜ 2 ⎟⎟ 0.891 = 0.891 0.109 ⎝⎝ ⎠+⎝ ⎠⎠ . 12.081 12.081 0.109 3.721 7.695 2 2 As shown in these identities, the overall Gini coefficient is a combination of inequalities among Blacks and among Whites, together with other components indicating not only the extent to which Blacks and Whites differ from each other, but also whether Blacks outperform Whites and vice versa. More spe- cifically, we can see that the Gini coefficient for Blacks is notably larger than that of Whites (12.081 vs. 9.685). In addition, the average longevity difference between Blacks and Whites (the term IB12 in (5)) is larger than the average longevity difference within Whites (11.416 vs. 9.685). In other words: the Permanyer et al. 13 Females Females 12 0.05 11 10 0.04 9 0.03 8 Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023 7 0.02 6 Absolute Gini coefficient 5 0.01 4 Theil index 0 3 Males Males 12 0.05 11 10 0.04 9 0.03 8 7 0.02 6 5 0.01 4 0 3 1970 1980 1990 2000 2005 2010 2017 1970 1980 1990 2000 2005 2010 2017 Year Theil decomposition Gini decomposition Total Between group Total Within group Blacks Black advantage Whites White advantage Fig. 4. Evolution of the Theil and Gini indices and their decompositions over time between 1970 and 2017 for the Black– White population in the United States. Source: Authors’ elaboration based on data from NCHS (2021) and US DHHS (2020). difference in longevity between a randomly chosen Black and a randomly chosen White is larger than the difference in longevity among two randomly chosen Whites. Breaking down the IB12 = 11.416 term in its group-specific advantage components, we can also see that the average longevity advantage of Blacks over Whites is 3.721, while that of Whites over Blacks is 7.695 (in such a way that 11.416 = 3.721 + 7.695). Thus, the average longevity advantage of Whites outliving Blacks (A12 = 7.695) is twice as large as the average longevity advantage of Blacks outliving Whites (A21 = 3.721). All these numbers are plotted in the bottom right panel of Figure 4 for the year 1970. Performing now the same decomposition exercise for the overall Gini index for males in 2017, we get      8.850 9.673 0.851 9.087 = 0.851 0.149 9.673 10.136 0.149 = ⎛⎛ ⎞ ⎛ ⎞⎞ 8.850 8.850    ⎜⎜ 2 5.942 3.731 ⎟ ⎜ 2 ⎟⎟ 0.851 = 0.851 0.149 ⎝⎝ ⎠+⎝ ⎠⎠ 10.136 10.136 0.149 3.731 5.942 2 2 As can be seen, lifespan inequality has declined both among Blacks (10.136) and among Whites (8.850), but especially among the former. We observe that the differences between Blacks and Whites have diminished as well (now IB12 = 9.673). The decomposition of the between-group compo- nent IB12 in its group-specific advantage components has also changed. In 2017, the average longevity advantage of male Whites outliving male Blacks has gone down to A12 = 5.942, while the average 14 J R Stat Soc Series A: Statistics in Society, 2023, Vol. 0, No. 0 longevity advantage of male Blacks outliving male Whites has remained at a similar level of A21 = 3.731. These results indicate that the age-at-death distributions of Blacks and Whites are becoming increasingly similar, with the advantage of the later over the former gradually declining over time. Very similar patterns can be identified for women as well (see upper right panel in Figure 4). Yet, this convergence in age-at-death distributions has not been a steady process throughout the study pe- riod. Periods of great progress—i.e., equalization of Black and White lifespan distributions—occurred Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023 in the 1970s and 1990s, whereas stagnation and even reversal occurred in the 1980s and early 2000s. We address some of the reasons underlying the uneven progress in the Discussion section. Summing up, the three approaches discussed in this paper offer complementary information about longevity inequality trends in the United States. First, according to the group-based means approach, the life expectancy gap between Blacks and Whites declined between 1970 and 2017. Second, the decomposition of the Theil index shows that lifespan inequality declined in the United States between 1970 and 2000 and stagnated from 2000 onwards, and that differences in life expectancy between Blacks and Whites only explain 1% of the total inequality in lifespans. Finally, the new decomposition approach suggested here provides more information and permits a more thorough interpretation of the inequalities between and within the two groups: (a) lifespan inequality has been declining both among Blacks and Whites between 1970 and 2017; (b) the aver- age longevity advantage of Whites outliving Blacks has declined over time; and (c) the average lon- gevity advantage of Blacks outliving Whites has remained fairly constant during that period—thus resulting in a decreasing White advantage in the age-at-death distribution. Outliving probabilities We conclude this empirical section by analysing the time trends in the performance probabilities defined in (13) and (14). Because of data availability, we first show the results for Blacks and Whites (irrespective of Hispanic origin) from 1970 to 2017 (Figure 5). Next, we examine these performance probabilities between non-Hispanic Whites, non-Hispanic Blacks and Hispanics be- tween 2006 and 2018 (Figure 6). As it can be seen in Figure 5, the probability that a randomly chosen White outlives a randomly chosen Black is higher than the opposite probability, but it tends to decline over time. For females, such probability was 0.604 in 1970, but declined to 0.536 in 2017. For males, such probability dropped from 0.610 to 0.567, but following a more erratic Females Males 0.60 Performance probabilities 0.55 0.50 0.45 0.40 0.35 1970 1980 1990 2000 2005 2010 2017 1970 1980 1990 2000 2005 2010 2017 Year Blacks over whites Whites over blacks Fig. 5. Over- and under-performance probabilities for Blacks and Whites between 1970 and 2017. Source: Authors’ elaboration based on data from NCHS (2021). Permanyer et al. 15 Females Males Performance probabilities 0.60 0.55 Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023 0.50 0.45 0.40 0.35 2006 2010 2014 2018 2006 2010 2014 2018 Year Non−Hisp whites over non−Hisp blacks Non−Hisp whites over Hispanic Non−Hisp blacks over Hispanic Non−Hisp blacks over non−Hisp whites Hispanic over non−Hisp whites Hispanic over non−Hisp blacks Fig. 6. Over- and under-performance probabilities for non-Hispanic Blacks, non-Hispanic Whites, and Hispanics between 2006 and 2018. Source: Authors’ elaboration based on data from NCHS (2021). trajectory over time. Symmetrically, the probability that a randomly chosen Black outlives a ran- domly chosen White has tended to increase over time. In 1970, such probabilities were 0.376 and 0.373 for females and males, respectively, while in 2017 they were 0.441 and 0.414. Figure 6 depicts the performance probabilities over time for non-Hispanic Blacks, non-Hispanic Whites, and Hispanics between 2006 and 2018. We observe that the probability that Hispanics outlive non-Hispanic Whites has tended to increase over time, from 0.542 in 2006 to 0.563 in 2018 for females, and from 0.529 to 0.548 for males. In addition, the probability that Hispanics outlive non-Hispanic Blacks is considerably high, with values around 0.6 for both sexes that have remained relatively stable over time. In this three-group comparison, Hispanics perform better than non-Hispanic Blacks and non-Hispanic Whites, with the latter two becoming increas- ingly similar over time. Discussion Analysts are often interested in measuring inter-individual health inequalities in settings where the populations under study are partitioned across socially relevant groups (for instance, defined across socioeconomic status, ethnic, religious, or racial lines). A simple and very popular way of measuring health inequalities in such settings is to assess groups’ performances by comparing the means of the corresponding health outcome—a group-based approach that ignores intra-group heterogeneity. Another common approach is to determine how much of the inter-individual variability can be ex- plained by the variable that partitions the population into mutually exclusive groups. For that pur- pose, analysts often choose the so-called ‘additively decomposable inequality measures’ (like the Theil index, or the Variance) that break down total inter-individual variability in two components: (a) the between-group component (which measures the amount of inequality we would observe if we counterfactually removed intra-group variability), and (b) the within-group component (which is a weighted sum of the inequalities within the different groups)—see (1). Unfortunately, the second approach to inequality measurement does not provide any information about the groups’ relative performance. In this paper, we propose a third approach that allows determining not only the health variability that might exist across individuals within and between groups, but also the relative per- formance of the different groups vis-a-vis each other. The decomposition method suggested here has 16 J R Stat Soc Series A: Statistics in Society, 2023, Vol. 0, No. 0 the advantage of integrating comparisons across individuals and social groups into a coherent whole and responds to the need of having a summary measure of health inequality ‘which gives an overall picture of health inequalities in the population while maintaining pertinent information on [group- based] health inequalities’ (Asada, 2010:3). It is important to clarify that in this paper we focus our attention on measures of inter-individual inequality (like the Theil index, the variance or the Gini coefficient) admitting mathematically exact Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023 decompositions in those settings where the populations under study are partitioned across social groups. Thus, we are not delving into other very popular approaches to explore health inequalities that either (a) do not have the individuals as the basic units of analysis or (b) are not aiming at pri- marily obtaining exact inequality decompositions. Among the former, we include those approaches where the health outcome variable is only meaningful at an aggregate level (e.g., hazard ratios, rela- tive risks, odd ratios, or mortality ratios applying to different population subgroups defined across geographical, social, cultural, or economic lines). Among the latter, we include a long and varie- gated list of models that aim at predicting individuals’ health outcomes on the basis of some inde- pendent variables (e.g., income level, educational attainment, employment status, place of residence, household type, age, sex, and so on and so forth). In those settings, total inter-individual variability of the outcome variable is partly (but not fully) explained by the variability of the inde- pendent variables, and the remaining part is a residual component that the model fails to explain. Inter alia, such methods allow incorporating multiple factors to predict the outcome variable and estimate their relative role in explaining total inter-individual variability.7 The exact inequality de- composition methods discussed in this paper offer alternative analytical perspectives that can com- plement the aforementioned approaches when assessing health inequalities. Interpreting the new decompositions The decomposition approach suggested in this paper applies to inequality measures that are based on making all pairwise comparisons across individual outcomes. Prominent members of this class include the AID index (or absolute Gini coefficient), or the (relative) Gini coefficient. While both the standard ‘additive decomposition approach’ of the Theil index or the variance and the new ap- proach proposed here have a ‘within-group’ and a ‘between-group’ component (see Appendix A), the meaning and interpretation of the latter differs completely when moving from one approach to the other.8 For standard additively decomposable measures, the between-group component has a single term that measures the inequality that would be observed in a hypothetical distribution sup- pressing variability within each of the G groups in which the population is partitioned. For the ‘pairwise comparisons’-based approach proposed in this paper and described in (9) and (12), the between-group component contains the G(G − 1) terms off the diagonal in the advantage matrix MA , indicating not only how ‘close’ or ‘far away’ any two group-specific health distributions are, but also the extent to which one is (dis)advantaged with respect to the other. In this regard, the new approach is much more informative than the currently existing ones as for the groups’ rela- tive performance—something that has been clearly illustrated in the empirical section of the paper. To further clarify the different kind of information that both approaches convey, let us consider two hypothetical scenarios. In the first one, suppose the G groups analyzed have exactly the same age-at-death distribution. In that scenario, additively decomposable measures would conclude that, since there are no differences in the group-specific life expectancies, the between-group com- ponent is zero. In contrast, all the terms off the diagonal in the distance matrix MD would be a constant number x > 0 equal to half the expected age-at-death difference between two individuals randomly chosen from the age-at-death distribution shared across all groups. Likewise, the terms off the diagonal in the advantage matrix MA would be all constant and equal to x/2. Suppose a second hypothetical scenario in which there is no within-group variability and all individuals 7 In the ‘exact decompositions’ setting discussed in this paper, this goal can be achieved inspecting the size of the between-group components ensuing from the corresponding population partitions. That is, each of the factors (e.g., race, sex, education, or income) generates a partition of the population, and the larger the size of the corresponding between-group component, the more relevant that factor is to explain total inter-individual variability. 8 Indeed, neither the absolute nor the relative Gini coefficients are additively decomposable in the sense described in (1). Attempts at breaking down the Gini coefficient along these lines results in a three-term decomposition, G = Gw + Gb + R, where Gw and Gb are the standard within-group and between-group components, and R is a residual component which is only zero in case the group-specific distributions do not overlap with each other (a circumstance that is very un- likely to occur in practice; see Lambert and Aronson 1993). Permanyer et al. 17 from a given group die at the same age—which naturally coincides with the group-specific life ex- pectancy at birth and can vary across groups. In that setting, additively decomposable measures collapse into a single term, measuring the extent of inequality among the different life expectan- cies. Alternatively, in the pairwise comparison approach advocated in this paper, the terms off g the diagonal in MD would measure the distance between life expectancy pairs |e0 − eh0 |, and the g g terms off the diagonal in MA would be equal to either |e0 − e0 | or 0, depending on whether e0 > h g g e0 or e0 < e0 (e0 , e0 being the life expectancy of groups g and h, respectively). Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023 h h h Given the fundamentally different ways in which the two types of decompositions are defined, it is not surprising that the results they generate vary considerably. The findings reported in the em- pirical section of the paper are very illustrative in this regard. While the between-group component of the Theil index is almost non-existent in the US Black–White comparison (it barely explains 1% of the total variation) and tells us nothing about the relative performance of the two groups vis-a-vis each other, the terms included in the between-group component of the Gini coefficient uncover new and valuable information about the longevity advantages of Whites over Blacks, and vice versa. Inter alia, they show that while the average longevity advantage of Blacks outliving Whites has remained fairly constant, the average longevity advantage of Whites outliving Blacks has declined between 1970 and 2017. The small explanatory power of the between-group compo- nent when using additively decomposable measures like the Theil index or the variance has already been identified in previous studies (e.g., Edwards, 2011; Permanyer et al., 2018; Permanyer & Scholl, 2019; Seaman et al., 2019; Smits & Monden, 2009, van Raalte et al., 2012). This suggests that the variables that are commonly used to partition populations into groups (e.g., educational attainment, country of residence, ZIP code) are relatively weak predictors of individuals’ longevity and that mortality is a highly stochastic process (Caswell, 2009). Yet, as our findings clearly illus- trate, the low predictive power of these groupings on individuals’ longevity does not mean that differences across groups are unimportant or non-existent; we can obtain very different insights adopting alternative measurement strategies like the one proposed here. The different approaches to inequality measurement discussed in this paper are extremely useful and can offer complemen- tary insights to better understand past, contemporary, and prospective health dynamics. The evolution of racial and ethnic inequalities in length of life in the United States The US population is racially and ethnically diverse, with each group characterized by its unique mor- tality regime. Black Americans have long been at a disadvantage compared with the majority White population (Hummer & Chinn, 2011). Our study period, beginning in 1970, captures only the past five decades of this long history of racial inequality. Overall, this period was characterized by progress, though uneven at times, toward greater equality in lifespan distributions between Blacks and Whites in the United States. We find that among White men and women, lifespan inequality declined steadily from 1970 to 2000 but stagnated since the turn of the 21st century and even increased in recent years. Among Blacks, lifespan inequality diminished substantially in the 1970s, stagnated in the 1980s, and declined steadily during the 1990s and 2000s. Around 2010, however, this trend came to a halt and since 2014 an upsurge in lifespan inequality is observed among Blacks. Applying the new decomposition approach proposed in this paper we can see how the compos- ition of lifespan inequality across and within racial groups has shifted dramatically over time. In 2017, the longevity advantage of Whites over Blacks plays a much less prominent role than it did in the 1970s, while the longevity advantage of Blacks over Whites has remained relatively stable over time. Overall, these trends have resulted in a decreasing White advantage in the age-at-death distribution. These findings are consistent with prior research: the disproportionate impact of HIV/AIDS on Black communities in the 1980s leading to increased mortality in that group (Kochanek et al., 1994); the greater reductions in Black mortality from homicide, HIV, and heart disease (first among women, followed by men) during the 1990s and 2000s (Firebaugh et al., 2014a; Harper et al., 2007); and the effect of the opioid epidemic on White Americans and its lagged effect on the Black population (Alexander et al., 2018; Sasson & Hayward, 2019). The White advantage relative to Blacks, however, is far from disappearing entirely and might have worsened since 2018, the last year in our time series. COVID-19 deaths in particular have taken a far greater toll on Blacks (as well as Hispanics) in the United States, potentially reversing years of progress toward narrowing the Black–White mortality gap (Andrasfay & Goldman, 2021). 18 J R Stat Soc Series A: Statistics in Society, 2023, Vol. 0, No. 0 Unlike US Blacks, Hispanic hold a survival advantage relative to the non-Hispanic White popu- lation, in spite of their lower socioeconomic status (Boen & Hummer, 2019; Markides & Eschbach, 2005). We replicate this finding but show in addition that from 2006 to 2018 the prob- ability that a randomly selected Hispanic would outlive his or her non-Hispanic White counterpart has increased. The Hispanic advantage over non-Hispanic Blacks was even greater but did not change substantially during those years. Explanations for the Hispanic survival advantage range Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023 from health behaviours to selection effects—both of healthy immigrants and unhealthy emigrants —but little attention has been given to changes in this advantage over time (Borrell & Lancet, 2012). As we have shown here, the Hispanic survival advantage is dynamic and warrants addition- al research to understand why it has changed over time. Final remarks While the decomposition approach suggested in this paper has many advantages, it is important to be aware of its limitations. Indeed, what is unarguably one of its most important strengths (the fact that it generates a detailed decomposition informing about many aspects of the health distribu- tion) might turn out to be a disadvantage when the number of groups (G) is relatively large. Interpreting the levels and trends of the G within-group and G(G − 1) between-group terms shown in (9) and (12) can be daunting if, for instance, one is interested in measuring health inequalities around the world—where the population is partitioned in around G ≈ 150 or 200 countries (see, for instance, Edwards, 2011; Permanyer & Scholl, 2019, Smits & Monden, 2009). In this line, one might further argue that when the number of groups becomes very large, the between-group com- ponent explains almost all the observed variability. While true, it is important to highlight two im- portant issues: (a) by definition, this will happen to any other inequality measure or decomposition method;9 and (b) the number of groups in which populations are partitioned is usually fixed and tends to remain unchanged across time or space. The decomposition approach proposed in this paper is both practical and very informative when the number of groups is relatively low, for ex- ample, when populations are partitioned by educational attainment, religion, ethnic groups, race, or income/wealth quartiles/quintiles. Addressing health inequalities is becoming a top priority for public health planners all over the world. The implementation of efficient and fair policies to curb health inequalities can greatly benefit from indicators that inform planners not only about the extent of inter-individual variabil- ity, but also about the relative performance of the groups into which populations might be parti- tioned. This is the case of the measures proposed in this paper, which can be very useful to identify the main drivers of inequality change over time. While we have illustrated their usefulness using age-at-death distributions, the same approach can be implemented with any other health outcome measured in a cardinal scale (e.g., length of life lived in different health states (e.g., ‘in good health’, ‘free from specific diseases or conditions’), height, frailty, grip strength, and so on), so the scope for practical applications is extraordinarily large. Funding The research leading to this publication was partially funded by the European Research Council under the European Union’s Horizon 2020 research and innovation programme (grants 2019-CoG-864616 and 2019-AdG-884328), and the Spanish Ministry of Science and Innovation R+D LONGHEALTH project (grant PID2021-128892OB-I00). I.S. acknowledges funding from the Israel Science Foundation (grant 2532/22). F.V. acknowledges funding from the Spanish State Research Agency under the Juan de la Cierva (grant IJC2019-039144-I) and Ramón y Cajal (grant RYC2021-033979-I) programmes. 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We begin with the Theil index, which is defined as     1 ω αx αx T(d) = dx log . ℓ0 x=0 e0 e0 When the population is partitioned in G groups, the Theil index can be decomposed as  G g  g  G g ng e e0 ng e 0 T(d) = Tb (d) + Tw (d) = 0 log + Tg . (A1) g=1 n e0 e0 g=1 n e0 The first part in (A1) is the between-group component, which is obtained assuming all individuals g in each group die at the same age e0 , so there is no within-group variation. The second term is the within-group component, which is a weighted sum of the within-group inequalities measured by Tg, the Theil index applied to group g. Another inequality index that is additively decomposable is the mean log deviation, which is defined as   1 ω e0 L(d) = dx log . ℓ0 x=0 αx When the population is partitioned in G groups, L(d) can be decomposed as  G    G ng e0 ng L(d) = Lb (d) + Lw (d) = log g + Lg , g=1 n e0 g=1 n where Lg is the mean log deviation applied to group g. Finally, the variance is the only absolute inequality index that is additively decomposable. It is defined as 1 ω V(d) = dx (αx − e0 )2 . ℓ0 x=0 22 J R Stat Soc Series A: Statistics in Society, 2023, Vol. 0, No. 0 When the population is partitioned in G groups, V(d) can be decomposed as  G ng g  G ng V(d) = Vb (d) + Vw (d) = (e0 − e0 )2 + Vg , g=1 n g=1 n where Vg is the variance applied to group g. Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023 Appendix B. Decomposition of the relative Gini coefficient The same approach used to decompose the absolute Gini coefficient Δ(d) can be applied to G(d), the relative Gini coefficient defined in (3). It is straightforward to check that (5) can be adapted to the relative context, expressed as  G g  G  g−1 gh G(d) = GW (d) + GB (d) = S̃ g ĨW + S̃ gh ĨB , (A2) g=1 g=2 h=1 where g g IW ĨW : =G(dg ) = g e0 gh gh IB ĨB : = g (e0 + eh0 )/2 n 2 eg g S̃ g : = 0 n e0 g ng nh (e0 + eh0 ) S̃ gh : = n2 e0 g e0 , eh0 being the life expectancy of groups g and h, respectively. The interpretation of (A2) is very similar to that of (5): The first part includes G within-group inequality terms and the second part includes G(G − 1)/2 between-group terms. The only difference now is that the within- and between-group inequality elements are defined in relative terms (i.e., the absolute variation in the numerator is put in relation with respect to the corresponding mean/life expectancy in the de- nominator). Like in the previous section, the between-group terms in (A2) measuring the distance between pairs of groups can be further broken down in two components as follows:   gh gh IB Agh + Ahg 2Agh 2Ahg ĨB = 2 g =2 g = g + g . (A3) e0 + e0h e0 + e0 h e0 + e0 e0 + eh0 h Defining 2Agh 2Ahg Ãgh : = g and Ãhg : = g , (A4) e0 + eh0 e0 + eh0 we can rewrite (A2) as  G g  G  g−1 G(d) = GW (d) + GB (d) = S̃ g ĨW + S̃ gh (Ãgh + Ãhg ), (A5) g=1 g=2 h=1 which is the ‘relative version’ of (9). In matrix notation, (A2) and (A5) can be expressed more com- pactly as ′ G(d) = p′ × M̃D × s = p′ × (M̃A + M̃A ) × s, Permanyer et al. 23 where p = (p1, …, pG) is, again, the vector of population shares across groups, and s = (s1, …, sG) is a column vector whose elements are defined as g ng e 0 sg : = . n e0 M̃D and M̃A are the ‘relative version’ of the distance and advantage matrices MD and MA given Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023 g gh g gh in (10) and (11); that is, replacing the IW , IB and Agh terms by the corresponding ĨW , ĨB and Ãgh elements defined in (A2)–(A4). Appendix C. The two-group decomposition of the absolute Gini coefficient Assume a population that is partitioned in two groups, for instance females (F) and males (M), with respective population sizes nF and nM , such that the total population is n = nF + nM . Following (2), given the group-specific age-at-death distributions dF = {dF0 , · · · , dFω } and dM = {dM 0 , · · · , dω }, the M overall lifespan inequality measured by the absolute Gini coefficient can be broken down as 1  ω  ω Δ(d) = da db |a − b| 2ℓ20 a=0 b=0   1  ω ω ω = 2 d0 db |0 − b| + d1 db |1 − b| + · · · + dω db |ω − b| 2ℓ0 b=0 b=0 b=0 1 = (d0 d0 |0 − 0| + d0 d1 |0 − 1| + d0 d2 |0 − 2| + · · · + d0 dω |0 − ω| + · · · 2ℓ20 +dω d0 |ω − 0| + dω d1 |ω − 1| + dω d2 |ω − 2| + · · · + dω dω |ω − ω|). (A6) Note that for any pair of ages a and b    nF · dFa + nM · dM nF · dFb + nM · dM da db = a b n n n2F · dFa dFb + n2M · dM d a b M + n F M · da db + nF nM · db da n F M F M (A7) = 2 · n In particular, when a = b n2F · dFa dFa + n2M · dM a da + 2nF nM · da da M F M da da = 2 . n Using (A6) in (A7), we get 1  n2F · dF0 dF0 + n2M · dM 0 d0 + 2nF nM · d0 d0 M F M Δ(d) = 2 2 |0 − 0| 2ℓ0 n n2F · dF0 dF1 + n2M · dM 0 d1 + nF nM · d0 d1 + nF nM · d1 d0 M F M F M + 2 |0 − 1| + · · · n n2F · dF0 dFω + n2M · dM 0 dω + nF nM · d0 dω + nF nM · dω d0 M F M F M + 2 |0 − ω| + · · · n n2F · dFω dF0 + n2M · dM ω d0 + nF nM · dω d0 + nF nM · d0 dω M F M F M + |ω − 0| n2 n2F · dFω dF1 + n2M · dM ω d1 + nF nM · dω d1 + nF nM · d1 dω M F M F M + 2 |ω − 1| + · · · n n2 · dF dF + n2M · dM  ω dω + 2nF nM · dω dω M F M + F ω ω 2 |ω − ω| · n 24 J R Stat Soc Series A: Statistics in Society, 2023, Vol. 0, No. 0 Re-arranging terms, 1 nF 2  F F  Δ(d) = d 0 d 0 |0 − 0| + ... + dF F 0 d ω |0 − ω| + ... + d F F ω d 0 |ω − 0| + ... + d F F ω dω |ω − ω| 2ℓ20 n n 2   M + dM d 0 0 M |0 − 0| + ... + d M M 0 ωd |0 − ω| + ... + d M M d ω 0 |ω − 0| + ... + d M M d ω ω |ω − ω| n Downloaded from https://academic.oup.com/jrsssa/advance-article/doi/10.1093/jrsssa/qnac001/7023973 by guest on 24 April 2023  2nF nM M F  + 2 d0 d0 |0 − 0| + ... + dM F M F M F 0 dω |0 − ω| + ... + dω d0 |ω − 0| + ... + dω dω |ω − ω| · n = S F IW F + S M IW M + S FM IBFM , F where IW is the within-group variation for females 1  ω  ω F IW : =Δ(dF ) = dF dF · |a − b|, 2ℓ0 a=0 b=0 a b 2 M IW is the within-group variation for males 1  ω  ω M IW : =Δ(dM ) = dM dM · |a − b|, 2ℓ0 a=0 b=0 a b 2 and IBFM the between-group variation 1  ω  ω IBFM : = dF dM · |a − b|. 2ℓ0 a=0 b=0 a b 2 Each of these three terms have respective weights n 2 n 2 nF nM F M SF : = , SF : = , and S FM : =2 2 . n n n