Socioeconomic mortality gradients are well established in most countries.^{1} ^{,} ^{2} They matter for many reasons, including unfairness and the idea that future gains in overall health might be best achieved by reducing inequalities.^{3} ^{,} ^{4} Gradients vary by context and over time in response to the timing and patterning of social, economic, behavioral, and health service influences.^{1} ^{,} ^{4} Describing how socioeconomic mortality gradients change over time, and then analyzing and interpreting why these changes occurred, is not only of explanatory utility, but it is also a powerful tool to predict socioeconomic mortality gradients in the near future, and by extension how to intervene to change them: “The basic premise … is that the measurement strategies applied to health inequalities have implications for setting policy goals and understanding the extent of progress toward the reduction of health inequalities.”^{5}

A key point of departure is answering the question “are inequalities decreasing or increasing over time?” There are several empirical and value judgments immediately embedded in this question that have been reviewed extensively elsewhere.^{5–9} In this article, we take the following positions. First, we focus on mortality rates rather than—say—gaps in survival probability or life expectancy. Second, we do not “pick” between absolute or relative measures of inequality—rather, we maintain that both are informative, and indeed should be arrayed on a spectrum. Third, we focus on inequalities across the whole population (i.e., the gradient); nominal social group comparisons (e.g., by ethnicity) are also important, and we have published on these extensively (e.g., references ^{10–12} ), but they are out of scope here. To operationalize this focus on gradients, and allow for differing distributions of populations by socioeconomic factors over time and between countries, we also adopt the slope and relative index of inequality approach, which requires ranking the population by socioeconomic position.^{8} ^{,} ^{13}

Presenting trends in health inequalities to researchers, policy makers, and stakeholders, to allow easy interpretation and a “holistic” assessment of trends over time in all of average mortality (health) rates, and absolute and relative inequalities, is challenging. Line graphs of trends in mortality rates for each socioeconomic group and for gaps between groups, and accompanying tables with tests of trend in absolute and relative inequalities, are common (e.g., references ^{6} ^{, } ^{14} ). However, it is challenging to convey multiple dimensions of analyses in one graphic—especially if presenting trends over time. Barros and Victora^{15} and Mackenbach et al.^{16} plot changes over time in absolute inequalities versus changes over time in relative inequalities. Kjellsson et al.^{7} present a graphic of overall mortality rates on the x axis, absolute inequality on the y axis, and relative inequality as contour lines. We suggest an alternative approach that emphasizes trends in socioeconomic mortality gradients over time, modifies the graphic to accommodate low- and high-mortality rate comparisons (e.g., young with old, suicide with cardiovascular disease [CVD]), and allows projections into the future to facilitate target setting.

The structure of this article is as follows. First, we present a typology of socioeconomic mortality gradient trends and accompanying graphical presentation—using a compass analogy (e.g., “trends heading northwest”). Second, we briefly describe the methods for, and apply the typology and graphical presentations to, two case studies: (a) gradients for 25- to 74-year olds in NZ from 1981 to 2011, and (b) selected European countries with at least 15 years of data (maximum 1980 to 2010 for Finland) as published by Mackenbach et al.^{16} or similarly calculated. Third, we extrapolate gradients in New Zealand from 2011 to 2031, both under “business as usual” (i.e., past trends projected to the future) and for an intervention that equalizes annual percentage reductions in CVD mortality, to explore the possible utility of the graphical presentations to assist target setting.

DESCRIPTIVE TYPOLOGY AND GRAPHICAL VISUALIZATION OF TRENDS IN SOCIOECONOMIC MORTALITY GRADIENTS
Trends in average mortality rates over time may be: falling (denoted as m↓ ; e.g., cardiovascular disease in most countries); stable (m– ; e.g., some cancers); or increasing (m↑ ; e.g., suicide in some contexts). Imagine that the population is then split into low- and high-socioeconomic groups, and that the low socioeconomic group has higher mortality rates (as is usually the case). The gap, or inequality, between mortality rates for this low and high socioeconomic group can be considered on both the absolute (a ) and relative (r ) scales, and again for increasing, stable, and decreasing trends over time; there are mathematically nine (3 × 3) possible combinations (i.e., denoted as a↑r↑ , a↑r– , a↑r↓ , and a–r↑ , a–r– , a–r↓ , and a↓r↑ , a↓r– , a↓r↓ ). However, when you bring together the three possible trends in average mortality rates and the nine possible trends in inequalities, there are not 27 (3 × 9) possible combinations as many are impossible. For example, when the average mortality rate is decreasing, one cannot have increasing absolute inequalities and decreasing relative inequalities. Rather, there are 13 possible combination of m, a, r trends over time, presented in Figure 1 .

FIGURE 1: Typology of population average and inequality trends in mortality (schematic figures: y axis = mortality rate, x axis = time; solid lines, low socioeconomic group trend; dashed lines, high socioeconomic group trends). The criteria for coding, and coloring were highly desirable must include reductions in the average mortality rate and absolute inequalities, and no increase in relative inequalities; desirable must include reductions in the average mortality rate and no increase in absolute inequalities; undesirable included any increasing average mortality rate, or stable average mortality with widening inequalities; and equivocal included the remaining types. Other perspectives and criteria may lead to a different classification. However, the purpose of this diagram and is to make it clear that (a) there is a continuum or spectrum from the bottom left to the top right; (b) that trends in all of average mortality rate, and absolute and relative inequalities, can be considered simultaneously. SEP indicates socioeconomic position. Figure is available in color online.

For each panel in Figure 1 , a stylistic graph is shown with trends in the high socioeconomic group mortality rate shown as the dashed line, and the low socioeconomic group mortality shown as the solid line. Each panel is labeled with the “m a r ” notation above. The most desirable type is labeled “m↓a↓r↓ ,” decreases in all of average mortality rates, and absolute and relative inequalities, and can be found at the bottom left of Figure 1 .

In this article, we adhere to the Mackenbach and Kunst^{8} conceptualization of the relative and slope index of inequality, namely that: socioeconomic groups are ordered and given a value for the midpoint of the cumulative rank (e.g., if the socioeconomic groups were exact quintiles, the five groups would be assigned rank scores of 0.1, 0.3, 0.5, 0.7, and 0.9, with 0 being the rank for person with the highest socioeconomic position and 1 for the lowest socioeconomic position); using linear regression of the mortality rates on this cumulative rank, the coefficient for the rank is the slope index of inequality (SII), and the relative index of inequality (RII) = [intercept + SII]/intercept. And there is a simple relationship between the average mortality rate, SII and RII (i.e., SII = 2 × m (RII − 1)/(1 + RII), where m = average mortality rate). (Of note, researchers often use a log linear [e.g., Poisson regression] such that the exponentiated coefficient is the RII. Furthermore, Moreno-Betancur et al.^{17} have recently argued that a log-link model to estimate the RII and an additive [or linear] Poisson regression for the SII is theoretically and statistically preferred. However, this comes at the cost of not easily being able to mathematically relate the average mortality rate, SII, and RII.)

Figure 2A presents our typologies on the graphic used by Kjellsson et al.^{7} , with average (i.e., all socioeconomic groups combined) mortality on the x axis, absolute inequalities (SII) on the y axis and relative inequalities (RII) as contour lines. However, this graphical presentation is problematic for differing mortality rates (e.g., age- and cause-specific) as the area available to plot in funnels to the origin. Log transformation of the average mortality rate x axis assists with visualization, but results still funnel to the origin (as one cannot have high absolute inequalities and low mortality rates). Additional log transformation of the SII y axis assists when the mortality rate among the low socioeconomic group is higher than the high socioeconomic group, but when the gradient is reversed (e.g., lung cancer early in the tobacco epidemic), negative SIIs are not possible to plot on a log scale. (eFigures 2–4; https://links.lww.com/EDE/B197 illustrate these above limitations, using the same data used in the case studies presented below.)

FIGURE 2: A, Schematics showing trend typologies outlined in

Figure 1 . Absolute inequality (SII;

y axis) and change in average mortality rate (age standardized) in population (

x axis), with contour lines for relative inequalities (RII), following Kjellsson, et al.

^{7} B, Schematics showing trend typologies outlined in

Figure 1 . B, Relative inequality (RII;

y axis, log scale) and change in average mortality rate (age standardized) in population (

x axis, log scale), with contour lines for labeled slope inequalities per 100,000 (SII). Each arrow represents one of the typologies in

Figure 1 , and is labeled accordingly. Because of the mathematical relationship of the average mortality rate, SII and RII, having any two of these values fixed determines the third. For A, therefore, any point on the

x (average mortality rate) versus

y (RII; relative inequality) plane determines the SII (absolute inequality) value; SII contour plots are included to allow the viewer to not only see progress over time in average mortality and relative inequality, but progress also in terms of absolute inequality (arrows travelling parallel to contour plots mean no change in absolute inequality, whereas travel across [or towards] contour plots means an increase or decrease in absolute inequality). The same logic applies to B, except that the SII is now on the

y axis and the RII is the contour line. Although the exact direction of travel of each arrow varies between A and B (e.g.,

m↑a-r↓ is due east in A, compared with southeast in B), the ordering of typologies around the “compass face” is identical. For both A and B, reverse inequalities can also be shown: in A by extending the

y axis down to include negative SIIs, and plotting mirror image RII contour plots (i.e., 1/RII) beneath the

x axis; in B by including (as many as necessary) values of the RII less than 1 on the

y axis, and plotting mirror image (across the horizontal RII = 1 line) SIIs (negative values of SII). Indeed, due to suicide rates actually being higher among high income adults in New Zealand in 1981–1984, such an extended variant of B is shown in

Figure 5 . If these plots are extended to include negative SIIs and RIIs less than 1, then the most desirable direction of travel becomes “to the origin” rather than southwest (where the origin is for an RII of 1.0 in B). Figure is available in color online.

Figure 2B places the logarithm of the RII on the y axis, and the log of average mortality rate for the population on the x axis, with the SII now as contour lines (i.e., all x –y coordinate points on a given contour line will have the same absolute inequality [SII], but varying mortality rate and RII as given by the x and y axes). This configuration has several advantages, namely: RII is always greater than zero and therefore can be plotted on the y axis; RII and average mortality rate do not have a structural correlation as does the SII and average mortality rate, meaning a greater area of the x –y plane is used in plots.

Consider a hypothetical example, labeled m↓a↓r↑ in Figure 2B . The average mortality rate for this example decreases from 400 to 200 per 100,000. At t _{0} expected mortality for the 100th percentile (high) income rank might be 267 per 100,000 and 533 per 100,000 for the 0th percentile (low) income rank, and at t _{1} 110 and 290, respectively. Here, absolute inequality decreases over time (SII decreases from 533 − 267 = 267 per 100,000 to 290 − 110 = 180 per 100,000; arrow or trend crosses over contour lines), but the relative inequality increases (RII increases from 533/267 = 2 to 290/110 = 2.64), with the net result being an arrow pointing northwest. As well as the hypothetical example above, 11 of the 12 remaining typologies in Figure 1 are also plotted in Figure 2 , for the same starting rates. (Type m–a–r– has no change over time to plot, so is not plotted.) Type m↓a↓r↓ , the most desirable, heads southwest. “Undesirable” types are shown as red lines, heading somewhere in the arc northwest to southeast (the worst, m↑a↑r↑ , heading northeast).

METHODS FOR CASE STUDIES
Ethics approval for the New Zealand Census-Mortality Study was given by the Central Regional Ethics Committee (WGT/04/10/093).

NZ Census Mortality Linked Data
For the period 1981–1999, mortality records for people alive on the previous census night and who died within 3 years of the 1981, 1986, 1991, and 1996 censuses are assembled. The same procedure is carried out following the 2001 and 2006 censuses, but using mortality records within 5 years of census night. Probabilistic record linkage methods were then used to link census and mortality records.^{10} ^{,} ^{11} ^{,} ^{14} ^{,} ^{18–20}

The NZ censuses collect categories of individual income, which we convert to equivalized household using a NZ-specific index accounting for the number of children and adults in the household.^{21} Missing individual income data on one or more adult household members generates a missing household income; 12.2% (1991 census) to 17.4% (2001 census) of individuals aged 25–74 years. Results were also calculated by quintile of small area deprivation^{22} (for comparative purposes for the main income analyses, as income may be effected by reverse causation due to poor health preceding death; plots presented as eFigures; https://links.lww.com/EDE/B197 ) and education (nil, school-only, and post-school qualifications; to allow comparison with European countries described further below).

Mortality rates were directly standardized by both age and ethnicity; ethnicity is a cause of socioeconomic position, and associated with mortality independent of socioeconomic position, hence is a confounder of socioeconomic mortality gradients. The World Health Organization (WHO) standard was used for age weights, cross-classified by NZ’s ethnic (Māori, Pacific, European/Other) distribution in 2001.

The SIIs and RIIs were calculated using weighted (by person time) linear regression with quintile groupings of census respondents by household income, with quintile-cumulative rank midpoints calculated separately by age group, but pooled across sex and ethnic groups. Statistically significant trends across the census cohorts (i.e., time) were tested for SIIs and the logarithm of RIIs. Further methodological detail, and a worked example, on calculating the SIIs and RIIs is in the eAppendix (https://links.lww.com/EDE/B197 ).

Projections
We fitted Poisson regressions for age- and ethnicity-standardized mortality rates, for 25- to 74-year olds combined, for the 1991–1994 to 2006–2011 cohorts (i.e., last four cohorts). Independent variables were year, midpoint on cumulative income rank for each quintile, and an interaction of year and income. From these regressions, we estimated a business as usual annual percentage change in mortality rate for the 0th and 100th percentiles of the income distribution, then projected out mortality rates by income, and hence slope and relative indices of inequality, to 2031. For the purposes of exploring how much business-as-usual projections of these indices can be altered, we specified an egalitarian scenario of 5% per annum reductions in CVD mortality for all income groups and estimated mortality rates, slope, and relative indices of inequality out to 2031 for CVD and for all-cause mortality.

European Education Mortality Gradients
We used all-cause mortality rates (35- to 79-year olds, WHO European standard), SIIs, and RIIs for multiple European countries from the period 1980–2010 as published elsewhere,^{16} ^{,} ^{23} selected by us to give geographic and typology variation. In brief, these estimates are based on national (in some cases regional) data on mortality by level of education (in three groups: primary and lower secondary; upper secondary; and postsecondary), mostly measured in a census follow-up. We also calculate New Zealand 35- to 74-year-old mortality rates with the same standard, and SIIs and RIIs, for comparison. (Unfortunately, 75- to 79-year-old deaths were not linked to New Zealand censuses till the 2000s, thus there is a slight incomparability by age between New Zealand and European countries.)

RESULTS AND PROJECTIONS NZ CASE STUDY
Trends in All-cause Mortality
Figure 3 shows the overall trends in all-cause mortality rates by income tertile by age group, using “standard” graphs. eTable 1 (https://links.lww.com/EDE/B197 ) shows the accompanying SIIs and RIIs (using income quintile rates as inputs to regressions). All-cause mortality in 25- to 74-year olds consistently declined throughout the whole study period—roughly in parallel by income tertile such that absolute inequalities (SII) showed no trend (P = 0.99) over time for males but relative inequalities (RII) increased from 1.93 to 2.80 (ratio increase per 10 years 1.17 [95% CI = 1.11, 1.22], eTable 1; https://links.lww.com/EDE/B197 ). The pattern was similar for females, but with absolute inequalities also increasing (217–281 per 100,000, increase per 10 years 34.5 per 100,000 [95% CI = 15.4, 53.6], eTable 1; https://links.lww.com/EDE/B197 ).

FIGURE 3: Trends in age-standardized mortality rates (per 100,000) by income tertile, New Zealand, 1981 to 2011. A, Males, (B) females. Ribbons around trend lines are 95% confidence intervals. Source:

http://www.uow.otago.ac.nz/NZCMS-CT-dataexplorer , eTable 2; (

https://links.lww.com/EDE/B197 ). Equivalent cause-specific mortality rate figures are shown in eFigure 1 (

https://links.lww.com/EDE/B197 ). 25- to 74-year-old mortality combined is shown in eFigure 1 (

https://links.lww.com/EDE/B197 ).

Figure 4 shows the graphical plots of all-cause mortality, with the arrowhead denoting the most recent 2006–2011 cohort. For 25- to 74-year-old males combined, the population is traveling northwest (increasing relative inequalities, and reducing average mortality rates, respectively), roughly parallel to the 640 per 100,000 SII contour line (stable absolute inequalities)—that is, a m↓a–r↑ typology. The female plot starts further to the west (lower average mortality rate), has a similar increase in relative inequalities but also moves closer to the 320 per 100,000 SII contour line (i.e., heading north-northwest, m↓a↑r↑ ). For 25- to 44-year olds, the arrows travel more northerly (large increases in relative and absolute inequalities), due to small falls in low-income mortality rates over time compared with larger falls for high-income rates (Figure 3 ). Specifically, the 25- to 44-year-old RIIs increased from 1.71 to 2.80 for males and 1.52 to 2.63 for females (eTable 1; https://links.lww.com/EDE/B197 , per 10-year ratio increases of 1.29 [95% CI = 1.15, 1.45] and 1.44 [1.29, 1.61] respectively). Importantly, no plots for any sex by age group are heading in the south to west quadrant; relative inequalities are increasing across all age groups.

FIGURE 4: Mortality inequality plots, by household income, for age-specific all-cause mortality for New Zealand 1981–1984 to 2006–2011. The arrows denote the most recent cohort (2006–2011), and all trajectories are heading in a northwest direction (i.e., decreasing mortality, stable or increasing absolute inequalities, and increasing relative inequalities;

m↓ a−/↑r↑ ). Actual RIIs and SIIs, and

P for trend, are shown in eTable 1 (

https://links.lww.com/EDE/B197 ). Using

P < 0.05 as an arbitrary cut-off for significant trends, then the majority of plots are

m↓a−r↑ . Exceptions are: 25–74 years females 65–74 years females =

m↓a↑r↑ ; 65–74 years males =

m↓a−r− . Figure is available in color online.

Trends in Cause-specific Mortality
A large amount of the all-cause mortality falls came from the declines in CVD, for both men and women, and again roughly in parallel for the lowest and highest income tertiles, resulting in stable absolute inequalities but increasing relative inequalities (e.g., RIIs increasing from 1.86 to 3.07 for males, and from 1.79 to 3.87 for females [both P for trend < 0.01]; eFigure 1; https://links.lww.com/EDE/B197 shows the “standard” cause-specific mortality rates as per the Figure 3 layout).

Figure 5 summarizes all of the cause-specific mortality trends in one graphic (ignore the dotted projection extensions for now). The CVD plots for both males and females travel a considerable distance west relative to the log mortality rate x axis, consistent with large percentage falls in CVD mortality for all income groups. Apart from the second data point (1986–1989) on the CVD plots, there is also a clear upward trend in relative inequalities, with the lines heading north with respect to the y axis. However, there is also a consistent reduction in absolute inequality: males are moving away from the 160 SII contour toward the 80 SII contour and females are moving away from 80 SII contour toward the 40 SII contour. Taking the study period as a whole both males and females CVD socioeconomic inequality could be classified as m↓a↓r↑ .

FIGURE 5: Mortality inequality plots, by equivalized household income, for cause-specific mortality for New Zealand 1981–1984 to 2006–2011. The arrows denote the most recent cohort of data (2006–2011). The dotted lines are projections to 2031: under BAU trends in annual percentage change in mortality rates by low and high income; for CVD scenario of 5% reduction in CVD mortality for all levels of income and BAU for other causes of death (

Table ). A, Males, 25–74 years. B, Females, 25–74 years. Actual RIIs and SIIs, and

P for trend, in eTable 1 (

https://links.lww.com/EDE/B197 ). CVD scenario = 5% per annum reduction for all income groups for CVD, with BAU for other causes of death. BAU indicates business as usual. Figure is available in color online.

TABLE: Annual percentage changes in 25- to 74-year-old Mortality Rates (Age-standardized, per 100,000) from 1991–1994 to 2006–2011 in New Zealand, and Projected Mortality Rates, SII and RII Under Business as Usual and for a CVD Mortality Scenario of 5% per Annum Reductions in Mortality for All Income Groups to 2031

Nonlung cancer exhibits only a small reduction in the population standardized rate, evident in the small movement west (Figure 5 ). However, the nonlung cancer plots clearly cross SII contour plots as they head north, with absolute inequality more than doubling (each labeled contour indicates a multiplier of two for the SII; 45 to 85 per 100,000 for males and 17 to 58 per 100,000 for females). Both male and female nonlung cancer is best classified as m↓a↑r↑ .

Female lung cancer exhibits a clear rise in both absolute and relative inequalities, with the plot tracking north and across the absolute inequality contour lines. There is also, apart from the last cohort, a steady movement east—an increase in the overall population rates (and see Figure 3 ; m↑a↑r↑ ). For males, with the line heading in a northwest direction, a m↓a–r↑ classification is most appropriate.

Unintentional injury for both males and females is heading west with no obvious trends in absolute or relative inequality (and as reflected by lack of trends in both RIIs and SIIs in eTable 1; https://links.lww.com/EDE/B197 ).

Male suicide is best assigned to the m↑a↑r↑ typology when one takes the study period as a whole. However, the direction of the trend line on Figure 5 changes over time—at first heading east, then north (consistent with a peak in suicide rates in the 1996–1999 cohort; Figure 3 ) and finally in a northwest direction. Female suicide changes from higher rates among higher-income groups (RII < 1 and SII < 0), to clearly increasing absolute and relative inequalities with no clear change in the average mortality rate (m–a↑r↑ ).

Projections
The Table shows annual percentage changes in mortality rates for the period covered by the four latest cohorts, then extrapolated out to 2031 assuming business as usual. Consistent with widening relative inequalities in all-cause mortality inequalities, the annual percentage changes are stronger (i.e., more negative) for the highest income percentile (−3.09% and −2.90% for males and females, respectively, compared with −1.92% and −1.13% for the lowest income percentile; Table ). If this pattern continues out to 2031, the relative index of inequality will increase to 3.66 and 3.94 for males and females, but the SIIs decrease to 338 and 262 per 100,000 (Table ), a northwest and west-northwest direction of travel for males and females, respectively (dotted line extensions in Figure 5 ).

When we posit a −5% annual percentage change for CVD for all sex by income groups (a hastening of decline for low income, but slowing for high income; otherwise business as usual annual percentage changes for other causes of death), the relative index of inequality for CVD remains constant (by definition) and slope indices of inequality fall more rapidly out to 2031 (Table , and due-west plots in Figure 5 ). The all-cause mortality trends improve a little with the plots bearing further west. For example, for males, the 2031 all-cause relative index of inequality would “only” increase to 3.39 and the slope index of inequality would decrease further to 292 per 100,000.

Cross National Comparisons Using Typology Plots
eFigure 2 (https://links.lww.com/EDE/B197 ) shows the cross-national comparisons of mortality in 35- to 79-year olds, using education as the measure of socioeconomic position. The most common pattern or typology is northwest, that is, m↓a–r↑. However, there are two notable exceptions: eastern European countries (Czech Republic, Slovakia, Hungary, and Lithuania) are heading north, with profound increases in both absolute and relative inequalities and little change in (high) average mortality; Spain (Barcelona) females are heading southwest.

DISCUSSION
This article lays out a typology and graphical presentation for trends in mortality rates and both absolute and relative inequalities. The graphs we present parsimoniously allow for comparing multiple subpopulations or countries on one graph—especially when average mortality rates vary (e.g., between age groups and causes of death). These graphs are a summary tool; it is still necessary, in our view, to build up to them from the actual trends in mortality rates over time by socioeconomic position (Figure 3 ), then the quantification of absolute and relative inequality measures and their trends over time (Table ), and then summarize inequality trends with the typology and graphical tool.

So what does a typology and graphical tool add? Is it worth going this extra step? We argue yes, for three reasons. First, a lot of information can be conveyed in one graphic—although it requires a reasonably sophisticated understanding, or at least practice, to immediately interpret the typologies and graphics. We offer the compass bearing descriptors as one approach. Second, the typology and graphic make the user think in terms of three aspects simultaneously: the change in average mortality rate, and the change in both absolute and relative inequalities. We believe this is a major advance over the oft heard debate among colleagues and policy makers as to “what matters most, relative or absolute inequalities?”; the answer is both, plus trends in average mortality. Third, the graphics may be useful for assisting target setting—a tool for presenting how much one can “change direction” under different plausible scenarios of changing future mortality by socioeconomic position. We presented a simple—but plausible—example in this article for CVD mortality. We also note that the European WHO Region has a goal of 25% reductions in inequalities by 2020,^{24} albeit gaps in life expectancy. However, if targets were set for, say, a 25% reduction in the absolute gap in mortality rates they could easily be visualized on our typology graphs as travelling west an amount equivalent to the distance between the contour lines. Further exploration and research using graphical tools to assist both monitoring and target setting is justified, building on the graphical layout we present in this article and those of other researchers (e.g., Kjellsson et al.^{7} variants shown in eFigures 2–4; https://links.lww.com/EDE/B197 ).

The general pattern for the New Zealand case study, and the cross-national comparisons, is of falling mortality rates, stable absolute inequalities, but increasing relative inequalities (i.e., m↓a–r↑ , or a northwest direction of travel in the typology graphs). There are exceptions to this general pattern, for example: among 25- to 44-year olds in New Zealand there have been increases in absolute (in addition to relative) inequality due to stasis in low-income mortality rates (m↓a↑r↑ ); among females, CVD mortality has fallen for all income groups in New Zealand, but while relative inequalities have increased absolute inequalities have decreased (m↓a↓r↑ ); in Eastern European countries, both absolute and relative inequalities have increased with stasis or modest falls only in average mortality (m–a↑r↑ or m↓a↑r↑ ).

The New Zealand case study presents mortality inequalities by equivalized household income, which has advantages of easy manipulation into quintiles (or other) groupings for calculating slope and relative indices of inequality. However, income may be prone to some reverse causation (i.e., poor health leading up to death causing a reduction in income). To bias trends over time, this amount of potential reverse causation would have to vary over time (e.g., due to changing income protection policies when sick over time (not the case in New Zealand), or changing proportionate causes of death over time (possible, in that CVD has reduced and cancer increased). However, the inequality plots by a small area (neighborhood) deprivation index using linked New Zealand census-mortality data were very similar (eFigure 5; https://links.lww.com/EDE/B197 ).

To conclude, we have presented a typology and graphical presentation that holistically conveys trends in average mortality rates, and both absolute and relative inequalities. The graphs can summarize and present information for multiple causes of death, age groups, and socioeconomic factors simultaneously. We encourage others to critique and further develop these approaches, both for monitoring and potentially assisting target setting.

ACKNOWLEDGMENTS
We gratefully acknowledge the many helpful comments on an earlier draft of this article from Sam Harper, Laura Howe, Anton Kunst, John Lynch, Neil Pearce, Dennis Petrie, and Nick Wilson.

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