## Abstract

The relationship between differential mortality rates and differences in life expectancy is well understood, but how changing differential rates translate into changing differences in life expectancy has not been fully explained. To elucidate the mechanism involved, this study extends existing decomposition methods. The extended method decomposes change in the sex gap in life expectancy at birth into three components capturing the effects of the sex difference in mortality improvement (ρ-effect), life table deaths density by age (f-effect), and remaining life expectancy by age (e-effect). These three effects oppose and augment each other, depending on relative change in sex-differential mortality rates. The new method is applied to period data for 35 countries and cohort data for 25 countries. The results demonstrate how the mechanism, involving the three effects, operates to determine change in the sex difference in life expectancy. We observe the pivotal importance of the f-effect, which is predominantly negative because of lower female mortality, in favoring narrowing rather than widening of the sex gap, in shifting the overall effect to younger ages, and in exaggerating fluctuations due to crisis mortality. The new decomposition provides a more detailed basis for substantive analyses examining change in differences in life expectancy.

## Introduction

Since the publication of John Graunt’s (1662/1939) pioneering work recognizing sex-differential morbidity and mortality, the sex difference in life expectancy at birth has received considerable attention in demography, actuarial studies, epidemiology, biology, sociology, and anthropology. In the twentieth century, the growing deficit in the life expectancy of males compared with females became of some public concern and “increasingly seemed to require explanation” (Nathanson 1984:191). Various studies have sought to explain the sex difference in mortality in terms of the underlying social, environmental, biological, genetic, and behavioral determinants (Lindahl-Jacobsen et al. 2016; Lopez 1995; Luy 2003; Preston and Wang 2006; Waldron 1983, 1986, 1995; Wingard 1982, 1984; Zarulli et al. 2018). Others have focused on decomposition methods to identify the age and cause of death contributions to the sex gap (Booth 2003; Booth et al. 2016; Canudas-Romo et al. 2015; Trovato and Lalu 2007; Vallin and Meslé 2001).

Clearly, a full understanding of the sex difference in mortality must incorporate an appreciation of how the difference changes over time. Industrialized countries have experienced similar trends in the sex gap in life expectancy at birth. For most countries, the sex gap widened after 1950 but has more recently narrowed, although the timing of the turning point varies considerably (Glei and Horiuchi 2007). Methods for the decomposition of change in the sex gap in life expectancy are relatively recent (Canudas-Romo 2003; Glei and Horiuchi 2007; Zhang and Vaupel 2009) and do not quantify the separate components of change. Further, decomposition of change in the sex gap in life expectancy has not been explored from a cohort perspective.

The aim of this study is to elucidate the mechanism involved in determining the magnitude of change in the sex gap in life expectancy at birth when mortality rates change. We first develop a method for the extended decomposition of change in the sex gap in life expectancy, identifying three related but separate effects for quantification. We then apply the new method to change in the sex gap in period and cohort life expectancy at birth to examine how the three effects change over time and birth cohort, and how they operate to jointly determine change in the sex gap. In the following sections, we derive the extended decomposition method and introduce the data used in the analysis. The results consist of period and cohort decompositions of temporal change in the sex gap in life expectancy at birth for selected populations (full results appear in the online appendix). Finally, we discuss the method and draw conclusions from the findings.

## Development of the Extended Decomposition Method

There are three existing methods for decomposing change in the sex gap in life expectancy at birth. The Canudas-Romo (2003) method separates change in the sex gap into the sex difference in the effect of mortality improvement and the effect of heterogeneity in mortality improvement at different ages. The Glei and Horiuchi (2007) method, discussed later, and the Zhang and Vaupel (2009) method are in fact identical, although they use different terms (see online appendix, section A1). All three methods are derived from the Vaupel and Canudas-Romo (2003) decomposition method.

Among the three methods, the Glei and Horiuchi method is the most often cited and most influential. In this method, change in the sex gap in life expectancy at birth is decomposed into change in mortality sex ratios and the sex difference in the mortality age pattern. The former is dependent on the sex difference in mortality improvement, whereas the latter is represented by the sex difference in the slope of the Gompertz model. The Gompertz slope is affected by two sources of change: in the distribution of life table deaths and in remaining life expectancy (Vaupel 1986; Vaupel and Canudas-Romo 2003). However, in the Glei and Horiuchi method, these two sources of change are not made explicit.

A similar constraint is found in the Canudas-Romo (2003) and Zhang and Vaupel (2009) methods, both of which preclude full consideration of all underlying effects. Given this limitation in existing methods, we use the Vaupel and Canudas-Romo (2003) decomposition of change in life expectancy as the starting point for the derivation of an extended decomposition of change in the sex difference in life expectancy at birth that explicitly captures each underlying effect.

### Vaupel and Canudas-Romo Decomposition

Vaupel and Canudas-Romo (2003) developed a method for the decomposition of change over time in life expectancy. Their method involves three age-specific components: mortality improvement rates (ρ(a,t)), the density function of the life table death distribution (f(a,t)), and remaining life expectancy (e(a,t)), where a denotes age, and t denotes time. The rate of mortality improvement directly measures change in mortality by age, and the deaths density function and remaining life expectancy capture the indirect effects of mortality change from the life table.

Denoting change over time, or the time-derivative of life expectancy, by a superscript dot and subscript t, the Vaupel and Canudas-Romo decomposition is
$ėt0t=∫0ωρatfateatda,$
1
where ω is the highest age at death. Note that the rate of mortality improvement is the time change in the force of mortality, μ(a,t), or its relative derivative with respect to time: $ρat=−∂μatμat∂t$. The remaining two components, the density function of death distribution and remaining life expectancy, are functions of the life table and therefore also depend on the force of mortality. Equation (1) shows that the magnitude of change in life expectancy is determined not only by change in the force of mortality but also by the shape of the age distribution of deaths and by the age pattern of remaining life. These three components are integral to a full understanding of change in life expectancy.

### Extended Decomposition of Change in the Sex Gap

The extended decomposition of change over time in the sex gap (female – male) in life expectancy at birth is based on Eq. (1). The new method first applies Eq. (1) to the time change in life expectancy by sex s, $ėt0ts$, obtaining ρ(a,t,s), f(a,t,s), and e(a,t,s) for each sex. These time changes in female and male life expectancy are then compared by taking the partial derivative with respect to s:
$ėt,s0ts=∂ėt0ts∂s=∂∂s∫0ωρatsfatseatsda=∫0ωρ̇satsfatseats+ρatsḟsatseats+ρatsfatsėsatsda.$
2

The right side of Eq. (2) has three terms, each including a partial derivative with respect to sex. The first term captures the effect of the sex difference in the rate of mortality improvement, ρ(a,t,s), and is called the “ρ-effect.” A positive (negative) ρ-effect stems from a faster (slower) mortality decline among females than among males. The second term captures the effect of the sex difference in the life table deaths density function, f(a,t,s), and is called the “f-effect.” A negative f-effect stems from a more dispersed male death distribution (which is the case when female life expectancy exceeds male life expectancy), and a positive f-effect would arise from a more dispersed female distribution. The third term captures the effect of the sex difference in the age pattern of remaining life expectancy, e(a,t,s), and is called the “e-effect.” Greater female remaining life expectancies produce a positive e-effect, whereas greater male remaining life expectancies would produce a negative value. Thus, the f-effect and e-effect have opposite signs. The “total effect”—the sum of the three terms—equals change in the sex gap in life expectancy. Note that if mortality deterioration occurs in one or both sexes, the signs of the f-effect and e-effect are reversed.

The extended method was originally derived using a discrete approach (Cui 2017), whereas Eq. (2) treats the variables t and s as continuous. The advantage of the continuous approach is that we can consider the derivative of a measure as the transition from one life expectancy to another (Andreev 1982; Andreev et al. 2002). Thus, we can regard the derivative of life expectancy as the transition between life expectancies at two time points for the same population (Eq. (1)) or as the transition between life expectancies for females and males at the same time. In the first case, the second derivative is with respect to s (Eq. (2)); in the second case, it is with respect to t. According to Clairaut’s theorem, the double derivative is not affected by the order of differentiation (Stewart 2011). The methods employed in estimating Eq. (2) for discrete data are presented in the online appendix, section A2.

The extended decomposition in Eq. (2) can also be applied to cohort data. Furthermore, it can be applied to any age range. We use the truncated cohort life expectancy at birth, defined as
$e0cT=∫0Tlxcdx=∫0Te−∫0xμacdadx,$
3
where T is the age at truncation, c denotes birth cohort, l(x,c) is the probability of surviving from birth to age x, and μ(a,c) is the force of mortality at age a. The radix l(0,c) = 1. Section A2 of the online appendix details the procedure for the discrete approach.

## Data

The data were obtained from the Human Mortality Database (HMD 2018). The HMD contains high-quality historical data combining vital statistics and census counts or official population estimates; standard methods applied to all populations over time ensure comparability (Wilmoth et al. 2017). For this analysis, countries with a population of less than 1 million in 2010 were omitted so as to avoid substantial random fluctuation.

The trend analysis was based on life expectancies calculated from sex-specific central death rates by single year of age for the years (or birth cohorts) 1880 to 2010. For the period analysis, we included the 35 countries with series starting in or before 1970 to ensure a time series of adequate length, which thus ranged from 41 to 131 years. For the cohort analysis, we included the 25 countries with series starting in or before 1960 with data for at least 11 annual birth cohorts, giving series of 11 to 81 years. By employing truncated cohort life expectancies from birth to ages 50, 70, and 90, we increased the number of countries included in the cohort comparisons. Table 1 lists the countries and time series covered.

For the decomposition analysis, we employed data for five-year periods and birth cohorts to reduce fluctuation. To reduce fluctuation in the age-contribution analysis, we combined single ages into five-year age groups.

## Results

Trend and decomposition analyses are presented for selected countries. For the period perspective, we selected England and Wales, Switzerland, France, the United States, Japan, and Russia; for the cohort perspective, we selected England and Wales and Switzerland. We also present an analysis of age contributions to the three effects, using the example of England and Wales for selected periods and birth cohorts. Full results are presented in the online appendix.

### Period Life Expectancy

Figure 1 shows the trends in the sex gap in life expectancy at birth from 1880 to 2010. Apart from the clearly visible effects of World War I (WWI) in 1914–1918, World War II (WWII) in 1939–1945, and the 1918 Spanish influenza epidemic (Ansart et al. 2009), the sex gap increased gradually over the first 90–100 years. Thereafter, the gap began to decrease in most countries. The timing of the turning point, or onset of a sustained narrowing of the sex gap, varies by several decades. Among the selected countries, the earliest turning point occurred in England and Wales (1970), followed by the United States (1976), Switzerland (1992), France (1993), Russia (2005), and Japan (2006).

Figure 2 presents the extended decomposition of change in the sex gap between five-year periods (excluding crisis mortality) for the six selected countries. Change in the sex gap—or total effect—is positive (fluctuations notwithstanding) before the turning point and negative after the turning point. The decomposition shows that the contribution due to differential mortality improvement by sex (ρ-effect) is similarly positive/negative in relation to the turning point. In contrast, the contributions due to sex differences by age in the density function of deaths (f-effect) and in remaining life expectancy (e-effect) tend not to change sign. The f-effect is generally negative, in line with the more dispersed male death distribution, contributing to narrowing of the sex gap. The e-effect is generally positive because females have higher remaining life expectancy, contributing to widening of the sex gap. The f-effect is larger than the e-effect, such that their combined effect contributes to narrowing.

Figure 2 also demonstrates that when the sex gap widens because of faster mortality improvement among females, the positive ρ-effect is dominant but partly counterbalanced by the negative f-effect, although it is augmented by the positive e-effect. However, when the sex gap narrows because of faster mortality improvement among males, the negative ρ-effect is relatively small but is augmented by the f-effect, which becomes dominant. Their combined contribution to narrowing of the sex gap is reduced relatively little by the positive e-effect.

For England and Wales, France, and Switzerland (fluctuations notwithstanding), the ρ-effect is generally increasing from 1880 to 1884, reaching a peak in the 1950s. However, for these three countries and the United States, the ρ-effect is then decreasing and becomes negative after the turning point. At the turning point, the positive ρ-effect is exactly counterbalanced by the combined f-effect and e-effect. In the most recent period, the ρ-effect is diminished. For Japan, the ρ-effect is always positive, consistent with the late turning point. For Russia, the erratic pattern is a result of substantial fluctuation in mortality change, especially for male mortality, which produces widely fluctuating f-effects. In the case of Switzerland, the f-effect and e-effect have the unusual feature of sharing the same sign for change between 1910–1914 and 1915–1919, which arises from the age distribution of deaths from the Spanish influenza epidemic (see the upcoming section, Age Contributions to Change in the Sex Gap).

### Cohort Life Expectancy

Figure 3 shows the sex gap in truncated life expectancies between birth and ages 50, 70, and 90 for England and Wales and for Switzerland. Given lower mortality among females and cumulative mortality sex discrepancies over age, the sex gap increases with age at truncation. The waves in the trends for England and Wales are due to relatively high young adult male mortality during the two world wars and the Spanish influenza epidemic. Male cohorts born around 1895 were exposed to service in WWI (with heavy loss of life) and to the Spanish influenza epidemic, and cohorts born around 1923 were exposed only to WWII, producing a smaller wave. This results in three turning points. For Switzerland, no such waves exist because the country remained neutral during the world wars and because excess male mortality at young adult ages from the Spanish influenza epidemic is spread over many birth cohorts. The sex gaps for 50$e0c$ and 70$e0c$ increase slightly up to the 1920 birth cohort and then decrease slowly, whereas the gap for 90$e0c$increases from 1880.

Figure 4 presents the decomposition of change in the sex gap in cohort life expectancies. As in the period perspective, the total effect and (for the most part) the ρ-effect are positive when the gap is widening and negative when the gap is narrowing, whereas the f-effect and e-effect are, respectively, negative and positive in most instances. The cases of 50$e0c$ and 70$e0c$ for Switzerland show that after 1920, the ρ-effect is positive, but the sex gap is narrowing. This occurs because the negative f-effect more than counterbalances the small ρ-effect and e-effect.

In the case of England and Wales, the f-effect is positive for change between birth cohorts born in 1880–1884 to 1890–1894 for the sex gaps in 50$e0c$, 70$e0c$, and 90$e0c$. This is due to increasing mortality among males. However, the ρ-effect decreases because the age pattern of male mortality increases and then decreases as a result of the age of different birth cohorts at the time of WWI. For change between cohorts born in 1890–1894 and 1900–1904, the ρ-effect is negative because males were decreasingly exposed to war, producing faster mortality improvement across cohorts than occurred for females.

For Switzerland, for all three sex gaps, when the positive ρ-effect is dominant, the contributions of the f-effect and e-effect tend to increase in size between birth cohorts born in 1880–1884 and 1910–1914. For later cohorts, the positive ρ-effect decreases, and the contributions of the f-effect and e-effect tend to decrease.

### Age Contributions to Change in the Sex Gap

Figure 5 shows the age contributions to change in the sex gap in life expectancy at birth for selected periods for England and Wales. The age pattern differs considerably among periods. For 1935–1939 to 1940–1944, the influence of increased male war deaths at young adult ages is evident in the large positive ρ-effect and f-effect and negative e-effect. These war deaths also result in the large negative f-effect at these ages for 1940–1944 to 1945–1949. The widening sex gap during 1955–1959 to 1960–1964 (Fig. 2) is due to contributions at almost all ages, and counterbalancing age contributions result in the turning point in the sex gap during 1965–1969 to 1970–1974. In later periods, narrowing of the sex gap is due to negative ρ-effects at older ages augmented by large negative f-effects. However, at the very oldest ages, small ρ-effects arise from similarly low rates of mortality improvement by sex, and positive f-effects occur because of the greater concentration of female deaths at these ages.

Figure 6 presents the age contributions to change in the sex gap in 70$e0c$ for selected birth cohorts for England and Wales. The concentration at young adult ages (15–19 to 30–34) reflects the influence of WWI, the Spanish influenza epidemic, and WWII. The war-related pattern in the effects shown in Fig. 5 for periods 1935–1939 to 1940–1944 and 1940–1944 to 1945–1949 is also evidenced in Fig. 6 in successive ages for birth cohorts 1910–1914 to 1915–1919 and 1915–1919 to 1920–1924.

## The Underlying Mechanism

The extended decomposition formulates change in the sex gap into the effects of the sex difference in mortality improvement (ρ-effect), the sex difference by age in the density function of deaths (f-effect), and the sex difference by age in remaining life expectancy (e-effect). These three effects are the fundamental building blocks of change in the sex difference in life expectancy and enhance understanding of the mechanism involved. Previous decompositions have not fully identified the three separate effects.

The mechanism at play derives from the calculations involved in the life table, but how these interact in determining change in the sex gap in life expectancy at birth is not immediately evident. Mortality change by sex can be considered as a perturbation to the life table (Væth et al. 2018; Wrycza and Baudisch 2012); the ρ-effect, f-effect, and e-effect change according to the perturbation.

Table 2 summarizes the underlying mechanism of change in the sex gap in life expectancy at birth when the sex gap is positive. (For a negative sex gap, it suffices to exchange the sexes in the table.) In general (given age aggregation), when mortality is improving for both sexes, faster improvement among females will produce a greater sex difference in mortality improvement, in the deaths density, and in remaining life expectancy. Thus, in most cases, the ρ-effect will be positive, the f-effect will be negative, and the e-effect will be positive. A slower improvement among females will produce a smaller sex difference in the mortality improvement, in the deaths density, and in remaining life expectancy; and in most cases, the ρ-effect will be negative, the f-effect will be negative, and the e-effect will be positive. In both cases, the combined f-effect and e-effect will be negative. Thus, when mortality is improving, the only situation in which widening occurs in the sex gap in life expectancy at birth is when the ρ-effect is positive and greater than the combined f-effect and e-effect. When the ρ-effect is positive and equal to the combined f-effect and e-effect, the turning point of change in the sex gap will occur. When the ρ-effect is positive but less than the combined f-effect and e-effect, or is zero or negative, the sex gap will narrow. When female mortality is improving but male mortality is deteriorating at a greater rate (such as during war), the ρ-effect and f-effect will be positive, and the e-effect will be negative.

If mortality were deteriorating for both sexes, the reversed signs of the f-effect and e-effect would produce a positive combined effect, leading to widening of the sex gap except when the ρ-effect is negative and greater than or equal to the combined f-effect and e-effect. If mortality were constant for both sexes, the ρ-effect, f-effect, and e-effect would be 0, and the sex gap would be constant.

## Discussion

In this study, we presented an extended decomposition method and applied it to period and cohort data to achieve a better understanding of the mechanism involved in determining change in the sex gap in life expectancy at birth. The three effects are directly interpretable. We make three observations based on our analysis.

First, because the f-effect is almost always negative and larger than the positive e-effect, it is easier to achieve narrowing of the sex gap than widening. For this reason, when mortality is improving faster for females, the turning point in the sex gap occurs before the rates of improvement are equalized (ρ-effect = 0). Further, when mortality is improving faster for males, the relatively rapid narrowing of the sex gap belies actual progress in equalizing mortality rates. Recognition of these facts is important in the evaluation of public health interventions because the turning point and speed of closure in the sex gap in life expectancy will give an overly optimistic impression.

Second, the age distributions of the three effects differ. In particular, given greater compression of the female death distribution and in the context of mortality improvement at increasing ages, the ρ-effect has an older distribution than the f-effect because the constraint on life table deaths to sum to 1 leads to greater differences at younger ages. Thus, the ρ-effect is greatest at an older age than the total effect. Again, this has significance for the evaluation of public health measures because the greatest change in the ρ-effect, due to sex differences in mortality improvement, occurs at ages that are older than the total effect would imply.

Third, in situations of crisis mortality, the f-effect is large and has the same sign as the ρ-effect, with only marginal counterbalancing by the e-effect, resulting in large fluctuations in the total effect. Thus, the significance of perturbations in mortality rates is grossly exaggerated in the sex gap in life expectancy.

These three observations demonstrate the pivotal role of the f-effect in determining change in the sex gap in life expectancy. They also indicate that the gap in life expectancy is suboptimal for the evaluation of public health programs and policies aimed at reducing sex differences in mortality. Rather, mortality rates portray a more relevant picture. In order to appreciate the three effects more clearly, Figs. S9S11 in the online appendix contrast one effect with the other two combined, highlighting the relevance of all three effects.

In our analysis, we addressed the concept of change over time in the sex gap in life expectancy by using a method that derives the sex difference in change over time in life expectancy. This is possible because of the interchangeability in the double derivative. The advantage of the extended decomposition is that it elucidates the underlying mechanism in terms of three easily interpretable effects. A focus on change in the sex gap, differentiating first with respect to sex and then with respect to time, would produce less interpretable effects referring to transitional mortality between male and female. Further, in using a continuous approach, the extended decomposition avoids the order residual (Dasgupta 1993; Kim and Strobino 1984), which arises from the covariances between the three effects (Cui 2017) when using a discrete approach (e.g., Kitagawa 1955). The continuous approach also produces more straightforward and interpretable effects than the discrete approach.

The sex gap in life expectancy is a relatively sensitive measure. This is both a limitation and a strength. In this analysis, we reduced fluctuation by using data for five-year age groups and five-year periods and birth cohorts, thereby enabling change to be examined with greater reliability and the underlying mechanism to be understood. An additional limitation stems from the approximations involved in empirical application of the continuous method to discrete data. However, other methods using derivatives and assuming continuous variables have shown only small discrepancies in their estimates (Canudas-Romo and Guillot 2015; Horiuchi et al. 2008).

## Conclusion

The novel decomposition method for the sex gap in life expectancy developed through this research builds on the work of Canudas-Romo (2003), Glei and Horiuchi (2007), and Zhang and Vaupel (2009). The three effects identified through the extended decomposition provide a basis for greater understanding of mortality change. We applied the new method to life expectancy at birth, but it is applicable to life expectancy at any age or age range. Further, it can be used to study differences between populations defined by any variable, such as sex, region, ethnicity, and socioeconomic status.

Existing studies on the sex gap in life expectancy have mainly focused on the decomposition of period life expectancy (Trovato and Lalu 2007; Vallin and Meslé 2001); very few have adopted the cohort perspective (Beltrán-Sánchez et al. 2015; Lindahl-Jacobsen et al. 2013). In the context of change in the sex gap in life expectancy, this study is the first to address cohort mortality and to analyze incomplete cohorts through truncated life expectancies.

Previous research has tended to focus on why mortality sex differences exist and change, drawing on external factors, such as health behavior and socioeconomic status. From a compositional perspective, change in sex-specific mortality rates is the perturbation that initiates change in the sex gap in life expectancy. Thus, the focus of our research is the mechanism of mortality change internal to the life table—in other words, how a perturbation is translated into change in the sex gap. The study elucidates this mechanism, leading to the observation that life expectancy is suboptimal for the evaluation of change in sex differences in mortality.

Finally, the extended decomposition provides a new and more detailed basis for future substantive analyses examining change in differences in life expectancy. The pivotal role of the f-effect warrants recognition and further investigation in substantive analyses.

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