## Abstract

This study examines proximate sources of change in first-marriage trajectories in the United States between 1960 and 2010. This was a period of tremendous social change: divorce became more common, people started marrying later or not marrying at all, innovations in medicine and changes in social and behavioral factors led to reduced mortality, inequality grew stronger and was reflected by more intense assortative mating, and the country underwent a massive educational expansion. Each of these factors influenced the formation and dissolution of first marriages over this period. This article extends the multiple-decrement life table to incorporate heterogeneity and assortative mating, which allows the quantification of how changes in the incidence of marriage, divorce, and mortality, along with changes in educational attainment and assortative mating, have shaped trends in first-marriage trajectories. The model is used to prove that stronger educational assortative mating leads to longer average durations of first marriage. Using data from multiple sources and this model, this study shows that although the incidence of divorce was the primary determinant of changes in first-marriage trajectories between 1960 and 1980, it has played a relatively smaller role in driving change in marital trajectories between 1980 and 2010. Instead, factors such as later age at first marriage, educational expansion, declining mortality, narrowing sex differences in mortality, and more intense educational assortative mating have been the major drivers of changes in first-marriage trajectories since 1980.

## Introduction

A couple’s trajectory into and out of first marriage is one of the most important factors structuring adult life in the contemporary United States. The overwhelming majority of Americans marry (Hendi 2015a), and marital life course characteristics play a role in determining well-being and shaping decisions about work, retirement, housing, health, and savings and investment (Waite 1995). At the aggregate level, marital trajectories are of interest to governments and organizations such as insurance companies, whose provision of goods and services are often contingent on the contours of married life.

Over the twentieth century, marital trajectories changed tremendously, driven partly by the rise of divorce but also by other proximate factors shaped by profound changes in social structure. Educational assortative mating intensified since the 1950s (Schwartz and Mare 2005); at the same time, the population as a whole became more educated. Marriage has been pushed to older ages or sometimes replaced entirely as cohabitation has become more common and accepted (Bumpass and Lu 2000; Smock 2000). Sharpening barriers in educational intermarriage, rising divorce, declining mortality, growing inequality, educational expansion, and increasingly delayed marriage have all worked in combination to truncate, lengthen, and punctuate the marital life course since the early 1960s.

The present study examines how one part of the marital life course—specifically, first-marriage trajectories—changed between 1960 and 2010 in the United States, and which proximate determinants were responsible for that change. I consider four summary measures of first marriages: (1) the probability of ever marrying, (2) the expected duration of first marriage, (3) the probability that a first marriage will end in divorce, and (4) the probability that a first marriage will end in widowhood.1 I compute these measures for each year since 1960 and decompose the change over time in the latter three measures into seven proximate determinants: (1) secular declines in mortality, (2) narrowing sex differences in mortality, (3) increasing age at first marriage, (4) rising rates of divorce, (5) increased matching between spouses on age, (6) the intensification of educational assortative mating, and (7) the rightward shift in the education distribution. This article provides new estimates and also creates the scaffolding to examine important factors driving changes in marriage trajectories. The broad picture resulting from this study is that although divorce was the key factor explaining changes in first-marriage trajectories between 1960 and 1980, it played a more muted role thereafter. The main factors explaining change since 1980 are increasing age at first marriage, declining mortality, narrowing sex differences in mortality, more intense educational assortative mating, and educational expansion.

This study makes four main contributions. First, it provides new estimates of first-marriage trajectory measures for each year between 1960 and 2010. These estimates are more informative than those in prior studies because they pertain to the post-2000 period and incorporate heterogeneity in social characteristics and corresponding demographic rates, changes in assortative mating, and changes in the education distribution for men and women. Second, this study introduces a methodological innovation to model first marriages: multiple-decrement life tables with assortative mating. This formal demographic model provides a consistent framework to examine how the explanatory factors act simultaneously to affect the four outcome measures. This contribution allows us to incorporate heterogeneity across couples (i.e., different rates by education) and assortative mating. Multiple-decrement models used in previous studies did not incorporate heterogeneity and typically assumed that marriages consist of the average man marrying the average woman. Third, because of its multiplicative form, the model allows us to decompose the first-marriage measures into contributions from different proximate determinants to explain precisely why they have changed over time. The fourth contribution is to use this demographic model to mathematically prove a new theoretical result: under relatively weak conditions, more intense educational assortative mating leads to a longer expected duration of first marriage. This formal demographic framework carves out space for future studies to make other theoretical predictions or statements about the interlinkages between social processes (such as divorce or assortative mating) and aggregate phenomena (such as income inequality).

## The Marital Life Course

The life courses of married couples are of interest to demographers because of the changing role of marriage as a central institution in American life. Two major questions are of interest. First, “Do Americans continue to marry?” And second, “What happens after a marriage has been contracted?” The four measures I consider are vital to answering these questions given that these specific measures are universal—that is, every individual must either marry or not, and every marriage will end in either divorce or death—and are the cornerstones of the marital life course. These measures are also important because they are correlates of well-being and determinants of behavior.

If longevity is the foremost measure of well-being for an individual, then for married individuals, having a living spouse is at least a close second. Having a living spouse is associated with better outcomes in many arenas of life, including financial status, health, and happiness (Waite 1995). Married men and women maximize their well-being by having more years to spend together. By contrast, divorce is often related to negative outcomes, including loss of income, and may present greater challenges to child-rearing (Waite 1995). The death of a spouse is often cited as the most stressful life event that an individual can experience (Holmes and Rahe 1967) and is associated with depression, financial loss, a long-term decline in happiness (Lucas et al. 2003), and even death of the surviving spouse (Elwert and Christakis 2008a, 2008b).

Marital trajectories also have welfare implications. Social security and private pensions are intimately linked with spousal life courses and often include payouts predicated on the circumstances surrounding the death of a spouse or marital dissolution. For example, the Social Security Administration authorizes survivorship benefits for bereaved spouses only if they meet a set of requirements, including surviving to a particular age and having been married for a minimum duration. Ceteris paribus, people who had longer marriages, did not divorce, and lost their spouses at older ages are better able to capture such benefits.

## Sources of Change in First Marriage

Despite much research on changing marital trajectories, little consensus exists regarding which distal phenomena are driving these changes. Typically, distal theories of marital change have foci falling into three broad categories: normative change, shifting economic circumstances, and technological and institutional change.

Several theories hypothesize that normative change is the main driver of changing marital trajectories. For example, the second demographic transition theory posits that changing value orientations ushered in an era of greater individualism, and this cultural shift was accompanied by increased premarital cohabitation, delayed or forgone marriage, and increased divorce (Lesthaeghe 2010; Van De Kaa 1987). Related frameworks suggest that feminist values led to changes in ideas about the necessity and favorability of marriage for young women and to greater acceptance of divorce as a mechanism to escape bad marriages.

Other theoretical perspectives emphasize changing economic circumstances as the main distal determinant of marital change. This literature has highlighted the roles of increased female labor force participation, greater educational requirements for competing on the job market, worsening economic conditions for low-status young men, greater inequality, and increased economic uncertainty in driving declines in marriage and increases in divorce (Goldin and Katz 2002; Preston and Richards 1975). Related theories have focused on the role of increasing cohort sizes and declining relative income across generations (Easterlin 1987). In this formulation, delayed marriage results from the gap between aspirations for standards of living formed in childhood and actual earnings in adulthood.

The third type of explanation suggests that marital change has been driven by shifting technological and institutional variables impinging on marriage. Legal and legislative institutions, such as no-fault and unilateral divorce laws, the diffusion of safe and effective birth control, and increased abortion, are hypothesized to delay marriage and increase divorce (Stevenson and Wolfers 2007, 2011). It seems likely that increased incarceration, particularly among young black men, led to delays and declines in marriage and perhaps even increased divorce. The rise of tertiary education likely led to delayed marriage and, because of accompanying increases in the intensity of marital search, may have also prevented quicker increases in divorce among the more-educated (Oppenheimer 1988). The impact of very recent technologies such as online dating is still unclear.

All these distal theories of changing marital trajectories likely hold some truth, and they clearly operate on different levels and are probably causally linked. The goal of this analysis is not to sort out how these distal mechanisms contributed to marital change. Rather, the goal is to decompose the contribution of proximate factors to changes in the length of first marriage and the mode of marital dissolution. These proximate determinants are the intermediaries between the aforementioned distal mechanisms and the outcomes. The seven proximate sources of change examined in this study are changes in (1) assortative mating on age, (2) assortative mating on education, (3) age at first marriage, (4) duration-specific incidence of divorce, (5) the education distribution, (6) overall mortality, and (7) sex differences in mortality.

### Assortative Mating on Age and Education

Assortative mating configures the makeup of families by leading like individuals to marry. Couples tend to be similar in terms of education and age (Mare 1991; Qian and Preston 1993). One major social transformation of the last century was the dramatic change in the demographic and educational composition of first marriages. Increasingly, men and women graduated from high school and attended college, and the gender gap in educational attainment narrowed considerably. The rise of secondary and tertiary education, accompanied by more egalitarian gender norms and increased social distance between socioeconomic groups, led spouses to become more similar on characteristics such as educational attainment and age (Mare 1991; Schwartz 2013; Schwartz and Mare 2005).

We expect assortative mating to affect outcomes patterned by age and education, of which the prime example is mortality. The difference in American life expectancy between the least and most educated is now more than 10 years (Hendi 2015b, 2017). The average spousal difference in mortality may not resemble the average sex difference in mortality because spouses look more like each other than a randomly selected male and female from the population. Thus, as the sorting process intensifies, the correlation in mortality between husbands and wives will increase, and between-couple differences in marital survivorship will increase.

An increase in assortative mating will increase the average duration of first marriage. The intuitive explanation is that stronger assortative mating leads to more highly correlated mortality between husbands and wives, and marital duration is typically increasing in this correlation. I provide a mathematical proof in the online appendix (section A) showing that under conditions that usually hold in real populations, stronger educational assortative mating leads to longer average durations of first marriage. There is no clear expectation about how more intense assortative mating affects the probability of widowhood. Because the average difference in mortality between men and women who are concordant on education exceeds the average difference in mortality between men and women who are discordant on education, more intense assortative mating may increase the probability of widowhood. The effect of stronger educational assortative mating on the probability that a first marriage will end in divorce is similarly ambiguous.

Assortative mating on age grew stronger over time albeit not to the same degree as educational assortative mating. Still, we might predict that stronger sorting on age would lead, on average, to longer marital durations and lower probabilities of widowhood.

### Age at First Marriage and Incidence of Divorce

Average age at first marriage for women increased by several years between 1960 and 2010, with much of the change occurring post-1980. These shifting age patterns are partly due to changes in beliefs about marriage, including beliefs about its necessity and its economic and emotional prerequisites (Edin and Kefalas 2005; Manning et al. 2007). As social norms and beliefs about marriage evolved, the prevalence of cohabitation rose (Axinn and Thornton 1993; Smock 2000). Cohabitation is now the modal type of first union (Bumpass and Lu 2000; Kennedy and Bumpass 2008), and evidence suggests that cohabitation both delays and supplants marriage (Sassler 2004; Seltzer 2000). As cohabitation delays marriage to older ages, the mean length of marriages will tend to decline. Economic changes may also have increased age at marriage, as women increasingly attended college and entered the formal labor force, and lower-status men’s economic position deteriorated (Oppenheimer 2000; Preston and Richards 1975).

Changes in marriage patterns were accompanied by widening educational differences in first-marriage rates. In 2010, women with less than a high school diploma had less than a 75 % chance of ever marrying, whereas women with a college degree had a 90 % chance of ever marrying. In the 1960s, there was hardly any educational gradient in first marriage (Hendi 2015a).

Divorce followed similar trends. In the mid-1960s and 1970s, divorce rates reached record highs; and by 2010, roughly one-half of first marriages were expected to end in divorce. These changes are patterned by education, although not as monotonically as for marriage. Women who are high school graduates or have some college are much more likely to divorce than college graduates (Hendi 2015a; Martin 2006). Once again, the rise in cohabitation may have driven some of the change (or stabilization) in the incidence of divorce (Bennett et al. 1988; Manning and Cohen 2012; Reinhold 2010). Marital duration and age at first marriage are inversely related to the likelihood that a first marriage will end in divorce (Kennedy and Ruggles 2014; Raley and Bumpass 2003).

The aforementioned changes in marriage and divorce may lead to more truncated first marriages. Later age at first marriage and higher rates of divorce lead to shorter average durations of first marriage. Because later marriage decreases the amount of exposure time for divorce, delayed marriage is predicted to decrease the probability a first marriage will end in divorce. Its effect on widowhood probability is less clear, but because it decreases the amount of time during which a couple can divorce, it may lead to higher chances of a first marriage ending in widowhood. Rising divorce rates obviously increase the likelihood of a first marriage ending in divorce, and because of the competing risk effect, decrease the chance of a first marriage ending in widowhood.

### Educational Expansion

Since 1960, the population has grown more educated. Whereas the modal type of marriage in 1960 was one in which both husband and wife had less than a high school education, the modal type of first marriage in 2010 was one in which both spouses had a college degree or more (see Fig. 1).

The effect of educational expansion on first marriages is less clear. The more-educated tend to divorce less, marry more, and die later. However, increased education likely delays marriage, leading to fewer years spent together. Also unclear is how educational expansion affects the probability that a wife will outlive her husband. Because more-educated people are less likely to divorce (Martin 2006), the shifting education distribution may lead to a lower divorce probability.

### Secular Declines and Narrowing Sex Differences in Mortality

The final factor affecting first marriages is changing mortality. Mortality has declined rapidly since 1960. The distribution of mortality decline across time, age, and education, however, has been uneven. Since the late 1970s, mortality declined more rapidly for the college-educated than for the less-educated and more rapidly for men than for women (Hendi 2017; Kitagawa and Hauser 1973). Declines in mortality typically increase the average duration of marriages because, all else being equal, couples can spend more time together. When mortality decreases for both sexes, couples stay alive longer, leading to greater exposure time for a potential divorce and resulting in a slightly higher probability of divorce due solely to greater exposure to divorce. And because death is a competing risk for divorce, this increase in the divorce probability will lead to a decrease in the probability of a marriage ending in widowhood or widowerhood. Narrowing sex differences in life expectancy should increase the mean length of marriage and decrease the widowhood probability. Because declining mortality and narrowing sex differences in mortality both increase the exposure time for divorce, both should result in a higher probability of divorce.

## A Model of First Marriages With Assortative Mating

This study introduces the multiple-decrement life table with assortative mating, which incorporates educational heterogeneity in demographic rates and educational assortative mating into the analyses of first-marriage trajectories. There are two key advantages to recognizing multiple events and incorporating heterogeneity and assortative mating. First, such a model more closely approximates reality. Not every marriage involves the average man and the average woman, and incorporating heterogeneity allows the researcher to acknowledge this fact empirically. Second, recognizing multiple events provides a richer picture of marriage dynamics. Marriage and divorce rates are not the only factors influencing the marital life course, so understanding how the incidence of mortality, marriage, and divorce and who-marries-whom interact aids an understanding of how and why first-marriage trajectories change over time.

### Mortality as the Sole Source of Marital Dissolution

To introduce the model, I present a simplified version that assumes everyone marries and incorporates mortality as the sole source of marital disruption. In the next subsection, I build in marriage and divorce, leading to the full model used in all analyses.

In the absence of divorce, the dimensions of first marriage are determined entirely by the timing of marriage and mortality. The age at which each spouse marries and dies determines the duration of the marriage, whether the wife outlives the husband, and the duration of widowhood. In prior studies, spousal life cycles have been modeled as a function of age alone (Goldman and Lord 1983). We can extend these life cycle models to account for other characteristics and incorporate assortative mating.

Consider a couple who marry when the wife is aged x and the husband is aged y. They have education levels {cf, cm}, where f and m denote the wife and husband, respectively. They are subject to a particular life table from marriage until death: the wife’s survivorship function is lf(a|cf), and the husband’s is lm(a|cm), where a is age. Their respective forces of mortality are μf(a|cf) and μm(a|cm). Assuming conditional independence between spouses’ mortality risks, the probability that a wife will become a widow at exact time t years after marriage is
$lfx+tcflfxcflmy+tcmlmycmμmy+tcm.$
The first two terms are the probabilities that the wife and husband will survive from marriage to t years after marriage, and the third term is the instantaneous probability that the husband will die exactly t years after marriage. This equation is the instantaneous probability that the wife will become a widow exactly t years after marriage. The probability that the marriage will end in widowhood is
$∫0∞lfx+tcflfxcflmy+tcmlmycmμmy+tcmdt.$
1
But not all husbands and wives get married at exact ages y and x, and couples also differ on educational attainment. The probability distribution of the characteristics of couples in the population is H(x, y, cf, cm). H gives the likelihood of observing a wife and a husband who got married at exact ages x and y and who have vectors of characteristics cf and cm. In other words, H is the function summarizing assortative mating and estimated using the observed characteristics of spouses. I compute the probability that a wife outlives her husband by taking a weighted average of the above integral equation (Eq. (1)), where the weights are supplied by H:
$PrWife Outlives Husband=∫Ωx,y,cf,cm∫0∞lfx+tcflfxcflmy+tcmlmycmμmy+tcmdtdHxycfcm,$
where $Ωx,y,cf,cm$ is the support for H.
An equivalent expression for the expected duration of marriage (years spent together) is
$EYears Spent Together=∫Ωx,y,cf,cm∫0∞lfx+tcflfxcflmy+tcmlmycmdtdHxycfcm.$

### Incorporating First Marriage and Divorce

I now incorporate rates of first marriage and divorce into the model. First, the density h corresponding to the H function—the joint distribution of husbands’ and wives’ ages at marriage and educational attainments—can be decomposed into four components:
$hτxycfcm=hτyxcmcf×hτxcmcf×hτcfcm×hτcm,$
where the first component is the density of the husband’s age at marriage given the wife’s age at marriage and both spouses’ education levels; the second is the density of the wife’s age at marriage given both spouses’ education levels; the third is the educational assortative mating function; and the fourth is the probability mass function for the husband’s education. Transition data on new marriages can be used to recover fairly accurate age and education-specific marriage rates to construct this decomposition.
In addition to the density function for spouses’ education levels and ages at marriage, the first-marriage measures consist of marriage, divorce, and mortality rates. The probability that a marriage survives to duration t is the probability that neither spouse dies and that the marriage does not end in divorce by duration t. The former component is as before:
$lfx+tcflfxcflmy+tcmlmycm,$
which can be rewritten as
$lfx+tcflfxcflmy+tcmlmycm=lmy+tcmlmycmlfx+tcflfxcf×lfy+tcflfxcf2=δxytcfcm×lfx+tcflfxcf2,$
so δ is a function of only the difference in mortality between males and females, and the second term is the probability that a marriage does not end in death as of duration t if both spouses had the same life tables. The survivorship function for divorce, pD(t|cf), is the associated single-decrement probability that a married woman with education cf has not divorced by marital duration t.
The expected duration of first marriage is
$∫Ωx,y,cf,cm∫0∞δxytcfcmlfx+tcflfxcf2×hτyxcmcf×hτxcmcf×hτcfcm×hτcm×pDtdtdxdydcfdcm$
2
and is interpreted as follows: if a cohort of 15-year-old boys and girls were subject to period mortality, first marriage, and divorce rates and followed the period distribution of education and assortative mating, they could expect to spend the estimated number of years in their first marriage.2 Because the marital duration and other measures use period rates, they summarize the average first marriage for a synthetic cohort of 15-year-olds. The initial composition of these life table cohorts is all 15-year-old girls differentiated by (eventual) educational attainment. Causes of decrement include marriage and mortality; for those who marry, additional causes are divorce, widowhood, and widowerhood. As can be seen in Eq. (2), these sources of decrement are conditioned on age and education. The remaining measures are constructed similarly.

Because the expected duration of marriage integral is a sum-product of seven factors, changes over time in this measure can be decomposed into the effect of changes in each factor (Das Gupta 1993). This decomposition clarifies how much of the change over time in, for example, duration of first marriage is due to changes in the seven aforementioned factors. The sex difference in mortality factor is included in the decomposition only for duration of first marriage and probability of divorce because the functional form for the probability of widowhood does not include an easily interpreted sex difference in the mortality factor. For this measure, a single mortality component capturing both changing sex differences and secular declines in mortality is used.

The multiple-decrement life table with assortative mating differs from prior life table analyses in several ways. The most prominent method for summarizing the marital life course, the multistate life table, focuses on transitions into and out of marriage (Schoen 1988; Schoen and Nelson 1974). This model can be used to track the subsequent lives of those who have experienced a marital transition and allows for reentry (e.g., remarriage). In contrast, multiple-decrement life tables track lives after marital transition but do not allow for remarriage (Preston et al. 2001). In using both of these models, researchers have tended to assume fixed characteristics of spouses (e.g., a two-year difference in spouses’ age at marriage), have not allowed for heterogeneity in transition rates (i.e., education-specific rates), and have typically assumed that the average man will marry the average woman (as opposed to assortative mating). The model presented in this study—the multiple-decrement life table with assortative mating—improves on the standard multiple-decrement model by incorporating heterogeneity in rates and by allowing assortative mating. This builds on the work of Goldman and Lord (1983), who examined how age patterns of marriage influence the incidence of widowhood. The main weakness of the present model is that it does not allow for remarriage and thus is informative only for first marriages.

The multiple-decrement life table with assortative mating has three key advantages. First, it incorporates assortative mating into a model of first marriages. Prior models assumed homogeneity across couples or accounted for changing ages at marriage (Goldman and Lord 1983; Schoen 1988; Schoen and Standish 2001). Second, it is more parsimonious than a full multistate model, which requires estimating a prohibitively large number of transitions to incorporate assortative mating. The new model uses data that are more commonly available and thus can capture trends over longer periods. Third, its mathematical tractability allows researchers to make theoretical predictions about the effects of assortative mating on aggregate outcomes, such as wealth accumulation and income inequality.

## Data and Estimation

I use data from nine sources: (1) life tables from the Human Mortality Database (HMD) for 1960–2010; (2) 1960 U.S. Census; (3) the March Current Population Survey (CPS) for 1962 and 1964–2001; (4) the June CPS Fertility and Marital History Supplements for 1971, 1976, 1980, and 1990 for marriage and the 1990 June CPS for divorce data; (5) vital statistics on marriage for 1960–1990; (6) the American Community Survey (ACS) for 2002–2013, including marriage and divorce transitions for 2008–2013; (7) the National Health Interview Survey (NHIS) for 1986–2009 linked to the National Death Index through 2011; (8) the National Longitudinal Mortality Study (NLMS) centered on 1983 with 11 years of mortality follow-up; and (9) 1960 education-specific mortality ratios from the Matched Records Study (Kitagawa and Hauser 1973). The 1960 census, March CPS, and NHIS data are from IPUMS and IHIS (Minnesota Population Center 2012; Ruggles et al. 2015). The estimation procedure requires data on education-specific rates of first marriage, divorce, and mortality and data on the joint distribution of spouses’ age and education.

Throughout this analysis, I focus on opposite-sex couples in which both spouses are aged 15 or older, incorporating age-specific marriage rates and duration-specific divorce rates by female education for all first marriages. Because data on divorced individuals do not permit identification of marriage number for both partners, marriage number is based on the wife’s data. Rate estimates do not condition on male education, which will typically understate the effects of educational assortative mating and thus understate marital durations in the life table. I include people aged 15–24 because in the 1960s, many people still married before age 20, and most women were married by age 25 (Hendi 2015a). I concentrate on first marriages because the likelihood that an individual will ever marry is the most salient dimension of marriage and is a leading indicator of marriage trends. Furthermore, in the 2009 ACS, roughly 75 % of ever-married people were married only once. Evidence on the accuracy of ACS marriage/divorce data is mixed, particularly at older ages (Elliott et al. 2010; O’Connell et al. 2007), although Elliott et al. (2010:16) noted that the data “. . . are valid and quite comparable to vital records tabulations at the state and national levels.”

Further details on the data and measures are provided in the online appendix (section B).

The decompositions of changes in the measures into the aforementioned seven components are carried out using mortality, first marriage, divorce, and joint density data centered on single-year age groups. The decomposition for duration of marriage, for example, has seven factors: (1) δ(x, y, t, cf, cm) is the sex difference in mortality; (2) (lf(x + t|cf) / lf(x|cf))2 represents secular trends in mortality; (3) hτ(y|x, cf, cm) is assortative mating on age; (4) hτ(x|cf, cm) is age at first marriage; (5) hτ(cf|cm) is educational assortative mating; (6) hτ(cm) is the education distribution; and (7) pD(t) is divorce. Following the notation and equations in Das Gupta (1993:15–16, eqs. 2.45–2.50) for a seven-factor decomposition, each of these seven factors can be denoted by A1, . . . ,A7 in Period 1 and a1, . . . ,a7 in Period 2, so that the change between the two periods in the mean duration of first marriage is:
$∫Ωx,y,cf,cm∫0∞a1a2...a7−A1A2...A7dtdxdydcfdcm.$
The contribution of factor i to the change between two periods is
$∫Ωx,y,cf,cm∫0∞Qiai−Aidtdxdydcfdcm,$
where ai and Ai are, respectively, the last period and first period values of factor i; and Qi is a function of the remaining factors in the two periods (Das Gupta 1993: eq. 2.50). The decomposition distributes interaction effects equally into direct effects. The most important interaction effect is likely the one between first marriage and divorce incidence, given that declining marriage could lead to declines (or increases) in divorce. The decomposition would attribute this interaction equally to both marriage and divorce factors even if more of the underlying change is attributable to marriage.

## Results

As shown earlier, Fig. 1 plots the joint distribution of husband’s and wife’s education in 1960 and 2010. The size of each block represents the relative size of each type of marriage, with the wife’s education on the y-axis and the husband’s education on the x-axis. In 1960, the modal couple consisted of a marriage where both spouses had less than a high school diploma. By 2010, the modal couple consisted of a marriage where both spouses had at least a college degree.

Mortality declined dramatically between 1960 and 2010, and it did so faster for some education groups than for others. Figure 2 plots mortality ratios for 1960–2010 by age and education group for men. (The equivalent figure for women looks highly similar.) The light circles represent earlier years, and the dark circles represent later years. Mortality ratios tend to be greatest in magnitude at the younger adult ages and asymptote toward 1 as age approaches 90. This is consistent with prior literature documenting a flattening of educational gradients in mortality with age in period data (Elo and Preston 1996; Hendi 2015b).

The mortality ratios of education groups diverged. They increased for those with less than a high school education and decreased for those with college or more. In the earlier years, male high school graduates had a fairly flat age profile in mortality ratios (i.e., mortality ratios were fairly close to 1 at all ages). In more recent years, their mortality ratios have climbed to much higher levels, well above 1. As the education distribution shifted rightward and the size of the high school graduate group shrank relative to the college graduate group, the mortality ratios for high school graduates increased at the prime adult ages. Trends in the mortality ratios for those with some college are less clear but have also increased at most ages over time.

The estimates plotted in Figs. 3, 4, and 5 are reported for each year in Table 1. Figure 3 plots the probability of ever marrying by education for 1960–2012. The estimates are interpolated for 1988–2007. The vital statistics estimates (indicated by diamonds) are situated in the center of the education-specific estimates, suggesting that these estimates of education-specific first-marriage probabilities are reasonable. In the 1960s, more than 88 % of single women could expect to eventually get married (based on period rates). College-educated women had the lowest likelihood of marriage, at roughly 83 % in 1960. With the rise of tertiary education among women, first-marriage rates declined through the 1960s and 1970s. By 2010, just over 80 % of single women could expect to get married. Furthermore, a clear educational gradient emerged. Whereas in 1960, college-educated women were the least likely to get married, by 2010, they were the most likely to eventually marry: 87 % of college-educated women could expect to get married in 2010 compared with 73 % to 79 % for women in the remaining three education groups. Women with less than a high school education were the least likely to marry.

Figure 4 plots the period probability that a first marriage will end in divorce by education. Divorce was relatively uncommon in the early 1960s. In the mid-1960s, approximately one-quarter of women on their first marriages could expect to divorce in all education groups. By 1970, that number was closer to 35 %; and by the 1980s, the divorce probability for first marriages was more than 48 % for three of the four education groups. For most of the period under consideration, college-educated women were the least likely to experience divorce from a first marriage. Divorce probabilities plateaued in the early 1980s, declined slightly in the late 1980s, and have stayed fairly constant since for those with less than high school or college or more and increased for the middle two education groups, leading to an educational divergence. Approximately 40 % of college-educated women can expect their first marriage to end in divorce as of 2010, compared with 53 % and 55 % for women with a high school diploma or some college, respectively. Women with less than a high school diploma have a divorce probability of 41 %.

The finding that the divorce probability is lower for those with less than a high school education than for high school graduates is somewhat surprising, so the estimate was subjected to further scrutiny. The most likely explanations for this finding are errors in the data, a misspecification of the mortality joint distribution, differential racial/ethnic makeup of the two groups, or changing selection into marriage. Education-specific divorce probabilities were recomputed for 2010 using the 2008–2012 ACS data and the variable-r procedures described in Preston (1987), producing results that replicate the pattern of lower probabilities for the less-educated group. This indicates that the mortality misspecification explanation is unlikely. The group with less than a high school education changed over time, consisting increasingly of foreign-born Hispanics who tend to have lower incidence of divorce. Table 2 shows associated single-decrement divorce probability estimates by education and Hispanic ethnicity. Although the greater representation of Hispanics among those with less than a high school education is clearly driving down the divorce probability, the probability is still lower in this group for non-Hispanics than for non-Hispanic high school graduates. These results suggest that the greater representation of foreign-born Hispanics combined with increased selection into marriage among those with less than a high school education are likely driving the divorce probability to be lower among the least-educated group.

Figure 5 plots the expected duration of first marriage and the probability that a first marriage will end in widowhood and divorce. In 1960, a 15-year-old could expect to spend 34.5 years in a first marriage. By 2010, that number had declined to 25.6 years—an almost nine-year decline. The decline was especially rapid between 1960 and 1980. Since 1980, the expected duration of first marriage varied less. This coincides with the less-rapid increases in divorce following 1980 and the plateauing of divorce rates for those with less than high school or college or more.

In 1960, the probability of first marriage ending in widowhood was 57 %; by 2010, that probability was 34 %. Similar to the trends in marital duration, the probability of a first marriage ending in widowhood decreased dramatically from 1960 through the late 1970s and then declined more gradually from 1980 through 2010.

In 1960, the probability of a first marriage ending in divorce was roughly 20 %; by 2010, it had climbed to 48 %. Most of the increase occurred between 1960 and 1980 (20 % to 49 %). Beginning around 1980, the divorce probability dipped and then increased more gradually.

Figures 6, 7 and 8 plot decompositions of changes over time in the first-marriage measures (tabulated in section C of the online appendix). Panel a of each figure shows decompositions in the change between 1960 and year t; panel b of each figure shows decompositions in the change between 1980 and year t. The decompositions for duration of first marriage and probability of a first marriage ending in divorce include seven factors: (1) changes over time in sex differences in mortality (SDM), (2) overall mortality (Mortality), (3) age at first marriage (Marriage), (4) assortative mating on age (AMA), (5) rates of divorce (Divorce), (6) assortative mating on education (AME), and (7) the education distribution (Education). The decompositions for the probability of widowhood do not include a separate sex differences in mortality component for reasons described earlier. For any given year, the bars in each decomposition sum to the change over time between 1960 (panel a) or 1980 (panel b) and year t in the relevant outcome.

Panel a of Fig. 6 graphs a decomposition of the change between 1960 and year t in the expected duration of first marriage between 1960 and 2010. Over this period, the expected duration of first marriage decreased by 8.9 years. Most of this decline was due to increases in rates of divorce and increasing age at first marriage. Declines in mortality, narrowing sex differences in mortality, the intensification of educational assortative mating, and the rightward shift of the education distribution offset further declines in the duration of first marriage. Panel b of Fig. 6 shows that since 1980, most of the change was due to factors other than divorce. Changes in divorce produced a 0.81-year increase in marital duration; narrowing sex differences in mortality, secular declines in mortality, stronger educational assortative mating, and educational expansion increased marital duration by 0.81, 1.58, 0.49, and 0.88 years, respectively. Increasing age at marriage and changes in assortative mating on age offset these increases by producing declines of 4.2 and 0.23 years, respectively.

Figure 7 plots decompositions of the change between 1960 or 1980 and year t in the probability of a first marriage ending in widowhood. Most of the change in the probability of widowhood can be attributed to increasing rates of divorce. Between 1960 and the mid-1980s, changes in mortality increased the probability of a first marriage ending in widowhood; between 1980 and 2010, changes in mortality led to a 4 % decrease in this probability. Assortative mating on age had the opposite pattern: changes in assortative mating on age reduced the probability of widowhood prior to the mid-1980s but prevented further declines in the probability of a first marriage ending in widowhood in more recent years. Changes in educational assortative mating and age at first marriage both led to higher probabilities of widowhood, whereas rightward shifts in the education distribution led to lower probabilities of a first marriage ending in widowhood. Again, the change between 1980 and 2010 was relatively unaffected by changes in divorce. Instead, declining mortality and educational expansion worked to decrease the widowhood probability, but delayed marriage and assortative mating on age and education partly offset this decrease.

Figure 8 shows decompositions of the change between 1960 or 1980 and year t in the probability of a first marriage ending in divorce. Unsurprisingly, this decomposition was dominated by the effect of changes in divorce rates. What is perhaps more surprising is that since 1980, changes in divorce had a relatively small impact on the divorce probability. Instead, rising age at first marriage and stronger assortative mating reduced the divorce probability; narrowing sex differences in mortality, declining overall mortality, and educational expansion partly offset this decrease.

## Discussion

This study provides a few surprising findings and several interesting findings. The first surprising finding is that although the probability that a first marriage will end in divorce increased monotonically between 1960 and 1980, it has not increased significantly between 1980 and 2010. This finding lies in contrast to other recent reports that the propensity for divorce has either increased dramatically or decreased. Period estimates indicate that the reality lies somewhere in between: the probability of a first marriage ending in divorce increased by approximately 1 %. We also find that changes in the incidence of divorce had a large impact on changes in the duration of first marriages between 1960 and 1980 but were far less important for explaining changes since 1980. This may be due to increasing cohabitation leading to more stable marriages, although the relationship between cohabitation and divorce has not been constant over time (Bennett et al. 1988; Manning and Cohen 2012; Reinhold 2010). The remaining factors—secular declines in mortality, narrowing sex differences in mortality, strengthening assortative mating on education, educational expansion, and delayed marriage—explain most of the change between 1980 and 2010.

Between 1980 and 2010, declining mortality and narrowing sex differences in mortality were responsible for a 2.4-year increase in average duration of first marriage, a 1 % increase in the probability of divorce, and a 4 % decrease in the probability of widowhood. This is likely related to declines in older age mortality given that mortality above age 50 was the primary source of mortality decline since the late 1970s. As Americans increasingly survive to older ages, events occurring at these ages will become increasingly important for understanding marital trajectories, suggesting that an important avenue for understanding changing marital life courses is the joint study of aging and processes related to marriage. This finding relates to recent work on “gray divorce” identifying recent increases in divorce at older ages (Brown and Lin 2012).

Between 1960 and 1980, sex differences in mortality had relatively little effect. This is partly related to increasing smoking-attributable mortality among men during the 1960s and 1970s (Preston et al. 2010), leading to little change in the sex difference in mortality over this period. However, since the 1980s, sex differences in mortality have narrowed dramatically because of declines in smoking among men and increases among women (Preston and Wang 2006) and declines in injury-related mortality among men (Centers for Disease Control and Prevention 2012; Ma et al. 2015). This has led to longer average marital durations, higher probabilities of divorce, and lower probabilities of widowhood. Furthermore, both causes of death are strongly patterned by education (Ho 2017; Ho and Fenelon 2015). More-educated people benefited more from the decline in smoking-related mortality than less-educated people, which means that the mean duration of first marriage for more-educated couples became relatively longer than that of less-educated couples. Because smoking-related mortality is increasingly concentrated among less-educated individuals, future declines in smoking-related mortality will have stronger effects on married people at the bottom of the distribution of socioeconomic status (Ho and Fenelon 2015).

Stronger assortative mating on education between 1960 and 2010 led to a 0.7-year increase in average marital duration, a 1.5 % increase in the probability of widowhood, and a 0.7 % decrease in the probability of divorce. Between 1980 and 2010, these effects were a 0.5-year increase, a 0.9 % increase, and a 0.2 % decrease, respectively. This is one of the first studies to show that assortative mating on education can have real effects. The previous literature has focused on identifying instances of assortative mating and describing changes in assortative mating. I find that the sharpening of socioeconomic boundaries in the choice of marital partners led not only to more differentiated marriage outcomes across education groups but also to increases in the mean duration of first marriage, decreases in the probability of divorce, and sharp increases in the probability of a first marriage ending in widowhood for the married population as a whole.

The mathematical proof in the online appendix (section A) shows that under relatively weak conditions, stronger assortative mating on education leads to longer expected durations of first marriage. This may have further implications for other social and economic processes, such as wealth accumulation and rising income inequality. These effects differ from those hypothesized in prior studies, which were concerned with intergenerational mobility as opposed to effects within one’s own lifetime.

Over the course of post-1960 educational expansion, the structuring effect of education played an increasingly important role in the marital life course. Educational expansion increased the average marital duration by 1.6 years since 1960 and 0.9 years since 1980. This is due partly to the fact that more-educated people are less likely to divorce and have lower mortality rates (which overcame the countervailing effect of more-educated people marrying later). I find not only that education-specific hazard ratios increased but also that the shift in the educational distribution itself had effects. Furthermore, the educational divergence in first marriage continues to grow, while the probability of divorce for the least-educated is now lower than for high school graduates.

Increasing age at marriage played the biggest role in explaining changes in first-marriage trajectories since 1980. People are marrying later, due partly to changing attitudes surrounding marriage and cohabitation (Axinn and Thornton 1993; Manning et al. 2007), and this is fundamentally altering marital trajectories. Since 1980, increases in age at marriage decreased the expected duration of first marriage by roughly 5.4 years. Similarly, later age at marriage led to a 1 % increase in the probability of widowhood and a 2.9 % decrease in the probability of divorce. These findings highlight the fact that focusing on rates alone can be misleading when trying to understand marriage processes. Although researchers have lamented the fact that divorce rates have increased since 1980, this increase has had a surprisingly muted effect on first-marriage trajectories, largely because increases in age at first marriage offset much of its impact.

The estimates presented in this analysis are based on a synthetic cohort and should be taken as summary measures of period conditions, not as forecasts for real birth or marriage cohorts. Synthetic cohort estimates are useful because documenting the marital life course for a real cohort would require waiting decades until that cohort dies out. The synthetic cohort estimates will correspond to actual cohorts if age- and duration-specific rates and assortative mating patterns stay fixed in the coming decades. If marriage rates continue to decline and divorce rates stay roughly constant, as has been the recent trend, then birth cohorts of 15-year-olds today can expect to have lower probabilities of marriage and divorce and shorter marital durations than the period measures reported here. Although it is difficult to predict future marriage outcomes, supplementary analyses (see Fig. B1 in the online appendix) suggest that according to a Hernes (1972) model forecast (Myrskyla and Goldstein 2013), 82.2 % of American women born in 1982 and 79.5 % born in 1987 can expect to have married by age 50.3 These forecasts are similar to the period estimates of ever marrying reported here.

One limitation of this study is that because of data constraints, it does not include cohabitation. Cohabitation transitions are difficult to measure and are available in only a few surveys, which do not cover the period or age range considered in this study. The exclusion of cohabitation has two types of effects: (1) selection into the study population, and (2) the effect of cohabitation on subsequent marital transitions, such as divorce. People who cohabit have different mortality, marriage, divorce, and education profiles from the population in general, and these differences have changed over time (Jose et al. 2010; Raley and Bumpass 2003). If cohabiting unions were counted as marriages, then rates of dissolution would be much higher, unions would be shorter, “divorce” probabilities would be higher, and divergence across education would be greater (Vespa 2014). To the extent that cohabitation has recently become a trial period for marriages, premarital cohabitation may lead to lower risks of divorce after marriage (Kulu and Boyle 2010; Manning and Cohen 2012; Reinhold 2010). The focus in this study is marriage, but future studies can extend these analyses to examine how changes in cohabitation influenced marital trajectories.

## Conclusion

This study traced out the evolution of American first-marriage trajectories between 1960 and 2010. Although almost all the change in first-marriage trajectories between 1960 and 1980 was due to the rise in divorce incidence, most of the post-1980 change was due to other factors. Chief among these were secular mortality declines, narrowing sex differences in mortality, increased assortative mating on education, educational expansion, and increasing age at first marriage. The probability of ever marrying appears to be declining to 80 %, while the length of first marriage and the probabilities of divorce and widowhood seem to be stabilizing. However, this apparent stagnancy masks great change in the underlying mechanisms structuring American marriage. This change is revealed by the decompositions presented within this article. I find that first marriages are multidimensional. Simply knowing the marriage or divorce rate isn’t enough; this study provides a fuller understanding of changes in first-marriage trajectories over the past five decades.

Another contribution of this study is its development of a methodological innovation used to model first marriages. The multiple-decrement life table with assortative mating allows incorporating heterogeneity and assortative mating in measures of first-marriage trajectories. This demographic model has wide applicability and provides a scaffolding for future research to study marriage processes in both theoretical and empirical respects. The model allows researchers to make theoretical predictions about the impact of changes in assortative mating or divorce on, for example, life cycle measures, such as income inequality or the duration of marriage. Future studies can employ the methods introduced here to make theoretical predictions and to undertake analyses of how social and demographic factors governing marriage are affecting aggregate outcomes.

## Acknowledgments

This work was supported by the National Institute on Aging [T32 AG000177] and the National Institute of Child Health and Human Development [T32 HD007242]. Thanks to Susan Brown, Irma Elo, Elizabeth Frankenberg, Michel Guillot, Jessica Ho, I-Fen Lin, and Sam Preston for helpful comments.

## Notes

1

Probability of widowerhood is fully determined as 1 minus the probability that a marriage will end in widowhood or divorce. For parsimony, figures on widowerhood are not included in the main text but are available on request.

2

Not all these 15-year-olds will eventually get married because in general, the hτ(x|cf, cm) function does not integrate to 1. To compute quantities referring only to the ever-married population (e.g., probability that a marriage will end in divorce), I normalize this function to sum to 1 and thus provide estimates that are conditional on marriage, which are concerned only with the ever-married population.

3

Martin et al. (2014) and Ruggles (2016) reported 70 % to 79 % probabilities, although they did not consider marriages for people in their 40s.

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