## Abstract

The lack of nationally representative data with detailed marriage histories in low- and middle-income countries (LMICs) impedes a comprehensive understanding of essential aspects of union dissolution, such as the timing of first union dissolution, in these countries. We propose a method for estimating quantum-adjusted measures of the timing of first union dissolution from incomplete marriage histories. This method, *indirect life table of first union dissolution* (ILTUD), estimates the first union survival function from a simple tabulation of ever-married women by duration since first union, classified by union dissolution status (intact vs. dissolved first union). It then uses the relationships between life table functions to generate the distribution of marriages ending each year ($\theta t$) for a given marriage cohort. Using this distribution, ILTUD generates quantum-adjusted first union survival rates from which the percentiles of first union dissolution are calculated. ILTUD estimates are consistent with estimates produced using traditional statistical methods, such as the Kaplan–Meier estimator. In addition, ILTUD is simple to implement and has minimal data requirements, which are available in most nationally representative surveys. Thus, the ILTUD method has the potential to broaden our understanding of union dissolution dynamics in LMICs.

## Introduction

The timing of reproductive and demographic events during the life course is consequential for an individual's subsequent socioeconomic, demographic, and health outcomes. For example, a union dissolution is a precursor to the formation of complex family structures, such as single-parent households and stepfamilies, which are associated with adverse socioeconomic and health outcomes for individuals who experience this event and their affected children (Adjiwanou et al. 2021; Ziol-Guest and Dunifon 2014). The timing of union dissolution defines the ages at which individuals who experience this event and their affected children are exposed to these living arrangements. Moreover, the timing of union dissolution is likely crucial in understanding fertility variation between individuals who experience marital dissolution and those who remain in intact first unions. Indeed, studies suggest that women who experience a union dissolution end up with lower fertility than those who remain in an intact first union (John and Adjiwanou 2022; Meggiolaro and Ongaro 2010; Thomson et al. 2012). These differences are assumed to arise from loss of exposure to regular sexual intercourse and other factors (Davis and Blake 1956; Griffith et al. 1985; John and Adjiwanou 2022). Thus, information about the timing of union dissolution provides essential insights into when this loss of exposure likely occurs during the reproductive lifespan. Therefore, assessing how long individuals are partnered before experiencing a union dissolution is relevant for studying union dissolution dynamics and their consequences for socioeconomic, demographic, and health outcomes.

However, nationally representative information about the timing of union dissolution is unavailable in most low- and middle-income countries (LMICs). This omission is mainly due to the limited availability of detailed marriage histories in the available data sources in these countries. For example, the Demographic Health Surveys (DHS) and Multiple Indicator Cluster Surveys (MICS), the most widely used and reliable sources of nationally representative nuptiality information in these countries, include information on whether a first union ended but not when (the date or age at which) the event occurred. This data gap makes using traditional statistical techniques, such as the Kaplan–Meier estimator (Kaplan and Meier 1958), implausible for analyzing the time to first union dissolution. Thus, the growing interest in understanding union dissolution variation and change in these countries (Cherlin 2017; Clark and Brauner-Otto 2015) underscores the need to develop approaches that can produce reasonable measures of union dissolution timing from these data.

Therefore, we propose a technique for estimating quantum-adjusted first union survival rates. This method, which we call *indirect life table of first union dissolution* (ILTUD), calculates the first union survival rates under assumptions from incomplete marriage histories. It then uses the relationships between life table functions to calculate the distribution of marriages ending each year for a given marriage cohort.

ILTUD is a synthetic cohort measure. A long-standing concern about such measures is the estimate distortions when rates are changing over periods or cohorts. Thus, we also suggest an extension of the ILTUD method (ILTUDext) for use when multiple surveys conducted 5 or 10 years apart are available and evidence suggests a substantial change in union dissolution rates over marriage cohorts. We discuss and demonstrate the application and robustness of ILTUD using World Fertility Survey (WFS) data. We use Malawi 2010 and 2015 DHS data to demonstrate the application of ILTUDext.

## Traditional Methods for Analyzing the Timing of Demographic Events

Two methods stand out for analyzing the timing of life course events in demographic analysis. The first is the Kaplan–Meier method, a statistical technique for estimating the proportion of individuals surviving the event of interest over time (Kaplan and Meier 1958). The second is the life table, which is equivalent to the Kaplan–Meier method in the demographic technique's toolbox, for its use in estimating survival rates. The life table technique was first used by John Graunt in 1662 to analyze mortality patterns. Since then, researchers have used life tables to assess the timing of different social and demographic events, including fertility, migration, and family dynamics (Andersson and Philipov 2002; Ní Bhrolcháin 1987; Vega and Brazil 2015).

The life table involves estimating survival rates from the distribution of events and censored observations at predefined time points. Table 1 shows the typical data matrix required for this estimation. The first column, *t*, represents the time intervals. The column labeled *n _{t}* depicts the number of observations at the beginning of interval

*t*. $\omega t$ and $\theta t$ denote the number of censored observations and events during interval

*t*, respectively. $S(t)$ indicates the survival rate during interval

*t*.

In Table 1, the values of $nt$, $\omega t,$ and $\theta t$ are related as

The survival rate at interval *t*,$S(t),$ is estimated as

One of the key advantages of the Kaplan–Meier and life table techniques is their versatility in returning meaningful variability measures for skewed time-to-event data distributions. Indeed, for such distributions, the knowledge of the median and other percentiles (e.g., 25th and 75th percentiles) at which the population at risk has experienced the event provides essential measures of the timing of the demographic event that uniquely complement other measures of central tendency, such as the mean.

However, the Kaplan–Meier and life table techniques can be used only when information about the timing of the event in question is available. In the absence of such information, indirect demographic methods are required to produce equivalent measures of the central tendency or variability. In demographic analysis, the singulate mean age at first marriage (SMAM) and its variants are the main approaches widely used in such situations. Hajnal (1953) proposed SMAM to estimate the mean age at first marriage in the absence or unreliability of data on age at first marriage. This method involves estimating the mean age at first marriage among individuals who would eventually marry by a certain age from the distribution of the proportion evermarried according to age. van de Walle (1968) aimed to develop a measure of the mean age at first marriage that is robust to age misreporting and thus proposed an alternative estimator to SMAM based on a stable population structure. Trussell (1976) showed that both van de Walle's (1968) estimator and his own refined estimator of that measure closely resemble the SMAM (but not the median age at first marriage, given the inherent differences in the mean and the median in nonnormal distributions). Nevertheless, SMAM remains a standard approach for estimating the mean age at first marriage when information about first marriage age or date is unknown.

Consequently, SMAM has been adapted in other demographic analyses—for example, in estimating the mean age at first birth and mean birth intervals (Booth 2001; Feeney and Feng 1993). Indeed, SMAM can easily be adapted to estimate the singulate mean duration at first union dissolution. In this case, one only needs to calculate an equivalent of SMAM from the distribution of the proportion of women still in intact first unions according to the duration since the first union. However, the first union dissolution rate distribution is right-skewed (e.g., see Jalovaara and Kulu 2018). Thus, measures of variability of the timing of first union dissolution that account for this skewness provide valuable complementary information to estimates that can be obtained by adapting the SMAM approach. The ILTUD method seeks to produce such measures by constructing a quantum-adjusted life table of first union dissolution using incomplete marriage histories from which variability measures, such as the median and other percentiles, can be derived.

## Synthetic Cohort Measures in Demographic Analysis

The derivation of the life table using Eqs. (1) and (2) implies a cohort analysis of survival probabilities. This analysis means that a group of individuals simultaneously enter the risk population of a given event and are followed up until everyone in the group has experienced the event. In mortality analysis, for example, this analysis would involve following up on individuals born in the same year until everyone dies. A life table constructed using this procedure is called a cohort life table. Its construction would take many years to complete and would generally reflect past experiences, thereby having little relevance for policy and planning. Therefore, life tables are often constructed as period life tables based on cross-sectional observations of the experiences of different cohorts. In contrast to the cohort life table, the period life table does not reflect any actual cohort's experience. Instead, it reflects the experience of a synthetic cohort (Preston et al. 2001). Synthetic cohort estimates are widely used and form a fundamental base of demographic knowledge. For example, the total fertility rate (TFR) and SMAM are synthetic cohort measures.

Synthetic cohort measures can be classified into two groups on the basis of their data input. On the one hand, synthetic cohort estimates can be directly estimated from reports of events that occurred during a delineated short interval. For example, the direct estimation of the TFR based on reports of births observed in the survey year or three years before the survey date falls into this category (hereafter, *synthetic cohort Type I*). In this case, the TFR reflects a synthetic cohort's experience based on rates corresponding to a clearly defined period. On the other hand, synthetic cohort estimates corresponding to a specified period can be indirectly calculated from lifetime reports. For example, TFR for a given period can be calculated indirectly from reports of lifetime fertility (children ever born). The SMAM uses reports of lifetime marital status to estimate the timing of marriage for a given period and thus falls into this category of a synthetic cohort measure (hereafter, *synthetic cohort Type II*).

A long-standing issue with the use of synthetic cohort Type II measures is estimate distortions when rates are changing (see, e.g., Zlotnik and Hill 1981). This concern arises because younger cohorts' reports correspond to the period closer to the survey date, whereas older cohorts' reports correspond to a longer interval relative to the survey date. In summary, with synthetic cohort Type II measures, lifetime reports from different cohorts observed in a specified period correspond to different periods. Thus, they can represent the underlying rates (of interest) for the period closer to the survey date only if the rates in question have remained unchanged across cohorts.

Demographic analyses address this problem of synthetic cohort Type II estimates by estimating synthetic cohort measures on the basis of data collected at two periods, usually from surveys conducted 5 or 10 years apart. This approach ensures that changing rates are considered when the synthetic cohort measure is estimated. For example, in fertility analysis, the technique involves calculating intersurvey fertility increments from reports of lifetime fertility collected from surveys conducted 5 or 10 years apart. The method was documented first by Arretx (1975) and later by the United Nations Population Division (1983) and Moultrie et al. (2013). The idea is that if two surveys are conducted five years apart, individuals aged *x* at the first survey would be aged *x* + 5 at the second survey. Thus, the difference in lifetime fertility between women aged *x* + 5 at the second survey and those aged *x* at the first survey represents the fertility of these women during the intersurvey period. These fertility increments are then modeled to produce the TFR corresponding to the intersurvey period, yielding period estimates free of distortions due to changing rates. Agarwala (1962) applied this technique to estimate intersurvey SMAM estimates, and Zlotnik and Hill (1981) demonstrated its application with mortality data. In the following sections, we first describe and demonstrate the ILTUD method. We then present an extension of the ILTUD method (ILTUDext) based on this principle of intersurvey synthetic cohort estimates.

## Method Description

### Estimation of Survival Rates of First Union Dissolution (*S*(*t*)) From Incomplete Marriage Histories

For cross-sectional data with complete marriage histories, the survival rate of first union can be derived using traditional statistical methods. The survival rate of first union at time *t*—*S*(*t*)—calculated from such methods refers to the probability of the first union surviving *t* years following its onset. Therefore, if first union survival rates have remained reasonably stable across different marriage cohorts, *S*(*t*) can be equally approximated by calculating the proportion of women in intact first unions among women who first married *t* years before the survey (ρ_{t}). Thus, it implies that we can estimate *S*(*t*) from a tabulation of ever-married women by duration since first union, classified by first union dissolution status (intact vs. dissolved first union). Such a tabulation does not require knowledge of when a first union ended and can, therefore, be produced from cross-sectional data with incomplete marriage histories.

We suggest using ρ_{t} as a measure of first union survival rates, *S*(*t*), for data with incomplete marriage histories. However, first union survival rates may vary across marriage cohorts, in which case the $\rho t's$ values cannot equal the underlying first union survival rates in a given period. Thus, the *S*(*t*) estimates we propose to derive using $\rho t's$ generally relate to synthetic cohort Type II estimates.

Plotting the values of $\rho t's$ against *t* should, therefore, yield a monotonically decreasing function (because ρ_{t} ≡ *S*(*t*)). However, $\rho t's$ estimates may not decrease monotonically because of limited sample size, data reporting errors, or random variation. We can address this issue by smoothing the distribution of $\rho t's$ to yield monotonically decreasing *S*(*t*) estimates. We suggest smoothing the ρ_{t} values by fitting a polynomial function. One could also consider smoothing ρ_{t} estimates using parametric models, such as the log-normal function.

### Estimating Quantum-Adjusted First Union Survival Rates

The ultimate goal of ILTUD is to use incomplete marriage histories to measure how long, on average, individuals are partnered before experiencing a first union dissolution. Because not every married woman experiences a union dissolution and the levels of union dissolution vary substantially over space and time, the first union dissolution timing measures that are fully adjusted for the level (quantum) of union dissolution are more meaningful. ILTUD uses *S*(*t*) estimates and the relationships between life table functions to produce such estimates.

For the estimation of first union survival rates, *t* in the classic life table data matrix (Table 1) represents the time since the onset of the first union. *n _{t}* is equivalent to the number of ever-married women at the beginning of interval

*t*. $\omega t$ denotes the number of individuals in intact first unions, and $\theta t$ corresponds to the number of marriages ending during interval

*t*$.$ Thus, the

*S*(

*t*) in Eq. (2) estimates the proportion of first unions that survive marriage interval

*t*. Essentially, it is the distribution of θ

_{t}that we need to understand how long, on average, individuals are partnered before experiencing a first union dissolution. We can compute the values of$\theta t's$by rearranging Eq. (2) as

Equation (3) reduces to

Hence, solving for $\theta t$ in Eq. (4), we have

Because $\omega t$ denotes the number of individuals in intact first unions during interval *t*, we can directly extract the values of $\omega t's$ from a tabulation of ever-married women by duration since first union, classified by union dissolution status. We can also directly observe the value of $n1=[0,1)$ because it indicates the number of ever-married individuals in the data set. Thus, Eqs. (1) and (5) suggest that if *S*(*t*) estimates are available, one can obtain the values of $\theta t's$ and $nt's(t>1)$ iteratively. ILTUD uses the smoothed $\rho t's$ described earlier to produce the distribution of $\theta t's$.

Once the distribution of marriages ending each year for a given marriage cohort ($\theta t's)$is produced using Eq. (5), the quantum-adjusted survival rate of the first union at duration *t* (*S**(*t*)) can be estimated using an equivalent of Eq. (2) with $\omega t's=0$—that is, *S**(*t*) is estimated as

where

With linear interpolation, *S**(*t*) estimates can be used to calculate summary measures (e.g., 25th, 50th, and 75th percentiles) of the timing of first union dissolution that are fully adjusted for the quantum of union dissolution.

## Method Application

We now illustrate the application and robustness of ILTUD using data from World Fertility Surveys collected in 12 countries in Africa (see Table 2). Because these data sets contain detailed marriage histories, they allow us to derive first union survival rates using traditional statistical methods, which we use as a benchmark to evaluate ILTUD estimates. The purpose of this illustration is twofold. First, we intend to demonstrate the step-by-step application of ILTUD using data from Egypt. Second, applying the method to each of the 12 countries, we underscore the idea that estimates derived using ILTUD relate to a synthetic cohort and are robust.

The Egypt data set contains a sample of 8,788 ever-married women, of whom 1,438 (16.4%) had a dissolved first union. The first step of ILTUD is to tabulate ever-married women by the number of years since the first union, according to first union dissolution status (intact vs. dissolved first union). We use the output of this tabulation to calculate the proportion of women in intact first unions ($\rho t's$) at each single-year interval of union duration. For example, 372 women first married [2, 3) years before the survey, of whom 354 were still in a first union. Thus,

The second step involves smoothing the $\rho t's$ values to produce monotonically decreasing first union survival rates. This analysis smoothed the ρ_{t} values corresponding to marriage duration of 0–30 years using a third-degree polynomial function. Figure 1 shows the observed and smoothed estimates of ρ_{t} values alongside *S*(*t*) estimates derived using the Kaplan–Meier method. The smoothed ρ_{t} estimates closely mirror those derived using the Kaplan–Meier approach.

The third step involves using the smoothed $\rho t's$ estimates and the distribution of women in intact first unions to calculate the number of marriages ending each year ($\theta t's)$ for a given marriage cohort. We first use Eq. (5) to produce $\theta t's$ and then use Eq. (1) to calculate the number of ever-married women at the beginning of each interval $(nt's)$. For example, we produce the number of unions expected to end during the first year of union (θ_{1}) by setting *n*_{1} = 8,788. The number of women in intact unions during this interval is 375. The fitted *S*(1) is 0.9712. Thus, using Eq. (5) yields 248 as the number of first unions that dissolved during the first year. Subtracting 375 and 248 from 8,788 gives the number of ever-married women at the beginning of the second year of union (*n*_{2} = 8,165). We then use Eq. (5) again to produce $\theta 2$, Eq. (1) to produce *n*_{3}, and so forth.

Finally, we use Eqs. (6) and (7) to estimate the first union survival rates among women whose first union ended. Figure 2 shows these estimates based on the ILTUD and Kaplan–Meier methods. ILTUD estimates are closely similar to those derived using the Kaplan–Meier method. Using ILTUD, we find that half the women whose first union ended experienced the event within 5.2 (95% CI = 4.4–6.3) years of the first union. The Kaplan–Meier method also yields an estimate of 5.2 (95% CI = 4.6–5.8) years.

Table 2 summarizes the results of the application of ILTUD to each of the 12 countries considered. It shows the duration at which 25%, 50%, and 75% of first unions were dissolved among individuals whose first union ended, derived using the ILTUD and Kaplan–Meier methods. The estimates from these two methods are relatively similar for Benin, Cameroon, Egypt, Nigeria, Senegal, and Sudan. In these countries, the first union survival rates are comparable across marriage cohorts (Figure 3). Moreover, as would be expected (given that ILTUD is a synthetic cohort Type II measure), Table 2 and Figure 3 reveal that significant discrepancies between estimates derived using ILTUD and Kaplan–Meier methods exist in countries where first union survival rates for younger cohorts differ substantially from those for older marriage cohorts (e.g., Ghana and Morocco).

The results for Ghana and Morocco illustrate the long-standing concern about distortions in the synthetic cohort Type II measures when rates are changing. Therefore, we perform a sensitivity analysis based on assumed rates of change of first union dissolution to provide insights into levels of change in first union dissolution rates that are likely to be of less concern when using the ILTUD method (see the online appendix). Overall, the results suggest that if the proportion of first marriages ending within the first five years across marriage cohorts that are 5–9, 10–14, 15–19, 20–24, and 25–29 years apart has changed by −0.57% to 0.88%, −0.99% to 1.41%, −1.42% to 2.01%, −1.91% to 2.6%, and −2.43% to 2.99%, respectively, ILTUD is more likely to yield a reasonable estimate. The online appendix provides similar markers for longer interval durations.

Union dissolution rates have declined slowly or remained stable in some LMICs—particularly in sub-Saharan Africa (Clark and Brauner-Otto 2015). Thus, the ILTUD method remains relevant in such settings. However, these rates may have changed substantially elsewhere. Therefore, we propose an extension of the ILTUD method (ILTUDext) relevant to such conditions when multiple surveys conducted 5 or 10 years apart are available.

## ILTUD Extension: Using Multiple Surveys to Estimate the Quantum-Adjusted Timing of First Union Dissolution

The extension of our ILTUD method draws on the principles of indirect estimation of period fertility rates using lifetime fertility reports of women collected at two distinct periods—usually data from two censuses/surveys conducted 5 or 10 years apart (Arretx 1975; Moultrie et al. 2013; United Nations 1983). As highlighted earlier, these methods of measuring period fertility involve estimating intersurvey/census increments in lifetime fertility, which are used to measure the level of fertility attributable to the intersurvey period. We suggest following this logic when multiple surveys conducted 5 or 10 years apart are available to derive duration-specific first union survival rates corresponding to the intersurvey period. We then estimate summary measures of the quantum-adjusted timing of first union dissolution attributable to the intersurvey period, fully accounting for changing union dissolution rates. We use Malawi 2010 and 2015 DHS data to demonstrate the procedure.

The first step involves tabulating, for each survey, all ever-married individuals according to duration since the first union, classified by first union dissolution status (intact first union vs. dissolved first union). For surveys five years apart, the duration since the first union should be classified in five-year intervals (although one can also classify the data in single-year intervals if the sample size is large). Table 3 shows an example of this tabulation using the Malawi DHS data (see columns 1–5).

In the second step, we calculate the proportion of ever-married women in intact first unions for each five-year duration and survey period. Because of the data reporting problems we highlighted, these estimates should be smoothed (as discussed earlier). Table 3 also shows the observed and smoothed estimates for Malawi (see columns 6–9).

The third step involves calculating the intersurvey decrement in first union survival rates for a given marriage cohort. These estimates represent the standardized number of unions that ended during the intersurvey period for the cohort in question. For example, for surveys that are five years apart, we compare the proportion of individuals in intact first unions at interval duration 0–4 at the first survey with the corresponding proportion at interval duration 5–9 at the second survey to determine the change in the proportion of individuals in intact first unions. This value represents the number of intersurvey first union dissolutions for this cohort, standardized for a population of one person. Column 3 of Table 4 shows the marriage cohort–specific intersurvey decrements in first union survival rates.

In the fourth step, we convert the intersurvey decrements in first union survival rates into intersurvey probabilities of first union dissolution (Table 4, column 4), which we then convert into the intersurvey first union survival rates (column 5). For example, to calculate the intersurvey probabilities of first union dissolution for individuals who first married 0–4 years before the first survey, we divide the intersurvey decrement in the proportion of women in intact first unions for this cohort by the corresponding proportion (for the same cohort) at the first survey. The proportion of individuals at interval duration 0–4 during the second survey reflects the survival probability of the first union during the intersurvey period among individuals who first married 0–4 years before the second survey. Thus, subtracting this estimate from 1 yields this cohort's intersurvey probability of union dissolution. We assume that the intersurvey probabilities of union dissolution correspond to the middle period of the intersurvey period. Thus, we set *t* = 2.5 for individuals who first married 0–4 years before the second survey, *t* = 5 for individuals who married 0–4 years before the first survey and who were married for 5–9 years at the second survey, and so forth (columns 1 and 2 of Table 4). The intersurvey first union survival rates are derived successively by first setting *S*(0) = 1. Thus, *S*(2.5) = 1 – *P*(2.5), *S*(5) = *S*(2.5) – *S*(2.5) × *P*(5), *S*(10) = *S*(5) – *S*(5) × *P*(5), and so forth (where *P*(*t*) is an intersurvey probability of first union dissolution).

The fifth step involves fitting a polynomial function (as described earlier) to intersurvey first union survival rates. The fitted function is then used to produce *S*(*t*) estimates for single interval durations—that is, for *t* = 1, 2, 3, . . . (Table 5, column 3). Finally, the implied number of marriages ending each year that correspond to the intersurvey period can be calculated iteratively using Eqs. (1) and (5). The only issue to consider at this point is the implied intersurvey duration–specific distribution of censored observations: the distribution of women in intact first unions at different interval durations and the implied intersurvey number of ever-married individuals $(n1=[0,1)).$ Generally, these numbers can take any positive values that are consistent with the derived intersurvey first union survival rates (because, by definition, *S*(*t*) is intrinsically equivalent to the ratio of individuals in intact first unions to all ever-married individuals at interval *t*). The derived *S*(*t*) function could be considered the censored observation distribution. In this case, it is equivalent to having one individual at each interval. In general, any random number of ever-married individuals at a given interval duration would yield similar results; one needs only to multiply the derived intersurvey *S*(*t*) estimate with the generated random number of ever-married individuals. As discussed earlier, the value of $n1=[0,1)$ is the summation of all ever-married individuals across all interval durations. For Malawi, we assign 1,000 ever-married individuals to each interval. Column 6 of Table 5 shows the implied distribution of censored observations at each interval. Note that the numbers in columns 6–9 are meaningless in themselves because different assumed numbers of ever-married individuals would produce different distributions. We present them here because they are essential to produce accurate estimates in column 11 (i.e., the quantum-adjusted first union survival rates).

Column 11 of Table 5 shows the quantum-adjusted first union survival rates in Malawi between 2010 and 2015. The results suggest that if a given marriage cohort experienced the union dissolution rates observed between 2010 and 2015 in Malawi, 25% of women whose first union would eventually end within the first 30 years of marriage would experience the event within the first 0.7 years, 50% of the women would experience the dissolution within the first 2.3 years, and 75% would experience the dissolution within the first 5.6 years of their first marriage.

## Conclusion

Estimating the quantum-adjusted timing of first union dissolutions when such information is not directly available is a necessary next step for broadening our understanding of country and period differences in first union dissolution dynamics. In this article, we proposed and demonstrated our indirect life table of first union dissolution method for producing such estimates from incomplete marriage histories. ILTUD uses a simple tabulation of ever-married women by duration since the first union, classified by union dissolution status (intact vs. dissolved first union), to produce measures of first union dissolution timing that are fully adjusted for the level (quantum) of union dissolution. The key insight of this method is that one can calculate the survival function *S*(*t*) under assumptions from incomplete life course histories and then use relationships between life table functions to derive quantum-adjusted survival rates. We show that this method performs as well as standard statistical methods using complete histories.

ILTUD is a synthetic cohort Type II measure and thus produces robust period estimates under stable first union dissolution rates over marriage cohorts. In some LMICs, particularly in sub-Saharan Africa, union dissolution rates have declined slowly or remained stable (Clark and Brauner-Otto 2015). Therefore, we expect the ILTUD method to be relevant in such settings. An integral component of ITULD is its extension ILTUDext, which we suggest using when multiple surveys conducted 5 or 10 years apart are available and union dissolution rates have changed substantially across marriage cohorts. Thus, ILTUDext provides a robust solution for addressing concerns about distortions in ILTUD estimates when first union dissolution rates are changing. This extension increases the practicality of the ILTUD method in different settings, including those that might have experienced substantial changes in first union dissolution rates.

The key data input for ILTUD is the distribution of ever-married individuals by duration since the first union, classified by first union dissolution status. These data can be extracted from the most reliable and nationally representative sources of nuptiality information in most LMICs, such as the DHS and MICS. Indeed, these surveys are now conducted in more than 85 countries, and 668 such surveys (DHS and MICS combined) span more than 30 years of data collection. Thus, ILTUD provides an opportunity to advance scholarship on family dynamics in LMICs, where representative data on union dissolution timing is often lacking but incomplete marriage histories are available.

Nevertheless, a few caveats are noteworthy for applying the ILTUD method. First, the quality of the input data for ILTUD could be affected by reporting errors, such as the omission of early unions or misreporting of the date of the first marriage. Evaluations of marriage histories in developing countries suggest that these reporting errors are pronounced with increasing age or marriage duration (Chae 2016; Gage-Brandon 1995; Mensch et al. 2006). Therefore, we recommend estimating the first union survival rates on the basis of data from younger marriage cohorts. In this article, we considered women married for 0–30 years, but a lower limit—say, 25—could be ideal. Furthermore, because of data errors or limited sample size, the distribution of the proportion of individuals in intact first unions by marriage duration may require smoothing to produce reasonable estimates. We suggested using polynomial regression for such smoothing. However, other parametric or semiparametric models that adequately fit the data could be used.

Second, ILTUD methods assume equal migration and mortality rates for women in an intact first union and women whose first union dissolved. Although the assumption of equal migration is less likely to be problematic in LMICs, mortality could differ between these groups. Studies in high-income countries suggest that experiencing a union dissolution is associated with higher mortality (Bulanda et al. 2016; Franke and Kulu 2018; Metsä-Simola and Martikainen 2013). We have no reason to believe that this pattern is different in LMICs. Besides, in countries hit hard by HIV, individuals whose first union ended because of spousal death from HIV/AIDS are more likely to have died from HIV/AIDS themselves and are thus underrepresented in the survey. The differential mortality between these two groups could lead to an overestimation of first union survival rates, thus biasing the quantum-adjusted summary measures of the timing of first union dissolution. Although HIV mortality has declined in most LMICs (Granich et al. 2015), we advise using the ILTUD method with caution when evidence suggests substantial mortality differences between women in intact first unions and those who experienced union dissolution.

Notwithstanding these limitations, the ILTUD method opens up numerous demographic analysis possibilities in LMICs. Besides advancing scholarship on family dynamics in these countries, ILTUD is applicable for estimating the timing of other life events for which direct individual-level data on the event's timing are missing. Thus, ILTUD enhances demographic estimations in demographic subfields within and beyond family demography. For example, ILTUD could be used to analyze the timing of first contraceptive use in countries where data on the age at or date of first contraceptive use are not available. This information is generally not collected in the DHS or MICS. Thus, adapting the ILTUD method to perform such analysis could provide valuable insights into how the timing of first contraceptive use relates to other vital reproductive events, such as first sex, first marriage, and first birth.

## Acknowledgments

We are grateful for the feedback on an early version of this paper from Angela Carollo and Christian Dudel of the Max Planck Institute for Demographic Research. Their insights helped us tremendously to refine the message of this manuscript. We further thank Ben Wilson of Stockholm University, the three anonymous reviewers, and the Editors for their constructive comments.