## Abstract

In general, the use of indirect methods is limited to developing countries. Developed countries are usually assumed to have no need to apply such methods because detailed demographic data exist. However, the potentialities of demographic analysis with direct methods are limited to the characteristics of available macro data on births, deaths, and migration. For instance, in many Western countries, official population statistics do not permit the estimation of mortality by socioeconomic status (SES) or migration background, or for estimating the relationship between parity and mortality. In order to overcome these shortcomings, I modify and extend the so-called orphanhood method for indirect estimation of adult mortality from survey information on maternal and paternal survival to allow its application to populations of developed countries. The method is demonstrated and tested with data from two independent Italian cross-sectional surveys by estimating overall and SES-specific life expectancy. The empirical applications reveal that the proposed method can be used successfully for estimating levels and trends of mortality differences in developed countries and thus offers new prospects for the analysis of mortality.

## Introduction

To what extent does life expectancy differ by education level and occupation status? Are there mortality differences between migrants and nonmigrants? Does parity influence the survival of people and, if so, in what direction? Although the political and societal importance of these aspects is obvious, many countries cannot estimate the extent and trend of such mortality differences for their population because they lack sufficiently detailed statistical data. In such cases, one usually adopts experience from another country that seems comparable to the population of interest. Yet, although such empirical experience usually reveals that the direction of socioeconomic-, ethnicity-, or parity-specific mortality differences is the same, its extent and trend can vary considerably.

In the field of fertility research, similar constraints concerning the possibilities of demographic analysis with official population statistics have recently led to an increased use of survey data and event-history analysis. For mortality, however, the practicability of survey data is more limited, since direct analysis of mortality requires longitudinal data as well as long observation times and large sample sizes in order to provide a sufficient number of deaths. Thus, mortality researchers in developed countries are often faced with the same problem as demographers in developing countries, where demographic data are either nonexistent or of such low quality that they are not usable. In the absence of quality data for developing countries, the use of indirect estimation techniques based on survey data has been shown to provide reliable insights into demographic conditions and trends. In view of the shortcomings of the available information on mortality differences for many developed countries, the rare use of indirect methods outside the developing world is surprising. The typical indirect estimation based on the mortality of relatives of survey respondents has been used only for assessing levels and trends of mortality differences in the Russian Federation (Bobak et al. 2002, 2003; Murphy et al. 2006; Nicholson et al. 2005).

In order to improve the availability of information on specific mortality differences when no official data are available, I will modify and extend the so-called orphanhood method for indirect estimation of adult mortality from information on maternal and paternal survival to permit its application to populations of developed countries. Generally, the orphanhood method seems to be the most promising indirect method for estimating mortality differences because the survival status of respondents’ parents is included in many surveys and is likely to be reported more adequately than the survival status of other family members. There are several interesting data sources for applying the orphanhood method to investigate specific mortality differences that often cannot be analyzed with existing population statistics: above all, the Gender and Generations Programme (GGP), with more than 20 participating countries and a survey including several questions on various characteristics of respondents’ parents during the childhood of respondents; but also other surveys, like the Survey of Health, Ageing and Retirement in Europe (SHARE) and the U.S. Health and Retirement Study (HRS).

The article is structured as follows. The next section summarizes the theoretical background of the orphanhood method and the data used from two cross-sectional surveys in Italy, which is one of the countries with limited information about mortality by socioeconomic status from official statistics. Then I describe a modification of the orphanhood method and apply the method empirically to the survey data used. In order to evaluate the general approach of the orphanhood method and the functionality of the proposed variant for its application to populations from developed countries, overall mortality of Italy is estimated by means of survey information on orphanhood and compared with official statistics. Finally, the application to the estimation of life expectancy by education and occupation is tested by comparing the results obtained from the two independent surveys. For that purpose, I estimate life expectancy for the education groups no completed and primary, lower secondary, upper secondary, and tertiary education; and with regard to occupation for economically inactive persons, manual workers, nonmanual workers, self-employed persons, and professionals.

## Italian Multipurpose Survey and Orphanhood Method

The data to which the method derived in this paper is applied were taken from the Italian multipurpose surveys Family and Social Subjects carried out by the Italian National Statistical Institute (Istat) in the years 1998 and 2003 (Istat, Famiglia, Soggetti Sociali e Condizione Dell’infanzia 1998 and 2003). Both surveys belong to the system of cross-sectional surveys on Italian families and are representative for the Italian population at the subnational level (NUTS 1). The first survey was conducted in June 1998 and includes a total of 59,050 individuals from about 20,000 interviewed families. The 2003 survey belongs to the international GGP program and includes 49,451 individuals, with the majority of interviews carried out in November. The data contain information about whether respondents’ parents are still alive and, if so, their parents’ current age. Tables 1 and 2 summarize the corresponding numbers for the total population included in the 1998 and 2003 surveys, respectively, divided into five-year age groups from 20–24 to 60–64. Additional questions on the highest education level of the parents and about several characteristics of their job activities when the respondent was about 14 years old make the use of these surveys especially interesting for the study of socioeconomic mortality differences (see Luy et al. 2011).

The estimation of adult mortality from information on parents’ survival with the orphanhood method is the dominating tool for the indirect estimation of adult mortality levels in developing countries for which population statistics are lacking (see Bradshaw and Timæus 2006; United Nations 2006). Methodological descriptions can be found in the United Nations’ *Manual X* (Hill et al. 1983) or in some more recent publications (Hill 2006; Hill et al. 2005; Preston et al. 2001; Timæus 1991c; United Nations 2002). The demographic relationship between the proportion of orphaned persons and the mortality experiences of their parents was first described by Lotka (1939), who proposed to estimate the number of orphans from life table functions for adult survivorship. Later, Henry (1960) suggested a reversal of this approach in order to estimate adult mortality from the number of orphaned children when the underlying mortality and fertility schedules were known or assumptions could be drawn for applying specific mortality and fertility models. Brass and Hill (1973) further developed this idea, proposing methods to estimate life table survivorship probabilities from proportions of respondents of successive five-year age groups with mother or father alive based on a set of weighting factors (the Brass method). In the subsequent years, several scholars suggested successively improved and modified methods for estimating adult mortality from orphanhood data (Chackiel and Orellana 1985; Hill and Trussell 1977; Hill et al. 1983; Timæus 1991a, b, 1992; Timæus and Nunn 1997) or for using two sets of orphanhood data to estimate adult mortality in the time between the surveys (Timæus 1986; Zlotnik and Hill 1981).

The basic idea of the orphanhood method is that the age group of respondents represents the survival time of the mother (or father). Consequently, the proportion of respondents of a given age group with mother (or father) alive approximates a survivorship ratio from an average age at childbearing to that age plus the age of the respondents. The available methods model this relation using different theoretical patterns of fertility, mortality, and age composition to allow the conversion of a proportion with parent surviving into a life table survivorship probability, controlling for the actual pattern of childbearing (Hill 2001). Moreover, Brass and Bamgboye (1981) developed a general method for estimating the reference date of estimates derived from data on the survival of parents.

Demographers have not reached a clear consensus on the validity of applying the orphanhood method in developing countries, and such applications have had mixed success (a conclusion arrived at by Hill 1984; Timæus 1991c; Timæus and Graham 1989). Typical problems are seen in a possible adoption effect (respondents whose parents have died are likely to be reared by another adult and may not even know that this person is not their biological parent), multireporting (the frequency of reporting about each parent depends on his or her number of surviving children and thus is connected to both mortality and fertility levels of the family), selection effects (regarding fathers and mothers if there is a relationship between parity and mortality and regarding respondents if there is a relationship between parental and child mortality), and wrong reporting of respondents’ ages. Another critical issue is the specific choice of theoretical fertility and mortality models underlying the different approaches to convert a proportion of non-orphaned respondents into life table estimates that do not necessarily reflect the real prevailing demographic conditions of the studied population.

In developing countries, the use of such theoretical population models is necessary because no data exist about the basic fertility and mortality patterns. However, such basic data are well known for populations of developed countries, with age-specific fertility and mortality patterns being available in detail for both periods and cohorts. Adoption effects and wrong age reporting are unlikely to bias orphanhood-based estimates in modern populations from developed countries. Moreover, the biases caused by multireporting and various kinds of selection are to some extent mutually offsetting and thus are considered to be small and rather unimportant, as demonstrated in detail by Palloni et al. (1984).

## Modification of the Orphanhood Method

*n*,

*n*+ 4) with mother/father still alive is usually denoted by

*S*(

*n*), with

*n*= 20 for the age group 20–24,

*n*= 25 for the age group 25–29, and so forth. If the time of interview is denoted by

*T*, we know that respondents aged (

*n*,

*n*+ 4) were born, on average, in , with being the average age of respondents of that age group. At that time, their mothers/fathers were between α and β years old, with α and β being, respectively, the minimum and maximum ages at childbearing. Thus, the proportion of respondents whose mother/father survived until time

*T*approximates a weighted average of cohort survivorshipswhere

*x*represents the single ages at childbearing of respondents’ mothers/fathers,

*p*

_{x}is the corresponding cohort life table survival probability to age

*x*, and the weights

*w*

_{x}are the proportions of

*x*-year-old mothers/fathers at time (i.e., at the moment of respondents’ birth). These weights result fromwhere

*N*

_{x}(respective

*N*

_{a}) and

*f*

_{x}(respective

*f*

_{a}) denote the number of

*x*-year-olds (

*a*-year-olds) and the fertility rate at age

*x*(age

*a*) at the time of respondents’ birth. Hence, the extent to which the cohorts of parents contribute to the value of

*S*(

*n*) depends on two factors: (1) the age distribution

*N*

_{x}of the parents’ cohorts at the time of respondents’ birth, and (2) the fertility schedule composed by the age-specific fertility rates

*f*

_{x}at that time.

Equation 1 shows that *S*(*n*) represents an average survival experience of β – α + 1 cohorts over a time span of calendar years. In principle, the survival time of fathers should be calculated 0.75 years longer to account for the gestation period. Whereas mothers’ deaths occur only during or after childbirth, fathers may die at any time since conception. Although the incorporation of such a prolonged survival time of fathers is technically possible, I did not adjust for the gestation period because the effect of an inclusion is minor for the estimation of life expectancy.^{1}

*S*(

*n*) into period survival probabilities for a specific reference period

*t*(

*n*). I defined this reference period as the average calendar year in which deceased parents of respondents aged (

*n*,

*n*+ 4) died. Thus,with

*D*(

*n*)

_{y}being the number of deceased mothers/fathers of respondents aged (

*n*,

*n*+ 4) during year

*y*after the birth of respondents;

*y*= 1 for the first year after respondents’ birth,

*y*= 2 for the second year, and so forth, until in the last year of the observed life span of respondents’ parents. This approach differs from the traditional method for the determination of reference periods in which the reference time is defined as the time point when the cohort survivorship equals the survivorship of a period life table given certain assumptions regarding the pattern of mortality and the nature of mortality changes (Brass and Bamgboye 1981). My approach is driven by the idea that the best time point represented by the mortality of respondents’ parents is the calendar year that centers the deaths that occurred; this approach was also suggested by Chackiel and Orellana (1985), who showed that such an empirical estimate of the reference year provides better results than an estimate using theoretical models.

^{2}

*t*(

*n*), I converted the empirical values for derived from the surveys into period survivorship probabilities from age 30 to age 33 +

*n*.

^{3}The procedure combines two transformations. The first is the transformation of survival of the cohorts of respondents’ parents from an average age at childbearing to —reconstructed from official mortality data with the weights derived from the survey data—into cohort survival from age 30 to 33 +

*n*(first fraction on the right side of Eq. 4). The second is the transformation of this cohort survival into period survival from age 30 to 33 +

*n*for the corresponding reference period

*t*(

*n*) (second fraction on the right side of Eq. 4). Thus,with

*p*

_{x}and

*l*

_{x}representing, respectively, the cohort and period life table survival probability to age

*x*; variables with hats refer directly to the parents of survey respondents, whereas variables without hats refer to the entire population (i.e., in our empirical application, the total Italian population). Hence is the estimated period life table survival probability to age 30 of respondents’ parents, and

*l*

_{30}is the survival probability to age 30 of the official period life table. Finally, Eq. 4 can be simplified to

By using Brass’s logit life table model (Brass 1971, 1975) with the official life table for the determined reference period as the standard, the derived survivorship probabilities can be transferred into complete life tables from age 30, with the parameter α of Brass’s logit life table model being determined by the estimated survivorship probability from age 30 to age 33 + *n*. If specific information about the pattern of mortality of the analyzed population subgroup is available, this information can be used to determine the corresponding parameter β of Brass’s logit life table model (see, e.g., Ngom and Bawah 2004; or Stewart 2004). If no information about the specific mortality pattern is available, Brass’s β should be set to 1.0, assuming that the mortality pattern is equal to that of the total population. The complete conversion of empirical values for into period survivorship probabilities from age 30 to age 33 + *n* is demonstrated with a numerical example in Appendix 1.

As described in the introduction, the application of indirect methods for populations of developed countries becomes interesting when the mortality of specific subgroups is analyzed. Thus, in the practical application of the orphanhood method, respondents will be divided into subgroups by characteristics of their parents. These subgroups differ with regard to not only the mortality of their parents but also their parents’ age distribution—that is, to the weights that depend on the age-specific number of parents at the time of respondents’ birth and the corresponding fertility rates (see Eq. 2). In surveys with complete information on all data required, the weights can be derived directly from the survey data. However, some surveys, like the Italian multipurpose surveys, contain the ages of only parents still alive instead of including information on age at childbearing of all parents. In such cases, the weights can be approximated from the information on age at childbearing of the parents still alive at the time of the interview when cohort life tables for the cohorts of respondents’ parents are available covering the time span between and *T* (see Appendix 2).

## Application of the Modified Orphanhood Method to the 1998 and 2003 Survey Data

The suggested method was applied to all age groups of respondents for which information on maternal and paternal orphanhood existed and the application of the method was possible (age groups 20–24, 25–29, 30–34, 35–39, 40–44, 45–49, 50–54, 55–59, and 60–64). In contrast to the situation in developing countries, the functionality and efficiency of the proposed method can be tested when it is applied to a developed country like Italy. In the first part of this section, I employ the modified orphanhood method (MOM) as well as the most common traditional methods introduced by Brass and Hill (1973; the Brass method) and Timæus (1992) using the data for the entire population from the Italian multipurpose surveys of 1998 and 2003 in order to compare the results with the life tables for Italy from the Human Mortality Database (HMD 2009), which are reconstructed from official Italian population statistics. The reference periods for the traditional estimates were derived using the method of Brass and Bamgboye (1981), with the adjusted ages at childbearing as described in Appendix 2.

Before comparing the orphanhood-based estimates to the HMD life table data, we have to consider how orphanhood-based estimates of life expectancy should look compared to results from life tables for the total population. By nature, orphanhood-based estimates refer exclusively to parous women and men (with surviving children). Several studies have shown that women with children have lower mortality than nulliparous women, although mortality among parous women seems to increase at higher parities (Butt et al. 2009; Doblhammer 2000; Green et al. 1988; Grundy and Tomassini 2005; Hurt et al. 2006; Kitagawa and Hauser 1973; Kvåle et al. 1994; Le Bourg 2007; Lund et al. 1990; Spence and Eberstein 2009). However, because parities of four or more children are the minority among the parous Italian population, the positive effect of having children can be assumed to dominate the negative effect seen at higher parities. With regard to men, the few existing studies report different findings. Friedlander (1996) found no relationship between male reproduction and survivorship, whereas Grundy and Kravdal (2010) reported similar effects of fertility history on all-cause mortality for both sexes. Furthermore, childbearing in Italy occurs almost exclusively among married women and men, whose lower mortality as compared with unmarried persons has been shown in many studies and for many populations (e.g., Goldman and Hu 1993; Hu and Goldman 1990; Rogers 1995).

Aside from these causal effects, there is also a structural effect leading to better survival for women and men with children. Regarding the total number of years lived, and thus life expectancy, deaths at younger ages have a stronger impact than deaths at older ages. The closer deaths occur to the beginning of the reproductive life span, the more likely they affect childless individuals. Thus, the population of parents necessarily shows better longitudinal survival than all individuals at reproductive ages at the beginning of the observation period. Consequently, both the causal effects and the structural effect should entail a higher life expectancy for the population of parents than for the total population. This holds independently from the aspect of multireporting and has already been demonstrated empirically. Using Austrian micro-census data and official death statistics, Festy (1995) showed that the proportion of maternal orphans is in fact lower than suggested by the corresponding survival ratios reconstructed from official mortality statistics for the total population.

Panels a and b of Fig. 1 illustrate the orphanhood-based estimates for life expectancy at age 35 for men derived from the multipurpose surveys of 1998 and 2003, respectively.^{4} The main results and conclusions are the same for the applications to both surveys. In more recent years (i.e., with estimates referring to younger respondents and consequently based on the survival experiences of younger cohorts of parents), the Brass method performs well: it provides life expectancy estimates above the level for the total population, as reflected by the life tables from the HMD. The same holds for the Timæus method. However, the estimates obtained with the Timæus method are higher above the HMD level than are those obtained by the Brass method. For estimates referring to the three latest periods of both survey data (based on respondents aged 25–29, 30–34, and 35–39), the Brass method provides results close to the MOM. On the other hand, the life expectancy estimates derived from the Timæus method are closer to those derived from the MOM for the estimates based on respondents aged 40–44. That both traditional methods perform quite well for younger parents is due to the fact that the mortality pattern in younger ages does not differ substantially between the Italian population and the mortality models used for these variants of the orphanhood method. However, the higher the age groups of respondents, and accordingly the longer the survival time of parents, the more the results obtained by the Brass method deviate from the real trend in life expectancy, leading to life expectancy estimates distinctly below the level of the total Italian population. These deviations are a consequence of the larger differences between the Brass general standard (which is the mortality pattern underlying the Brass method) and the mortality of the Italian population at higher ages.

The MOM, however, provides estimates that are as expected according to the theoretical considerations at the beginning of this section. Most estimates lie above the level for the total Italian population and differ to a reasonable extent between 0.5 and 1.0 years to a maximum of 1.7 years in life expectancy at age 35. Only the estimates based on information from respondents aged 20–24, 50–54, 55–59, and 60–64 from the 2003 survey (the earliest three periods and the latest) result in slightly lower life expectancy levels than the HMD data. The MOM estimates based on information from younger respondents seem to deviate somewhat more strongly from the HMD data than the MOM estimates based on information from older respondents. This might be explained by the fact that the number of respondents with deceased fathers is lower among younger age groups than among higher age groups (see Tables 1 and 2), making the estimates more sensitive to arbitrary bias attributable to low case numbers.

The MOM estimates for female life expectancy are more or less consistent with those for males (see panels c and d of Fig. 1). The two traditional approaches, however, perform considerably worse than they did in the application to data on male mortality. This holds above all for the Timæus method and the method of Brass and Bamgboye (1981) for the derivation of reference years. Regarding the estimates based on the 1998 survey, the three life table estimates with the Brass method that are below the HMD level are those derived from the orphanhood information of the three oldest age groups of respondents: 50–54, 55–59, and 60–64. Thus, according to the reference date estimates obtained by applying the Brass/Bamgboye method, the three oldest age groups provide mortality estimates for the earliest as well as for the latest period. Similar problems occur with the 2003 data, where the three most recent estimates refer to information from age groups 25–29, 50–54, and 55–59. The reason for the failure of the traditional methods when applied to Italian women is the low level of female mortality in developed countries that is even more different from the mortality models used for these variants of the orphanhood method than among males.

Among the results for female life expectancy at age 35 obtained by the MOM, there are only a few outliers in the 2003 data. The one referring to the time 1990.2 (*e*_{35} = 49.3 years) is based on information from respondents aged 60–64—that is, those who were born during the years 1938–1942. Interestingly, also in the 1998 survey—although hardly visible in panel c of Fig. 1—respondents of the same birth cohorts (here, aged 55–59) reported a somewhat lower mortality level of their mothers compared with the respondents of the neighboring age groups. Thus, there might be a specific adoption effect among respondents who lived their early childhood in wartime years, which could affect not only the estimates for life expectancy but also the derived reference periods when the number of surviving mothers becomes very small. Of course, this hypothesis requires further investigation. Nevertheless, because the bias in information about maternal survival of respondents aged 60–64 in the 2003 survey is obvious, it seems better to exclude them from more detailed analysis of mortality trends and differences. The same holds for the estimates based on respondents aged 25–29 and 30–34, which clearly deviate from the trend depicted by all other estimates from the two surveys.^{5}

Although the case numbers of the Italian multipurpose surveys are comparatively high, the graphs in Fig. 1 reveal that the orphanhood-based estimates for life expectancy are subject to fluctuations. This could become a problem when population subgroups with lower case numbers are analyzed. In order to better control for this possible bias when analyzing mortality differences, one can fit an appropriate smoothing function to all available point estimates (see also Murphy et al. 2006). Figure 2 shows the estimated linear trends of male and female life expectancy at age 35—which fit the MOM estimates obtained from the two Italian multipurpose surveys very well—for the complete observation period. The graphs elucidate the functionality of the proposed method. Both aspects required for concluding that the method provides reliable mortality estimates are reflected accordingly by the estimated trends: the higher level of life expectancy compared with the official data and the trend of rising life expectancy as shown in the HMD data. The somewhat higher differences among women might be due to larger effects of reproduction on mortality for women compared with men.

The data on paternal orphanhood from the two surveys further allow a test of whether the proposed method can provide reliable results for analyzing the mortality of subgroups because estimates for the period 1984–1990 can be obtained from both the 1998 and the 2003 surveys (in contrast to the data on maternal orphanhood, in which the reference periods of estimates from the two surveys do not overlap; see Fig. 1).^{6} The dot plots in Figs. 3 and 4 display the corresponding estimates for life expectancy at age 30 by education level and occupation, respectively.^{7} Note that the information on SES refers to the education and occupation of the fathers at the time when respondents were about 14 years old, which is one specific feature of the surveys of the GGP program and the Italian multipurpose surveys. As already shown in panels a and b of Fig. 1, the 2003 survey provides somewhat lower estimates for life expectancy than the 1998 survey for the years 1984–1990, which also holds true for every population subgroup displayed in Figs. 3 and 4. This is, however, no problem for the application of the method. First, all population subgroups are affected similarly by this effect. Second, because the values for life expectancy obtained by the orphanhood method are, by nature, difficult to assess, it is preferable to analyze the differences in life expectancy between specific subgroups instead of analyzing the absolute values.

Concerning the applicability of the MOM, most important is whether the two surveys provide similar results regarding the patterns of education- and occupation-specific differences in life expectancy at age 30 for the period 1984–1990. Figure 3 shows that this holds very well for the education levels analyzed. The data from both surveys provide the same order from the lowest to the highest levels of life expectancy—that is, lower secondary education, primary education, upper secondary education, and tertiary education. Even differences between the groups are comparable. Both surveys also provide the same ranking of occupation groups by life expectancy (Fig. 4), with similar differences between them. Professionals show the highest life expectancy, followed by nonmanual workers, self-employed men, and manual workers. Economically inactive men have the lowest life expectancy, which is six to seven years below the level of manual workers. The similarity of the results obtained from the two independent surveys indicates that the MOM provides stable results even for the analysis of population subgroups.

## Discussion

In this article, I modified the orphanhood method, which was originally designed to be applied to populations of developing countries, for estimating mortality differences in populations from developed countries. The modified method can be used to reconstruct a time series of female and male survivorship estimates and thus to analyze time trends of mortality differences from survey information on the proportion of respondents of different ages with mother/father still alive. Italy was chosen for demonstrating the proposed method for two reasons. First, Italy is one of those developed countries where data and knowledge on socioeconomic mortality differences are limited. Second, Italy already provides two large comparable and nationally representative cross-sectional surveys with the necessary information to test the applicability of the method.

The main conceptual difference between the traditional variants of the orphanhood method and the MOM is that the weighting factors and regression coefficients of the traditional methods are based on theoretical population models, independent of the period to which the estimates refer. By contrast, the MOM is based exclusively on empirical data from official statistics for the population analyzed. Only the specific feature of the Italian multipurpose surveys—that information on age at childbearing is missing for deceased parents—required the use of theoretical fertility models (see Appendix 2). However, even these modeled fertility schedules are derived from the real age-specific fertility pattern prevailing at the time of respondents’ birth. Another difference is that the traditional methods for reference period estimation use the theoretical time lag between period- and cohort-type life tables to display the same survivorship ratio. In the MOM, the reference periods are derived from the average dates of death of the deceased parents.

Although the reported survival of respondents’ parents is cohort survival, I followed the approach of the traditional variants of the orphanhood method to estimate past period mortality. Because of the differences between period- and cohort-type mortality schedules, the reported cohort survival of respondents’ parents has a different age pattern than the estimated period life table. This difference becomes stronger in ages at which the survivorship curve turns downward. Consequently, estimates based on younger ages of respondents are more robust than estimates based on older respondents. According to the empirical applications, 60–64 should be the highest age group to apply the orphanhood method because—in addition to the turn of the survivorship curve—at older ages, the case numbers of respondents with parents still alive become too low.

The lower mortality level of a population of parents compared with the total population as necessarily obtained by orphanhood-based estimates does not reduce or qualify the applicability of the proposed method. It is important to note that the aim of the MOM is not to provide a new or alternative estimate for the life expectancy of the total population. The aim is to estimate differences in life expectancy that cannot be analyzed otherwise because of the lack of statistical data. The overestimation of life expectancy can be assumed to affect population subgroups equally, which enables a comparison of the differences between them.^{8} The results obtained from applying the MOM to the empirical survey data suggest that the method can in fact provide important insights into trends and levels of mortality differences. The estimated differences in life expectancy by education for Italian men are very similar to those reported for males of other Western countries with better data availability, like Belgium (Deboosere et al. 2009), Finland (Shkolnikov et al. 2006; Valkonen 2006), or the United States (Crimmins and Saito 2001; Molla et al. 2004).

Nevertheless, as Timæus (1991c) outlined in detail, indirect methods like the orphanhood method always entail several drawbacks. Indirect methods can provide only broad measures of the overall level and trend in adult mortality. They are inherently unable to detect short-term trends or abnormal age patterns of mortality within adulthood. A further limitation of the orphanhood method is that it yields estimates of mortality that refer to dates well before the survey was conducted. Deaths of parents occur over a period extending back to when respondents were born in the case of mothers, and about nine months earlier for fathers. The younger the respondents, the more recent are the derived mortality estimates. But even the estimates based on respondents aged 20–24 refer to a period about 8.5 years prior to the survey. This problem could be overcome when two or more surveys containing the necessary information for the orphanhood method and spaced by approximately 5 or 10 years are available so that synthetic cohorts can be constructed. The Italian multipurpose surveys of 1998 and 2003 offer this possibility. Using the MOM estimates for the intersurvey method described in United Nations’ *Manual X* (Hill et al. 1983) and excluding the female outliers from the 2003 survey yields estimates for overall life expectancy at age 35 for the year 2001 of 44.4 years for men and 48.6 years for women. The corresponding values derived from the HMD data for the total Italian population are 43.2 and 48.6 years, respectively. Thus, even in this variant, the orphanhood method seems to be a promising estimation technique; this is especially true with regard to the results for men, for whom the intersurvey method provides a similar overestimation of life expectancy as do the MOM estimates for the surveys of 1998 and 2003 presented in panels a and b of Fig. 1.

On the other hand, the orphanhood method and other indirect methods have some advantages over direct methods. First, they permit the deriving of life tables and thus the estimation of life expectancy. In many cases and for most users of demographic data, such as policy makers, differences in life expectancy are more important and more informative than differences in standardized death rates or relative risks, which are usually calculated when surveys or linked census data are used for the analysis of mortality. Years of life represent the most easily understandable unit of measurement of mortality differences. Variations in standardized death rates or relative risks are more difficult to assess, especially for laypersons, since large differences in these measures do not necessarily reflect large differences in actual lifetime. Second, indirect methods typically provide trends of demographic conditions that are derived from a single cross-sectional survey, whereas direct methods usually provide but one estimate from cross-sectional data for a specific year. A third advantage is that the information used is based on respondents’ lifetime experience, and thus fairly precise estimates of the proportions of respondents with living parents (or other relatives) can be obtained even from surveys of moderate size. Knowing the general functionality of this method, researchers can collect information on interesting aspects of mortality quite easily and at a moderate cost by including a few simple questions into existing or planned survey programs. Information about the level and trend of mortality differences is often sufficient to provide a basis for forecasting and the allocation of resources.

Finally, the empirical applications of the proposed modification of the orphanhood method provide important information for its traditional variants. The fact that the traditional approaches of Brass and Hill (1973) and Timæus (1992) and the method of reference date estimation by Brass and Bamgboye (1981) do not perform well in the application to Italian survey data does not mean that they do not work for populations of developing countries. The results presented in this article reveal that the underlying mortality models of these methods are too different from the mortality of a developed population to be useful here. In developing countries with high mortality levels, they should be more appropriate. However, the theoretical considerations and empirical outcomes regarding the lower mortality levels of parous women and men suggest that these relations hold in every population. This effect does not appear to be adjusted for in the traditional methods that were developed to obtain estimates of adult mortality for the total population. A further adjustment of the traditional methods to overcome this underestimation of the overall mortality level might be useful.

## Acknowledgments

I thank Paola Di Giulio for preparing the data of the Italian multipurpose surveys and for running specific computer programs; Graziella Caselli and Griffith Feeney for fruitful discussions; and two anonymous reviewers for their careful reading and very helpful comments and suggestions on earlier versions of this article.

#### Appendix 1

*n*is demonstrated with the example of mortality of mothers with primary education of respondents aged 40–44 from the Italian multipurpose survey of 1998. Of all 3,219 respondents of that age group who declared that their mother has primary education, 2,635 reported that their mothers are still alive, and 584 reported that their mothers are already deceased. From these numbers, I calculated the proportion of respondents aged 40–44 with mother alive: . From the birth years of respondents and their mothers I could derive further that at the time of respondents’ birth, 0.10% of the mothers were 15 years old, 0.37% were 16 years, 0.79% were 17 years, 1.34% were 18 years, and so forth.

^{9}Applying these proportions as weights (with , , and so forth) to the cohort life tables for the total Italian female population (reconstructed from official population statistics) enabled me to calculate the proportion of women alive with the same age structure as respondents’ mothers but with the cohort mortality of Italian women (representing the denominator of Eq. 5):

*q*

_{x}for single ages of the Italian period life tables for women in 1986 and 1987, I estimated the single

*q*

_{x}for 1986.58 by linear interpolation, producing a life table with a life expectancy at age 30 of 50.55 years. The survivorship probability from age 30 to 33 + 40 of this life table is 0.7932, representing the enumerator on the right-hand side of Eq. 5. Thus, the survivorship probability from age 30 to 73 for the mothers with primary education of respondents aged 40–44 from the 1998 survey could be estimated from Eq. 5 by

By again using Brass’s logit life table model, I shifted the reference life table with the survivorship probability from age 30 to 73 of 0.7932 to a life table with the estimated survivorship probability of 0.8363, producing a Brass parameter α of 0.1431 and an estimate for life expectancy at age 30 of 52.28 years (with the Brass parameter β = 1.0). Changing the Brass parameter β to 0.90674 (see footnote 7) provides the final survivorship function for women with primary education, with an estimated life expectancy at age 30 of 51.92 years.

#### Appendix 2

The Italian multipurpose survey allows the determination of the age at childbearing only for those parents who are still alive at the time of the survey. Because the weights in Eq. 5 refer to the ages at childbearing of all parents at the time of respondents’ birth, the corresponding schedule of age-specific fertility rates (in the following short schedule) has to be estimated. Therefore, I used a three-step procedure (which is necessary only when the weights cannot be derived directly from the survey data):

From Eq. 2, I reconstructed the

*w*_{x}for all age groups of respondents from ages 20–24 to 60–64 with*N*_{x}and*f*_{x}taken from official Italian population statistics for single calendar years from 1933 to 1982 (i.e., the birth years of survey respondents), averaged for the five calendar years in which the respondents of a specific five-year age group were born.^{10}The series of*f*_{x}were smoothed with the fertility model proposed by Schmertmann (2003), setting α = 14 (the youngest age at which fertility rises above age zero). The other parameters of the model were estimated from the empirical*f*_{x}of official Italian statistics. For instance, for mothers of respondents aged 40–44 from the 1998 survey, the fertility rates resulted in*P*= 26.0 (the age at which fertility reaches its peak level),*f*(*P*) = 0.1473 (fertility rate at age*P*), and*H*= 36.3 (the age above*P*at which fertility falls to half of its peak level). The resulting fertility schedule and the given*N*_{x}of the years 1953–1957 (birth years of respondents) led to an average age at childbearing of (i.e., the average age at childbearing of the entire Italian population at that time).In the next step, I modified the age-specific fertility rates

*f*_{x}from Step 1 to a new schedule with an average age at childbearing of years, which is the age at childbearing of parents still alive at the time of respondents’ birth; hats indicate that the parameters refer to the analyzed subpopulation of the survey, and the asterisks indicate that they refer to parents still alive at the time of the interview (for the example of mothers with primary education of respondents aged 40–44 from the 1998 survey, ). For that purpose, the relational Gompertz fertility model of Brass (1981) was used to shift the age-specific fertility rates*f*_{x}from Step 1 along the age axes by varying Brass’s α and setting Brass’s β = 1.0 (in the current example, using Brass’s α = 0.1619 shifts the*f*_{x}schedule from Step 1 to a new set of , with ). In cases where and deviated by more than one year, I used a combination of the relational Gompertz fertility model and the Schmertmann fertility model. The Schmertmann model was used to shift the basic age-specific fertility rates*f*_{x}along the age axes by systematic changes of parameters*P*and*H*. The system of changes was oriented on the number of years which differed from . For each year that was higher or lower than ,*P*and*H*were increased by 2.0 and 1.0, or decreased by 1.0 and 2.0 years, respectively. The other two parameters of the Schmertmann model were always kept constant at α = 14 and*f*(*P*) as given by the basic fertility schedule of the total Italian population. By using the relational Gompertz fertility model, keeping Brass’s β = 1.0 constant and varying Brass’s α, the fertility schedule was then further modified to provide a set of age-specific fertility rates , leading to an age at childbearing of years.^{11}Applying the fertility rates from Step 2 and the

*N*_{x}from Step 1 to the series of cohort life tables for the Italian population yields a specific survival function for the cohorts of parents of respondents aged (*n*,*n*+ 4) with an average age at childbearing of years. From this survival function, I derived an approximate estimate of the age at childbearing of all parents, , by assuming that refers only to the survivors at the end of the observation time (time of interview) and by adding the deceased individuals to the calculation with their ages at childbearing. Then I shifted the fertility rates*f*_{x}from Step 1 as described in Step 2 to provide the final schedule, leading to an age at childbearing of years (for the example of mothers with primary education of respondents aged 40–44 from the 1998 survey, the estimated ). Finally, this fertility schedule and the*N*_{x}for the entire Italian population from Step 1 were used to determine the weights by applying Eq. 2.

## Notes

^{1}

The gestation period has two opposing effects on the survivorship ratio of respondents’ parents: (1) an increase of the survival time by 0.75 years, and (2) a decrease of the initial age of the survivorship ratio by 0.75 years. While the first effect leads to an increase of mortality, the latter causes the risk of dying to decrease. Consequently, the effects cancel each other out to some extent and thus are expected to be negligible.

^{2}

The number of deceased parents during year *y* after respondents’ birth, *D*(*n*)_{y}, can be estimated from the real cohort survival of the population analyzed, which is known for most developed countries (see Appendix 1).

^{3}

The reported proportions of mothers and fathers alive reflect survival experiences of approximately *n* + 2.5 years. I use a time span of *n* + 3 years in order to provide a survivorship estimate that is directly comparable to the official period life tables.

^{4}

Life expectancy at age 35 was chosen because this is the youngest age for which all methods can be employed.

^{5}

Alternatively, two or more five-year age groups could be combined to form age groups with higher case numbers. This flexibility regarding the size of the age intervals is another major advantage of the MOM over the traditional variants of the orphanhood method.

^{6}

The period was extended by one year to 1991 only for the estimates of tertiary education level and of the occupation group *professional*.

^{7}

The estimates were derived using the MOM, and the resulting survivorship probabilities were transferred into complete life tables from age 30 with Brass’s logit life table model. Values for the Brass parameter β were estimated from education- and occupation-specific death rates for age groups 18–29, 30–44, 45–54, 55–64, and 65–74 published by Istat (2001) for the years 1991–1992. Brass’s β was set to 1.0 only for economically inactive and self-employed women and men because the death rates available from Istat are not fully comparable owing to different compositions of the occupation groups as compared with the multipurpose survey.

^{8}

Murphy et al. (2006) arrived at a similar conclusion in their indirect analysis of mortality by education level in Russia based on information about survival of spouses and siblings.

^{9}

Because the Italian multipurpose surveys entail the age at childbearing only for those parents who are still alive at the time of the survey, these proportions had to be estimated from the available information on parents still alive (see Appendix 2).

^{10}

For males, I assumed the female fertility rates for ages 15–49 to be shifted by four years to ages 19–53. For the case of France, Caselli and Vallin (2006) showed that this assumption fits empirical reality in populations from developed countries very well for the distant past and approximately well for more recent years.

^{11}

A fertility schedule leading to the average age at childbearing of years could also be derived directly from the Schmertmann model. I chose the described combination of the Schmertmann and Gompertz models because I wanted to keep a systematic relation between the Schmertmann parameters *H* and *P*, which is not possible when using the Schmertmann model alone. This is, however, a personal preference, with minor impacts on the final orphanhood-based estimates for life expectancy. In principle, any fertility model can be used to implement Step 2.

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