Estimating Mortality Differences in Developed Countries From Survey Information on Maternal and Paternal Orphanhood

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.

Appendices

Appendix 1

The complete conversion of empirical values for $\hat{S}(n)$ into period survivorship probabilities from age 30 to age 33 + 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: $\hat{S}(40) = 2,635/\left( {2,635 + 584} \right) = 0.8186$. 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 ${\hat{w}_x}$ (with ${\hat{w}_{{15}}} = 0.0010$, ${\hat{w}_{{16}}} = 0.0037$, 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):

$$ \sum\limits_{{15}}^{{49}} {{{\hat{w}}_x} \cdot \left( {\frac{{{p_{{x + 42.5}}}}}{{{p_x}}}} \right) = 0.7764} \;. $$

With Brass’s logit life table model, I transformed this reconstructed cohort survival function to a new survival function with a survival rate of 0.8186—the empirical value $\hat{S}(40)$ from the survey data (setting Brass’s parameter β = 1.0). From this survival function, I derived that the average time of death of respondents’ mothers was 11.92 years prior to the end of the observed survival time (the term in parentheses in Eq. 3). Thus, the reference time for the period estimate was 1998.5 – 11.92 = 1986.58. From the probabilities of dying 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

$$ {\left( {\frac{{{{\hat{l}}_{{73}}}}}{{{{\hat{l}}_{{30}}}}}} \right)_{{1986.58}}} = 0.8186 \cdot \frac{{0.7932}}{{0.7764}} = 0.8363. $$

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 ${\hat{w}_x}$ 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 ${\hat{f}_x}$ (in the following short ${\hat{f}_x}$ schedule) has to be estimated. Therefore, I used a three-step procedure (which is necessary only when the weights ${\hat{w}_x}$ cannot be derived directly from the survey data):

• 1. 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 $\bar{x} = 29.24\,{\text{years}}$ (i.e., the average age at childbearing of the entire Italian population at that time).

• 2. In the next step, I modified the age-specific fertility rates f _x from Step 1 to a new $\hat{f}_x^{*}$ schedule with an average age at childbearing of ${\hat{\bar{x}}^{*}}$ 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, ${\hat{\bar{x}}^{*}} = 28.38$). 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 $\hat{f}_x^{*}$, with ${\hat{\bar{x}}^{*}} = 28.38$). In cases where $\bar{x}$ and ${\hat{\bar{x}}^{*}}$ 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 ${\hat{\bar{x}}^{*}}$ differed from $\bar{x}$. For each year that ${\hat{\bar{x}}^{*}}$ was higher or lower than $\bar{x}$, 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 $\hat{f}_x^{*}$, leading to an age at childbearing of ${\hat{\bar{x}}^{*}}$ years.¹¹

• 3. Applying the fertility rates $\hat{f}_x^{*}$ 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 ${\hat{\bar{x}}^{*}}$ years. From this survival function, I derived an approximate estimate of the age at childbearing of all parents, $\hat{\bar{x}}$, by assuming that ${\hat{\bar{x}}^{*}}$ 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 ${\hat{f}_x}$ schedule, leading to an age at childbearing of $\hat{\bar{x}}$ years (for the example of mothers with primary education of respondents aged 40–44 from the 1998 survey, the estimated $\hat{\bar{x}} = 29.29$). Finally, this fertility schedule and the N _x for the entire Italian population from Step 1 were used to determine the weights ${\hat{w}_x}$ by applying Eq. 2.