3. Application of the Modified DeMoivre Method to Bulgaria

3.1 Estimation of the modified DeMoivre hazard

The Bulgarian mortality pattern for males and females during 1992–93 has already been depicted in Figure 1. In this Section we use the modified DeMoivre hazard function in combination with unobserved heterogeneity in order to evaluate whether the differential increase in mortality by age between males and females in Bulgaria can plausibly be explained by a selection process in which the higher male mortality level leads to a quicker selection of the male population towards low-frailty individuals.

Our analyses are based on an unique database for Bulgaria which is based on a linkage between the death records for the period 5th of December 1992 – 31st of December 1993 and the census on 4th of December 1992. This data-set is the first comprehensive population-based data on mortality in Bulgaria that includes a broad array of socioeconomic information for individuals who are at risk of death. The linkage of the death certificates to the census records is carried out using a personal identification number (PID) included in all official records of an individual in Bulgaria. The linkage between the census and the death registration is of high quality, and in total 92.67 per cent of all death certificates are linked to the census records. Among the linked deaths, 95.07 per cent are based on the PID number. Only 4.93 per cent of the deaths are linked using other identification variables (e.g., place and region of residence, birth day, sex, education, marital status) because the PID number is missing or incomplete. A relatively small fraction of 7.33 per cent of all deaths could not be linked to the census records. [Note 3]

The subsequent analyses include all individuals in Bulgaria who are at least 40 years old at census and are below age 100 either at death or on 31st December 1993. Our data thus comprise 1.87 million males of whom 57,221 die during the observation period, and 2.09 million females of whom 45,831 die during the 13 months after the census.

We first estimate a standard piecewise-constant hazard model (with constant hazards in two-year age intervals) separately for males and females in order to obtain a nonparametric estimate for the observed mortality pattern. The respective estimates have already been depicted in Figure 1(a), and they will also be included in subsequent Figures for comparison with our parametric estimates.

In Table 1 we report the results of different parametric specifications. Model 0 is a standard Gompertz model without unobserved heterogeneity that allows for male-female differences in both parameters a and b. These estimates have been used for the fitted Gompertz hazard curve in Figure 1(a). The estimates reveal that the ‘level-parameter’ a for females is only a fraction of about 0.27 of the respective parameter for males, while the ‘slope-parameter’ b for females exceeds that of males by 0.025. The latter difference implies that the relative increase of mortality rates by each year of age is 2.5 percentage points higher for females than for males and it leads to the — already discussed — strong convergence between the male and female mortality rates in Bulgaria.

Models 1–4 in Table 1 include a Gamma-distributed frailty and are based on a modified DeMoivre hazard function. Except for Model 4, where the maximum attainable age w is estimated from the data, these models assume a w which is equal to Madame Calment’s age at death. The parameters a, b, and s² are specified as in equations (9–11) using sex as the only covariate.

The simplest Model 1 estimates the male and female mortality pattern using identical parameter values b and s² for both sexes, while the parameter a is allowed to vary between males and females in order to capture the different mortality levels. That is, the model allows for different levels of mortality by sex, but it assumes an equal ‘slope-parameter’ b across sex. The model therefore attempts to explain the differential increase in mortality with age merely by the selection hypothesis, i.e., the fact that the male population faces a more rapid selection towards low-frailty individuals due to the higher overall male mortality level. Most importantly, the model yields an estimate of ² = exp(-.6186) = 0.54, indicating a quite substantial heterogeneity in the population. According to this estimate, about 28 per cent of the population at age 40 have a frailty of z £ .5 and 9.7 per cent have a frailty of z ³ 2.

Figure 4(a) shows that this model traces the convergence between the observed male and female mortality rates with increasing age quite well (full lines), despite the fact that the mortality rates for a constant frailty z = 1 increase in a parallel fashion (dashed-dotted line). Hence, Model 1 contributes a substantial part of the observed convergence between male and female mortality rates to the differential strength of the selection process in the male and female population (a formal measurement of the fit of this model and a comparison with alternative Gompertz specifications are provided in the sensitivity analysis in Section 3.3 below).

Model 2 in Table 1 provides an extension of the above model and incorporates a potentially different degree of heterogeneity between the male and female populations. Possible reasons for such a differential variance in unobserved frailty could be a greater variation in life styles (such as smoking habits or other risky behaviors) among the male as compared to the female population. Indeed, the estimates in Table 1 suggest that the male variance of frailty is s_male² = exp(-.4876) = 0.61, while the corresponding variance for females is 35% smaller (s_female² = 0.40). This implies that about 30 per cent of males, but only 22 per cent of females, have a low frailty of z £ .5, and more than 10 per cent of males, but about 8 per cent of females, have a high frailty of z ³ 2 at age 40.

The resulting fit of the Model 2 is depicted in Figure 4(b). Since this model allows for an additional parameter, it fits the Bulgarian mortality pattern slightly better than our earlier model. Differences in the observed slope of the mortality pattern in Model 2 are again only due to differences in the overall mortality level and in subsequent differences in the selection process in a heterogeneous population. The present model therefore suggests that the male population may be more heterogeneous with respect to various biological or socioeconomic determinants of mortality. The specific investigation of this issue is beyond the scope of the present paper, but it is feasible on the basis of our data that includes a broad range of covariates about individuals. Most important in the present context is that Model 2 further confirms our argument that differential selection process provides a plausible explanation for the male-female difference in the increase of mortality with age.

An alternative generalization of our initial estimation is provided in Model 3 in Table 1, where the parameter b, instead of the variance s², is allowed to vary across sexes. The coefficients show that the relative difference in the ‘slope-parameter’ b between males and females is substantially reduced by incorporating unobserved heterogeneity as compared to the Gompertz model without any frailty considerations. This finding is again consistent with an important male-female difference in the strength of selection towards low-frailty individuals in the population. The fit of this model is given in Figure 4(c), where this model performs slightly better than the two earlier models. However, this improvement is not surprising since the specification of a separate slope-parameter b for males and females provides a direct modelling of the differential mortality increase between sexes.

Finally, model 4 in Table 1 estimates the maximum attainable age w in addition to the remaining parameters a, b, and s². While our earlier models were based on a predetermined w of 122.45 years, the present estimate reveals a w of slightly above 104 years. Moreover, the variance of the unobserved frailty has increased to s² = 1.10 (with s² = 1.10, 11 per cent of the population have a frailty of z £ .1, 41 per cent have a frailty of z £ .5, and 14 per cent have a frailty of z ³ 2). While these parameter estimates yield a ‘best-fitting model’ in Figure 4(d), the estimate for w is not plausible and it depends strongly on the age at which the data are censored. For instance, if the data include individuals who survive to age 105 or 110, then the respective estimates for w increase respectively to 108 and 111.5. and that for s² decline to 0.91 and 0.79. The estimates for the parameters a and b are relatively insensitive to changes in the age range above 100. Hence, while Model 4 provides the best fit of the Bulgarian mortality data using a modified DeMoivre hazard with no sex-differences in the slope parameter b or the variance of frailty s², the estimates of this model about w cannot be interpreted in terms of a maximum life-span because the estimation was deliberately censored at age 100 in order to focus on the convergence of male and female mortality in the age-range 40–100. Without survivors to very old ages, however, an estimation that assumes a plausible value for w as in Models 1–3 seems preferable to the direct estimation of w from the data. It is beyond the scope of this paper to apply the modified DeMoivre hazard explicitly to reliable mortality data at ages 100+, but these future applications provide a possibility to estimate interpretable and realistic values for the maximum attainable age w and the changes of this limit to life-span over time.

3.2 Comparison with piecewise-constant hazard with unobserved frailty

In order to assess the empirical plausibility of the modified DeMoivre hazard model we compare the above results with a nonparametric estimation of the hazard curve. This nonparametric alternative is feasible if the mortality hazards for males and females, conditional on the frailty z, differ only by a factor of proportionality. In this case it is possible to combine Gamma-distributed relative frailty with a piecewise-constant hazard function m^PW (x) that does not impose parametric restrictions on the shape of the mortality pattern [Note 4]. This nonparametric estimation of the baseline-hazard m^PW (x) can then be compared to the modified DeMoivre hazard m^MD(x) in order to assess the implications of the parametric assumptions used in the previous section.

The left graph in Figure 5 shows the observed male and female mortality level in Bulgaria along with the estimated baseline hazard m^PW (x) for individuals with a constant frailty z = 1 and the corresponding observed hazard ^PW (x) obtained from the piecewise-constant estimation. The model fits the observed male and female mortality pattern relatively well and the fit is comparable to our earlier Model 4 in Figure 4.

The primary question regarding the estimation of this piecewise-constant frailty model in Figure 5(a) is whether the respective estimates for the variance of the frailty distribution and the increase of the mortality rates for a constant frailty z = 1 are consistent with our knowledge about human mortality. In order to investigate this issue, we compare in Figure 5(b) the estimated baseline hazards m(x) obtained from the piecewise-constant and the modified DeMoivre model.

The graph reveals that the piecewise-constant specification yields the fastest increasing baseline hazard across all estimated models, and it also yields the highest variance s² for unobserved frailty (s² = 1.57). The differences are largest between the piecewise-constant model and the DeMoivre models with w = 122.45 (Models 1 and 2 in Figures 4 and 5b), while it is only modest when compared to the DeMoivre model where w is estimated from the data (Model 4 in Figures 4 and 5b). Since the last model already implied an implausible value of only 104 years for the highest attainable age w, we also consider the increase of the baseline hazard obtained from the piecewise-constant estimation as too steep. Similarly, the variance of the frailty distribution in the piecewise-constant model seems unrealistically high since it suggests that about 19 per cent of the initial population have a frailty of z £ 0.1 and about 15 per cent have a frailty of z ³ 2. However, Iachine et al. [1998] obtained similarly large values for the variance of unobserved frailty from twin data.

While the nonparametric estimation of the baseline hazard with a piecewise-constant model certainly has its virtues, it can also lead to estimates of the baseline-hazard that are implausible. The modified DeMoivre model with w set to 122.5 years provides a possibility to restrict the increase of the baseline hazard with age to values that are consistent with observed survival to very old ages. Moreover, the differences between the observed and estimated mortality pattern in the simplest DeMoivre model in Figure 4(a) suggest alternative specifications that are not feasible with the piecewise-constant estimation. For instance, in Models 2 and 3 in Figure 4 we allow the variance of the frailty distribution or the slope-parameter b to be different between males and females. Both extensions substantially improve the fit of the model. In our opinion these extensions provide a more plausible, and probably also more accurate description of the mortality dynamics than the piecewise-constant analysis. The modified DeMoivre function therefore provides a very suitable hazard function for the application of frailty models to adult ages, and it allows specifications of the parameters that are not available with nonparametric estimations.

3.3 Sensitivity analysis of estimated parameters

In this Section we provide a sensitivity analysis in order to investigate the extent to which the coefficients obtained from the modified DeMoivre hazard function depend on the choice of the maximum attainable age w. Our earlier estimates of Models 1–3 in Table 1 and Figure 4 were based on w = 122.45, i.e., Madame Calment’s age at death. In order to evaluate the sensitivity of our estimates with respect to the choice of w, we reestimate Models 1 and 2 in Table 1 using a w that ranges from 105 to 150 years (we do not report the sensitivity analysis for Model 3 since it leads to similar results).

The top-left graph in Figure 6 shows the relative deviation of the estimated coefficients a_male, a_female, b, and s² as a function of w. The top-right graph in this Figure shows the goodness-of-fit of the Model 1 as a function of w. The goodness-of-fit is calculated as 1 - RSS/SST , where RSS is the residual sum of squares on the logarithmic scale and SST is the total sum of squared deviations from the mean on the logarithmic scale.

The Figure shows that the parameters a and b are relatively insensitive to the specification of w, while the estimated variance of frailty depends quite strongly on the choice of w. The latter is not very surprising since w determines the convexity of the hazard function m^MD(x) in equation (5), i.e., the hazard for individuals with a constant frailty z = 1. A low w implies a quite strongly increasing hazard m^MD(x) with age. This subsequently results in a higher estimate for s² in order for the model to fit to the observed mortality rate.

The top-right graph in Figure 6 shows a goodness-of-fit analysis of Model 1, i.e., a model that allows only for level-differences in mortality across sex but no differences in the ‘slope-parameter’ b The dash-dotted lines in this graph reveal on one hand the fit of the Gompertz model with separate parameters a and b for males and females (i.e., Model 0 in Table 1), and on the other hand the Gompertz model with b constrained equal across sexes. The fit of Model 1 is between these two benchmarks, and it tends to decrease the higher is the value for w. For low values of w, Model 1 fits almost as good as Model 0 with no parameter restrictions across sex. With increasing w this goodness-of-fit declines. Ultimately it approaches the lower dash-dotted line because a rising w renders the modified DeMoivre hazard more and more like a Gompertz model. However, for values of w below 140 years the goodness-of-fit of Model 1 is closer to the Gompertz model with separate slope-parameters for males and females than that of the Gompertz model with equal b for both sexes.

The sensitivity of the estimates for s², therefore, is not as severe as the top-left graph in Figure 6 may suggest. First, the model with w = 122.45, i.e., Madame Calment’s age at death, provides the best-fitting which is based on a maximum attainable age w that is at least as high as the highest age lived by any person so far. Second, for a quite broad range of plausible choices for w, say, between 115 and 130 years, the main conclusion of Model 1 remains unaltered: a substantial part of the male-female differences in the slope of the observed mortality pattern can be explained by a differential strength of the selection process towards low-frailty individuals that is caused by differences in the overall level of mortality between males and females.

The bottom-left and bottom-right graph in Figure 6 show the corresponding sensitivity analysis for Model 2 in Table 1. Similar to Model 1, the estimates for a and b are not very sensitive with respect to the choice of w, while the estimates for s_male² and s_female² change substantially with w. The analysis in the bottom-right graph, however, reveals that these changing estimates for s²-parameters leave the goodness-of-fit of the model almost unaffected.

The choice of w in Model 2 is thus not essential for the main conclusion of the analysis: A modified DeMoivre hazard model with only one ‘slope-parameter’ b for both males and females provides a very good description of the Bulgarian mortality pattern. Moreover, this model attributes a substantial part of the differential male-female increase in mortality by age to a differential strength of the selection process which is caused by the higher overall level of male mortality.

Our preferred choice for w in analyses with the modified DeMoivre hazard is w = 122.45 years based on Madame Calment’s age at death. While the specific estimates for s² are sensitive to this choice, the primary conclusion resulting from the incorporation of frailty in the analysis of Bulgarian mortality is very robust with respect to this specification.