Reasons to suspect that µ(t) trajectories are special in RF regions compared with SFJ emerge upon the very first attempts to analyze data on mortality rates in RF in terms of GGM. Using GGM to extract C estimates from data related to SFJ (Golubev, 2023) yielded values ranging from 0.005 to decreasing down to marginally above zero throughout the 20th century. Hereinafter, the estimates of C derived according to the GGM will be designated as C0. In Table 1, the negative values of C0 estimates are encountered too regularly to think of them as of mere accidents. Although the higher 95% CI margins are above zero in almost all such cases, they all relate to men, none to women; and, even without that, C0 in women are always higher than in men, although it is commonly believed that men expose themselves to the external hazards more than women do. Altogether, this looks as an odd regularity even though any single case may be judged as insignificant statistically.
Table 1
Estimates of GGM parameters derived from demographic data related to different administrative regions of the Russian Federation (RF)
| Men | Women |
| Mean | 95% CI | Mean | 95% CI |
NCFD*, 2018 |
С0 | -0.00043 | -0.00172 | 0.00087 | 0.00087 | 0.00039 | 0.00136 |
µ0 | 0.00019 | 0.00011 | 0.00026 | 0.000003 | 0.000001 | 0.000004 |
γ | 0.076 | 0.071 | 0.081 | 0.124 | 0.118 | 0.131 |
FEFD**, 2018 |
С0 | 0.00112 | -0.00128 | 0.00351 | 0.00250 | 0.00165 | 0.00335 |
µ0 | 0.000358 | 0.000189 | 0.000527 | 0.000010 | 0.000004 | 0.000016 |
γ | 0.072 | 0.067 | 0.079 | 0.110 | 0.103 | 0.118 |
Moscow, 2018 |
С0 | -0.00036 | -0.00194 | 0.00123 | 0.00106 | 0.00052 | 0.00161 |
µ0 | 0.00033 | 0.00016 | 0.00051 | 0.000015 | 0.000002 | 0.000009 |
γ | 0.064 | 0.058 | 0.071 | 0.112 | 0.105 | 0.120 |
Saint Petersburg, 2018 |
С0 | -0.00106 | -0.00225 | 0.00012 | 0.00118 | 0.00070 | 0.00160 |
µ0 | 0.000465 | 0.000328 | 0.000602 | 0,000010 | 0,000006 | 0,000014 |
γ | 0,063 | 0,060 | 0,067 | 0,106 | 0,101 | 0,112 |
Moscow, 1989*** |
С0 | -0.00031 | -0.00139 | 0.00077 | 0.00068 | 0.00025 | 0.00111 |
µ0 | 0.000324 | 0.000253 | 0.000394 | 0.000016 | 0.000012 | 0.000019 |
γ | 0.073 | 0.071 | 0.076 | 0.108 | 0.105 | 0.110 |
*Features the highest life expectancy in RF; **Features the lowest life expectancy in RF; *** the first year available in RFMD |
For finding a cause of this oddity, LAR trajectories shown in Fig. 3, where the aging-related segments of the trajectories shown in Fig. 2 are presented at a higher resolution, will first be examined.
In SFJ, LAR trajectories tend to increase in the age interval from 25 to 45 years, to stay relatively constant from 45 to 65 years, and to feature a hump in the age interval from 65 to 95 years. Reasons were suggested in (Golubev 2023) to regard the LAR trajectory between ca. 25 and 65 years as a manifestation of the effect of C0 on the early LAR (a downward bias, which is most prominent at the earliest ages and gradually decreases to reach zero at later ages), whereas the late-life hump of LAR, as a manifestation of changes in the biological aging rate. The patterns derived from data related to NCFD and Moscow (the latter is not shown because it is quite similar to the former), which are the two regions featuring the highest LE in the RF, show prominent LAR peaks in the middle of lifespan, whereas LAR trajectories are almost flat within this age range in SFJ (even quite flat in men in Japan (Golubev 2023)), and in the early age interval, wherein LAR only increases in SFJ. As a result, three major peaks may be distinguished in LAR trajectories in NCFD. For convenience, they will be called the early, middle and late peaks (or humps) hereinbelow. In FEFD, there are only two major peaks in LAR trajectories, probably because the middle peak is either markedly shrunk or merged with one of the two other peaks. These trajectories are also peculiar in having a pronounced dip in the middle of lifespan. Both NCFD and FEFD (and Moscow, not shown) have a common peculiarity, which is a bizarre gender-related difference between the late-age LAR trajectories.
An approach to making ends meet in situations where C0 according to GMM are negative is prompted by the consideration that positive C0 make lnµ(t) trajectories slightly concave (running above the tangent to any specific point). Therefore, C0 may become negative if the trajectories are convex, even slightly. Such convexity may emerge if an increase in C(t) is not distributed evenly, but is confined to the middle of a lifespan. This indeed seems to be the case in RF (Brainerd 2021); and the middle humps in the apparent LAR trajectories in NCFD (Fig. 3) are consistent with that. Noteworthy, in FEFO, there is a dip there, which is consistent with a concave lnµ(t) trajectory, which in its turn is consistent with a positive C0.
A generic model of such a situation is shown in Fig. 4.
When the external hazards that are irresistible irrespective of age are virtually absent, LAR is constant (open circles series in Fig. 4). The presence of evenly distributed irresistible hazards will make LAR underestimated at earlier ages. The degree of this bias will asymptotically approach zero (the blue line) with increasing age because the share of C0 in µ(t) decreases when the Gompertzian term of GMM increases exponentially. Reasons to believe that evenly distributed hazards captured by C0 do exist are provided by the publications where causes of death partitioning revealed that mortality rates associated with some of the causes, such as road accidents, remain fairly constant throughout adult human lifespan (Dolejs 1997, Newman and Easteal 2017). Supplementing C0 with a small increment confined to a segment in the middle of lifespan will somewhat elevate the respective segment of the lnµ(t) trajectory, and, even with such a small elevation, LAR trajectory becomes distorted markedly. Importantly, when µ(t) corresponding to this situation is treated according to GMM, the estimates of C related to the presence of the increment become negative. The initial C0, to which the increment is added, is + 0,0004; yet, the apparent C0 is −0,0016. This negative value is achieved with a modification of µ(t) trajectory, which is even hardly discernable visually in a double linear plot.
To modify C with an increment having a peak about the middle of a lifespan and approaching zero at its extremes, several functions were tried, including probability density functions (PDF) of the normal, lognormal, gamma-, Weibull, and Wald distributions. None was found to provide for a peak looking symmetric on a semilogarithmic scale and being sufficiently broad to lead to a negative estimate of C when included in a GGM model. An appropriate shape of an increment can be achieved by convoluting a PDF such as those above with an exponential PDF (a decreasing exponent). In Fig. 4, the convolution of the PDFs of the gamma- and exponential distributions is used. This function (exponentially modified gamma-distribution, EMGD) was suggested earlier for modeling cell cycle and gene transcription time variabilities based on certain mechanistic premises, which allow ascribing physical (biological) meaning to EMGD parameters (Golubev 2016, 2017). Here, EMGD is found useful for purely descriptive purposes, at the least, for showing the possibility that an increment in C confined to a middle segment of lifespan may decrease the apparent C0 even when the lifelong exposure to hazards captured by C(t) is actually increased.
To present the notations used below, EMGD components may be written as
$$E\left(t\right)=\frac{1}{k}{\text{e}}^{-\frac{1}{k}} , G\left(t\right)=\frac{{t}^{c-1}{\text{e}}^{\frac{-1}{b}}}{{b}^{c}\left(c\right)}, \text{a}\text{n}\text{d} Q\left(t\right)=E\left(t\right)*G\left(t\right)$$
where Γ(c) is a gamma-function, and \(*\) denotes convolution.
The closed-form formula for Q(t), which was introduced in (Golubev 2016, 2017) and is used for modeling here, is too cumbersome to show it. Instead, the effects of arbitrary changes in EMGD parameters made to adjust them to particular situations, such as discussed herein, are shown in Fig. 5, which evidences a high versatility of EMGD, its plot being amenable to translocation along the X-axis and to drastic modifications of its shape due to appropriate changes in EMGD parameters (the left column of Fig. 5).
Figure 5. The main principles of reconstructing the salient features of the adult-age LAR trajectories shown in Fig. 3. The left column shows how changes in EMGD parameters can modify EMGD plot position and shape. The right column shows how the resulting changes in C(t) and/or µ(t) (I) are translated into age-associated changes in LAR (II and III) and µ(t) (IV). In the right column, the time scale is shifted by 25 years relative to that in the left column because aging is assumed to start at about 25 years and GGM is applied to the lifespan interval starting at this age.
The upper right plot (I) shows the lnC(t) trajectories that are produced due to multiplying Q(t) by factors appropriate for constructing LAR humps having peaks at different ages.
The LAR plot (II) shows the superposition of LARs each having only one peak of the three observed in NCFD, the GMM parameters being set to values within ranges derived from human data (e.g. Table 1): µ0 = 0.0001, g = 0.1, and C0 = 0.0001.
Importantly, the transient increments sufficient for modifying LAR to the extents comparable to those seen in Fig. 3 are produced by multiplying Q(t) by factors that have to be increased with increasing t, although the under-the-peak areas of all unmultiplied Q(t) are similar. This is because the increments produce their effects against the exponentially increasing Gompertzian component of GGML and, therefore, must roughly match this increase. Besides that, the increase in this component virtually nullifies the relative contributions of the far ends of the right shoulders of EMGD peaks to changes in LAR at later ages.
At differences from the upper LAR plot (II), which shows the superposition of the three LAR trajectories produced by the respective increments in C0, the lower LAR plot (III) shows not a superposition, but the combined effect of all three Q(t) included in the Model I:
F(t) = d{ln[C0 + A1Q(t, k1, b1, c1) + A2Q(t, k2, b2, c2) + A3Q(t, k3, b3, c3) + µ0exp(gt)]}/dt
where: F stands for LAR; Ai stands for a multiplier at Q; and AiQ(t, ki, bi, ci) = Ci(t).
A notable feature of the interaction between LAR humps derived from AiQi and Ai+1Qi+1 is that a decrement in LAR related to the descending part of a previous AiQi can decrease the manifested effect of the subsequent Ai+1Qi+1, which nevertheless remains apparent.
The second LAR trajectory (dotted line) in the lower LAR plot (III) is produced by substituting the increment in C(t) confined to late ages with an increment in g(t) confined to a similar late age interval according to the Model II:
F(t) = d{ln[C0 + A1Q(t, k1, b1, c1) + A2Q(t, k2, b2, c2) + µ0exp((g0 + A4Q(t, k4, b4, c4)t)]}/dt
where: the component that makes the two models different is underlined and g0 + A4Q(t, k4, b4, c4) = g(t).
The late-age LAR humps produced by both models are virtually identical. In Model II, this hump is also influenced by the middle hump whose aftereffect makes the later hump somewhat lower than that modeled separately (not shown).
The resulting lnµ(t) trajectory is shown in the lower right plot (IV) where it is compared with the lnµ(t) trajectory at the same C0, which is not supplemented with any increments (pure GMM). The slight elevation of the middle part of the trajectory, which makes the trajectory concave, is sufficient to produce C0 = −0.00009 (95% CI: −0.000019, 0.000000).
The crucially important question that emerges upon examining the lower LAR plot (IV) is whether the late-age LAR hump is produced by an increment in g(t), as it was suggested earlier (Golubev, 2023), on in C(t)? In both cases EMGD parameters may be adjusted to make LAR humps strikingly similar. One argument in favor of the first possibility follows from examining the upper right plot (I): an increment in C(t) required to produce a realistically looking late-life LAR hump must be unrealistically huge.
To see how true may it be when it comes to modelling LAR derived from demographic data, the parameters of the two models were adjusted so as to fit LAR models to LARs derived from data related to men and women in NCFD (Fig. 6).
Given that adjusting the fifteen parameters of each model had to be carried out manually to begin with and, therefore, better fits could be achieved by efforts more meticulous and time consuming, and that EMGD may not provide for the most accurate description of each and every C(t) and g(t) hump, the correspondence between the modeled and data-derived trajectories looks suggestive, at the least. Moreover, the parameters of the two models are successfully adjusted so as to make the respective two trajectories of the late-life LAR hump virtually identical in each case. The question is then, once again, what differences between the models determine which of them is preferential? One argument in favor of Model II is suggested by comparing the trajectories of C(t) and g(t) required to model the late-life LAR hump (Fig. 7).
The main point illustrated in Fig. 7 is that to model the late hump of a LAR, the required increments in g(t) must be within 10% of g0, whereas in C(t), they must be tens, even hundreds, times higher than C0. That is, increases in the exposure of a population to the irresistible hazards required to produce the late-life LAR hump must be so high that it is unreasonable to explain the hump with them; therefore, the acceleration of biological aging after ca. 65 years is a more plausible explanation.
The second argument in favor of increases in g(t) rather than in C(t) as the cause of the late-life LAR humps follows from comparing the age-at-death (lifespan) distributions constructed according to the two models and derived from demographic data (Fig. 8)
Taken together, the above makes it reasonable to use only Model II for comparisons between NCDO, FEFO and SFJ (Fig. 9).
With account for that model parameters were adjusted manually, there is no doubt that better fits are possible and, most probably, the best fits may be found among them by applying sophisticated techniques developed for multiparametric sloppy models (Evangelou, Wichrowski et al. 2022, Jagadeesan, Raman et al. 2023, Quinn, Abbott et al. 2023).
Another source of uncertainty in relations between modeled and data-derived LAR trajectories is the effects of smoothing procedures. It was shown (Golubev 2023) that LOESS, as it is implemented in TC2D at automatically set optimal smoothing levels of about 14–16% in specific cases and applied to initial lnµ(t) data series related to the total age range, reproduces, upon differencing, the result of the more commonly used moving averaging procedure applied to calculated LAR data series. Hereinafter, other smoothing techniques where applied, at difference from hereinbefore, to only the age segment of interest. Figure 10 compares the results of LAR modeling and of data-derived LAR smoothing using Fast Fourier Transform (FFT), Savitzky-Golay Filtering (SGF) and Kaiser-Bessel Filtering (KBF) applied to data in the age segment 25 to 95 years. FFT and KBF were used at smoothing levels 45% and 34.4%, respectively, which were set automatically for data related to NCDF women. The automatic setting for SGF (19.9%) produced excessively winding trajectories, and therefore the level of 35% was set manually. All other datasets were smoothed using the same settings irrespective of the automatically generated suggestions, which could differ between specific cases.
The discrepancies between the results of using different smoothing procedures are not unexpected; however, they do not obscure neither the major trends evident in modeled LAR trajectories fitted to the results of a still other smoothing procedure, which is LOESS, nor the gender- and population-related differences between the trends. The undulations produced by KBF and FFT (but not SGF) upon applying them to SFJ data may be related to that data series related to three different counties were combined in this case. When the undulations are reduced by increasing the level of smoothing, the general trends still remain evident. The limited experience gained so far is in favor of SGF (dark thin trajectories in Fig. 10) for lnµ(t) smoothing. In the present work, LOESS was used in the continuation of the previous study.
Developing the most adequate approaches to smoothing of lnµ(t) data series and to fitting of LAR model parameters may be a trend for future developments. As of today, even with what has been achieved so far, Figs. 9 and 10 prompt some robust inferences, which are discussed in the next section.