Bull Math Biol (2014) 76:2144–2174 DOI 10.1007/s11538-014-0005-0 ORIGINAL ARTICLE
Age Trajectories of Mortality from All Diseases in the Six Most Populated Countries of the South America During the Last Decades Josef Dolejs
Received: 23 January 2014 / Accepted: 25 July 2014 / Published online: 15 August 2014 © Society for Mathematical Biology 2014
Abstract Age trajectories of total mortality represent an irreplaceable source of information about aging. In principle, age affects mortality from all diseases differently than it affects mortality from external causes. External causes (accidents) are excluded here from all causes, and the resultant category “all-diseases” is tested as a helpful tool to better understand the relationship between mortality and age. Age trajectories of alldiseases mortality are studied in the six most populated countries of the South America during 1996–2010. The numbers of deaths for specific causes of death are extracted from the database of WHO, where the ICD-10 revision is used. The all-diseases mortality shows a strong minimum, which is hidden in total mortality. Two simple deterministic models fit the age trajectories of all-diseases mortality. The inverse proportion between mortality and age fits the mortality decreases up to minimum value in all six countries. All previous models describing mortality decline after birth are discussed. Theoretical relationships are derived between the parameter in the first model and standard mortality indicators: Infant mortality, Neonatal mortality, and Postneonatal mortality. The Gompertz model extended with a small positive quadratic element fit the age trajectories of all-diseases mortality after the age of 10 years. Keywords
Mortality · Age · All diseases · External causes · South America
1 Introduction Age trajectories of human mortality are usually studied for total mortality or for a specific set of causes of death (e.g., Arbeev et al. 2011; Feichtingera et al. 2004; Lin
J. Dolejs (B) Department of Informatics and Quantitative Methods, University of Hradec Králové, Rokitanského 62, 500 03 Hradec Králové, Czech Republic e-mail: [email protected]
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Table 1 International statistical classification of diseases and related health problems, 10th revision Chapter
Blocks
Title
I
A00-B99
Certain infectious and parasitic diseases
II
C00-D48
Neoplasms
III
D50-D89
Diseases of the blood and blood-forming organs and certain disorders involving the immune mechanism
IV
E00-E90
Endocrine, nutritional and metabolic diseases
V
F00-F99
Mental and behavioral disorders
VI
G00-G99
Diseases of the nervous system
VII
H00-H59
Diseases of the eye and adnexa
VIII
H60-H95
Diseases of the ear and mastoid process
IX
I00-I99
Diseases of the circulatory system
X
J00-J99
Diseases of the respiratory system
XI
K00-K93
Diseases of the digestive system
XII
L00-L99
Diseases of the skin and subcutaneous tissue
XIII
M00-M99
Diseases of the musculoskeletal system and connective tissue
XIV
N00-N99
Diseases of the genitourinary system
XV
O00-O99
Pregnancy, childbirth and the puerperium
XVI
P00-P96
Certain conditions originating in the perinatal period
XVII
Q00-Q99
Congenital malformations, deformations and chromosomal abnormalities
XVIII
R00-R99
Symptoms, signs and abnormal clinical and laboratory findings, not elsewhere classified
Diseases
Non-biological causes XIX
S00-T98
XX
V01-Y98
Injury, poisoning and certain other consequences of external causes External causes of morbidity and mortality
XXI
Z00-Z99
Factors influencing health status and contact with health services
XXII
U00-U99
Codes for special purposes
and Liu 2007; Lutz et al. 1998; Riggs 1992; Robine et al. 2012; Shkolnikov et al. 2012; Su and Sherris 2012; Vaupel et al. 1998; Yashin et al. 2008; Zheng et al. 2011). They represent an irreplaceable source of information about the relationship between mortality and age, and it is well known that external causes affect total mortality as the set of causes with a fractionally age-independent mortality rate (e.g., Makeham 1860; Vaupel et al. 1998; Willemse and Koppelaar 2000). The most recent WHO definition of death causes is in Table 1, and external (or non-biological) causes represent the last four chapters. External causes affect total mortality, especially within the age range of 5–30 years, but lose significance over the age of 40 (e.g., de Beer 2012; Granados and Ionides 2011; Heligman and Pollard 1980; Preston et al. 2001 or see Fig. 2). One typical example of age trajectory of mortality from all non-biological causes is in Fig. 1 in the log-log scale and concurrently in the semi-logarithmic scale in Fig. 2. Specific fractions of non-biological causes of total deaths are between 0.01 and 0.14 in the interval [0,1) year, between 0.05 and 0.18 in the interval [0,5) years, between
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Brazil 2010 in the log-log scale
Logarithm of mortality per 10 5 per 1 year
1000000
All causes (ICD10: A00-Z99) All diseases (ICD10: A00-R99) Non-biological causes (ICD10: S00-Z99)
100000
__ 10000
___
Gompertz model Model = μ 1 /x
1000
100
10
logarithm of age [years] 1st day
1
1-7 days
7-28 days 28days-365days 1-2 years 10-15 years
-8
Fig. 1 Age trajectories of mortality in Brazil in the log-log scale in 2010
Brazil 2010 in the semi-logarithmic scale
Logarithm of mortality per 10 5 per one year
1000000
All causes (ICD10: A00-Z99) All diseases (ICD10: A00-R99) Non-biological causes (ICD10: S00-Z99)
100000
__
10000
___
Gompertz model Model = μ 1 /x
1000
100
10
age [years] 1
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Fig. 2 Age trajectories of mortality in Brazil in the semi-logarithmic scale in 2010
0.32 and 0.79 in the interval [5,30) years and between 0.04 and 0.16 in the interval [30,95) years in the six countries during the studied period 1996–2010. It is evident that the age trajectory of mortality from non-biological causes distinctively differs
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from the age trajectory of all-causes mortality (de Beer 2012; Heligman and Pollard 1980; Vaupel et al. 1998; Willemse and Koppelaar 2000). In addition, it can be assumed that the mechanism which is responsible for the relationship between mortality and age differs between all-diseases and non-biological causes. The possibility to apply the category all-diseases as a meaningful tool to describe and understand the relationship between mortality and age is examined here. Age trajectories of mortality from all-diseases in Argentina, Brazil, Chile, Colombia, Peru and Venezuela are analyzed in all calendar years when the ICD10 classification of causes was used. The population of the six countries represented about 92 % of the South America in July 2013 and about 5 % of the world’s population. The exclusion of non-biological causes from total mortality can be realized only if information about specific causes of death is known in convenient age categories. Such information is available in the mortality database of the World Health Organization (World Health Organization 2013). The database usually uses four age categories in the first year of life, enabling the construction of detailed age trajectory of mortality in the first year. 2 Materials and Methods The numbers of deaths in the six countries for detailed causes of death in specific age categories are extracted from the file “Mortality, ICD-10” available in the mortality database of World Health Organization (World Health Organization 2013). The ICD10 classification of causes of death is used in the database (Table 1; World Health Organization 1997), and the database usually uses the following 26 age intervals: [0,24) h, [1,7), [7,28), [28,365) days, [1,2), [2,3), [3,4), [4,5), [5,10), [10,15),… [90,95) years. Unfortunately, the first age interval is also [0,1) year or the second interval is [1,5) years in a specific country in a specific calendar year. If these two age categories are used, such calendar years are excluded (see the first column in Table 2). The resulting age trajectories of mortality from all-diseases are constructed for the calendar years using the classification ICD10, four age categories in the first year and four age categories in the range of 1–5 years. The numbers of living people are obtained from the US Census Bureau (Bureau of the Census 2013). The set “all-diseases” is constructed from the first 18 chapters of ICD10 (Table 1; World Health Organization 1997). The excluded set “non-biological causes” contains the last four chapters. In fact, no deaths are in the last two chapters for any age category, in any country and in any calendar year, and the chapters have no meaning. Arithmetic mean of age limits in age category is used as a representative point in all calculations, and all analyses were completed in 2014. 3 Results The first calendar year for Argentina, 2003, is shown as an example in Fig. 3 in the log-log scale and also in Fig. 4 in the semi-logarithmic scale. The first and the last calendar years of every country are shown in the log-log scale in the “Appendix” (see all odd-numbered Figs. 9, 11, 13, 15, 17, 19, 21, 23, 25, 27) and also in the semi-
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Table 2 Results of the linear model in the log-log scale in the age interval [0,A) years Year
A
Slope
95 % CI
μ1
R¯ 2
Rb2
Rb2 – R¯ 2
C¯ 2a
C¯ 2b
C¯ 1c
Result
ln(x) ln(x)2 c/x
γ Argentina 2003 15
−1.064 −1.154, −0.973 160.9 0.9880 0.9858 −0.00225 1.6
223.8
2.4
c/x
2005 10
−1.068 −1.158, −0.977 128.9 0.9898 0.9871 −0.00271 4.8
293.0
2.7
c/x
2006 10
−1.067 −1.160, −0.974 124.1 0.9892 0.9866 −0.00261 4.7
274.7
2.5
c/x
2007 10
−1.063 −1.162, −0.964 126.2 0.9877 0.9857 −0.00197 7.5
321.5
1.9
Q
2008 10
−1.074 −1.172, −0.974 123.1 0.9879 0.9847 −0.00313 5.3
265.1
2.7
c/x
2009 10
−1.063 −1.159, −0.967 123.7 0.9884 0.9864 −0.00206 4.6
254.9
2.0
c/x
2010 10
−1.082 −1.180, −0.984 114.3 0.9884 0.9841 −0.00428 6.9
319.4
3.5
Q
2006 10
−1.062 −1.164, −0.959 148.6 0.9868 0.9851 −0.0017
6.0
262.9
1.6
c/x
2007 10
−1.069 −1.170, −0.966 138.9 0.9872 0.9847 −0.0025
6.7
289.3
2.1
Q
2008 10
−1.069 −1.163, −0.973 139.1 0.9889 0.9862 −0.0027
5.4
287.2
2.5
c/x
Brazil
2009 10
−1.072 −1.156, −0.988 137.7 0.9913 0.9878 −0.0034
5.7
370.0
3.8
c/x
2010 10
−1.079 −1.147, −0.998 131.7 0.9941 0.9896 −0.0045
4.2
452.6
6.8
c/x
Chile 1997 15
−1.011 −1.115, −0.906 112.1 0.9822 0.9840
0.0019
0.8
112.6 −0.3
c/x
1998 15
−1.019 −1.109, −0.928 116.0 0.9869 0.9880
0.0011
0.6
164.6 −0.1
c/x
1999 15
−1.033 −1.148, −0.917 103.2 0.9792 0.9805
0.0013
0.7
97.3
0.1
c/x
2000 15
−1.047 −1.164, −0.929
0.0004
0.9
94.9
0.5
c/x
2001 10
−1.046 −1.132, −0.958
94.1 0.9902 0.9896 −0.0007
1.6
193.5
1.1
c/x
2002 10
−1.048 −1.120, −0.974
90.3 0.9930 0.9919 −0.0012
1.2
247.2
2.0
c/x
94.0 0.9791 0.9794
2003 10
−1.073 −1.171, −0.974
79.7 0.9880 0.9850 −0.0031
0.8
131.3
2.6
c/x
2004 10
−1.083 −1.172, −0.994
83.2 0.9905 0.9858 −0.0047
1.2
180.8
4.5
c/x
2005
5
−1.116 −1.228, −1.004
69.4 0.9884 0.9793 −0.0091
1.5
120.7
6.0
c/x
2006 10
−1.087 −1.158, −1.014
76.8 0.9938 0.9882 −0.0055
0.6
234.5
7.7
L
2007
−1.121 −1.225, −1.015
68.1 0.9898 0.9798 −0.0101
1.7
141.9
7.4
c/x
5
2008
5
−1.132 −1.221, −1.041
61.6 0.9926 0.9802 −0.0124
0.8
154.1 12.3
L
2009
5
−1.133 −1.239, −1.026
63.7 0.9898 0.9775 −0.0122
1.8
143.7
L
8.8
Colombia 1997 15
−0.979 −1.059, −0.898 175.0 0.9887 0.9895
0.0008
1.8
244.8
0.02 c/x
1998 15
−1.025 −1.104, −0.945 195.3 0.9899 0.9904
0.0006
1.3
250.8
0.18 c/x 0.60 c/x
Peru 1999 10
−0.963 −1.049, −0.876 216.1 0.9886 0.9886
0.0000
4.9
265.6
2000 10
−0.954 −1.027, −0.880 194.4 0.9916 0.9903 −0.0013
7.2
437.5
1.84 Q
2002 15
−0.957 −1.032, −0.880 163.9 0.9894 0.9885 −0.0009
1.3
240.4
1.40 c/x
2003 15
−0.974 −1.043, −0.904 174.0 0.9915 0.9917
0.0003
1.7
319.2
0.39 c/x
2004 15
−0.964 −1.040, −0.886 169.7 0.9892 0.9890 −0.0002
1.3
238.7
0.84 c/x
2005 15
−0.977 −1.052, −0.901 145.6 0.9899 0.9905
1.5
261.0
0.15 c/x
123
0.0006
Age Trajectories of Mortality from All Diseases
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Table 2 continued Year
A
Slope
95 % CI
μ1
R¯ 2
Rb2
Rb2 – R¯ 2
C¯ 2a
C¯ 2b
C¯ 1c
Result
ln(x) ln(x)2 c/x
γ 2006 10
−0.983
−1.078, −0.886 130.4 0.9866 0.9880
0.0014 7.9
309.6 −0.21 Q
2007 10
−0.976
−1.072, −0.879 124.2 0.9862 0.9874
0.0011 8.9
327.6 −0.05 Q
Year
Slope γ 95 % CI
A
μ1
R¯ 2
Rb2
Rb2 – R¯ 2
C¯ 2a
C¯ 2b
C¯ 1c
Result
Venezuela 1996 15
−1.002
−1.097, −0.907 255.8 0.9851 0.9867
0.0017 3.7
248.2 −0.3
c/x
1997 15
−1.018
−1.108, −0.927 229.4 0.9868 0.9880
0.0012 4.4
301.5 −0.1
c/x
1998 10
−1.023
−1.144, −0.901 210.7 0.9802 0.9822
0.0020 6.2
188.9 −0.2
c/x
1999 15
−1.011
−1.110, −0.910 212.2 0.9837 0.9854
0.0017 2.7
201.3 −0.3
c/x
2000 15
−1.018
−1.117, −0.918 204.1 0.9841 0.9855
0.0014 3.2
220.2 −0.2
c/x
2001 15
−1.012
−1.105, −0.918 205.7 0.9858 0.9872
0.0015 2.4
216.7 −0.3
c/x
2002 10
−1.051
−1.156, −0.944 174.0 0.9856 0.9851 −0.0005 4.9
216.9
c/x
0.9
2003 10
−1.026
−1.155, −0.896 213.2 0.9775 0.9797
0.0022 8.8
213.9 −0.2
2004 15
−1.035
−1.123, −0.946 194.3 0.9878 0.9881
0.0002 2.4
251.5
0.5
c/x
2005 15
−1.066
−1.169, −0.962 163.4 0.9843 0.9823 −0.0021 1.9
185.2
1.8
c/x
Q
2006 10
−1.072
−1.167, −0.975 154.6 0.9887 0.9857 −0.0030 6.4
314.5
2.7
c/x
2007 10
−1.076
−1.174, −0.976 149.6 0.9880 0.9846 −0.0034 6.5
301.9
2.9
Q
A is the upper age limit of the age category when mortality reaches the minimal value, C I 95% confidence interval; the parameter μ1 is per 105 persons per one year, the adjusted coefficient of determination R¯ is calculated in the linear model (2) for one predictor and n points, and the coefficient of determination Rb2 is calculated in the model (3); C¯ p = C p − 2∗ (k − p + 1)/(n − k − 3), where n is the number of points, k is the number of candidate regressors in a full model without constant (k = 2 in the quadratic model and k = 1 in the linear model), p is the number of parameters in a submodel ( p = 2 in the submodel model (2) and p = 1 in the submodel 3); “Q” means the quadratic full model (1); “L” means the linear submodel (2), and “c/x” means the submodel (3); C¯ 2a is the Mallow’s statistic in the submodel “L” with the predictor ln(x); C¯ 2b is the Mallow’s statistic in the submodel with the predictor ln(x).ln(x); C¯ 1c is the Mallow’s statistic in the submodel “c/x”(“L” is the full model in the second step and the quadratic model should be rejected in the first step in these cases); the last column contains result models according to the Gilmour test for P < 0.05 (Gilmour 1996)
logarithmic scale (see all even-numbered Figs. 10, 12, 14, 16, 18, 20, 22, 24, 26, 28). Mortality from all-diseases reaches the minimal value in the three different age [4,5), [5,10) and [10,15) years. The upper age limit “A” of the specific age category in which mortality reaches the minimal value is in the second column in Table 2. The age axis is divided into two parts so that age trajectories are analyzed separately in the interval [0,A) years and in the interval [A,95) years. Age “A” is the upper limit of age category where mortality reaches the minimal value. The resulting age trajectories of mortality from all-diseases show the linear dependence in the log-log scale in 0–A years in all calendar years and in all countries. For example, Heligman and Pollard (1980) or Siler (1979) supposed an exponential model of total mortality decline with age after birth. The supposed exponential model declines linearly in the semi-logarithmic scale, but data show the nonlinear
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Argentina in 2003 in the log-log scale
Logarithm of mortality per 10 5 per 1 year
1000000
All diseases 2003 100000
Model = μ 1 /x and Gompertz model
10000
1000
100
10
logarithm of age [years] 1st day
1
1-7 days 7-28 days
28days-365days1-2 years
10-15 years
-8
Fig. 3 Age trajectory of all-diseases mortality fitted by the two models in Argentina in the log-log scale in 2003
Argentina in 2003 in the semi-logarithmic scale
Logarithm of mortality per 10 5 per one year
1000000
100000
All diseases 2003 Model = μ 1 /x and Gompertz model
10000
1000
100
10
age [years] 1
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Fig. 4 Age trajectory of all-diseases mortality fitted by the two models in Argentina in the semi-logarithmic scale in 2003
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convex trajectories in the semi-logarithmic scale. The nonlinear convex decline in the semi-logarithmic scale among those aged 0–A years is also ascertained visually for all cases. On the other hand, linear age trajectories of mortality are visually found out in the semi-logarithmic scale in all calendar years and in all countries in the second interval [A,95) years. The approximate linear data of all-causes mortality and all-diseases mortality in Brazil in 2010 correspond to the Gompertz exponential model, shown in the semi-logarithmic scale in Fig. 2. The data for all-diseases and for all-causes are identical over the age of 40, while the data are already linear by the age of 10 for the category of all-diseases. This approximate linearity is ascertained in all cases. 3.1 Age Interval 0–A Years At first, the linearity in the log-log scale is tested in the following full model using the method of least squares (MLS): ln(μ(x)) = constant + γ .ln (x) + δ.ln (x) .ln (x)
(1)
The null hypothesis Ho : δ = 0 is not rejected in all cases (with the two-sided P values >0.05 in any country in any calendar year). The slope γ is significant in all cases (the two-sided P values xmin . The first day can be chosen as xmin , and the theoretical number of deaths within the first 24 h can be estimated using a single point of mortality at an age of 12 h. Mortality rate at 12 h = 1/730 years is μ(x = 1/730) = 730. μ1 . Simultaneously, it is the number of deaths per one year per one living 365.D1 /L, where D1 is the absolute number of deaths within the first day and L is the number of live births. Consequently, 365.D1 /L = 730. μ1 and D1 /L = 2. μ1 . Furthermore, the analytic survival function S(x) in model (3) could be applied for x ≥ xmin and it is valid: −dS(x) dx (x) = −d ln[S(x)] dx ⇒ ln[S(x)] = − μ(x)dx because μ(x) = S μ1 , for x ≥ xmin ⇒ ln[S(x)] if γ = −1 then μ(x) = x = −μ1 ln(x) + c, for x ≥ xmin if S(xmin ) = 1 ⇒ ln[S(xmin )] = 0 and ⇒ c = μ1 · ln(xmin )
μ1 and S (xn ) = xmin x ⇒ ln[S(x)] = μ1 · ln xmin x =
−μ1
x xmin for x ≥ xmin
(10)
Because S(x) = (365.x)−μ1 for x ≥ xmin = 24 h= 1/365 year, Infant mortality (the number of deaths divided by L live birth within the age interval [0,1) year) is as follows:
1q0 = D1 /L + D2 /L = 2.μ1 + 1 − S (x = 1 year) = 2.μ1 + 1 − 365−μ1 , (11) where D2 is the number of deaths in the age interval [1,365) days (the approximation D2 /L/D2 /(L − D1 ) is assumed here).
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Parametr μ 1 in Chile during the period 1997-2009
160
μ 1 per 100 000 per one year
140
120
100
80
60
40
I. - the values calculated from Infant mortality (Instituto Nacional de Estadísticas, Chile) and using numerical solution of the equation (10) II. - the values from Table 2 calculated using MLS in the model (3) and the WHO data
20
0
1997
1999
2001
2003
2005
2007
2009
Fig. 8 Two sets of parameter μ1 calculated from infant mortality and in the model (3)
Similarly, Postneonatal mortality, within the age interval [28/365,1) years, is given as follows:
PNM = 1 − S (x = 1 year) − 1 − S (x = 28/365 years) = (28)−μ1 − (365)−μ1 (12) and Neonatal mortality within the age interval [0,28/365) years is as follows: NNM = 2.μ1 + 1 − S (x = 28/365 years) = 2.μ1 + 1 − (28)−μ1
(13)
Conversely, the estimation of parameter μ1 could be calculated using a numerical solution of one of the formulas (11), (12) or (13) if one of the three values of 1q0 , PNM and NNM is known. For example, parameter μ1 is numerically calculated here using equation (11) and data of Infant mortality 1q0 downloaded from the Instituto Nacional de Estadísticas, Chile, for calendar years 1997–2009 (Instituto Nacional de Estadísticas, Chile 2013). The values are the time series I. in Fig. 8, and the second time series II. represents the values calculated in the model (3) (the 6th column in Table 2 for Chile). The values calculated in the model (3) for all diseases are systematically lower because external causes are not negligible in Chile even in the range of 0–1 year. 3.4 Age Interval A–95 Years Age trajectories of mortality from all-diseases are approximately linear in the semilogarithmic scale over age A. At first, the linearity in the semi-logarithmic scale is tested in the following full model using MLS in all of cases:
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Table 3 Results of the two models in the semi-logarithmic scale in the second age interval [A,95) years Year
n
δ
P-value
β
0.49
0.091
ln(x)
ln(x)2
R¯ q2
2 R¯ q2 − R¯ lin
4.3
0.9988
0.9986 −0.0001
C¯ 2d
C¯ 2f
Result
Argentina 2003
16
7.5 334.6
Q
2005
17
0.18
0.090
4.3
0.9978
0.9975
0.0002
11.0 226.8
Q
2006
17
0.12
0.089
4.5
0.9978
0.9974
0.0003
11.7 217.5
Q
2007
17
2008
17
0.0001
2009
17
2010
17
0.0001
Year
n
δ
0.04
0.077
6.0
0.9978
0.9975
0.0007
14.0 208.1
Q
0.14
0.088
4.6
0.9974
0.9971
0.0003
11.5 189.8
Q
0.08
0.087
5.2
0.9984
0.9982
0.0003
12.5 300.6
Q
0.03
0.075
6.1
0.9977
0.9974
0.0009
14.8 188.9
Q
P-value
β
2 R¯ lin
R¯ q2
2 R¯ q2 − R¯ lin
C¯ 2d
Result
μ0
C¯ 2f
Brazil 2006
17
0.0002
0.001
0.069
8.3
0.9981
0.9978
0.0022
25.2 191.5
Q
2007
17
0.0002
0.001
0.069
8.1
0.9982
0.9979
0.0021
25.1 199.0
Q
2008
17
0.0002
0.001
0.069
8.2
0.9982
0.9979
0.0021
25.0 199.3
Q
2009
17
0.0002
0.001
0.067
8.7
0.9983
0.9981
0.0025
29.9 208.4
Q
2010
17
0.0002
0.001
0.067
8.5
0.9984
0.9982
0.0026
32.0 216.9
Q
16
0.0002
0.0001
0.075
4.5
0.9991
0.9990
0.0020
36.0 293.4
Q
Chile 1997 1998
16
0.0002
0.0001
0.075
4.1
0.9988
0.9987
0.0022
31.4 216.8
Q
1999
16
0.0003
0.0001
0.068
5.0
0.9992
0.9991
0.0036
63.6 262.9
Q
2000
16
0.0002
0.001
0.074
4.1
0.9991
0.9989
0.0024
39.7 268.4
Q
2001
17
0.0002
0.0001
0.071
4.8
0.9992
0.9991
0.0032
60.8 386.0
Q
2002
17
0.0003
0.0001
0.068
5.0
0.9988
0.9986
0.0038
49.7 240.8
Q
2003
17
0.0002
0.0001
0.071
4.5
0.9991
0.9990
0.0033
60.2 367.0
Q
2004
17
0.0003
0.0001
0.065
5.2
0.9992
0.9991
0.0050
89.8 322.4
Q
2005
18
0.0003
0.001
0.062
5.2
0.9982
0.9979
0.0068
64.5 185.4
Q
2006
17
0.0003
0.0001
0.070
4.1
0.9994
0.9994
0.0036
94.2 534.3
Q
2007
18
0.0004
0.0001
0.060
5.3
0.9986
0.9985
0.0076
90.4 229.8
Q
2008
18
0.0004
0.0001
0.058
5.4
0.9979
0.9977
0.0078
65.0 152.9
Q
2009
18
0.0003
0.0001
0.062
5.1
0.9985
0.9983
0.0062
69.8 233.6
Q
Colombia 1997
16
0.0006
0.0001
0.029
16.25
0.9985
0.9982
0.0172
142.8
34.0
1998
16
0.0006
0.0001
0.033
14.41
0.9985
0.9983
0.0155
137.1
42.6
1999
17
0.0008
0.0001
0.004
31.0
0.9933
0.9924
0.0390
86.7
9.2
2000
17
0.0007
0.0001
0.007
27.2
0.9933
0.9924
0.0362
81.4
9.6
Q
2002
16
0.0009
0.0001 −0.01
37.6
0.9967
0.9962
0.0410
160.3
7.7
Q
2003
16
0.0009
0.0001 −0.01
39.4
0.9969
0.9965
0.0410
170.6
7.8
Q
Peru Q
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J. Dolejs
Table 3 continued Year
n
δ
P-value
β
μ0
2 R¯ lin
R¯ q2
2 R¯ q2 − R¯ lin
C¯ 2d
C¯ 2f
Result
2004
16
0.0008
0.0001 −0.01
37.2
0.9972
0.9968
0.0398
179.7
7.5
Q
2005
16
0.0008
0.0001
27.3
0.9971
0.9966
0.0336
147.8
7.1
Q
0.003
2006
17
0.0007
0.0001
0.013
19.1
0.9955
0.9948
0.0312
100.6
12.0
Q
2007
17
0.0007
0.0001
0.017
16.4
0.9955
0.9949
0.0290
95.3
13.4
Q
Year
n
δ
P-value
β
ln(x)
ln(x)2
R¯ q2
2 R¯ q2 − R¯ lin
C¯ 2d
C¯ 2f
Result
Venezuela 1996
16
0.0002
0.0001
0.068
9.3
0.9993
0.9992
0.0018
39.8 383.4
Q
1997
16
0.0002
0.0001
0.068
9.1
0.9995
0.9994
0.0016
49.2 567.4
Q
1998
17
0.0001
0.0002
0.071
8.3
0.9994
0.9993
0.0014
42.1 685.4
Q
1999
16
0.0002
0.0001
0.071
8.1
0.9994
0.9993
0.0014
34.4 462.1
Q
2000
16
0.0002
0.0001
0.069
8.4
0.9994
0.9993
0.0015
39.3 484.7
Q
2001
16
0.0002
0.0001
0.070
8.1
0.9995
0.9994
0.0015
42.9 520.8
Q
2002
17
0.0002
0.0001
0.066
8.5
0.9989
0.9988
0.0024
40.2 331.6
Q
2003
17
0.0002
0.0001
0.063
9.5
0.9991
0.9990
0.0035
62.4 350.7
Q
2004
16
0.0003
0.0001
0.060
9.7
0.9987
0.9985
0.0036
42.0 166.3
Q
2005
16
0.0002
0.0002
0.062
9.3
0.9989
0.9987
0.0032
42.7 198.2
Q
2006
17
0.0002
0.0001
0.066
8.3
0.9990
0.9988
0.0028
44.7 321.4
Q
2007
17
0.0002
0.0001
0.068
7.7
0.9994
0.9993
0.0022
56.3 576.4
Q
P-value is the two-sided P-value of the t test of parameter δ, the parameter µ0 is per 105 persons per 1 year, 2 in the linear submodel (8) is calculated for one predictor and the adjusted coefficient of determination R¯ lin n points, the adjusted coefficient of determination R¯ q2 in the quadratic full model (7) is calculated for two predictors and n points; if P-value is less than 0.05, then the linear Gompertz model without δ is used for the calculation of the parameters β and μ0 .; C¯ p = C p − 2∗ (k − p + 1)/(n − k − 3), where k = 2 is the number of candidate regressors in the full model (7); p = 2 is the number of parameters in submodels with the predictor x and x 2 ; C¯ 2d is the Mallow’s statistic in the linear Gompertz submodel (8) with x; C¯ 2f is the Mallow’s statistic in the submodel with the single predictor x 2 . The last column contains result model according to the Gilmour test for P < 0.05, where “Q” means the full model (7) (Gilmour 1996)
ln (μ(x)) = constant + β.x + δ.x.x
(14)
The null hypothesis Ho : δ = 0 is not rejected in Argentina in 5 out of 7 cases (with the two-sided P values >0.05), and the parameter β is significant in these 5 cases (with the two-sided P values
Age Trajectories of Mortality from All Diseases in the Six Most Populated Countries of the South America During the Last Decades Josef Dolejs
Received: 23 January 2014 / Accepted: 25 July 2014 / Published online: 15 August 2014 © Society for Mathematical Biology 2014
Abstract Age trajectories of total mortality represent an irreplaceable source of information about aging. In principle, age affects mortality from all diseases differently than it affects mortality from external causes. External causes (accidents) are excluded here from all causes, and the resultant category “all-diseases” is tested as a helpful tool to better understand the relationship between mortality and age. Age trajectories of alldiseases mortality are studied in the six most populated countries of the South America during 1996–2010. The numbers of deaths for specific causes of death are extracted from the database of WHO, where the ICD-10 revision is used. The all-diseases mortality shows a strong minimum, which is hidden in total mortality. Two simple deterministic models fit the age trajectories of all-diseases mortality. The inverse proportion between mortality and age fits the mortality decreases up to minimum value in all six countries. All previous models describing mortality decline after birth are discussed. Theoretical relationships are derived between the parameter in the first model and standard mortality indicators: Infant mortality, Neonatal mortality, and Postneonatal mortality. The Gompertz model extended with a small positive quadratic element fit the age trajectories of all-diseases mortality after the age of 10 years. Keywords
Mortality · Age · All diseases · External causes · South America
1 Introduction Age trajectories of human mortality are usually studied for total mortality or for a specific set of causes of death (e.g., Arbeev et al. 2011; Feichtingera et al. 2004; Lin
J. Dolejs (B) Department of Informatics and Quantitative Methods, University of Hradec Králové, Rokitanského 62, 500 03 Hradec Králové, Czech Republic e-mail: [email protected]
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Table 1 International statistical classification of diseases and related health problems, 10th revision Chapter
Blocks
Title
I
A00-B99
Certain infectious and parasitic diseases
II
C00-D48
Neoplasms
III
D50-D89
Diseases of the blood and blood-forming organs and certain disorders involving the immune mechanism
IV
E00-E90
Endocrine, nutritional and metabolic diseases
V
F00-F99
Mental and behavioral disorders
VI
G00-G99
Diseases of the nervous system
VII
H00-H59
Diseases of the eye and adnexa
VIII
H60-H95
Diseases of the ear and mastoid process
IX
I00-I99
Diseases of the circulatory system
X
J00-J99
Diseases of the respiratory system
XI
K00-K93
Diseases of the digestive system
XII
L00-L99
Diseases of the skin and subcutaneous tissue
XIII
M00-M99
Diseases of the musculoskeletal system and connective tissue
XIV
N00-N99
Diseases of the genitourinary system
XV
O00-O99
Pregnancy, childbirth and the puerperium
XVI
P00-P96
Certain conditions originating in the perinatal period
XVII
Q00-Q99
Congenital malformations, deformations and chromosomal abnormalities
XVIII
R00-R99
Symptoms, signs and abnormal clinical and laboratory findings, not elsewhere classified
Diseases
Non-biological causes XIX
S00-T98
XX
V01-Y98
Injury, poisoning and certain other consequences of external causes External causes of morbidity and mortality
XXI
Z00-Z99
Factors influencing health status and contact with health services
XXII
U00-U99
Codes for special purposes
and Liu 2007; Lutz et al. 1998; Riggs 1992; Robine et al. 2012; Shkolnikov et al. 2012; Su and Sherris 2012; Vaupel et al. 1998; Yashin et al. 2008; Zheng et al. 2011). They represent an irreplaceable source of information about the relationship between mortality and age, and it is well known that external causes affect total mortality as the set of causes with a fractionally age-independent mortality rate (e.g., Makeham 1860; Vaupel et al. 1998; Willemse and Koppelaar 2000). The most recent WHO definition of death causes is in Table 1, and external (or non-biological) causes represent the last four chapters. External causes affect total mortality, especially within the age range of 5–30 years, but lose significance over the age of 40 (e.g., de Beer 2012; Granados and Ionides 2011; Heligman and Pollard 1980; Preston et al. 2001 or see Fig. 2). One typical example of age trajectory of mortality from all non-biological causes is in Fig. 1 in the log-log scale and concurrently in the semi-logarithmic scale in Fig. 2. Specific fractions of non-biological causes of total deaths are between 0.01 and 0.14 in the interval [0,1) year, between 0.05 and 0.18 in the interval [0,5) years, between
123
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J. Dolejs
Brazil 2010 in the log-log scale
Logarithm of mortality per 10 5 per 1 year
1000000
All causes (ICD10: A00-Z99) All diseases (ICD10: A00-R99) Non-biological causes (ICD10: S00-Z99)
100000
__ 10000
___
Gompertz model Model = μ 1 /x
1000
100
10
logarithm of age [years] 1st day
1
1-7 days
7-28 days 28days-365days 1-2 years 10-15 years
-8
Fig. 1 Age trajectories of mortality in Brazil in the log-log scale in 2010
Brazil 2010 in the semi-logarithmic scale
Logarithm of mortality per 10 5 per one year
1000000
All causes (ICD10: A00-Z99) All diseases (ICD10: A00-R99) Non-biological causes (ICD10: S00-Z99)
100000
__
10000
___
Gompertz model Model = μ 1 /x
1000
100
10
age [years] 1
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Fig. 2 Age trajectories of mortality in Brazil in the semi-logarithmic scale in 2010
0.32 and 0.79 in the interval [5,30) years and between 0.04 and 0.16 in the interval [30,95) years in the six countries during the studied period 1996–2010. It is evident that the age trajectory of mortality from non-biological causes distinctively differs
123
Age Trajectories of Mortality from All Diseases
2147
from the age trajectory of all-causes mortality (de Beer 2012; Heligman and Pollard 1980; Vaupel et al. 1998; Willemse and Koppelaar 2000). In addition, it can be assumed that the mechanism which is responsible for the relationship between mortality and age differs between all-diseases and non-biological causes. The possibility to apply the category all-diseases as a meaningful tool to describe and understand the relationship between mortality and age is examined here. Age trajectories of mortality from all-diseases in Argentina, Brazil, Chile, Colombia, Peru and Venezuela are analyzed in all calendar years when the ICD10 classification of causes was used. The population of the six countries represented about 92 % of the South America in July 2013 and about 5 % of the world’s population. The exclusion of non-biological causes from total mortality can be realized only if information about specific causes of death is known in convenient age categories. Such information is available in the mortality database of the World Health Organization (World Health Organization 2013). The database usually uses four age categories in the first year of life, enabling the construction of detailed age trajectory of mortality in the first year. 2 Materials and Methods The numbers of deaths in the six countries for detailed causes of death in specific age categories are extracted from the file “Mortality, ICD-10” available in the mortality database of World Health Organization (World Health Organization 2013). The ICD10 classification of causes of death is used in the database (Table 1; World Health Organization 1997), and the database usually uses the following 26 age intervals: [0,24) h, [1,7), [7,28), [28,365) days, [1,2), [2,3), [3,4), [4,5), [5,10), [10,15),… [90,95) years. Unfortunately, the first age interval is also [0,1) year or the second interval is [1,5) years in a specific country in a specific calendar year. If these two age categories are used, such calendar years are excluded (see the first column in Table 2). The resulting age trajectories of mortality from all-diseases are constructed for the calendar years using the classification ICD10, four age categories in the first year and four age categories in the range of 1–5 years. The numbers of living people are obtained from the US Census Bureau (Bureau of the Census 2013). The set “all-diseases” is constructed from the first 18 chapters of ICD10 (Table 1; World Health Organization 1997). The excluded set “non-biological causes” contains the last four chapters. In fact, no deaths are in the last two chapters for any age category, in any country and in any calendar year, and the chapters have no meaning. Arithmetic mean of age limits in age category is used as a representative point in all calculations, and all analyses were completed in 2014. 3 Results The first calendar year for Argentina, 2003, is shown as an example in Fig. 3 in the log-log scale and also in Fig. 4 in the semi-logarithmic scale. The first and the last calendar years of every country are shown in the log-log scale in the “Appendix” (see all odd-numbered Figs. 9, 11, 13, 15, 17, 19, 21, 23, 25, 27) and also in the semi-
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Table 2 Results of the linear model in the log-log scale in the age interval [0,A) years Year
A
Slope
95 % CI
μ1
R¯ 2
Rb2
Rb2 – R¯ 2
C¯ 2a
C¯ 2b
C¯ 1c
Result
ln(x) ln(x)2 c/x
γ Argentina 2003 15
−1.064 −1.154, −0.973 160.9 0.9880 0.9858 −0.00225 1.6
223.8
2.4
c/x
2005 10
−1.068 −1.158, −0.977 128.9 0.9898 0.9871 −0.00271 4.8
293.0
2.7
c/x
2006 10
−1.067 −1.160, −0.974 124.1 0.9892 0.9866 −0.00261 4.7
274.7
2.5
c/x
2007 10
−1.063 −1.162, −0.964 126.2 0.9877 0.9857 −0.00197 7.5
321.5
1.9
Q
2008 10
−1.074 −1.172, −0.974 123.1 0.9879 0.9847 −0.00313 5.3
265.1
2.7
c/x
2009 10
−1.063 −1.159, −0.967 123.7 0.9884 0.9864 −0.00206 4.6
254.9
2.0
c/x
2010 10
−1.082 −1.180, −0.984 114.3 0.9884 0.9841 −0.00428 6.9
319.4
3.5
Q
2006 10
−1.062 −1.164, −0.959 148.6 0.9868 0.9851 −0.0017
6.0
262.9
1.6
c/x
2007 10
−1.069 −1.170, −0.966 138.9 0.9872 0.9847 −0.0025
6.7
289.3
2.1
Q
2008 10
−1.069 −1.163, −0.973 139.1 0.9889 0.9862 −0.0027
5.4
287.2
2.5
c/x
Brazil
2009 10
−1.072 −1.156, −0.988 137.7 0.9913 0.9878 −0.0034
5.7
370.0
3.8
c/x
2010 10
−1.079 −1.147, −0.998 131.7 0.9941 0.9896 −0.0045
4.2
452.6
6.8
c/x
Chile 1997 15
−1.011 −1.115, −0.906 112.1 0.9822 0.9840
0.0019
0.8
112.6 −0.3
c/x
1998 15
−1.019 −1.109, −0.928 116.0 0.9869 0.9880
0.0011
0.6
164.6 −0.1
c/x
1999 15
−1.033 −1.148, −0.917 103.2 0.9792 0.9805
0.0013
0.7
97.3
0.1
c/x
2000 15
−1.047 −1.164, −0.929
0.0004
0.9
94.9
0.5
c/x
2001 10
−1.046 −1.132, −0.958
94.1 0.9902 0.9896 −0.0007
1.6
193.5
1.1
c/x
2002 10
−1.048 −1.120, −0.974
90.3 0.9930 0.9919 −0.0012
1.2
247.2
2.0
c/x
94.0 0.9791 0.9794
2003 10
−1.073 −1.171, −0.974
79.7 0.9880 0.9850 −0.0031
0.8
131.3
2.6
c/x
2004 10
−1.083 −1.172, −0.994
83.2 0.9905 0.9858 −0.0047
1.2
180.8
4.5
c/x
2005
5
−1.116 −1.228, −1.004
69.4 0.9884 0.9793 −0.0091
1.5
120.7
6.0
c/x
2006 10
−1.087 −1.158, −1.014
76.8 0.9938 0.9882 −0.0055
0.6
234.5
7.7
L
2007
−1.121 −1.225, −1.015
68.1 0.9898 0.9798 −0.0101
1.7
141.9
7.4
c/x
5
2008
5
−1.132 −1.221, −1.041
61.6 0.9926 0.9802 −0.0124
0.8
154.1 12.3
L
2009
5
−1.133 −1.239, −1.026
63.7 0.9898 0.9775 −0.0122
1.8
143.7
L
8.8
Colombia 1997 15
−0.979 −1.059, −0.898 175.0 0.9887 0.9895
0.0008
1.8
244.8
0.02 c/x
1998 15
−1.025 −1.104, −0.945 195.3 0.9899 0.9904
0.0006
1.3
250.8
0.18 c/x 0.60 c/x
Peru 1999 10
−0.963 −1.049, −0.876 216.1 0.9886 0.9886
0.0000
4.9
265.6
2000 10
−0.954 −1.027, −0.880 194.4 0.9916 0.9903 −0.0013
7.2
437.5
1.84 Q
2002 15
−0.957 −1.032, −0.880 163.9 0.9894 0.9885 −0.0009
1.3
240.4
1.40 c/x
2003 15
−0.974 −1.043, −0.904 174.0 0.9915 0.9917
0.0003
1.7
319.2
0.39 c/x
2004 15
−0.964 −1.040, −0.886 169.7 0.9892 0.9890 −0.0002
1.3
238.7
0.84 c/x
2005 15
−0.977 −1.052, −0.901 145.6 0.9899 0.9905
1.5
261.0
0.15 c/x
123
0.0006
Age Trajectories of Mortality from All Diseases
2149
Table 2 continued Year
A
Slope
95 % CI
μ1
R¯ 2
Rb2
Rb2 – R¯ 2
C¯ 2a
C¯ 2b
C¯ 1c
Result
ln(x) ln(x)2 c/x
γ 2006 10
−0.983
−1.078, −0.886 130.4 0.9866 0.9880
0.0014 7.9
309.6 −0.21 Q
2007 10
−0.976
−1.072, −0.879 124.2 0.9862 0.9874
0.0011 8.9
327.6 −0.05 Q
Year
Slope γ 95 % CI
A
μ1
R¯ 2
Rb2
Rb2 – R¯ 2
C¯ 2a
C¯ 2b
C¯ 1c
Result
Venezuela 1996 15
−1.002
−1.097, −0.907 255.8 0.9851 0.9867
0.0017 3.7
248.2 −0.3
c/x
1997 15
−1.018
−1.108, −0.927 229.4 0.9868 0.9880
0.0012 4.4
301.5 −0.1
c/x
1998 10
−1.023
−1.144, −0.901 210.7 0.9802 0.9822
0.0020 6.2
188.9 −0.2
c/x
1999 15
−1.011
−1.110, −0.910 212.2 0.9837 0.9854
0.0017 2.7
201.3 −0.3
c/x
2000 15
−1.018
−1.117, −0.918 204.1 0.9841 0.9855
0.0014 3.2
220.2 −0.2
c/x
2001 15
−1.012
−1.105, −0.918 205.7 0.9858 0.9872
0.0015 2.4
216.7 −0.3
c/x
2002 10
−1.051
−1.156, −0.944 174.0 0.9856 0.9851 −0.0005 4.9
216.9
c/x
0.9
2003 10
−1.026
−1.155, −0.896 213.2 0.9775 0.9797
0.0022 8.8
213.9 −0.2
2004 15
−1.035
−1.123, −0.946 194.3 0.9878 0.9881
0.0002 2.4
251.5
0.5
c/x
2005 15
−1.066
−1.169, −0.962 163.4 0.9843 0.9823 −0.0021 1.9
185.2
1.8
c/x
Q
2006 10
−1.072
−1.167, −0.975 154.6 0.9887 0.9857 −0.0030 6.4
314.5
2.7
c/x
2007 10
−1.076
−1.174, −0.976 149.6 0.9880 0.9846 −0.0034 6.5
301.9
2.9
Q
A is the upper age limit of the age category when mortality reaches the minimal value, C I 95% confidence interval; the parameter μ1 is per 105 persons per one year, the adjusted coefficient of determination R¯ is calculated in the linear model (2) for one predictor and n points, and the coefficient of determination Rb2 is calculated in the model (3); C¯ p = C p − 2∗ (k − p + 1)/(n − k − 3), where n is the number of points, k is the number of candidate regressors in a full model without constant (k = 2 in the quadratic model and k = 1 in the linear model), p is the number of parameters in a submodel ( p = 2 in the submodel model (2) and p = 1 in the submodel 3); “Q” means the quadratic full model (1); “L” means the linear submodel (2), and “c/x” means the submodel (3); C¯ 2a is the Mallow’s statistic in the submodel “L” with the predictor ln(x); C¯ 2b is the Mallow’s statistic in the submodel with the predictor ln(x).ln(x); C¯ 1c is the Mallow’s statistic in the submodel “c/x”(“L” is the full model in the second step and the quadratic model should be rejected in the first step in these cases); the last column contains result models according to the Gilmour test for P < 0.05 (Gilmour 1996)
logarithmic scale (see all even-numbered Figs. 10, 12, 14, 16, 18, 20, 22, 24, 26, 28). Mortality from all-diseases reaches the minimal value in the three different age [4,5), [5,10) and [10,15) years. The upper age limit “A” of the specific age category in which mortality reaches the minimal value is in the second column in Table 2. The age axis is divided into two parts so that age trajectories are analyzed separately in the interval [0,A) years and in the interval [A,95) years. Age “A” is the upper limit of age category where mortality reaches the minimal value. The resulting age trajectories of mortality from all-diseases show the linear dependence in the log-log scale in 0–A years in all calendar years and in all countries. For example, Heligman and Pollard (1980) or Siler (1979) supposed an exponential model of total mortality decline with age after birth. The supposed exponential model declines linearly in the semi-logarithmic scale, but data show the nonlinear
123
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Argentina in 2003 in the log-log scale
Logarithm of mortality per 10 5 per 1 year
1000000
All diseases 2003 100000
Model = μ 1 /x and Gompertz model
10000
1000
100
10
logarithm of age [years] 1st day
1
1-7 days 7-28 days
28days-365days1-2 years
10-15 years
-8
Fig. 3 Age trajectory of all-diseases mortality fitted by the two models in Argentina in the log-log scale in 2003
Argentina in 2003 in the semi-logarithmic scale
Logarithm of mortality per 10 5 per one year
1000000
100000
All diseases 2003 Model = μ 1 /x and Gompertz model
10000
1000
100
10
age [years] 1
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Fig. 4 Age trajectory of all-diseases mortality fitted by the two models in Argentina in the semi-logarithmic scale in 2003
123
Age Trajectories of Mortality from All Diseases
2151
convex trajectories in the semi-logarithmic scale. The nonlinear convex decline in the semi-logarithmic scale among those aged 0–A years is also ascertained visually for all cases. On the other hand, linear age trajectories of mortality are visually found out in the semi-logarithmic scale in all calendar years and in all countries in the second interval [A,95) years. The approximate linear data of all-causes mortality and all-diseases mortality in Brazil in 2010 correspond to the Gompertz exponential model, shown in the semi-logarithmic scale in Fig. 2. The data for all-diseases and for all-causes are identical over the age of 40, while the data are already linear by the age of 10 for the category of all-diseases. This approximate linearity is ascertained in all cases. 3.1 Age Interval 0–A Years At first, the linearity in the log-log scale is tested in the following full model using the method of least squares (MLS): ln(μ(x)) = constant + γ .ln (x) + δ.ln (x) .ln (x)
(1)
The null hypothesis Ho : δ = 0 is not rejected in all cases (with the two-sided P values >0.05 in any country in any calendar year). The slope γ is significant in all cases (the two-sided P values xmin . The first day can be chosen as xmin , and the theoretical number of deaths within the first 24 h can be estimated using a single point of mortality at an age of 12 h. Mortality rate at 12 h = 1/730 years is μ(x = 1/730) = 730. μ1 . Simultaneously, it is the number of deaths per one year per one living 365.D1 /L, where D1 is the absolute number of deaths within the first day and L is the number of live births. Consequently, 365.D1 /L = 730. μ1 and D1 /L = 2. μ1 . Furthermore, the analytic survival function S(x) in model (3) could be applied for x ≥ xmin and it is valid: −dS(x) dx (x) = −d ln[S(x)] dx ⇒ ln[S(x)] = − μ(x)dx because μ(x) = S μ1 , for x ≥ xmin ⇒ ln[S(x)] if γ = −1 then μ(x) = x = −μ1 ln(x) + c, for x ≥ xmin if S(xmin ) = 1 ⇒ ln[S(xmin )] = 0 and ⇒ c = μ1 · ln(xmin )
μ1 and S (xn ) = xmin x ⇒ ln[S(x)] = μ1 · ln xmin x =
−μ1
x xmin for x ≥ xmin
(10)
Because S(x) = (365.x)−μ1 for x ≥ xmin = 24 h= 1/365 year, Infant mortality (the number of deaths divided by L live birth within the age interval [0,1) year) is as follows:
1q0 = D1 /L + D2 /L = 2.μ1 + 1 − S (x = 1 year) = 2.μ1 + 1 − 365−μ1 , (11) where D2 is the number of deaths in the age interval [1,365) days (the approximation D2 /L/D2 /(L − D1 ) is assumed here).
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Parametr μ 1 in Chile during the period 1997-2009
160
μ 1 per 100 000 per one year
140
120
100
80
60
40
I. - the values calculated from Infant mortality (Instituto Nacional de Estadísticas, Chile) and using numerical solution of the equation (10) II. - the values from Table 2 calculated using MLS in the model (3) and the WHO data
20
0
1997
1999
2001
2003
2005
2007
2009
Fig. 8 Two sets of parameter μ1 calculated from infant mortality and in the model (3)
Similarly, Postneonatal mortality, within the age interval [28/365,1) years, is given as follows:
PNM = 1 − S (x = 1 year) − 1 − S (x = 28/365 years) = (28)−μ1 − (365)−μ1 (12) and Neonatal mortality within the age interval [0,28/365) years is as follows: NNM = 2.μ1 + 1 − S (x = 28/365 years) = 2.μ1 + 1 − (28)−μ1
(13)
Conversely, the estimation of parameter μ1 could be calculated using a numerical solution of one of the formulas (11), (12) or (13) if one of the three values of 1q0 , PNM and NNM is known. For example, parameter μ1 is numerically calculated here using equation (11) and data of Infant mortality 1q0 downloaded from the Instituto Nacional de Estadísticas, Chile, for calendar years 1997–2009 (Instituto Nacional de Estadísticas, Chile 2013). The values are the time series I. in Fig. 8, and the second time series II. represents the values calculated in the model (3) (the 6th column in Table 2 for Chile). The values calculated in the model (3) for all diseases are systematically lower because external causes are not negligible in Chile even in the range of 0–1 year. 3.4 Age Interval A–95 Years Age trajectories of mortality from all-diseases are approximately linear in the semilogarithmic scale over age A. At first, the linearity in the semi-logarithmic scale is tested in the following full model using MLS in all of cases:
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Age Trajectories of Mortality from All Diseases
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Table 3 Results of the two models in the semi-logarithmic scale in the second age interval [A,95) years Year
n
δ
P-value
β
0.49
0.091
ln(x)
ln(x)2
R¯ q2
2 R¯ q2 − R¯ lin
4.3
0.9988
0.9986 −0.0001
C¯ 2d
C¯ 2f
Result
Argentina 2003
16
7.5 334.6
Q
2005
17
0.18
0.090
4.3
0.9978
0.9975
0.0002
11.0 226.8
Q
2006
17
0.12
0.089
4.5
0.9978
0.9974
0.0003
11.7 217.5
Q
2007
17
2008
17
0.0001
2009
17
2010
17
0.0001
Year
n
δ
0.04
0.077
6.0
0.9978
0.9975
0.0007
14.0 208.1
Q
0.14
0.088
4.6
0.9974
0.9971
0.0003
11.5 189.8
Q
0.08
0.087
5.2
0.9984
0.9982
0.0003
12.5 300.6
Q
0.03
0.075
6.1
0.9977
0.9974
0.0009
14.8 188.9
Q
P-value
β
2 R¯ lin
R¯ q2
2 R¯ q2 − R¯ lin
C¯ 2d
Result
μ0
C¯ 2f
Brazil 2006
17
0.0002
0.001
0.069
8.3
0.9981
0.9978
0.0022
25.2 191.5
Q
2007
17
0.0002
0.001
0.069
8.1
0.9982
0.9979
0.0021
25.1 199.0
Q
2008
17
0.0002
0.001
0.069
8.2
0.9982
0.9979
0.0021
25.0 199.3
Q
2009
17
0.0002
0.001
0.067
8.7
0.9983
0.9981
0.0025
29.9 208.4
Q
2010
17
0.0002
0.001
0.067
8.5
0.9984
0.9982
0.0026
32.0 216.9
Q
16
0.0002
0.0001
0.075
4.5
0.9991
0.9990
0.0020
36.0 293.4
Q
Chile 1997 1998
16
0.0002
0.0001
0.075
4.1
0.9988
0.9987
0.0022
31.4 216.8
Q
1999
16
0.0003
0.0001
0.068
5.0
0.9992
0.9991
0.0036
63.6 262.9
Q
2000
16
0.0002
0.001
0.074
4.1
0.9991
0.9989
0.0024
39.7 268.4
Q
2001
17
0.0002
0.0001
0.071
4.8
0.9992
0.9991
0.0032
60.8 386.0
Q
2002
17
0.0003
0.0001
0.068
5.0
0.9988
0.9986
0.0038
49.7 240.8
Q
2003
17
0.0002
0.0001
0.071
4.5
0.9991
0.9990
0.0033
60.2 367.0
Q
2004
17
0.0003
0.0001
0.065
5.2
0.9992
0.9991
0.0050
89.8 322.4
Q
2005
18
0.0003
0.001
0.062
5.2
0.9982
0.9979
0.0068
64.5 185.4
Q
2006
17
0.0003
0.0001
0.070
4.1
0.9994
0.9994
0.0036
94.2 534.3
Q
2007
18
0.0004
0.0001
0.060
5.3
0.9986
0.9985
0.0076
90.4 229.8
Q
2008
18
0.0004
0.0001
0.058
5.4
0.9979
0.9977
0.0078
65.0 152.9
Q
2009
18
0.0003
0.0001
0.062
5.1
0.9985
0.9983
0.0062
69.8 233.6
Q
Colombia 1997
16
0.0006
0.0001
0.029
16.25
0.9985
0.9982
0.0172
142.8
34.0
1998
16
0.0006
0.0001
0.033
14.41
0.9985
0.9983
0.0155
137.1
42.6
1999
17
0.0008
0.0001
0.004
31.0
0.9933
0.9924
0.0390
86.7
9.2
2000
17
0.0007
0.0001
0.007
27.2
0.9933
0.9924
0.0362
81.4
9.6
Q
2002
16
0.0009
0.0001 −0.01
37.6
0.9967
0.9962
0.0410
160.3
7.7
Q
2003
16
0.0009
0.0001 −0.01
39.4
0.9969
0.9965
0.0410
170.6
7.8
Q
Peru Q
123
2158
J. Dolejs
Table 3 continued Year
n
δ
P-value
β
μ0
2 R¯ lin
R¯ q2
2 R¯ q2 − R¯ lin
C¯ 2d
C¯ 2f
Result
2004
16
0.0008
0.0001 −0.01
37.2
0.9972
0.9968
0.0398
179.7
7.5
Q
2005
16
0.0008
0.0001
27.3
0.9971
0.9966
0.0336
147.8
7.1
Q
0.003
2006
17
0.0007
0.0001
0.013
19.1
0.9955
0.9948
0.0312
100.6
12.0
Q
2007
17
0.0007
0.0001
0.017
16.4
0.9955
0.9949
0.0290
95.3
13.4
Q
Year
n
δ
P-value
β
ln(x)
ln(x)2
R¯ q2
2 R¯ q2 − R¯ lin
C¯ 2d
C¯ 2f
Result
Venezuela 1996
16
0.0002
0.0001
0.068
9.3
0.9993
0.9992
0.0018
39.8 383.4
Q
1997
16
0.0002
0.0001
0.068
9.1
0.9995
0.9994
0.0016
49.2 567.4
Q
1998
17
0.0001
0.0002
0.071
8.3
0.9994
0.9993
0.0014
42.1 685.4
Q
1999
16
0.0002
0.0001
0.071
8.1
0.9994
0.9993
0.0014
34.4 462.1
Q
2000
16
0.0002
0.0001
0.069
8.4
0.9994
0.9993
0.0015
39.3 484.7
Q
2001
16
0.0002
0.0001
0.070
8.1
0.9995
0.9994
0.0015
42.9 520.8
Q
2002
17
0.0002
0.0001
0.066
8.5
0.9989
0.9988
0.0024
40.2 331.6
Q
2003
17
0.0002
0.0001
0.063
9.5
0.9991
0.9990
0.0035
62.4 350.7
Q
2004
16
0.0003
0.0001
0.060
9.7
0.9987
0.9985
0.0036
42.0 166.3
Q
2005
16
0.0002
0.0002
0.062
9.3
0.9989
0.9987
0.0032
42.7 198.2
Q
2006
17
0.0002
0.0001
0.066
8.3
0.9990
0.9988
0.0028
44.7 321.4
Q
2007
17
0.0002
0.0001
0.068
7.7
0.9994
0.9993
0.0022
56.3 576.4
Q
P-value is the two-sided P-value of the t test of parameter δ, the parameter µ0 is per 105 persons per 1 year, 2 in the linear submodel (8) is calculated for one predictor and the adjusted coefficient of determination R¯ lin n points, the adjusted coefficient of determination R¯ q2 in the quadratic full model (7) is calculated for two predictors and n points; if P-value is less than 0.05, then the linear Gompertz model without δ is used for the calculation of the parameters β and μ0 .; C¯ p = C p − 2∗ (k − p + 1)/(n − k − 3), where k = 2 is the number of candidate regressors in the full model (7); p = 2 is the number of parameters in submodels with the predictor x and x 2 ; C¯ 2d is the Mallow’s statistic in the linear Gompertz submodel (8) with x; C¯ 2f is the Mallow’s statistic in the submodel with the single predictor x 2 . The last column contains result model according to the Gilmour test for P < 0.05, where “Q” means the full model (7) (Gilmour 1996)
ln (μ(x)) = constant + β.x + δ.x.x
(14)
The null hypothesis Ho : δ = 0 is not rejected in Argentina in 5 out of 7 cases (with the two-sided P values >0.05), and the parameter β is significant in these 5 cases (with the two-sided P values
