A demographic approach to competing risks analysis

Introduction

Let us consider any subject, whether it is mechanical or biological; it will fail at last. Now the concept is that, though the subject may fail due to one cause, it may be affected by many causes, which are risks. This terminology is said to have competing risks. As causes competing with each other to fail the subject. Here we are considering only the biological aspect, that is, the deaths of humans. More specifically, one can reasonably assume that in the human biosystem, right from the day when a person is born, all the risks that are capable of taking the life of an individual start competing with each other, and a particular risk ultimately succeeds in taking away the life of an individual either in the presence of a subset of the set of risks, depending upon the individual's setup. Now consider the deaths of individuals and the construction of the life table. The life table is a statistical tool that is very helpful to many fields like actuaries, economists, etc. Generally, we have two types of life tables: cohort and current life tables. The construction of the cohort life table is time-consuming, and we also do not get all the data. The use of the current life table nearly gives us the same result as the cohort (Dublin and Spiegelman, 1941) [12] , so it has become quite common to use the current life table, which considers the current mortality pattern to calculate the life table. Many authors worked on the construction of the life table with different methods, like King (1914) [16] , Reed and Merrel (1939) [ 17] , Greville (1943) [13] , Keyfitz (1966) [15] , and Chiang (1968) [6] . Choosing the method is based on the availability of data and the conditions. Using a life table, we can construct multiple-decrement (MD) and associated single-decrement (ASD) life tables, as MD life table provides the probability of dying from a certain cause of death in the presence of other causes and ASD life table answers the questions concerning what would be the life expectancy if certain causes were eliminated (Sharma and Choudhury, 2023) [18] . So, the concept of multiple decrements is nothing but the competing risks where more than two causes act. Also, using life tables, one can construct cause eliminated life tables, and from that, we can see the effect of a change in life expectancy or a gain in life expectancy.Here we are considering causes acts independently. Many authors (Cornfield, 1957; Chiang, 1960, 1961, 1972, 1978 and 1991; Keyfitz, 1966; Wong, 1977; Arias, and Tejada-Vera, 2023) [11, 4-5, 7-9, 15, 20, 1] also worked on the competing risks using the life table approach with the elimination of some causes. Here we were considering the construction of a life table, a multiple-decrement life table, and competing risks with different probabilities (crude, net, and partial crude). As cause eliminated life tables are used to assess the change in life expectancy in which a particular cause of death has been hypothetically elimination. Many causes (cardiovascular disease (CVD), cancer, tuberculosis (TB), malaria, chronic respiratory disease, the human immunodeficiency virus and acquired immune deficiency syndrome (HIV/AIDS), etc.) play a major role in human life. In this paper, we are considering these causes as cause of deaths and calculating the overall life expectancy and life expectancy when a specific cause is eliminated, and also the crude, net, and partial crude probabilities of dying.

Description about life expectancy

In India life expectancy at birth is about 50.5 and 49 years for male and female respectively in 1970-74. And in 2010, the life expectancy being increasing with 65.4 and 68.8 for male and female respectively and in 2018, 68.8 and 71.4 for male and female respectively (Sample registration survey report 2018). Now we are going to considering 2019 population and deaths.

Description about decrement

Generally, in survival studies of human population, decrement refers as lost. In construction of ordinary life table we are going to consider the loss of an individual is due to death, where we are not considering cause of death. While in construction of decrement life tables we are considering the cause of death of an individual and constructing multiple decrement life table.

Description about elimination

Generally, we see, what is the probability that a person dying from a specific cause. Now the concept of elimination, if we eliminate a specific cause then we can be able to see what will be the life expectancy, either it may be increasing or decreasing. From this we can be able to see how much this cause is affecting the human beings. Below is the flowchart for the causes we were considered.

Methods and Materials

The data for the study was collected from IHME, Global Burden, 2019 for India. This dataset contains age and sex wise deaths and populations, and we have considered causes of death such as malaria (m), chronic respiratory disease (CH), tuberculosis (TB), total cancers (TC), lip and oral cavity cancer (LO), HIV/AIDS, and cardiovascular disease (CVD). Now, age-specific and age-cause-specific death rates have to be calculated using the given data. First, we construct an abridged life table using the current life table method for population 2019. Several authors (King, 1914; Reed and Merrell, 1939; Greville, 1943; Sirken, 1964; Keyfitz, 1966; Chiang, 1968; and Coale and Demeny, 1983) [16,17,13,19,15,6,10] derived the formula for calculating the probability of dying through the observed death rate at each age. [17] , Chiang (1968) [6] , etc, here we are using Keyfitz (1966) [15] method that is (3).

Columns of Life

This method is applicable if we have equal width of interval 5. For less than 5 year age group Coale and Demeny (1983) [10] [10]

𝐿 𝑤 𝑥 = 𝑇 𝑤 = 𝑙 𝑤 𝑒 𝑤 (12) Using these equations we can construct ordinary life table for males and females.

Multiple

Cause Elimination life table

Here we are eliminating certain diseases, and checking the change in life expectancy. This table considers mortality experience due a specific cause (s) i,e, people have been exited from the cohort by a specific cause (s). In constructing the associated single decrement life table, the constant of proportionality for decrement other than specific cause in the age group (𝑥, 𝑥 + 𝑛), 𝑅 𝑥 -𝑖 𝑛 is computed using below formula, here -𝑖 refers as this cause is eliminated. (17) The probability of surviving between the age group (𝑥, 𝑥 + 𝑛) in absence of specific cause 𝑖 can be estimated using following formula (Chiang 1978) [8] 𝑝 𝑥 -𝑖 𝑛 = ( 𝑝 𝑛 𝑥 )

Competing risk approach

Generally competing risks means their exist more than two causes of failure. Here we can see the effect of one cause in presence and absence of the other causes (Chiang, 1960) [4] .

The study of survival and the applications of life table methodology requires understanding of these below probabilities.

a) The Crude Probability: The probability of deaths from a specific cause in presence of all other risks acting in a population.𝑄 𝑥𝛿 = 𝑝(an individual alive at time 𝑥will die in the interval (𝑥, 𝑥 + 𝑛) from cause 𝛿 in the presence of all other risks in the population)

b) The Net Probability: The probability of death if a specific risk is the only risk in effect in the population or conversely, the probability of death if a specific risk is eliminated from the population.

𝑞 𝑥𝛿 = 𝑝 (an individual alive at 𝑥will die in the interval (𝑥, 𝑥 + 𝑛) if 𝛿 is the only risk acting on the population); 𝑞 𝑥.𝛿 = 𝑝 (an individual alive at 𝑥will die in the interval (𝑥, 𝑥 + 𝑛 ) if 𝛿 is eliminated as a risk death)

c) The partial crude probability: The probability of death from a specific cause when another risk (risks) is eliminated from the population. 𝑄 𝑥𝛿.1 = 𝑝 (an individual alive at 𝑥 will die in the interval (𝑥, 𝑥 + 𝑛) from 𝛿 if cause 1 is eliminated from the population) 𝑄 𝑥𝛿.12 = 𝑝 (an individual alive at 𝑥 will die in the interval (𝑥, 𝑥 + 𝑛) from 𝛿 if cause 1 and cause 2 is eliminated from the population)

Note 1. Generally𝑞 𝑥𝛿 > 𝑄 𝑥𝛿 , Because of the absence of the competing risks, net probability is always greater than the corresponding crude probability. 2. If two risk 1 and 2 are such that 𝑄 𝑥1 > 𝑄 𝑥2 . Then 𝑞 𝑥1 > 𝑞 𝑥2 and 𝑞 𝑥.1 < 𝑞 𝑥.2 .

From these concepts we can be able to see the effect of one cause in presence and absence of other causes.

Decomposition of life expectancy (DLE) at birth difference

When a specific cause is eliminated from the decrement process, there has been change in the mortality in the all age groups that can be lead to gain or loss in the life expectancy. The difference in the life expectancy at birth with the presence and absence of the specific cause i.e. gain at birth can be decomposed into specific age groups to determine which age groups probably most contributed to the total gain using the procedure proposed by Arriaga, 1984 [2] .

The first term at the right side of the equation refers to the direct effect of a change in mortality rates between ages 𝑥and 𝑥 + 𝑛 that consequence to change in the life years within the specific age group, whereas the second term refers to the sum of both the indirect and interaction effects of contributions resulting from the number of person-years to be added because of additional survivors at age 𝑥 + 𝑛exposed to the new mortality conditions. The equation used for the openended interval is as follows

Gain in Life expectancy

The gain in life expectancy (Chiang 1991) [9] is given by

Where 𝑒 𝑥 ̂ is total life expectancy at age (𝑥, 𝑥 + 𝑛) 𝑒 𝑥.𝑗 ̂ is the life expectancy at age (𝑥, 𝑥 + 𝑛) after eliminating cause 𝐽.

Results and Discussions

As we have estimated life expectancy for male and female. And also calculated crude, net and partial crude probability, which can be given in below. Table 1 explains the abridged life table for the 2019 male population. We considered deaths and population for the year 2019 and calculated an age-specific death rate. Now the important aspect is the calculation of the fraction of years lived by l x individuals i,e, a n x . We have used the Keyfitz (1966) [15] method for the calculation of the a n x which can be calculated using current population and deaths, and we compared it with Chiang (1968, 1972) [6][7] constants; they are similar. And for the infant and child, we have used Coale and Demeny (1983) [10] values. From Table 1, we can say the life expectancy at birth is 69.63 years for the 2019 male population. Table 2 explains the life expectancy of males and females, in which we can see that females have a higher life expectancy as compared to males, as shown in Figure 1. The life expectancy of a female at birth is 72.24 years, whereas that of a male is 69.63 years; hence, we can say females have a higher life expectancy than males.

Figures 2 and3 show the life expectancy of males and females in total, and after eliminating the specified causes, here we can observe that if we eliminate all seven causes, we have a high life expectancy compared to the rest. 4 explains the life expectancy for females in that if we eliminate diseases like chronic respiratory disease and CVD, then they will get an additional 1.8 and 4.31 years. From this, we can say CVD affects more male than female populations. And if we eliminate all seven selected causes, the male population will have an additional 14.7 years of life, while females will have 12.4 additional years of life. From this, we can say these selected causes affect more males than females. Table 5 explains the competing risks approach. Here, we can say the crude probability of dying for the specific cause of malaria is 0.000188, which is acting in the presence of all causes. And for people aged 50 and older, we can see that CVD and chronic respiratory diseases have a high impact.

From this, we can say that older people are more affected by CVD and chronic respiratory diseases.

From Table 6, we can see the net probability when a specified cause was eliminated. The probability of dying at birth is 0.0299996. When we eliminate malaria and TB, we can see a slight decrease in the probability of dying, i.e., 0.029811 and 0.029882, respectively. And at age 50, the probability of dying in the presence of all causes is 0.042636, and in the absence of chronic respiratory disease, total cancers, TB, and CVD is 0.039516, 0.037152, 0.038988, and 0.027842. From this, we can say that for people aged 50 and older, these causes were affecting them more. Similarly, at age 80, the probability of dying is 0.439822, and in the absence of chronic respiratory disease and CVD, it is 0.363036 and 0.325438. From this, we can say these two causes were affecting more people aged 80 and older. Here, we can say that in the absence of competing risks, the effect changes. Table 7 explains the partial crude probability when malaria and chronic respiratory disease are eliminated. This table considers the probability of dying for different causes when malaria is eliminated. Here we see that if we eliminate malaria as a risk of death, then it will not be affected by the rest of the selected causes. This can be seen in Figure 4.

Similarly, if we eliminate chronic respiratory disease, we can see that at ages 80 and older, the probability of dying increases for the rest of the selected causes. And similarly, tables 8, 9, and 10 explain the probability of dying in the absence of total cancer, CVD, chronic respiratory disease, lip and oral cavity cancer, TB, HIV, and CVD, respectively. If we eliminate these causes, we cannot see much change in younger and middle-aged people, i.e., people aged 0-75; only those people get affected by these causes whose age is 80+. Figure 4 explains the crude and partial crude probabilities of dying with respect to the specified causes. Here we can say, as earlier explained, that the probability of dying is changing after 80+ years.

Similarly, figure 5 explains the effect on specified causes when each of the causes is eliminated separately and compared with the crude probability of the respective specific cause. This figure can be used to show the effect on a specified cause when the rest of the causes are eliminated separately. From this figure, we can say that in the absence of CVD, the probability of dying for the causes of chronic respiratory disease, total cancers, lip and oral cavity cancer, and TB, i.e., plots (b), (c), (d), and (e), has increased after age 80+. From this, we can say that the contribution of CVD causes is greater at older ages.

From tables 5 and 6, for malaria, the crude probability of dying is 0.000188, and the net probability of dying is 0.029811. From this, we can say that in the absence of competing risks, the net probability is high. Table 11 explains the decomposition of life expectancy at birth. The gain in life expectancy at birth when CVD is eliminated is 4.9 years, which means a person gains 4.9 years more life if they were not affected by CVD. This gain in life expectancy can be decomposed using the equation (33). And after decomposition, we can see that the majority of the gain is from older people, which means older people get affected more. Similarly, when we consider malaria, infants and children's lives contribute more to life expectancy; from this, we can say this cause has a high impact on these ages. Table 12 explains the percentage gain in life expectancy when the specific causes are eliminated. In the absence of CVD, there will be a 10.7% increase in life expectancy when compared with the presence of CVD at age 25. Similarly, in the absence of chronic respiratory disease at age 80+, life expectancy increases by 21.4% as compared to the presence of the disease.

References

1. Cornfield J. Estimation of the probability of developing a disease in the presence of competing risks. American Journal of Public Health and the Nations Health. 1957;47(5):601-607.
2. Dublin LI, Spiegelman M. Current versus generation life tables. Human Biology. 1941;13(4):439-458.
3. Greville TNE. Short Methods of Constructing Abridged Life Tables, Record of American Institute of Actuaries. 1943;32(65):29-43.
4. Keyfitz N. A life table that agrees with the data. Journal of the American Statistical Association. 1966;61(314):305-312.
5. King G. On a short method of constructing an abridged mortality table. Journal of the Institute of Actuaries. 1914;48(3):294-303.
6. Reed LJ, Merrell M. A Short Method for Constructing an Abridged Life Table. American Journal of Hygiene. 1939;30:33-62.
7. Sharma M, Choudhury L. Partial Elimination of Cause of Death under Dynamic Set Up and Its Applications. Thailand Statistician. 2023;21(1):137-147.
8. Sirken MG. Comparisons of Two Methods of Constructing Abridged Life Tables by reference to a "standard" table. Vital and Health statistics. Ser. 1, Programs and Collection Procedures. 1964;4(2):1-11.
9. Wong O. A competing-risk model based on the life table procedure in epidemiological studies. International Journal of Epidemiology. 1977;6(2):153-159.