Calculation of Life-Time Death Probability due Malignant Tumors Based on a Sampling Survey Area in China

There are at least two methods used to find the probability of death by cancer: the one is cumulative death probability (Sun, 2008), the other one is probability additive formula (Zhang, 2010). The former should be applied in medical and non-medical fields, but affected by the cut-off time. Therefore we recommend the latter method to calculate life-time death probability. Cancer incidence and mortality have increased sharply in China in recent years. The 2012 Chinese Cancer Registry Annual Report, published by the National Cancer Registry, pointed out that the probability of one person’s life-time death caused by malignant tumor was 13% in China (Inoue et al., 2000; Hao et al., 2012), which was calculated based on a cumulative death probability from 0-74-year-old residents. It does not including a person’s whole life time, as someone maybe lived more than 74 year old and it does not consider the risk of death competition. In this article, we recommend different method to calculate the probability of one person’s life-time death caused by a malignant tumor: probability additive formula. This method considers the risk of death competition, and a variety of death causes. The aim of this study is to calculate the probability of life time death caused by a malignant tumor by probability additive formula.


Introduction
There are at least two methods used to find the probability of death by cancer: the one is cumulative death probability (Sun, 2008), the other one is probability additive formula (Zhang, 2010).The former should be applied in medical and non-medical fields, but affected by the cut-off time.Therefore we recommend the latter method to calculate life-time death probability.
Cancer incidence and mortality have increased sharply in China in recent years.The 2012 Chinese Cancer Registry Annual Report, published by the National Cancer Registry, pointed out that the probability of one person's life-time death caused by malignant tumor was 13% in China (Inoue et al., 2000;Hao et al., 2012), which was calculated based on a cumulative death probability from 0-74-year-old residents.It does not including a person's whole life time, as someone maybe lived more than 74 year old and it does not consider the risk of death competition.In this article, we recommend different method to calculate the probability of one person's life-time death caused by a malignant tumor: probability additive formula.This method considers the risk of death competition, and a variety of death causes.The aim of this study is to calculate the probability of life time death caused by a malignant tumor by probability additive formula.

Data
All data (age-specific mortality) was from the third retrospective investigation of death cause in the sample area in China (attached list 1, P52-53 ), (Chen, 2008).This survey was covering 213 counties throughout the country, conducted by health ministry of the people's republic of China.

Statistical analysis
Based on an abridge life table (Sun et al., 2006), the probability additive formula was taken to calculate the probability of one person's life-time death caused by a malignant tumor.The series of formulas is as follows: (3) (4) In the end, P i 0 of age group 0 is representative of one person's life-time death probability (Zhou et al., 1991).

Results
The death probability (P i 0 ) during a life time is actually a composition index with the summation of each probability of death equal to 1.It is not affected by population composition since it is based on the life table (Sun et al., 2006), which is not influenced by the population composition.However, the ordinary proportional mortality (22.32%, is the number of deaths due to malignant tumor divided by the total number of deaths) tends to be affected by population composition (Chen, 2008).It should be noted that the proportional mortality is consistent with (P i 0 ) when the population composition equals the survival number l c on the life table.
By using probability additive formula and based on abridge life table (Sun et al., 2006)

Discussion
Cumulative mortality widely used by lots of references (Inoue et al., 2003;Kim, 2007;Andersen et al., 2012;Latouche et al., 2013), can indicate the dangerous degree of death cause and be uninfluenced by population composition.Cumulative risk belongs to a probability index, that is, it assumes no other causes of death.The calculative formula of the death probability of certain person during special age group (commonly 0~74 years) is ).It can be regarded as a strength index without the characteristics of an indicator of composition.The drawback of cumulative risk is that it cannot calculate the probability of death by during a person's life time due to complications with statistical theory.In this case, although there is only one cause of death, the death probability of everyone is 100% in the end; the only The 2012 China Cancer Registration Report concluded that the cumulative death probability of Chinese residents between the ages 0 to 74 is 13%.The third death retrospective investigation also reported (Chen, 2008) that the cumulative death probability of Chinese residents between the age of 0 to 74 due to malignant tumors is 14.13%.Taking into account the existence of multiple causes of death, in the context of competing risks, the death probability of one person during life time was 18.73%, which was calculated by the probability additive formula.Therefore, death probability 18.73% is more objective than 13% (Hao et al., 2012) or 14% (Chen, 2008) to reflect the cancer hazard level.
It can be seen from Table 2 that the mortality (26.26/10 million) and cumulative rate of liver cancer (2.72%), was higher than the mortality (24.71/10 million) and cumulative rate (2.63%) of stomach cancer, respectively, but the death probability of liver cancer (3.25%) is lower than that of gastric cancer (3.62%).The main reason is that the mortality is affected by age composition and not considering competing risks.Secondly, the liver cancer mortality of age 40 to 45 (per 21.45/10 million) was more than twice the mortality caused by stomach cancer (8.21/10 million); while at higher age groups (80 to 84 year old), the mortality of stomach cancer (per 296.76/10 million)was higher than that of the liver cancer (per 187.55/10 million).
The people dying from liver cancer tend to be younger, while most people who die from gastric cancer tend to be older.Therefore, one person's life-time death probability of liver cancer was lower than that of gastric cancer, considering the competitive death risks.
Malignant tumor is still the main cause of death in one's life time and the most common causes of cancer death were lung cancer, gastric cancer, liver cancer, esophageal cancer, colorectal and anal cancer, which was different from the result of the common causes of cancer death were lung cancer, liver cancer and colorectal cancer in Association of Southeast Asian Nations (Kimman et al, 2012).Therefore, targeted forms of cancer prevention and treatment strategies should be work out to improve people's health and prolong life in China.
By the way, the additive formula of probability would be also used to calculate the life-time incidence probability.The formula of life-time incidence probability of malignant tumors is P i 0 = (S x=0 w I i x )÷l 0 , I i x =f i x l x is for survival number on life table.
we found one person's life-time death probability caused by malignant tumor in China is 18.73% (shown in Table 1), which was different from 13% calculated by cumulative death probability, published by 2012 Chinese Cancer Registry Annual Report.On the abridge life table, χ is age group, m c is age-specific mortality, m i c is age-specific mortality caused by tumor, q c is age-specific death probability, l c is age-specific survival number, d c is age-specific death toll, r i c is age-specific proportional mortality caused by tumor, d i c is age-specific death toll caused by tumor, S x w d i x is all-age death toll caused by tumor, P i c is age-specific life-time death probability caused by tumor, P i 0 is one person's life-time death probability caused by tumor.

Table 2 . Main Death Caused by Different Malignant tumors on Sampling Areas in China
:http://dx.doi.org/10.7314/APJCP.2014.15.10.4307Calculation of Life-Time Death Probability due to a Malignancy Based on a Sampling Survey Area in China difference is the length of life time. DOI