## Chapter 3: Competing Risks to Breast Cancer Mortality

**Abstract**

Abstract Background: Simulation models analyzing the impact of treatment interventions and screening on the level of breast cancer mortality require an input of mortality from causes other than breast cancer, or competing risks. Methods: This chapter presents an actuarial method of creating cohort life tables using published data that removes breast cancer as a cause of death. Results: Mortality from causes other than breast cancer as a percentage of all-cause mortality is smallest for women in their forties and fifties, as small as 85% of the all-cause rate, although the level and percentage of the impact varies by birth cohort. Conclusion: This method produces life tables by birth cohort and by age that are easily included as a common input by the various CISNET modeling groups to predict mortality from other causes. Attention to removing breast cancer mortality from all-cause mortality is worthwhile, because breast cancer mortality can be as high as 15% at some ages. The objective of the breast base–case study is to examine the incidence of breast cancer by stage and mortality from breast cancer by birth cohort by single year of age. The previous chapters established consistent inputs for the breast base–case for the dissemination of mammography and adjuvant therapy, as well as the underlying risk of developing breast cancer. This chapter focuses on developing inputs for the mortality component for causes other than breast cancer. A woman may die in a given year and her death would be attributable to one or more of many causes of death. The all-cause mortality rate is the probability of dying from any cause. Deaths due to breast cancer occur in the presence of other causes, or competing risks. In actuarial science, the theory of competing risks is referred to as multiple decrement theory, where a woman's death from breast cancer in the presence of other causes is called a decrement. In a practical sense, a woman with breast cancer may die from breast cancer or may die of a cause other than breast cancer prior to the time when she would have died from breast cancer. A methodologically simple way of describing mortality is with a life table. Life tables indicate an age (x) and a probability of death between ages (x) and (x + 1). This chapter explains how an all-cause cohort life table is partitioned into breast cancer mortality and mortality due to causes other than breast cancer. There are different ways of expressing mortality rates. One way describes the probability that a person exact age (x) dies before age (x + 1) and is calculated by relating the number of deaths to the number of persons of exact age (x), called qx. A second way to define a mortality rate relates the deaths during that age to the average number of persons living at that age and is called a central death rate, or mx. Both definitions are sometimes referred to as mortality rates in the literature, which is used in a particular study is left to the reader to determine. Over a 1-year age interval, the differences in definitions generally are not material, but the differences may be larger over a wider interval. Life tables define mortality rates according to the first definition, or by qx. The goal of this chapter is to develop cohort life tables for causes other than breast cancer from year of birth 1900 until calendar year 2000, the last year for the modeling under CISNET. We do not distinguish mortality by race in this study. CISNET modelers have access to breast cancer mortality rates, but also require mortality from causes other than breast cancer. Data are available for all-cause mortality probabilities by age and birth year, whereas breast cancer central mortality rates are available for grouped ages and by calendar year. These data are summarized in the “Data” section. The central death rates by calendar year are converted to probabilities, converted to a birth cohort basis, and extrapolated to ages and years of birth where the data are unavailable. An assumption that the life table ends at age 100 is included. The annual probability of dying from breast cancer is subtracted from the annual all-cause mortality probability to yield the annual probability of dying from a cause of death other than breast cancer. DATA All-cause cohort life tables were obtained from the Berkeley Mortality Database Web site (http://www.demog.berkeley.edu/∼bmd/states.html) (2). These tables are based on data from the Social Security Administration used in the 1998 Trustees Report. The Social Security Administration data did not contain annual probabilities of death by single year of age and for all years of birth 1900 and later. Researchers at University of California–Berkeley extended the Social Security Administration data to create life tables by single year of age for years of birth 1900 to 2000, inclusive. Breast cancer central mortality rates for all races in the presence of other causes were obtained from the National Center for Health Statistics (NCHS) in 5-year age intervals (0–4, 5–9, 10–14, 15–19, 20–24, 25–29, 30–34, 35–39, 40–44, 50–54, 55–59, 60–64, 65–69, 70–74, 80–84, ≥85) and by calendar year of death (1950–1999) (NCHS, http://www.cdc.gov/nchs) (3). These mortality rates were calculated as the number of deaths from breast cancer divided by the population in the 5-year age group as of July 1 of each calendar year. All-cause central mortality rates by each 5-year age group were also provided for 1969–1999, as the rates for 1950–1968 were not readily available. At the time of this study, only data through calendar year 1999 were available. The breast cancer central mortality rates were adjusted to annual probabilities and converted from a calendar year of death basis to a cohort basis as explained in the “Methods” section. METHODS The objective of the analysis is to create cohort life tables for deaths due to causes other than breast cancer. The Berkeley cohort life tables provide annual probabilities that a female age (x) will die before age (x + 1) for x = 0, …, 119 by year of birth 1900, …, 2000. We assume for this study that women will die from some cause prior to age 100. Mortality rates are needed for CISNET through calendar year 2000. These tables are referred to as the all-cause cohort mortality tables and denoted \(q^{(\mathrm{{\tau}})}_{x,{\,}BY},\) the all-cause mortality probability for a person aged (x) dying before age (x +1) born in year BY. The NCHS all-cause and breast cancer central death rates (by age intervals by calendar year) are converted to single-year-of-age central death rates by year of birth. Next, these central death rates are converted to annual probabilities. The NCHS central rates (breast cancer and all-cause) were assumed to be accurate at the center of the corresponding age interval: ages 2, 7, 12, … 82, 92. The last category, assumed centered at age 92, is inclusive of ages 85 and beyond. For ages 3–91, rates were linearly interpolated between the central ages to yield rates for the single ages. For ages 0 and 1, the rates were extrapolated from age 2; and for ages 93–99 we set the rates equal to those at age 92. The life table that is created for this study assumed that all women died no later than age 100. Figure 1 is a graph of central death rates per 100 000 women by age, where the lines in the graph represent selected calendar years. The rates for younger ages, before 30, are close to zero, whereas the maximum rate is approximately 200 per 100 000 at the highest ages. These calendar-year rates were converted into central death rates by year of birth, noting that the birth year is equal to the calendar year minus the age at death. Fig. 1. View largeDownload slide NCHS breast cancer central death rates per 100 000 for selected calendar years. Fig. 1. View largeDownload slide NCHS breast cancer central death rates per 100 000 for selected calendar years. The breast cancer central death rates are denoted as \(m_{x,BY}^{(bc)},\) i.e., the central death rate at age (x) in birth year BY from breast cancer in the presence of other causes. The all-cause central death rates are labeled as \(m_{x,BY}^{(\mathrm{{\tau}})}.\) These \(m_{x,BY}^{(\mathrm{{\tau}})}\) are defined as the number of all deaths at age (x) divided by the number of years lived by women (x) to (x + 1), or \[m_{x,BY}^{(\mathrm{{\tau}})}{=}\frac{d_{x,BY}^{(\mathrm{{\tau}})}}{{{\int}_{0}^{1}}l_{x{+}t,BY}^{(\mathrm{{\tau}})}dt},\] [3.1]whereas \(m_{x,BY}^{(bc)}\) is defined as the number of breast cancer deaths at age (x) divided by the number of years lived by women aged (x) to (x + 1), or \[m_{x,BY}^{(bc)}{=}\frac{d_{x,BY}^{(bc)}}{{{\int}_{0}^{1}}l_{x{+}t,BY}^{(\mathrm{{\tau}})}dt},\] [3.2]where the \(d{\,}_{x,^{({\bullet})}BY}\) and \(l{\,}_{x,^{({\bullet})}BY}\) symbols are the deaths and number of lives at age x and birth year BY, respectively, and the superscript is either (τ) or (bc). The denominators in practice are sometimes approximated by the number in the population midway through the interval. Let \(q_{x,BY}^{(bc)}\) be the probability of dying from breast cancer in the presence of other causes between age (x) and (x + 1) for those in birth year BY. These probabilities are subtracted from the all-cause mortality probabilities \(q^{(\mathrm{{\tau}})}_{x,{\,}BY}\) to obtain probabilities from which we then create cohort life tables for deaths due to causes other than breast cancer by birth cohort. Assuming that deaths are uniformly distributed within a year of age, then the following two relationships hold: \[m_{x,BY}^{(\mathrm{{\tau}})}{=}\frac{q_{x,BY}^{(\mathrm{{\tau}})}}{1{-}0.5q_{x,BY}^{(\mathrm{{\tau}})}}\] [3.3] \[m_{x,BY}^{(bc)}{=}\frac{q_{x,BY}^{(bc)}}{1{-}0.5q_{x,BY}^{(\mathrm{{\tau}})}}\] [3.4] Let \(q_{x,BY}^{({-}bc)}\) be the probability of dying between age (x) and (x + 1) from any cause other than breast cancer from birth year BY. The goal is to calculate \(q_{x,BY}^{({-}bc)}\) for x = 18, …, 99 by year of birth BY = 1900 … 1982 (CISNET modelers assumed that breast cancer mortality is equal to zero for ages below 18), with a restriction that x + BY ≤ 2000, as we need mortality rates only through the calendar year 2000. For calendar years 1969–1999, both NCHS all-cause central death rates and breast cancer central death rates are available. Equation [3.3] is used to solve for \(q^{(\mathrm{{\tau}})}_{x,{\,}BY}\) , whereas equation [3.4] is used to solve for \(q_{x,BY}^{(bc)}\) and together they yield the following: \[q_{x,BY}^{(bc)}{=}\frac{m_{x,BY}^{(bc)}}{1{+}0.5m_{x,BY}^{(\mathrm{{\tau}})}}\] [3.5] For calendar years 1950–1968 NCHS all-cause central death rates were not available so equation [3.4] is used to calculate \(q_{x,BY}^{(bc)}\) : \[q_{x,BY}^{(bc)}{=}m_{x,BY}^{(bc)}(1{-}0.5q_{x,BY}^{(\mathrm{{\tau}})})\] [3.6]where \(m_{x,BY}^{(bc)}\) is the NCHS breast cancer central death rate, whereas \(q^{(\mathrm{{\tau}})}_{x,{\,}BY}\) is the annual probability from the Berkeley Database. Equation [3.5] is preferred as the central rates in the numerator and denominator are from the same source, whereas in equation [3.6], the source of data for the central death rates differs from the all-cause mortality probability. Using equation [3.6] is acceptable if the NCHS all-cause mortality probabilities are similar to the Berkeley all-cause mortality probabilities. Figure 2 shows a histogram of the ratio of NCHS all-cause mortality probabilities for calendar year of death 1969–1999 to Berkeley mortality probabilities by year of birth and age. Ratios near to 1 show a good fit. The smallest ratio is 0.49 from birth year 1900 and age at death 99, whereas the largest ratio is 1.11. Ninety-seven percent of the ratios are between 0.9 and 1.1. The ratios from the very early birth cohorts and older ages at death vary substantiallyfrom 1, whereas approximately from year of birth 1907 and later the ratios are close to 1. Some reasons for these differences are our assumption that the life table ends at age 100, whereas the Berkeley tables end at age 119, and that the Berkeley tables are smoothed versions of the NCHS rates developed for use in the cohort life tables. For our purposes, we conclude that the NCHS all-cause mortality probabilities approximate the Berkeley values. Fig. 2. View largeDownload slide Frequency of ratios of NCHS all-cause mortality to Berkeley all-cause mortality. Fig. 2. View largeDownload slide Frequency of ratios of NCHS all-cause mortality to Berkeley all-cause mortality. The NCHS breast mortality data which was readily available at the time this work was conducted was for calendar years 1950–1999. These data yield a band in the matrix shown in Fig. 3, A, where the sum of the year of birth and the age (x) equals the calendar year. Fig. 3, B demonstrates for a portion of the entire matrix shown in Fig. 3, A, the available data (dark shaded area) and data that had be extrapolated (light shaded area) to provide coverage between ages 18 and 99 for birth cohorts 1900 to 1982, with estimates only needed through calendar year 2000. For example, for year of birth 1981, only \(q_{18,^{(bc)}1981}\) are available; for year of birth 1980, \(q_{18,^{(bc)}1980}\) and \(q_{19,^{(bc)}1980}\) are available; and for year of birth 1900, \(q{\,}_{x,^{(bc)}1900}\) for x = 50,…,99 are available. Fig. 3. View largeDownload slide A) Data matrix of annual probabilities by age and year of birth. B) Shaded data matrix of available annual probabilities by age and year of birth (shown for birth years 1900–1943 as a subset of entire matrix). The darker area indicates the data that are available for calendar years 1950–1999. The lighter area indicates data that are estimated using neighboring cells from the same age at the top left for ages 18–49 at earlier years of birth, as well as the bottom right for each year of birth to extend the data to include calendar year 2000. Fig. 3. View largeDownload slide A) Data matrix of annual probabilities by age and year of birth. B) Shaded data matrix of available annual probabilities by age and year of birth (shown for birth years 1900–1943 as a subset of entire matrix). The darker area indicates the data that are available for calendar years 1950–1999. The lighter area indicates data that are estimated using neighboring cells from the same age at the top left for ages 18–49 at earlier years of birth, as well as the bottom right for each year of birth to extend the data to include calendar year 2000. The probability of dying from breast cancer is generally near zero (1/1 000 000 for ages near 18 and 1/1000 for ages near 90). The data needed are shaded in gray. The matrix was completed to create rates for the blank cells by assuming that the rates for a given age were constant for neighboring years of birth and the changes in rates between birth cohorts is minimal. Thus \(q_{99,^{(bc)}1901}\) for year of birth 1901 is assumed to be equal to \(q_{99,^{(bc)}1900}\) for year of birth 1900. Similarly, \(q_{18,^{(bc)}{\,}BY}\) for years of birth BY = 1900 to 1931 are the same as \(q_{18,^{(bc)}1932}\) for 1932. The idea was to fill in the above table so that all cells where the year of birth plus the age is less than or equal to calendar year 2000 (the last year of the base–case analysis) had some value. Using these results for \(q_{x,BY}^{(bc)}\) , then \(q_{x,BY}^{({-}bc)}\) = \(q^{(\mathrm{{\tau}})}_{x,{\,}BY}\) − \(q_{x,BY}^{(bc)},\) where \(q^{(\mathrm{{\tau}})}_{x,{\,}BY}\) is from the Berkeley cohort tables. We create a life table for ages 18–99 for years of birth 1900–1982. RESULTS The ratio of \(q_{x}^{({-}bc)}/q_{x}^{(\mathrm{{\tau}})}\) is calculated for ages 18 and older and shown in Figure 4 by age. The various lines in the figure indicate the calendar year of birth. The impact of removing deaths from breast cancer is nonexistent at the youngest ages and negligible at the highest ages. The graph shows that the largest decrease in all-cause mortality after removing breast cancer as a cause of death is at age 40 (86% of the all-cause mortality). Over all the data, the largest impact was at age 43 for year of birth 1947 (85% of all-cause mortality). Also, more recent years of birth show a larger impact of removing breast cancer as a cause of death than that from earlier years of birth. Fig. 4. View largeDownload slide Ratio of the annual probability of death with breast cancer removed to the annual probability of all-cause mortality. Fig. 4. View largeDownload slide Ratio of the annual probability of death with breast cancer removed to the annual probability of all-cause mortality. CONCLUSION In summary, this chapter documents the method used to create cohort life tables for causes of death other than breast cancer. Life tables provide a simple method for describing mortality, and these tables are easily incorporated for use in simulation modeling. This analysis also found that the reduction in mortality when removing breast cancer as a cause of death can be as much as 15%. Breast cancer differs from other adult cancers as the age of diagnosis is younger than that for other cancers such as lung, colorectal, and prostate. Calculating competing-cause mortality, removing breast cancer as a cause of death, is worthwhile for use in studies where competing-cause mortality is needed. References (1) Bowers N, Gerber H, Hickman J, Jones D, Nesbitt C. Actuarial mathematics, 2nd ed. Schaumburg (IL): Society of Actuaries; 1997. Google Scholar (2) Berkeley Mortality Database. Available at: http://www.demog.berkeley.edu/∼bmd/states.html. [Last accessed: September 12, 2006.] Google Scholar (3) National Center for Health Statistics. Available at: http://www.cdc.gov/nchs. [Last accessed: September 12, 2006.] Google Scholar © The Author 2006. Published by Oxford University Press. All rights reserved. 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