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Commentary: Secular trends in the context of competing risks

Commentary: Secular trends in the context of competing risks Cancer mortality is increasing while the mortality rates of ischaemic heart disease (IHD) and cerebro-vascular disease (CVD) are decreasing, so it is natural to speculate that the increase in cancer mortality may partly be explained by new cancer cases among those avoiding or surviving IHD and CVD. Llorca and Delgado-Rodríguez present an approach based on Markov chains to evaluate this question of competing risks.1 The statistical theory behind the use of Markov chains in the analyses of competing risks is well described2 (textbox). The merit of their paper is therefore to illustrate that the different approaches previously used on this subject in the epidemiological literature can be unified. They analyse the secular trend of cancer mortality from 1981 to 1994 in Spain in the context of competing risks from IHD and CVD using both an ‘elimination model’2 (‘What would have happened if IHD and/or CVD had been eradicated’) and a ‘constant model’3 (‘What would have happened if the mortality of IHD and/or CVD had remained constant’). Under both of these scenarios, the estimated lifetime risks of cancer are very much the same as observed in 1994. Based on these findings it is tempting to conclude that the secular trends of IHD and CVD have had little influence on the secular trends of cancer mortality from 1981 to 1994. Textbox: A discrete time Markov chain is a sequence of random variables (Xn, n = 0, 1, 2, 3,…) with a dependency structure defined by the Markov property. Typically n is representing a discrete time variable and Xn the state at time n (e.g. disease status at time n). The Markov property implies that future events are conditionally independent of the past given the present, i.e. the state at times n+1, n+2, … are independent of the state at times 1 to n – 1 given knowledge of the state at time n. The probability of being in state j at time n+1 when being in state i at time n is called a transition probability. If the time variable is continuous rather than discrete one would use continuous time Markov processes (Xt, t ≥ 0), with t being a continuous time variable. The ideas behind the two approaches are very similar, however, as there is no smallest unit of time in the latter, one defines the continuous time Markov processes by transition rates rather than transition probabilities. The transition rate from state i to state j at time t is the hazard rate by which Xt changes from state i to j. It might seem puzzling that one has to perform a simulation of different hypothetical scenarios in the evaluation of the past. If the question was whether an increase in cancer mortality from 1981 to 1994 was due to an increase in smoking, say, this could—in a cohort with smoking information—be evaluated by examining the secular trend among non-smokers only. However, the constant model can be viewed in the same way as an ‘everything-else-equal’ comparison (i.e. within the stratum of people where new survivors due to decrease in IHD and CVD since 1981 are excluded). In 1981 this stratum is just the total population, in 1994 this is estimated using the higher mortality rates of 1981. If cancer mortality is increasing even in this stratum, then it is unlikely that competing risks are the cause, because in this stratum the competing risk effect is constant. In other words, evaluating historical secular trends in the context of competing risks, as in the paper by Llorca and Delgado-Rodríguez, does not have to be interpreted using different hypothetical scenarios, nor does one model have to be judged as more unrealistic than others. When evaluating past secular trends the most attractive model is the constant model because it gives an ‘everything-else-equal’ comparison based on external information. The flexible continuum of models presented by the authors is, in this situation, of little use. By contrast, when the question is about secular trends in the future, the realism and multitude of models are important considerations and the framework described by Llorca and Delgado-Rodríguez is well suited for this purpose with the elimination model as the chief example. Another model mentioned in the epidemiological literature which could also be included in this framework is the cause-delay model.4 In the cause-delay model one assumes that each generation approaches the age-specific force of mortality from IHD and CVD later than observed currently, e.g. the current mortality-specific rate for people aged 50–54 years, say, is applied in the future scenario for people aged 55–59 years. A crucial assumption in these analyses is that one can estimate the cancer mortality among survivors using the general age-specific cancer mortality, i.e. the assumption of ‘independent competing risks’.5 If the assumption is not fulfilled then it will often induce an overestimation of the beneficial effect of decreasing competing risks because of an underestimation of the mortality rate among the survivors. This would be the case if there is an ascertainment bias that has made some people more prone to have a specific cause notified on their death certificate. Therefore one should always interpret the results as the most optimistic estimate. The assumption of ‘independent competing risks’ cannot be evaluated from the data essentially because one cannot identify those avoiding IHD and CVD. If it was possible to identify the people that would have developed IHD and CVD, had it not been eradicated, say, one could compare the cancer mortality in this group with the rest, and thereby check the assumption that the age-specific cancer mortality is the same in these two groups. This is the classical non-identifiability aspect of competing risks.5 As mentioned above, the assumption of ‘independent competing risks’ implies that the age-specific cancer mortality (hazard) rates remain the same during the study period. In other words, a person has the same day-to-day risk of dying from cancer irrespective of competing risks, but only the lifetime cancer risk and the expected life length will be affected by competing risks. Therefore when the authors observe that the 5-year mortality rates are higher when IHD and CVD are eliminated, one has to bear in mind that the difference does not reflect that the age-specific cancer mortality (hazard) rate would be higher if IHD and CVD were eliminated. Rather the difference reflects residual confounding in the sense that if IHD and CVD were eliminated, women would on average get older and as cancer mortality generally increases with age one would observe a relatively higher average mortality rate within a 5-year age group. The use of the word ‘rate’ in this context is therefore misleading and only life expectancy and probability of death from cancer should be presented. A more satisfactory approach would be to use continuous time Markov processes6 (textbox). In order to apply the discrete time Markov chain the authors have to categorize the time variable (i.e. age) and use the average ‘transition’ rate for each age group to calculate the transition probability. The misinterpretation, described above, occurs because the average transition rate cannot be interpreted as a transition rate due to the confounding by age. In a continuous time Markov process the age-specific hazard rates (i.e. the transition rates) are a generic part of the model and therefore the model does not create the illusion that the age-specific rates are changed under different regimes when assuming independent competing risks. In conclusion, the use of Markov chains, as presented by Llorca and Delgado-Rodríguez, is a flexible framework in the evaluation of competing risks. Using continuous time Markov processes would, however, be an even more direct approach. Furthermore it is important to recall that results should always be interpreted with due attention to the underlying unverifiable assumption of ‘independent competing risks’. References 1 Llorca J, Delgado-Rodríguez M. Competing risks analysis using Markov chains: impact of cerebrovascular and ischaemic heart disease in cancer mortality. Int J Epidemiol 2001 ; 30: 99 –101. 2 Chiang CL. Competing risks in mortality analysis. Annu Rev Public Health 1991 ; 12: 281 –307. 3 Rothenberg RB. Competing mortality and progress against cancer. Epidemiology 1994 ; 5: 197 –203. 4 Olshansky SJ. Pursuing longevity: delay vs. elimination of degenerative diseases. Am J Public Health 1985 ; 75: 754 –57. 5 Tsiatis AA. Competing risks. In: Armitage P, Colton T (eds). Encyclopedia of Biostatistics. Vol. 1. Chichester: Wiley, 1998, pp.824–34. 6 Chiang Cl. Introduction to Stochastic Processes in Biostatistics. Wiley, New York: Wiley, 1968, Ch. 11. © International Epidemiological Association 2001 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Journal of Epidemiology Oxford University Press

Commentary: Secular trends in the context of competing risks

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References (6)

Publisher
Oxford University Press
Copyright
© International Epidemiological Association 2001
ISSN
0300-5771
eISSN
1464-3685
DOI
10.1093/ije/30.1.102
Publisher site
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Abstract

Cancer mortality is increasing while the mortality rates of ischaemic heart disease (IHD) and cerebro-vascular disease (CVD) are decreasing, so it is natural to speculate that the increase in cancer mortality may partly be explained by new cancer cases among those avoiding or surviving IHD and CVD. Llorca and Delgado-Rodríguez present an approach based on Markov chains to evaluate this question of competing risks.1 The statistical theory behind the use of Markov chains in the analyses of competing risks is well described2 (textbox). The merit of their paper is therefore to illustrate that the different approaches previously used on this subject in the epidemiological literature can be unified. They analyse the secular trend of cancer mortality from 1981 to 1994 in Spain in the context of competing risks from IHD and CVD using both an ‘elimination model’2 (‘What would have happened if IHD and/or CVD had been eradicated’) and a ‘constant model’3 (‘What would have happened if the mortality of IHD and/or CVD had remained constant’). Under both of these scenarios, the estimated lifetime risks of cancer are very much the same as observed in 1994. Based on these findings it is tempting to conclude that the secular trends of IHD and CVD have had little influence on the secular trends of cancer mortality from 1981 to 1994. Textbox: A discrete time Markov chain is a sequence of random variables (Xn, n = 0, 1, 2, 3,…) with a dependency structure defined by the Markov property. Typically n is representing a discrete time variable and Xn the state at time n (e.g. disease status at time n). The Markov property implies that future events are conditionally independent of the past given the present, i.e. the state at times n+1, n+2, … are independent of the state at times 1 to n – 1 given knowledge of the state at time n. The probability of being in state j at time n+1 when being in state i at time n is called a transition probability. If the time variable is continuous rather than discrete one would use continuous time Markov processes (Xt, t ≥ 0), with t being a continuous time variable. The ideas behind the two approaches are very similar, however, as there is no smallest unit of time in the latter, one defines the continuous time Markov processes by transition rates rather than transition probabilities. The transition rate from state i to state j at time t is the hazard rate by which Xt changes from state i to j. It might seem puzzling that one has to perform a simulation of different hypothetical scenarios in the evaluation of the past. If the question was whether an increase in cancer mortality from 1981 to 1994 was due to an increase in smoking, say, this could—in a cohort with smoking information—be evaluated by examining the secular trend among non-smokers only. However, the constant model can be viewed in the same way as an ‘everything-else-equal’ comparison (i.e. within the stratum of people where new survivors due to decrease in IHD and CVD since 1981 are excluded). In 1981 this stratum is just the total population, in 1994 this is estimated using the higher mortality rates of 1981. If cancer mortality is increasing even in this stratum, then it is unlikely that competing risks are the cause, because in this stratum the competing risk effect is constant. In other words, evaluating historical secular trends in the context of competing risks, as in the paper by Llorca and Delgado-Rodríguez, does not have to be interpreted using different hypothetical scenarios, nor does one model have to be judged as more unrealistic than others. When evaluating past secular trends the most attractive model is the constant model because it gives an ‘everything-else-equal’ comparison based on external information. The flexible continuum of models presented by the authors is, in this situation, of little use. By contrast, when the question is about secular trends in the future, the realism and multitude of models are important considerations and the framework described by Llorca and Delgado-Rodríguez is well suited for this purpose with the elimination model as the chief example. Another model mentioned in the epidemiological literature which could also be included in this framework is the cause-delay model.4 In the cause-delay model one assumes that each generation approaches the age-specific force of mortality from IHD and CVD later than observed currently, e.g. the current mortality-specific rate for people aged 50–54 years, say, is applied in the future scenario for people aged 55–59 years. A crucial assumption in these analyses is that one can estimate the cancer mortality among survivors using the general age-specific cancer mortality, i.e. the assumption of ‘independent competing risks’.5 If the assumption is not fulfilled then it will often induce an overestimation of the beneficial effect of decreasing competing risks because of an underestimation of the mortality rate among the survivors. This would be the case if there is an ascertainment bias that has made some people more prone to have a specific cause notified on their death certificate. Therefore one should always interpret the results as the most optimistic estimate. The assumption of ‘independent competing risks’ cannot be evaluated from the data essentially because one cannot identify those avoiding IHD and CVD. If it was possible to identify the people that would have developed IHD and CVD, had it not been eradicated, say, one could compare the cancer mortality in this group with the rest, and thereby check the assumption that the age-specific cancer mortality is the same in these two groups. This is the classical non-identifiability aspect of competing risks.5 As mentioned above, the assumption of ‘independent competing risks’ implies that the age-specific cancer mortality (hazard) rates remain the same during the study period. In other words, a person has the same day-to-day risk of dying from cancer irrespective of competing risks, but only the lifetime cancer risk and the expected life length will be affected by competing risks. Therefore when the authors observe that the 5-year mortality rates are higher when IHD and CVD are eliminated, one has to bear in mind that the difference does not reflect that the age-specific cancer mortality (hazard) rate would be higher if IHD and CVD were eliminated. Rather the difference reflects residual confounding in the sense that if IHD and CVD were eliminated, women would on average get older and as cancer mortality generally increases with age one would observe a relatively higher average mortality rate within a 5-year age group. The use of the word ‘rate’ in this context is therefore misleading and only life expectancy and probability of death from cancer should be presented. A more satisfactory approach would be to use continuous time Markov processes6 (textbox). In order to apply the discrete time Markov chain the authors have to categorize the time variable (i.e. age) and use the average ‘transition’ rate for each age group to calculate the transition probability. The misinterpretation, described above, occurs because the average transition rate cannot be interpreted as a transition rate due to the confounding by age. In a continuous time Markov process the age-specific hazard rates (i.e. the transition rates) are a generic part of the model and therefore the model does not create the illusion that the age-specific rates are changed under different regimes when assuming independent competing risks. In conclusion, the use of Markov chains, as presented by Llorca and Delgado-Rodríguez, is a flexible framework in the evaluation of competing risks. Using continuous time Markov processes would, however, be an even more direct approach. Furthermore it is important to recall that results should always be interpreted with due attention to the underlying unverifiable assumption of ‘independent competing risks’. References 1 Llorca J, Delgado-Rodríguez M. Competing risks analysis using Markov chains: impact of cerebrovascular and ischaemic heart disease in cancer mortality. Int J Epidemiol 2001 ; 30: 99 –101. 2 Chiang CL. Competing risks in mortality analysis. Annu Rev Public Health 1991 ; 12: 281 –307. 3 Rothenberg RB. Competing mortality and progress against cancer. Epidemiology 1994 ; 5: 197 –203. 4 Olshansky SJ. Pursuing longevity: delay vs. elimination of degenerative diseases. Am J Public Health 1985 ; 75: 754 –57. 5 Tsiatis AA. Competing risks. In: Armitage P, Colton T (eds). Encyclopedia of Biostatistics. Vol. 1. Chichester: Wiley, 1998, pp.824–34. 6 Chiang Cl. Introduction to Stochastic Processes in Biostatistics. Wiley, New York: Wiley, 1968, Ch. 11. © International Epidemiological Association 2001

Journal

International Journal of Epidemiology – Oxford University Press

Published: Feb 1, 2001

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