Competing risks analysis using Markov chains: impact of cerebrovascular and ischaemic heart disease in cancer mortality

Javier Llorcaa and Miguel Delgado-Rodrígueza,b

a Division of Preventive Medicine and Public Health, University of Cantabria, School of Medicine, Santander, Spain.

Reprint requests to: Javier Llorca, Division of Preventive Medicine and Public Health, University of Cantabria School of Medicine, Av. Cardenal Herrera Oria s/n, 39011-Santander, Spain. E-mail: llorcaj{at}medi.unican.es

Abstract

Background A decrease in cerebrovascular disease (CVD) and ischaemic heart disease (IHD) mortality can produce an increase in mortality from other causes, even cancer. This problem is called the competing risks problem.

Methods A Markov chain is used to analyse the interrelation between CVD, IHD and cancer mortalities in Spanish women in 1981 and 1994. We compare the results using two models: discarding CVD and IHD mortality (the elimination model) and substituting CVD and IHD 1981 mortality rates in 1994 figures (the constant model).

Results Removing mortality from CVD and IHD increases cancer mortality rates in women aged >=70, and the probability of death from cancer rises from 10.7% to 13.3%. In the second model, the use of CVD and IHD 1981 mortality rates in 1994 data yields slightly lower mortality rates and so the impact of CVD and IHD mortality changes in the period 1981 to 1994 is negligible except in elderly women.

Conclusions Although IHD and CVD mortality have decreased in all age groups of Spanish women from 1981 to 1994, this has not had a great impact on cancer mortality.

Keywords Mortality, ischaemic heart disease, cerebrovascular disease, cancer, epidemiological methods

Accepted 11 December 1999

Ischaemic heart disease (IHD) mortality rates began to decrease in Spain in 1975,1,2 several years later than in other developed countries.3,4 Cerebrovascular disease (CVD) shows a similar trend. These changes have coincided with an increase in cancer mortality rates in elderly women, mainly due to breast cancer. How much of the increase in cancer mortality is due to the decrease in IHD and CVD mortality? This question relates to the competing risks concept. To analyse this several non-parametric approaches have been proposed and in this paper we deal with two of them: the elimination model from Chiang5 and the constant model from Rothenberg.6

Elimination model

Under the assumptions of independence between competing causes of death and proportionality of their forces of mortality, Chiang has shown the relation between the crude probability, the net probability, and the partial crude probability of death defined as probabilities of death from the cause {sigma} in the presence of all the other causes of death, in the absence of any other competing causes of death and in the absence of one specified cause of death, respectively.5,7

Constant model

Rothenberg considered elimination unrealistic and opposed the assumption of constant competing risks.6 In his paper, he analyses the effect of decreasing mortality from IHD on lung cancer mortality and provides lung cancer mortality rates if IHD mortality had remained constant. His method consists of calculating the number of survivors consequent to a decrease in IHD mortality, and applying the actual lung cancer mortality rates to survivors. As a result he obtained the number of deaths due to lung cancer among those who survive IHD. He concluded that the trend in IHD mortality has little impact on lung cancer.

To interpret both models, let us consider the impact of IHD mortality (the competing cause) on cancer mortality (the ‘competed’ cause). The elimination and constant models provide answers to two different questions: under the elimination model one can ask, ‘If an innovative treatment avoids mortality from IHD, what will happen to cancer mortality?’ (This question refers to the consequences of a measure applied to the competing cause.) Under the constant model one can ask, ‘What is the real evolution of cancer mortality when other conditions are unchanged?’ or ‘Have our measures against cancer mortality been effective when other conditions are unchanged?’ (This question refers to the consequences of a measure applied to the ‘competed’ cause.)

In this paper a Markov chain transition matrix is constructed using actual mortality rates and it is manipulated in several ways to obtain both the Chiang and Rothenberg models. With the mortality rates from a specified cause set to zero we obtain the elimination model, and substituting older mortality rates for current ones we obtain the constant model. In the two cases the impact on both mortality rates and survivors are measured.

The main goals of this paper are (1) to provide a general tool to tackle different non-parametric approaches to the competing risks problem, and (2) to measure the impact of IHD and CVD on cancer mortality rates in Spanish women aged >=40 years.

Methods

We construct a Markov chain8 with 14 states: 10 transient states corresponding to age groups 40–44, 45–49, ..., >=85; and four absorbent states corresponding to death from CVD, IHD, cancer, and other causes. Transition probabilities are estimated using actual mortality rates from 1994; for example, the probability of going from age group 50–54 to CVD is five times the CVD age-specific mortality rate, the probability of going from 50–54 to another age group is one to five times the whole mortality rate if the group is aged 55–59, or zero otherwise. This Markov chain allows us to estimate the life expectancy and the lifetime risk of death from cancer in women living in 1994.

Next, we manipulate the transition matrix by making CVD and IHD mortality rates zero. Then, the other transition probabilities must be recalculated using the equations described by Chiang. The new transition matrix is formed by both the cancer and other causes of death mortality rates that would be produced if CVD and IHD are removed as causes of death. So we can estimate the life expectancy and the lifetime risk of death from cancer in women living in 1994 once deaths from CVD and IHD have been removed.

Finally, we introduce the actual CVD and IHD mortality rates for 1981. Then, the rest of the transition probabilities are recalculated in a similar way thus obtaining the cancer and the other causes of death mortality rates that would occur in women living in 1994 if CVD and IHD mortality rates were unchanged from 1981. Once again, the life expectancy and the lifetime risk of death by cancer would be estimated under this assumption.

Results

The actual mortality rates from all-causes, CVD, IHD, and cancer are displayed in Table 1Go. Cancer mortality has decreased from 1981 to 1994 in the 50–74 age groups and increased in younger and older women. The IHD, CVD, and all-cause mortalities decreased in all age groups. As a consequence, lifetime probability of dying of cancer increased from 8.2% to 10.7%, and life expectancy increased by 0.4 years


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Table 1 Mortality rates (x105) from all causes, ischaemic heart disease (IHD), cerebrovascular disease (CVD), and cancer in Spanish women
 
Applying the elimination model to the 1994 figures (i.e. after removing mortality due to CVD, IHD, or both combined) an increase in the cancer mortality rate in elderly women >=70 years is obtained (Table 2Go). This increase is more important with increasing age (up to 2% in women >=85 years) and is mainly due to CVD. The lifetime probability of death of cancer increases to 13.3%.


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Table 2 Life expectancy, probability of death from cancer, and mortality rates from cancer in 40 year old Spanish women after avoiding mortality from cerebrovascular disease (CVD) and ischaemic heart disease (IHD)
 
Adjusting cancer mortality for CVD and IHD mortality in 1981 (the constant model), yields opposing results (Table 3Go): cancer mortality rates decrease with age compared to the actual figures for 1994; so a small part (from 1 per 100 000 in 70–74 year old women up to 12 per 100 000 in >=85 year old women) of the actual increase in cancer mortality trend is due to the decrease in mortality from CVD and IHD. Probability of death from cancer also decreases a little (from the actual 10.7% to 10.4%). Finally, life expectancy after adjusting for CVD and IHD is intermediate between those of 1981 and 1994, showing that half of the improvement in life expectancy is due to the decrease in CVD and IHD mortality.


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Table 3 Life expectancy, probability of death from cancer, and mortality rates from cancer in 40 year old Spanish women adjusted for 1981 cerebrovascular disease (CVD) and ischaemic heart disease (IHD) mortality
 
Discussion

Analyses of competing risks usually eliminate5,7,9 a specific cause to estimate the effect on mortality from other causes. Rothenberg believes that this assumption is unrealistic and proposes a method for analysing competing risks when mortality from competing causes remains constant.6 The approach in this paper considers changes in competing causes of death as a continuum, and it provides a unified tool to analyse it, including the possibility of simulating any other competing situation substituting one or more transition probabilities.

When cause-specific mortality rates set to zero are used the elimination model is employed; yielding the results displayed in Table 2Go. When cause-specific mortality rates are substituted with those from a previous cohort the constant model is employed; yielding the results displayed in Table 3Go. These two models are particular examples of this scheme, and a great diversity of other situations also exist.

Competing risks analysis is usually applied assuming independence between causes of death. Chiang suggests that independence can be accepted if the analysed causes are sufficiently distant. Although there is no rule to measure such a distance, a useful criterion would be to consider as independent causes which do not share any common risk factors. In our particular example, we assume that the unique common risk factor is tobacco smoking (risk factor for both IHD and several cancers). Nevertheless, smoking prevalence in Spanish women and mortality from tobacco-related cancers (including lung, laryngeal, and cervix uteri cancer) are low, so assuming independence cannot produce a large error. Such an assumption would be mistaken in Spanish men.

The elimination model shows that CVD and IHD mortality are responsible for a loss of 0.2 years in life expectancy in women. On the other hand, CVD and IHD mortality avoid an important increase of probability of death from cancer (from 10.7 to 13.3%), although the impact on cancer mortality rates is negligible under 70 years.

How many women would die from cancer in 1994 if CVD and IHD mortality had not changed since 1981? The constant model answers this question: 10.4% instead of the actual figure of 10.7%. The same model allows comparison of cancer age-specific mortality rates in 1981 (fifth column in Table 1Go) and 1994 after adjusting for the IHD and CVD rates (right column in Table 3Go). This comparison shows that the vast majority of the increase in cancer mortality between 1981 and 1994 has not been produced by the competing effects of the IHD and CVD mortalities. Therefore, decreases in CVD and IHD mortality have not played a relevant role in preventing a decrease in cancer mortality in Spanish women from1981 to 1994, except in elderly women.

Summarizing, Markov chains allow the use of several models (elimination, constant or other models) of competing risks, mixing effects on both mortality rates and survivors. Although IHD and CVD mortality have decreased in all age groups of Spanish women from 1981 to 1994, this has not had a great impact on cancer mortality.

Notes

b Current address: Division of Preventive Medicine and Public Health, University of Jaen, Jaén, Spain. Back

References

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