Affiliations of author: Institute of Social and Preventive Medicine, University of Geneva Medical School, and Quality of Care Unit, Geneva University Hospitals, Switzerland.
Correspondence to: Thomas V. Perneger, M.D., Ph.D., Quality of Care Unit, Geneva University Hospitals, CH-1211 Geneva 14, Switzerland (e-mail: thomas.perneger{at}hcuge.ch).
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INTRODUCTION |
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In this commentary, I suggest that the evidence from these studies is interpreted incorrectly, because of misunderstandings about the meanings of relative risk versus absolute risk and the meanings of statistical interaction versus biologic interaction.
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RELATIVE RISK VERSUS ABSOLUTE RISK |
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From these results, Castelao et al. (4) concluded, "when comparable numbers of cigarettes are smoked, the risk of bladder cancer may be higher in women than in men." This message was reproduced in the Journal's Memo to the Media and was highlighted by the Associated Press ("Women smokers may face a higher risk of bladder cancer than men who smoke the same amount") (5). These summary statements are wrong, because of the unfortunate substitution of "risk" for "relative risk, as compared with nonsmokers of the same sex." When case patients and control subjects are matched on sex, as they were in this and previous studies (14), relative risks of cancer for women versus men cannot be estimated. The statistically significant interaction term only means that the relative risk of cancer associated with smoking differs between men and women.
To demonstrate why this distinction is important, let us reconstruct the disease incidence patterns in the underlying population (Table 1). This data reconstruction can be done for the study by Castelao et al. (4) because both case patients and control subjects were representative of the underlying general population. To define the populations at risk, we start by assuming that there were equal numbers of men and women in the population (say, 0.5 million men and 0.5 million women followed for 10 years; however, any large number would lead to the same conclusion), and then we apply the prevalence rates of "ever smoking" reported among control subjects by Castelao et al. (4), i.e., 66.8% (788 male control subjects who ever smoked/1180 total male control subjects) among men and 55.1% (184 female control subjects who ever smoked/334 total female control subjects) among women (Table 1
, A). Dividing the numbers of cancer patients in each sex-smoking category (Table 1
, B) by the corresponding numbers of person-years produces estimates of incidence rates of bladder cancer (Table 1
, C). Reassuringly, these rough estimates are comparable to population-based incidence rates reported elsewhere (6). These estimates contradict the authors' conclusions that women are at greater risk of bladder cancer than menin fact, the incidence of bladder cancer is more than three times higher in men than in women, whether in ever smokers or in never smokers (Table 1
, D).
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Relative risks tell only part of the story. Relative risks are obtained by dividing each absolute risk (or incidence rate) by that of the reference category: for a global analysis, women who never smoked (Table 1, D) and, for a sex-stratified analysis, never smokers of either sex (Table 1
, E). Only the latter can be estimated from a sex-matched casecontrol study. However, in the reconstructed population experience, we can also compute the absolute risks of bladder cancer that are attributable to male sex, ever smoking, or both (Table 1
, F). A new notion emerges: The increase in the risk of bladder cancer associated with male sex is more than twice as large among smokers as that among nonsmokers, or, adopting the perspective of sex-matched studies, the risk attributable to smoking is almost three times larger in men than in women (Table 1
, G). These results are impressive, particularly if one is a male smoker.
Both the relative risk approach and the absolute risk approach constitute legitimate viewpoints on a complex reality; neither is right or wrong. Each serves a different purpose. For instance, evidence exists that results presented in relative risk format are more likely to induce behavior change in individuals (7). On the other hand, absolute risk data may be more relevant for public health decisions (8).
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STATISTICAL INTERACTIONS |
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![]() | [1] |
If the equality holds, the multiplicative model is correct, and there is no interaction. If the equality does not hold, an interaction exists. In our example (Table 1, D), multiplying the relative risk of male sex among nonsmokers (relative risk = 3.59) by the relative risk of smoking in women (relative risk = 2.81) predicts a relative risk of 10.09 in men who smoke. Because the observed value (relative risk = 8.79) is lower, the required equality does not hold, and a negative interaction is said to exist between male sex and smoking on a multiplicative scale. This finding was given much emphasis in the professional and lay media.
Now if the additive risk model is used, the reasoning is similar, but "product" is replaced by "sum," and "relative risk" is replaced by "attributable risk." The additive model is written as:
![]() | [2] |
In our example (Table 1, F), the absolute risk attributable to smoking among women (6.06 cases of bladder cancer per 105 person-years) is added to the risk attributable to male sex among nonsmokers (8.65 cases of bladder cancer per 105 person-years), to yield an expected 14.71 additional cases per 105 person-years in men who smoke, compared with women who do not. Because the observed risk difference (26.03 cases per 105 person-years) is much greater, there is a positive interaction between smoking and the male sex on an additive scale.
The fact that statistical interactions can be contradictory, depending on the risk scale used, is not unique to the problem being examined in this commentary. It is arithmetically impossible to make up a situation where there is neither a multiplicative nor an additive interaction between two risk factors, except for the trivial case where one of the presumed "risk factors" does not influence the risk of disease. Whether the interaction is statistically significant or not is only a question of statistical power.
So far we have considered only two risk scales, the additive and the multiplicative. The multiplicative scale is equivalent to an additive scale after a logarithm transformation. The additive model described in equation 2 simplifies to
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and the multiplicative model described in equation 1 simplifies to
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which becomes, after logarithm transformation,
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Logarithms aside, equations 3 and 5 are identical. It turns out that the logarithm is a special case of a family of power transformations (10), of the general form:
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and
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The simple additive scale also is a special case, where p = 1. Under certain conditions (each risk factor must increase the incidence of disease, whether the other risk factor is present or absent), a value for p can be found that will eliminate interaction altogether. By trial and error, for the bladder cancer data (Table 1, C, C), interaction disappears for p = 0.12. Indeed, T0.12(29.37) = T0.12(11.99) + T0.12(9.40) T0.12(3.34). Thus, the same risk data are compatible with a positive interaction (on an additive scale), a negative interaction (after logarithm transformation), and no interaction (after transformation of power = 0.12).
The message is loud and clear: If an "interaction" can be reversed or eliminated by statistical manipulation, it must be of a statistical nature, not of a biologic nature. To recount: A statistical interaction exists when the observed risk patterns of disease in populations do not fit predictions from a particular statistical model. In contrast, biologic interaction refers to causal pathways of disease in individuals. The two notions, despite the unfortunate shared name, operate on different conceptual planes.
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CAUSAL DISEASE PATHWAYS |
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Several sufficient causes may exist for the same disease, and each usually requires several components. Let us imagine that smoking and male sex (or rather specific carcinogens linked with these labels) belong only to separate sufficient causes of bladder cancer: Smoking requires one set of conditions to produce bladder cancer, and male sex requires a fully independent set of conditions. In this case, smoking-related bladder cancer and male-sex-related bladder cancer develop independently, even in male smokers, and the risks of bladder cancer attributable to each risk factor should be additive (11). Since the additive model is obviously wrong (Table 1, C), it is possible that both smoking and male sex may contribute to at least one sufficient cause of bladder cancerin other words, that they interact in a biologic sense. Note that it is not the negative multiplicative interaction, but the positive additive interaction, that leads to this possibility. But there is at least one alternative explanation: Smoking may be merely associated with one of the unmeasured components of the sufficient cause involving male sex (or vice versa). For instance, if alcohol consumption was a component cause of bladder cancer associated with male sex and if smokers tended to drink more than nonsmokers, a positive additive interaction would appear between smoking and male sex.
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A WAY OUT? |
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NOTES |
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REFERENCES |
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1 Risch HA, Howe GR, Jain M, Burch JD, Holowaty EJ, Miller AB. Are female smokers at higher risk for lung cancer than male smokers? A case control analysis by histologic type. Am J Epidemiol 1993;138:28193.[Abstract]
2 Zang EA, Wynder EL. Cumulative tar exposure. A new index for estimating lung cancer risk among cigarette smokers. Cancer 1992;70:6976.[Medline]
3
Zang EA, Wynder EL. Differences in lung cancer risk between men and women: examination of the evidence. J Natl Cancer Inst 1996;88:18392.
4
Castelao JE, Yuan JM, Skipper PL, Tannenbaum SR, Gago-Dominguez M, Crowder JS, et al. Gender- and smoking-related bladder cancer risk. J Natl Cancer Inst 2001;93:53845.
5 The Associated Press. Smoking study brings mixed reaction. Accessed on April 3, 2001, on New York Times Web site: www.nytimes.com/aponline/national/AP-Smoking-Bladder-Cancer.html
6 Hartge P, Harvey EB, Linehan WM, Silverman DT, Sullivan JW, Hoover RN, et al. Unexplained excess of bladder cancer in men. J Natl Cancer Inst 1990;82:163640.[Abstract]
7 Edwards A, Elwyn G, Covey J, Matthews E, Pill R. Presenting risk informationa review of the effects of "framing" and other manipulations on patient outcomes. J Health Commun 2001;6:6182.[Medline]
8 Kleinbaum DG, Kupper LL, Morgenstern H. Epidemiologic research. Principles and quantitative methods. New York (NY): Van Nostrand Reinhold; 1982.
9 Greenland S, Rothman KJ. Concepts of interaction. In: Rothman KJ, Greenland S, editors. Modern epidemiology. 2nd ed. Philadelphia (PA): Lippincott-Raven; 1998. p. 32942.
10 Box GE, Cox DR. An analysis of transformations (with discussion). J R Stat Soc B 1964;26:21152.
11 Rothman KJ, Greenland S. Causation and causal inference. In: Rothman KJ, Greenland S, editors. Modern epidemiology. 2nd ed. Philadelphia (PA): Lippincott-Raven; 1998. p. 728.
Manuscript received April 24, 2001; revised August 17, 2001; accepted August 28, 2001.
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