0895-4356/90$3.00+ 0.00 Copyright 0 1990Pergamon Press plc
J Clin Epidemiol Vol. 43, No. 9, pp. 875-879, 1990 Printed in Great Britain. All rights reserved
ESTIMATING THE BENEFITS OF CHOLESTEROL LOWERING: ARE RISK FACTORS FOR CORONARY HEART DISEASE MULTIPLICATIVE? JONATHAN S. SILBERBERG* Division of Cardiology, Royal Victoria Hospital, Departments of Cardiology and Epidemiology & Biostatistics, McGill University, Montreal, Canada (Received in revised form 14 March 1990)
Abstract-It is often stated that the major risk factors for coronary heart disease (CHD)-smoking, high blood pressure and high serum cholesterol-are not merely additive but act together such that each multiplies the effects of the others. Economic analyses in which the benefits of risk factor modification are estimated often reflect this. This paper explains how predictive models based on the simplest form of the multiple logistic function inevitably predict greater benefit from cholesterol lowering in those in whom other risk factors are adverse; this results from the model itelf, rather than the data. CHD death rates from the screenee population of the Multiple Risk Factor Intervention Trial are examined: these suggest that the relationship between cholesterol and both other major risk factors is closer to additive than to multiplicative. When the benefits of cholesterol lowering are estimated, a model based on additive risk, specifying product (“interaction”) terms, is to he preferred. Cholesterol
Additive risk
Coronary heart disease
INTRODUCTION
Multiplicative
risk
cholesterol lowering have used a predictive model based on the multiple logistic function, and it is frequently overlooked that the mathematical model has effects of its own, distinct from the data. This paper explains how an exponential relationship between serum cholesterol and risk, and apparent greater gain from cholesterol lowering in smokers and hypertensives, are inevitable results of predicting events using the simplest form of the multiple logistic model, and uses data from the Multiple Risk Factor Intervention Trial (MRFIT) screenee population to examine the premise that risk factors for coronary heart disease are multiplicative.
Increasingly, formal analyses of the costs and benefits of new interventions are being requested by third party payers [l]. Although such analyses should never be used alone to make decisions regarding treatment, they have considerable appeal to decision-makers [2] and have been used for comparison between treatment options [3]. Given this influential role, it is essential that such analyses be free of artefact. In the case of coronary heart disease (CHD), several analyses have claimed substantially greater gains from cholesterol lowering in smokers and hypertensives than in those whose only adverse risk marker is serum cholesterol [4-91. Most efforts to estimate the benefits of
MATHEMATICAL MODELS FOR PREDICTING EVENTS .
‘All correspondence should he addressed to: Dr J. S. Silberberg, Discipline of Medicine, University of Newcastle, N.S.W. 2308, Australia.
Any mathematical function can be fitted to any data; how well the model fits can then
875
816
JONATHAN %SILBERBERG
be examined by a goodness-of-fit statistic. The investigator is free to select a multivariate model, and may choose, inter alia, between one based on additive or on multiplicative risk. Ever since it was first developed at the time of the Framingham study [lo, 1l] the multiple logistic model has been the model most often chosen for risk estimation, and numerous such studies of the relationship between cholesterol and cardiovascular risk have been conducted [reviewed in Ref. 121. Even though the Framingham investigators cautioned [13] that the risk estimates rested heavily on the logistic model (with its built-in exponentiality), several authors “observed” an exponential relationship between cholesterol and risk; subsequently it appears that the relationship is better approximated as linear throughout most of the range of serum cholesterol [ 141. Similarly, differences between the rates predicted by the model and those observed have sometimes been quite striking. The model was used to estimate coronary disease risk based on the Medical Research Council trial in mild hypertension. The estimated benefit of cholesterol lowering was greater in smokers [9]; the model would similarly predict the benefits of blood pressure lowering to be greater in smokers. In contrast, the observed benefit of antihypertensive therapy was greater in non-smokers [15]. The limitations of the logistic model in the analysis of epidemiologic data have been clearly pointed out [16]. It is necessary to consider the nature of this model, and the assumptions inherent in its use. It assumes that the relationship between risk and the variables of interest and their coefficients is as described by the logistic function, and that this relationship is of the form
(1) where P is the probability of the event, e is the base of the natural logarithm, the X’s stand for the variables of interest, the /I’s are the regression coefficients for these variables, and a is the intercept. This model is nonlinear with respect to risk. However it can be rewritten in terms of the logit transformation, thus
In this form, the model is linear, and in the absence of product (“interaction”) terms, the right-hand side is additive: since the effects of various factors are added together and then exponentiated, the effects become multiplicative in the natural scale, and the effect of any continuous variable becomes exponential. Familiarity with the multiple logistic model, the convenience of risk being constrained to between 0 and 1, and the ready availability of computer programs have led to its almost exclusive use with case-noncase data. But in its simple form (without product terms) it cannot examine departures from multiplicativity, and even when such “interaction” terms are specified, their interpretation is not straightforward. As an alternative to the logistic model, a general linear model for risk has been described for some time. Thus P=a
+/?,X,+&XZ+...flnXn.
(3)
It is possible under this relation for the risk estimate to be less than zero or greater than one, and thus some have considered this model unsuitable [17]; however providing that realistic limits are set for the X-variables of interest (such as within the range of the original data), the zero/one boundaries will not always be exceeded. Simplicity under this model leads to a straight line relation between risk and the factor of interest and an additive relation between markers of risk; furthermore, the interpretation of product terms is straightforward [18]. More recently, relative risk models utilizing maximum likelihood methods which provide for additive risk and offer alternatives to the usual exponential form have been described [19-211. Computer programs have to be individualized, however, and the models are complex, requiring a large number of events for the examination of interactions. Such a function, fitted to a large body of data, could be used in economic analyses to better estimate the benefits of risk factor modification. Ongoing examination of risk functions, utilizing the MRFIT screenee population data, is currently in progress (Kjelsberg M: personal communication). At the time of model fitting, several considerations may underlie the choice between a model based on additive or on multiplicative risk. In some situations, an underlying biologic model of disease causation may indicate one mathematical model to be more plausible. Thus based on the multistage theory of carcinogenesis, there is reason to believe that causes of cancer may
Cholesterol,
CHD and MultiplicativeRisk
interact multiplicatively, depending on when exposure occurs relative to other causes [22]; in this case fitting the multiplicative model seems appropriate. With regard to coronary heart disease, although the sites at which the major risk factors act to cause disease and the possible sites of interaction between them have been described [23-251, whether joint effects are likely to be additive, multiplicative, or neither, has not been addressed in theory. In the absence of such biologic insights, a further desideratum of model fitting is that the model be as simple as possible, in terms of the number of parameters [26]: this implies a minimum number of product terms. The salient difference between models based on additive and those based on multiplicative risk is that in the former, the rate difference is held to be uniform across strata of covariates, whereas in the latter the rate ratio is held to be uniform. The initial choice of model should be based on consideration as to which parameter-rate difference or rate rat&--is more reasonably viewed as invariant against the background rate
P81There is reason to consider a priori that the rate ratio associated with hypercholesterolaemia is not the same in smokers as in non-smokers. Where multiple causes are operative, rates are high, and the proportion of cases attributable to any particular cause is less than in the lowrate domain, where single causes are usually responsible. In view of the mathematical relation between aetiologic fraction and rate ratio, it is thus unlikely that the rate ratio associated with a particular cause is consistent across strata with very different rates. This view is supported by a study of cigarette smoking and nonfatal myocardial infarction [27], in which is was evident that the rate ratio approached unity as the reference risk score increased. When using a fitted model to estimate benefits, an advantage of a model based on additive risk concerns the public health significance of “synergism”. Defined as interdependence of effects (at the individual level), it reflects the situation in which joint exposure to two or more factors results in a greater number of cases than exposure to the sum of the separate factors. The public health significance of joint effects is thus most clearly described under the additive risk model [28-301, where the coefficient for the product term addresses synergism or antagonism inherently [ 181.
811
JOINT EFFECTS OF CHOLESTEROL, SMOKING AND BLOOD PRESSURE
While multivariate regression functions are “robust” in summarizing effects over numerous strata, the clearest way to examine the joint effects of multiple factors is by cross-tabulation. Tables 1 and 2 examine the risk of coronary death within 6 years among primary screenees of the Multiple Risk Factor Intervention Trial (MRFIT) [31], according to level of serum cholesterol and the two other major risk factors. In these tables, only the risk associated with the upper 20% of the serum cholesterol distribution (“hypercholesterolaemia”) is examined, and the rates for the remaining 80% are pooled; but the analysis could be performed for all strata of serum cholesterol. Risk factors are examined two at a time, with further stratification according to the status of the third risk factor. (In this way, only the two-way joint effects are examined, not the three-way effects.) By definition [32], risks are multiplicative if the product of relative risks for each characteristic alone equals the relative risk for subjects manifesting both characteristics. Risks are additive if the excess relative risk (relative risk minus 1) for individuals manifesting two adverse characteristics equals the sum of excess relative risks for each characteristic alone. Comparison of the observed risk of CHD death with those expected under each of these assumptions indicates that the joint actions of high serum cholesterol and either smoking or hypertension are closer to additive than to multiplicative, with three of four “expected” excess relative risks falling within the asymptotic 95% confidence limits of the observed excess relative risk. In contrast, none of the four relative risks expected under multiplicativity fall within the confidence intervals of observed relative risks. The fit under the additive assumption is best when all three characteristics are adverse (observed excess risk 2.11, sum of individual excess risks 2.41; and observed, 2.57, vs sum, 3.0). While the MRFIT screenee population represents the largest cohort ever followed, it is only one source of data. Unfortunately, the published tables from the Framingham study [33] are not stratified in enough depth to allow two-way analysis as above. Similar analysis of these data and those of the other major observational cohort studies would be most instructive.
878
JONA~N S. SILBERBERG Table 1. Six-year coronary disease death rates and relative risks according to strata of risk factors in 356,222 screenees of MRFIT* [hypertension (diastolic BP > 90) and/or hypercholesterolaemia (z 245 mg%)]
Cholesterol > 245
Rate (per 1000)
No Yes
2.58 6.96
Yes
No
5.38
Yes
Yes
11.80
DBP > 90 Non smokers No No
Sum (no, yes + yes, no) Product (no, yes + yes, no)
Relative risk (95% CI) 1.0 2.7 (2.3,3.2) 2.1 (1.8,2.5) 4.58 (3.8,5.5)
Excess relative risk (95% CI)
1.7 (1.3,2.2) (0.81ll.5) 3.58 (2.8,4.5) 2.80
5.67
Smokers No No
No Yes
6.32 13.22
Yes
No
12.76
Yes
Yes
21.57
1.0 ,1.t:.,
Sum (no, yes + yes, no) Product (no, yes * yes, no)
(1.82.023) 3.41’ (2.9,4.0)
co.;.: 4) 1:0i (0.8,1.3) 2.41 (1.9,3.0) 2.11
4.22
*Risks are additive if the observed excess relative risk (yes, yes) equals the sum of excess relative risks (no, yes and yes, no). Risks are multiplicative if the observed relative risk equals the product of relative risks (no, yes and yes, no). 95% CI = Asymptotic 95% confidence interval. Table 2. Six-year coronary disease death rates and relative risks according to strata of risk factors in 356,222 screenees of MRFIT* [smoking and/or hypercholesterolaemia (> 245%)]
Smoker
Cholesterol >245
No hypertension No No
Rate (per 1000)
No Yes
2.58 6.96
Yes
No
6.32
Yes
Yes
13.2
Sum (no, yes + yes, no) Product (no, yes * yes, no) Hypertension No No
No Yes
Relative risk (95% CI)
Excess relative risk (95% CI)
1.0
2.70 (2.3,3.2) 2.45 (2.1,2.8) 5.13 (4.4,6.0)
1.70 (1.3,2.2) 1.45 (1.1, 1.8) 4.13 (3.4,5.0) 3.15
6.62 5.38 11.8
Yes
No
12.76
Yes
Yes
21.57
Sum (no, yes + yes, no) Product (no, yes z yes, no)
::;o
(1.8,2.6) 2.37 (2.0,2.8)
(0.k21 6) 1.37 (1.0, 1.8)
(3:,‘4.8)
(2.3;‘3 8) 2.5;
5.21
*Footnote as in Table 1.
CONCLUSION
AND IMPLICATIONS
When predictive models are used to estimate benefits of risk factor modification, these should be carefully chosen and their intrinsic effects
identified, since it is essential that the estimates which result reflect the data rather than the model. The foregoing analysis indicates that the risk of CHD death associated with high serum cholesterol and the two other major risk factors
Cholesterol, CHD and Multiplicative Risk
879
is greater than additive, but it is largely the use of the multiple logistic model which has led to the notion that these effects are multiplicative: analyses so based may have overestimated the benefits of cholesterol lowering in smokers and hypertensives. This is particularly relevant at the present time given the need for evaluation of the HMG-CoA reductase inhibitors, the most efficacious and also expensive agents yet available.
14* Martin MJ, Hulley SB, Browner WS, Kuller LH,
Acknowledgemenrs-Thanks are due to Olli Miettinen and Rick Grobbee for helpful comments.
19.
15. 16.
17. 18.
20. REFERENCES 1. Leaf A. Cost effectiveness as a criterion for Medicare coverage. N Engl J Med 1989; 321: 898900. 2. Office of Technology Assessment. The Impllcatlons of Cost-Effectiveness Analysis of Medical Technology. Washington, D.C.: Congress of the United States: 1980. 3. Weinstein MC, Stason WB. Cost-effectiveness of interventions to prevent or treat coronary heart disease. Ann Rev Publlc Health 1985; 6: 41-63. 4. Oster G, Epstein A. Primary prevention of coronary heart disease: the economic benefits of lowering serum cholesterol. Am J Public Health 1986; 76: 647-656. 5. Oster G, Epstein A. Cost-effectiveness of antihyperlipemic therapy in the prevention of coronary heart disease. JAMA 1987; 258: 2381-2387. 6. Kinosian B, Eisenberg J. Cutting into cholesterol. Cost effective alternatives for treating hypercholesterolemia. JAMA 1988; 259: 2249-2254. 7. Taylor W, Pass T, Shephard D, Komaroff A. Cholesterol reduction and life expectancy. A model incorporating multiple risk factors. Ann Intern Med 1987; 106: 605-614. 8. Gotto A. Coexistent hypercholesterolemia and hypertension in CHD risk: clinical aspects and considerations. Lipid Rev 1988; 2: 17-23. 9. Rose G. Review of primary prevention trials. Am Heart J 1986; 114: 1013-1017. 10. Walker S, Duncan D. Estimation of the probability of an event as a function of several independent variables. Biometrika 1967; 54: 167. 11. Truett J, Cornfield J, Kannel W. A multivariate analysis of coronary heart disease in Framingham. J Cbron Dis 1967; 20: 511-524. 12. Tyroler H. Review of lipid-lowering clinical trials in relation to observational epidemiologic studies. Circulation 1987; 76: 515-522. 13. Department of Health, Education and Welfare. The Framingham Study: an Epidemiologic Investigation of Cardiovascular Disease. Washinaton. D.C.: U.S. Govt Printing Office; 1968. -
21. 22.
23. 24. 25.
26.
27.
28. 29. 30. 31.
32.
33.
Wentworth D. Serum cholesterol, blood pressure and mortality: implications from a cohort of 361,662 men. Lancet 1986; ii: 933-936. Medical Research Council Working Party. MRC trial of treatment of mild hypertension: principal results. Br Med J 1985; 291: 97-104. Greenland S. Limitation of the logistic analysis of epidemiologic data. Am J Epidemiol 1979; 110: 693-698. Gordon T, Kannel W. Multiple risk functions for predicting coronary heart disease: the concept, accuracy and application. Am Heart J 1982; 103: 1031-1039.~ Miettinen 0. Theoretical Enidendoloer. New York: John Wiley; 1985; 233-237. _ -_ Thomas D. General relative risk models for survival time and matched case-control analvsis. Biometrles 1981; 37: 673-686. Breslow N, Storer B. General relative risk functions for case-control studies. Am J Enidemiol 1985: 122: 149-162. Lubin J, Gaffey W. Relative risk models for assessing the joint effects of multiple factors. Am J Iud Med 1988; 13: 149-167. Armitage P, Doll R. A two-stage theory of carcinogenesis in relation to the age distribution of human cancer. Br J Cancer 1957; XI: 161-169. Ross R. Atherosclerosis: a problem of the biology of arterial wall cells and their interactions with blood components. Arteriosclerosis 1981; 1: 293-311. Ross R. The pathogenesis of atherosclerosis-an update. N Engl J Med 1986; 314: 488-500. Dzau VJ. Mechanism of the interaction of hypertension and hypercholesterolemia in atherogenesis: the effects of antihypertensive agents. Am Heart J 1988; 116: 1725-1728. Walter SD, Holford TR. Additive, multiplicative and other models for disease risks. Am J Epidemiol 1978; 108: 341-346. Miettinen OS, Neff RK, Jick H. Cigarette-smoking and nonfatal myocardial infarction: rate ratio in relation to age, sex and predisposing conditions. Am J Epidemiol 1976: 103: 30-36. Blot WJ, Day NE (Letter). Synergism and antagonism: are they equivalent? Am J Epidemlol 1979; 110: 99-100. Rothman KJ. Occam’s razor pares the choice among statistical models. Am J Epidemlol 1978; 108: 347-349. Koopman J. Interaction between discrete causes. Am J Epidemiol 1981; 113: 716-724. Stamler J, Wentworth D, Neaton J for the MRFIT Research Group. Is relationship between serum cholesterol and risk of premature death from coronary heart disease continuous and graded? JAMA 1986; 256: 2823-2828. Cole P, McMahon B. Attributable risk percent in case-control studies. Br J Prev Sot Med 1971; 25: 242-244. U.S. Department of Health & Human Services. The Framingham Study: An Epidemiologic Investigation of CardiovascularDisease, NIH Publication No. 87-2703; 1987.
J Clin Epidemiol Vol. 43, No. 9, pp. 875-879, 1990 Printed in Great Britain. All rights reserved
ESTIMATING THE BENEFITS OF CHOLESTEROL LOWERING: ARE RISK FACTORS FOR CORONARY HEART DISEASE MULTIPLICATIVE? JONATHAN S. SILBERBERG* Division of Cardiology, Royal Victoria Hospital, Departments of Cardiology and Epidemiology & Biostatistics, McGill University, Montreal, Canada (Received in revised form 14 March 1990)
Abstract-It is often stated that the major risk factors for coronary heart disease (CHD)-smoking, high blood pressure and high serum cholesterol-are not merely additive but act together such that each multiplies the effects of the others. Economic analyses in which the benefits of risk factor modification are estimated often reflect this. This paper explains how predictive models based on the simplest form of the multiple logistic function inevitably predict greater benefit from cholesterol lowering in those in whom other risk factors are adverse; this results from the model itelf, rather than the data. CHD death rates from the screenee population of the Multiple Risk Factor Intervention Trial are examined: these suggest that the relationship between cholesterol and both other major risk factors is closer to additive than to multiplicative. When the benefits of cholesterol lowering are estimated, a model based on additive risk, specifying product (“interaction”) terms, is to he preferred. Cholesterol
Additive risk
Coronary heart disease
INTRODUCTION
Multiplicative
risk
cholesterol lowering have used a predictive model based on the multiple logistic function, and it is frequently overlooked that the mathematical model has effects of its own, distinct from the data. This paper explains how an exponential relationship between serum cholesterol and risk, and apparent greater gain from cholesterol lowering in smokers and hypertensives, are inevitable results of predicting events using the simplest form of the multiple logistic model, and uses data from the Multiple Risk Factor Intervention Trial (MRFIT) screenee population to examine the premise that risk factors for coronary heart disease are multiplicative.
Increasingly, formal analyses of the costs and benefits of new interventions are being requested by third party payers [l]. Although such analyses should never be used alone to make decisions regarding treatment, they have considerable appeal to decision-makers [2] and have been used for comparison between treatment options [3]. Given this influential role, it is essential that such analyses be free of artefact. In the case of coronary heart disease (CHD), several analyses have claimed substantially greater gains from cholesterol lowering in smokers and hypertensives than in those whose only adverse risk marker is serum cholesterol [4-91. Most efforts to estimate the benefits of
MATHEMATICAL MODELS FOR PREDICTING EVENTS .
‘All correspondence should he addressed to: Dr J. S. Silberberg, Discipline of Medicine, University of Newcastle, N.S.W. 2308, Australia.
Any mathematical function can be fitted to any data; how well the model fits can then
875
816
JONATHAN %SILBERBERG
be examined by a goodness-of-fit statistic. The investigator is free to select a multivariate model, and may choose, inter alia, between one based on additive or on multiplicative risk. Ever since it was first developed at the time of the Framingham study [lo, 1l] the multiple logistic model has been the model most often chosen for risk estimation, and numerous such studies of the relationship between cholesterol and cardiovascular risk have been conducted [reviewed in Ref. 121. Even though the Framingham investigators cautioned [13] that the risk estimates rested heavily on the logistic model (with its built-in exponentiality), several authors “observed” an exponential relationship between cholesterol and risk; subsequently it appears that the relationship is better approximated as linear throughout most of the range of serum cholesterol [ 141. Similarly, differences between the rates predicted by the model and those observed have sometimes been quite striking. The model was used to estimate coronary disease risk based on the Medical Research Council trial in mild hypertension. The estimated benefit of cholesterol lowering was greater in smokers [9]; the model would similarly predict the benefits of blood pressure lowering to be greater in smokers. In contrast, the observed benefit of antihypertensive therapy was greater in non-smokers [15]. The limitations of the logistic model in the analysis of epidemiologic data have been clearly pointed out [16]. It is necessary to consider the nature of this model, and the assumptions inherent in its use. It assumes that the relationship between risk and the variables of interest and their coefficients is as described by the logistic function, and that this relationship is of the form
(1) where P is the probability of the event, e is the base of the natural logarithm, the X’s stand for the variables of interest, the /I’s are the regression coefficients for these variables, and a is the intercept. This model is nonlinear with respect to risk. However it can be rewritten in terms of the logit transformation, thus
In this form, the model is linear, and in the absence of product (“interaction”) terms, the right-hand side is additive: since the effects of various factors are added together and then exponentiated, the effects become multiplicative in the natural scale, and the effect of any continuous variable becomes exponential. Familiarity with the multiple logistic model, the convenience of risk being constrained to between 0 and 1, and the ready availability of computer programs have led to its almost exclusive use with case-noncase data. But in its simple form (without product terms) it cannot examine departures from multiplicativity, and even when such “interaction” terms are specified, their interpretation is not straightforward. As an alternative to the logistic model, a general linear model for risk has been described for some time. Thus P=a
+/?,X,+&XZ+...flnXn.
(3)
It is possible under this relation for the risk estimate to be less than zero or greater than one, and thus some have considered this model unsuitable [17]; however providing that realistic limits are set for the X-variables of interest (such as within the range of the original data), the zero/one boundaries will not always be exceeded. Simplicity under this model leads to a straight line relation between risk and the factor of interest and an additive relation between markers of risk; furthermore, the interpretation of product terms is straightforward [18]. More recently, relative risk models utilizing maximum likelihood methods which provide for additive risk and offer alternatives to the usual exponential form have been described [19-211. Computer programs have to be individualized, however, and the models are complex, requiring a large number of events for the examination of interactions. Such a function, fitted to a large body of data, could be used in economic analyses to better estimate the benefits of risk factor modification. Ongoing examination of risk functions, utilizing the MRFIT screenee population data, is currently in progress (Kjelsberg M: personal communication). At the time of model fitting, several considerations may underlie the choice between a model based on additive or on multiplicative risk. In some situations, an underlying biologic model of disease causation may indicate one mathematical model to be more plausible. Thus based on the multistage theory of carcinogenesis, there is reason to believe that causes of cancer may
Cholesterol,
CHD and MultiplicativeRisk
interact multiplicatively, depending on when exposure occurs relative to other causes [22]; in this case fitting the multiplicative model seems appropriate. With regard to coronary heart disease, although the sites at which the major risk factors act to cause disease and the possible sites of interaction between them have been described [23-251, whether joint effects are likely to be additive, multiplicative, or neither, has not been addressed in theory. In the absence of such biologic insights, a further desideratum of model fitting is that the model be as simple as possible, in terms of the number of parameters [26]: this implies a minimum number of product terms. The salient difference between models based on additive and those based on multiplicative risk is that in the former, the rate difference is held to be uniform across strata of covariates, whereas in the latter the rate ratio is held to be uniform. The initial choice of model should be based on consideration as to which parameter-rate difference or rate rat&--is more reasonably viewed as invariant against the background rate
P81There is reason to consider a priori that the rate ratio associated with hypercholesterolaemia is not the same in smokers as in non-smokers. Where multiple causes are operative, rates are high, and the proportion of cases attributable to any particular cause is less than in the lowrate domain, where single causes are usually responsible. In view of the mathematical relation between aetiologic fraction and rate ratio, it is thus unlikely that the rate ratio associated with a particular cause is consistent across strata with very different rates. This view is supported by a study of cigarette smoking and nonfatal myocardial infarction [27], in which is was evident that the rate ratio approached unity as the reference risk score increased. When using a fitted model to estimate benefits, an advantage of a model based on additive risk concerns the public health significance of “synergism”. Defined as interdependence of effects (at the individual level), it reflects the situation in which joint exposure to two or more factors results in a greater number of cases than exposure to the sum of the separate factors. The public health significance of joint effects is thus most clearly described under the additive risk model [28-301, where the coefficient for the product term addresses synergism or antagonism inherently [ 181.
811
JOINT EFFECTS OF CHOLESTEROL, SMOKING AND BLOOD PRESSURE
While multivariate regression functions are “robust” in summarizing effects over numerous strata, the clearest way to examine the joint effects of multiple factors is by cross-tabulation. Tables 1 and 2 examine the risk of coronary death within 6 years among primary screenees of the Multiple Risk Factor Intervention Trial (MRFIT) [31], according to level of serum cholesterol and the two other major risk factors. In these tables, only the risk associated with the upper 20% of the serum cholesterol distribution (“hypercholesterolaemia”) is examined, and the rates for the remaining 80% are pooled; but the analysis could be performed for all strata of serum cholesterol. Risk factors are examined two at a time, with further stratification according to the status of the third risk factor. (In this way, only the two-way joint effects are examined, not the three-way effects.) By definition [32], risks are multiplicative if the product of relative risks for each characteristic alone equals the relative risk for subjects manifesting both characteristics. Risks are additive if the excess relative risk (relative risk minus 1) for individuals manifesting two adverse characteristics equals the sum of excess relative risks for each characteristic alone. Comparison of the observed risk of CHD death with those expected under each of these assumptions indicates that the joint actions of high serum cholesterol and either smoking or hypertension are closer to additive than to multiplicative, with three of four “expected” excess relative risks falling within the asymptotic 95% confidence limits of the observed excess relative risk. In contrast, none of the four relative risks expected under multiplicativity fall within the confidence intervals of observed relative risks. The fit under the additive assumption is best when all three characteristics are adverse (observed excess risk 2.11, sum of individual excess risks 2.41; and observed, 2.57, vs sum, 3.0). While the MRFIT screenee population represents the largest cohort ever followed, it is only one source of data. Unfortunately, the published tables from the Framingham study [33] are not stratified in enough depth to allow two-way analysis as above. Similar analysis of these data and those of the other major observational cohort studies would be most instructive.
878
JONA~N S. SILBERBERG Table 1. Six-year coronary disease death rates and relative risks according to strata of risk factors in 356,222 screenees of MRFIT* [hypertension (diastolic BP > 90) and/or hypercholesterolaemia (z 245 mg%)]
Cholesterol > 245
Rate (per 1000)
No Yes
2.58 6.96
Yes
No
5.38
Yes
Yes
11.80
DBP > 90 Non smokers No No
Sum (no, yes + yes, no) Product (no, yes + yes, no)
Relative risk (95% CI) 1.0 2.7 (2.3,3.2) 2.1 (1.8,2.5) 4.58 (3.8,5.5)
Excess relative risk (95% CI)
1.7 (1.3,2.2) (0.81ll.5) 3.58 (2.8,4.5) 2.80
5.67
Smokers No No
No Yes
6.32 13.22
Yes
No
12.76
Yes
Yes
21.57
1.0 ,1.t:.,
Sum (no, yes + yes, no) Product (no, yes * yes, no)
(1.82.023) 3.41’ (2.9,4.0)
co.;.: 4) 1:0i (0.8,1.3) 2.41 (1.9,3.0) 2.11
4.22
*Risks are additive if the observed excess relative risk (yes, yes) equals the sum of excess relative risks (no, yes and yes, no). Risks are multiplicative if the observed relative risk equals the product of relative risks (no, yes and yes, no). 95% CI = Asymptotic 95% confidence interval. Table 2. Six-year coronary disease death rates and relative risks according to strata of risk factors in 356,222 screenees of MRFIT* [smoking and/or hypercholesterolaemia (> 245%)]
Smoker
Cholesterol >245
No hypertension No No
Rate (per 1000)
No Yes
2.58 6.96
Yes
No
6.32
Yes
Yes
13.2
Sum (no, yes + yes, no) Product (no, yes * yes, no) Hypertension No No
No Yes
Relative risk (95% CI)
Excess relative risk (95% CI)
1.0
2.70 (2.3,3.2) 2.45 (2.1,2.8) 5.13 (4.4,6.0)
1.70 (1.3,2.2) 1.45 (1.1, 1.8) 4.13 (3.4,5.0) 3.15
6.62 5.38 11.8
Yes
No
12.76
Yes
Yes
21.57
Sum (no, yes + yes, no) Product (no, yes z yes, no)
::;o
(1.8,2.6) 2.37 (2.0,2.8)
(0.k21 6) 1.37 (1.0, 1.8)
(3:,‘4.8)
(2.3;‘3 8) 2.5;
5.21
*Footnote as in Table 1.
CONCLUSION
AND IMPLICATIONS
When predictive models are used to estimate benefits of risk factor modification, these should be carefully chosen and their intrinsic effects
identified, since it is essential that the estimates which result reflect the data rather than the model. The foregoing analysis indicates that the risk of CHD death associated with high serum cholesterol and the two other major risk factors
Cholesterol, CHD and Multiplicative Risk
879
is greater than additive, but it is largely the use of the multiple logistic model which has led to the notion that these effects are multiplicative: analyses so based may have overestimated the benefits of cholesterol lowering in smokers and hypertensives. This is particularly relevant at the present time given the need for evaluation of the HMG-CoA reductase inhibitors, the most efficacious and also expensive agents yet available.
14* Martin MJ, Hulley SB, Browner WS, Kuller LH,
Acknowledgemenrs-Thanks are due to Olli Miettinen and Rick Grobbee for helpful comments.
19.
15. 16.
17. 18.
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