200
LETTERS TO THE EDITOR
Both these points are extensively discussed in the above letters. They need no further elaboration. As to the sacrosanctity of the logistic model, we were happy to read in a recent paper that "biological processes are not identifiable from epidenuologic data" (7, p. 346). Perhaps the next time that we try to describe the hazards of epidemiologic methods, we might use examples in which all of the combatants are defunct, such as the 1889 writings of a French author who proposed that cancer was contagious and transferable by inoculation, given the way it clusters in Normandy (8).
4
5.
6. 7 8
REFERENCES
the c u e of "poppers" in the early epidemic of the acquired immunodeficiency syndrome (AIDS) " Am J Epidemiol 1990:131.197 Gnmson RC Re- "An autopsy of epidemiologic methods the c u e of "poppers" in the early epidemic of the acquired immunodeficiency syndrome (AIDS)." Am J Epidemiol 1990-.131198-9. Vondenbroucke JP, Pordoel VPAM. An autopsy of epidemiologic methods: the cose of "poppers" in the early epidemic of the acquired immunodeficiency syndrome (AIDS). Am J Epidemiol 1989:129:455-7 Vondenbroucke JP On not being born a native speaker of English. Br Med J 1989-^98:40-2. Greenland S Modelling and variable selection in epidemiologic analysis. Am J Public Health 1989:79-340-9. JAMA 100 years ago: contagious character and transferobdity of cancer by inoculation JAMA 1989;261-3250
1 Mannor M, Dubin N. Re: "An autopsy of epidamiologic methods the c u e of "poppers" in the early epidemic of the acquired immunodeficiency syndrome (AIDS) Am J Epidemiol 1990:131195-6. 2. Heuel P A Re- "An autopsy of epidemiologic methods: the c u e of "popperi" in the early epidemic of the acquired immunodeficiency syndrome (AIDS)." Am J Epidemiol 1990:131.196-7. 3. Kramer MD Re: "An autopsy of epidemiologic methods:
RE: "GENERAL RELATIVE
RISK REGRESSION MODELS FOR EPIDEMIOLOGIC STUDIES"
The interesting article by Moolgavkar and Venron (1) compares three different families of models which can be used to test the additivity or multiplicativity of relative risk regression models. These two investigators disclosed some problems that can arise in a multivanate situation when the coding of a dichotomous variable is reversed, X = 0 or 1 being replaced by Z =» 1 - X. They showed that for two of these families of models, initially proposed by Thomas (2) and by Breslow and Storer (3), the maximum likelihood estimations of the "shape" parameter were quite different under the two situations; but for the family of models proposed by Guerrero and Johnson (4), the "shape" parameter was constant under such a transformation. We would like to draw attention to a problem not treated by the authors, the eventuality of a different effect of the variables included in the model. We will also approach the question of the power. The three families of models, in the way they were used, share a common hypothesis which was not expressed. All of them assume that to measure the effect of two factors on a relative risk, odds ratio, or hazards ratio R, a single transformation linearizes the relation, i.e., there is a transformation / which is a linear function of Y for each level of X and the two regression lines are parallel:
f\R(o,
/[fid,
This might not be the case, and in many instances it may be useful to consider the possibility that one effect is additive while the other one is multiplicative. The Thomas approach (2) can be used to handle such a situation, using
w)
R( x, y) = [exp |(1 - n)yy + i-
yL + yy)'+
Jan P. Vandenbroucke Department of Clinical Epidemiology Leiden University Hospital Budding ICO-P P.O. Box 9600 2300 RC Leiden The Netherlands
Pi]*
with two shape parameters, X expressing the way the
risk depends on X and i± expressing the way it depends on Y. For both of them, 0 corresponds to a multiplicative and 1 to an additive effect. Note that since this expression takes into account the binary nature of X, the roles of (i, 0) and (y, y) are not symmetrical. We used such a family of models to analyze the data from the Los Angeles study of endometnal cancer, published in appendix 3 of Breslow and Day (5) and already used in the paper by Moolgavkar and Venzon (1). The two risk factors considered were history of gallbladder disease (X) and duration of estrogen use (Y). Each of the 57 cases was matched with four or five controls. Table 1 gives the maximum likelihood and the corresponding parameters obtained under different constraints: The underlined values TABLE 1
Results of the analysis of the endometrial cancer data from appendix 3 of Breslow and Day (5) Likelihood
-69.28 - 6 9 77 - 7 3 01 -73.10 -73.22 -70.61 -70.53
Xt 0.82
o§ 0.57 0 1 0 1
1.04* 105* 0 0 0 1 1
* p < 0.01 (likelihood ratio test). t Shape parameter for the effect of history of gallbladder disease. X Shape parameter for the effect of duration of estrogen use. § Underlined values are not maximum likelihood estimates.
201
LETTERS TO THE EDITOR TABLE
2
Number of simulations for which the effect of a uanable appeared significantly not multiplicative* Effect of the qualitative variable X
Effect of the quantitative variable V
Multiplicative Multiplicative Additive Additive
Multiplicative Additive Multiplicative Additive
100 100 100 100
Simulations with effect of X not multiplicative
Simulationi with effect of V not multiplicative
3 2 27 11
6 19 2 12
' Shape parameter aignLficantly 5* 0 according to the likelihood ratio test, p < 0.05.
were not estimated but imposed. One can see that the maximum likelihood was reached for values of the shape parameters "near" to 1 (additmty), but if M =* 1 04 was significantly different from 0 (p < 0.01), X = 0.82 was not significantly different from 0. Finally, the two models where the effect of Y ia additive and that of X is either additive or multiplicative both appeared appropriate. We also reversed the coding of the history of gallbladder disease, using Z =• 1 - X in place of X. The model, again, is not invariant under such a transformation, and one cannot expect to get the same estimations of the shape parameters. However, another difficulty appeared: Although Moolgavkar and Venzon quoted Breslow and Storer, who stated that "it is important wherever possible to code the X variables so that the corresponding /3 coefficients are positive" (3, p. 150) they did not treat that problem any further. Actually, in the example studied, the logarithm of negative expressions would appear in the calculus of the likelihood if a constraint were not imposed on the coefficient 0' of Z = 1 — X; the maximum likelihood is reached on the limit of /?'. Thus, the estimations of all the parameters cannot be considered as satisfactory, and the problem of the different estimations of the shape parameters is not as relevant as it seemed. The fact that, for a given value of ii, the likelihood remained practically unchanged when X — 0 (multiplicative effect of X) was replaced by X = 1 (additive effect of X) seemed to indicate that this procedure is not powerful. Actually, when one considers the four simple models obtained by combining the additivity or multiplicativity hypothesis for two risk factors, one dichotomous (X) and the other quantitative (Y), one can notice that for three of them, 1 + 0X + y Y, e*x * yY , and f3X + erY, the first-order approximations are identical. It is only for the combination of a multiplicative effect of X and an additive effect of Y, e** (1 + 750, that the first-order approximation is different. Thus, one should worry about the power of the testsdealing with the type of effects of the variables. In order to tackle this question, we simulated 100 data sets under each of the four hypothesis obtained by combining the assumptions of additivity or multiplicativity for the effects of two variables. Each data set included 100 "cases" with four controls matched to each case. Among the controls, the dichotomous variable had a Bernoulli distribution with parameter p =• 0.10; a normal accessory variable, A, with mean 45 and standard deviation 8, was used for matching, and the quantitative variable Y was exponential with its parameter varying linearly according to A, from V4 for A = 15 to '/» for a = 75. The values for the cases
were obtained under the assumption of a relative risk of R (X =» 1, V =• 0) » 7.39 and R (X = 0, Y = 3) = 1.35 (3 was the median of Y for the controls). The likelihood ratio test was used to determine whether a parameter was significantly different from 0 at the level a •» 0 05. Table 2 gives the number of data sets for which the shape parameters were significantly different from 0. The estimation of the power varied from 11 percent (95 percent confidence interval (CI) 4.8-17.2) to 27 percent (95 percent CI 18.2-35.8). This confirms that the procedure is not powerful. By contrast, in the least favorable situation, i.e, the effects of X and Y are additive, both (3 and 7 were significantly different from 0 in each of the 100 simulated data sets. In conclusion, although the problem disclosed by Moolgavkar and Venzon may be p u l l i n g we think it is of minor importance when, for one of the two possible codings, the maximum likelihood depends on the constraints imposed on a parameter. Practically, when it appears that an interaction term might be eliminated by freeing oneself from the hypothesis of multiplicative effects, we suggest that one focus interest on the interacting variables. Thomas' approach may be used to appreciate separately how the risk depends on the two interacting variables, while keeping the hypothesis of a multiplicative effect for the other variables. Obviously, the aim would be less to estimate some shape parameters than to test whether a partly additive model would be more parsimonious than a purely multiplicative one. REFERENCES 1. Moolgavkar SH, Venion DJ General relative rule regrwsion models for epidemiologic studies. Am J Epidemiol 1987,126.949-61 2. Thoma» DC. General relative-nsk model* for survival time and matched case-control analysis. Biometrics 198U7-673-86 3. Breslow NE, Storer BE. General relative nsk functions for case-control studies. Am J Epidemiol 1986,122 14962. 4 Guerrero VM, Johnson RA. Use of the Box-Coi transformation with binary response models. Biometnlta 1982.-69-309-14. 5. Breslow NE, Day NE. Statistical methods in cancer research. Vol 1. The analysis of case-control studies. Lyon. International Agency for Research on Cancer, 1980.
Michel Chavance INSERM U169 16 Avenue Paul Vaillant-Couturier 94807 Vdlejuif France
LETTERS TO THE EDITOR
Both these points are extensively discussed in the above letters. They need no further elaboration. As to the sacrosanctity of the logistic model, we were happy to read in a recent paper that "biological processes are not identifiable from epidenuologic data" (7, p. 346). Perhaps the next time that we try to describe the hazards of epidemiologic methods, we might use examples in which all of the combatants are defunct, such as the 1889 writings of a French author who proposed that cancer was contagious and transferable by inoculation, given the way it clusters in Normandy (8).
4
5.
6. 7 8
REFERENCES
the c u e of "poppers" in the early epidemic of the acquired immunodeficiency syndrome (AIDS) " Am J Epidemiol 1990:131.197 Gnmson RC Re- "An autopsy of epidemiologic methods the c u e of "poppers" in the early epidemic of the acquired immunodeficiency syndrome (AIDS)." Am J Epidemiol 1990-.131198-9. Vondenbroucke JP, Pordoel VPAM. An autopsy of epidemiologic methods: the cose of "poppers" in the early epidemic of the acquired immunodeficiency syndrome (AIDS). Am J Epidemiol 1989:129:455-7 Vondenbroucke JP On not being born a native speaker of English. Br Med J 1989-^98:40-2. Greenland S Modelling and variable selection in epidemiologic analysis. Am J Public Health 1989:79-340-9. JAMA 100 years ago: contagious character and transferobdity of cancer by inoculation JAMA 1989;261-3250
1 Mannor M, Dubin N. Re: "An autopsy of epidamiologic methods the c u e of "poppers" in the early epidemic of the acquired immunodeficiency syndrome (AIDS) Am J Epidemiol 1990:131195-6. 2. Heuel P A Re- "An autopsy of epidemiologic methods: the c u e of "popperi" in the early epidemic of the acquired immunodeficiency syndrome (AIDS)." Am J Epidemiol 1990:131.196-7. 3. Kramer MD Re: "An autopsy of epidemiologic methods:
RE: "GENERAL RELATIVE
RISK REGRESSION MODELS FOR EPIDEMIOLOGIC STUDIES"
The interesting article by Moolgavkar and Venron (1) compares three different families of models which can be used to test the additivity or multiplicativity of relative risk regression models. These two investigators disclosed some problems that can arise in a multivanate situation when the coding of a dichotomous variable is reversed, X = 0 or 1 being replaced by Z =» 1 - X. They showed that for two of these families of models, initially proposed by Thomas (2) and by Breslow and Storer (3), the maximum likelihood estimations of the "shape" parameter were quite different under the two situations; but for the family of models proposed by Guerrero and Johnson (4), the "shape" parameter was constant under such a transformation. We would like to draw attention to a problem not treated by the authors, the eventuality of a different effect of the variables included in the model. We will also approach the question of the power. The three families of models, in the way they were used, share a common hypothesis which was not expressed. All of them assume that to measure the effect of two factors on a relative risk, odds ratio, or hazards ratio R, a single transformation linearizes the relation, i.e., there is a transformation / which is a linear function of Y for each level of X and the two regression lines are parallel:
f\R(o,
/[fid,
This might not be the case, and in many instances it may be useful to consider the possibility that one effect is additive while the other one is multiplicative. The Thomas approach (2) can be used to handle such a situation, using
w)
R( x, y) = [exp |(1 - n)yy + i-
yL + yy)'+
Jan P. Vandenbroucke Department of Clinical Epidemiology Leiden University Hospital Budding ICO-P P.O. Box 9600 2300 RC Leiden The Netherlands
Pi]*
with two shape parameters, X expressing the way the
risk depends on X and i± expressing the way it depends on Y. For both of them, 0 corresponds to a multiplicative and 1 to an additive effect. Note that since this expression takes into account the binary nature of X, the roles of (i, 0) and (y, y) are not symmetrical. We used such a family of models to analyze the data from the Los Angeles study of endometnal cancer, published in appendix 3 of Breslow and Day (5) and already used in the paper by Moolgavkar and Venzon (1). The two risk factors considered were history of gallbladder disease (X) and duration of estrogen use (Y). Each of the 57 cases was matched with four or five controls. Table 1 gives the maximum likelihood and the corresponding parameters obtained under different constraints: The underlined values TABLE 1
Results of the analysis of the endometrial cancer data from appendix 3 of Breslow and Day (5) Likelihood
-69.28 - 6 9 77 - 7 3 01 -73.10 -73.22 -70.61 -70.53
Xt 0.82
o§ 0.57 0 1 0 1
1.04* 105* 0 0 0 1 1
* p < 0.01 (likelihood ratio test). t Shape parameter for the effect of history of gallbladder disease. X Shape parameter for the effect of duration of estrogen use. § Underlined values are not maximum likelihood estimates.
201
LETTERS TO THE EDITOR TABLE
2
Number of simulations for which the effect of a uanable appeared significantly not multiplicative* Effect of the qualitative variable X
Effect of the quantitative variable V
Multiplicative Multiplicative Additive Additive
Multiplicative Additive Multiplicative Additive
100 100 100 100
Simulations with effect of X not multiplicative
Simulationi with effect of V not multiplicative
3 2 27 11
6 19 2 12
' Shape parameter aignLficantly 5* 0 according to the likelihood ratio test, p < 0.05.
were not estimated but imposed. One can see that the maximum likelihood was reached for values of the shape parameters "near" to 1 (additmty), but if M =* 1 04 was significantly different from 0 (p < 0.01), X = 0.82 was not significantly different from 0. Finally, the two models where the effect of Y ia additive and that of X is either additive or multiplicative both appeared appropriate. We also reversed the coding of the history of gallbladder disease, using Z =• 1 - X in place of X. The model, again, is not invariant under such a transformation, and one cannot expect to get the same estimations of the shape parameters. However, another difficulty appeared: Although Moolgavkar and Venzon quoted Breslow and Storer, who stated that "it is important wherever possible to code the X variables so that the corresponding /3 coefficients are positive" (3, p. 150) they did not treat that problem any further. Actually, in the example studied, the logarithm of negative expressions would appear in the calculus of the likelihood if a constraint were not imposed on the coefficient 0' of Z = 1 — X; the maximum likelihood is reached on the limit of /?'. Thus, the estimations of all the parameters cannot be considered as satisfactory, and the problem of the different estimations of the shape parameters is not as relevant as it seemed. The fact that, for a given value of ii, the likelihood remained practically unchanged when X — 0 (multiplicative effect of X) was replaced by X = 1 (additive effect of X) seemed to indicate that this procedure is not powerful. Actually, when one considers the four simple models obtained by combining the additivity or multiplicativity hypothesis for two risk factors, one dichotomous (X) and the other quantitative (Y), one can notice that for three of them, 1 + 0X + y Y, e*x * yY , and f3X + erY, the first-order approximations are identical. It is only for the combination of a multiplicative effect of X and an additive effect of Y, e** (1 + 750, that the first-order approximation is different. Thus, one should worry about the power of the testsdealing with the type of effects of the variables. In order to tackle this question, we simulated 100 data sets under each of the four hypothesis obtained by combining the assumptions of additivity or multiplicativity for the effects of two variables. Each data set included 100 "cases" with four controls matched to each case. Among the controls, the dichotomous variable had a Bernoulli distribution with parameter p =• 0.10; a normal accessory variable, A, with mean 45 and standard deviation 8, was used for matching, and the quantitative variable Y was exponential with its parameter varying linearly according to A, from V4 for A = 15 to '/» for a = 75. The values for the cases
were obtained under the assumption of a relative risk of R (X =» 1, V =• 0) » 7.39 and R (X = 0, Y = 3) = 1.35 (3 was the median of Y for the controls). The likelihood ratio test was used to determine whether a parameter was significantly different from 0 at the level a •» 0 05. Table 2 gives the number of data sets for which the shape parameters were significantly different from 0. The estimation of the power varied from 11 percent (95 percent confidence interval (CI) 4.8-17.2) to 27 percent (95 percent CI 18.2-35.8). This confirms that the procedure is not powerful. By contrast, in the least favorable situation, i.e, the effects of X and Y are additive, both (3 and 7 were significantly different from 0 in each of the 100 simulated data sets. In conclusion, although the problem disclosed by Moolgavkar and Venzon may be p u l l i n g we think it is of minor importance when, for one of the two possible codings, the maximum likelihood depends on the constraints imposed on a parameter. Practically, when it appears that an interaction term might be eliminated by freeing oneself from the hypothesis of multiplicative effects, we suggest that one focus interest on the interacting variables. Thomas' approach may be used to appreciate separately how the risk depends on the two interacting variables, while keeping the hypothesis of a multiplicative effect for the other variables. Obviously, the aim would be less to estimate some shape parameters than to test whether a partly additive model would be more parsimonious than a purely multiplicative one. REFERENCES 1. Moolgavkar SH, Venion DJ General relative rule regrwsion models for epidemiologic studies. Am J Epidemiol 1987,126.949-61 2. Thoma» DC. General relative-nsk model* for survival time and matched case-control analysis. Biometrics 198U7-673-86 3. Breslow NE, Storer BE. General relative nsk functions for case-control studies. Am J Epidemiol 1986,122 14962. 4 Guerrero VM, Johnson RA. Use of the Box-Coi transformation with binary response models. Biometnlta 1982.-69-309-14. 5. Breslow NE, Day NE. Statistical methods in cancer research. Vol 1. The analysis of case-control studies. Lyon. International Agency for Research on Cancer, 1980.
Michel Chavance INSERM U169 16 Avenue Paul Vaillant-Couturier 94807 Vdlejuif France
