Letters to the Editor Epidemiol 1989; 130:1236-46. 3. Flegal KM, Brownie C, Haas JD. The effects of exposure misclassificauon on estimates of relative risk. Am J Epidemiol 1986;123:736-51. 4. Greenland S, Thomas D. On the need for the rare disease assumption in case-control studies. Am J Epidemiol 1982,116547-53. 5. Smith PG, Rodngues L, Fine PEM. Assessment of the protective efficacy of vaccines against common
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diseases using case-control and cohort studies. Int J Epidemiol 1984; 1387-93.
Ron Dewar Jack Siemiatycki Institut Armand-Frappier 531, boulevard des Prairies Ville de Laval, Quebec Canada H7V1B7
RE. "INTERPRETATION AND CHOICE OF EFFECT MEASURES IN EPIDEMIOLOGIC ANALYSES" Greenland (1,2) has argued that if one's objective is to study patterns of individual responses to exposure, then incidence proportions and densities are preferable to incidence odds. The deciding factor in this view seems to be that an observable difference in disease incidence proportions between an exposed and unexposed cohort has a simultaneous interpretation as an estimate of the average personal risk difference, assuming absence of confounding, whereas a disease incidence odds ratio does not have an analogous interpretation as an estimate of the average personal odds ratio. While I agree with this statement of fact, I do not share Greenland's larger view, that "logistic and log-linear models are useful [in epidemiologic inference] only insofar as they provide improved . . . estimates of incidence differences or ratios" (1, p. 761). Odds ratio non-collapsibility is one difficulty, of course, but another arises in the stochastic risk model by Greenland's use of arithmetic averages in a clearly multiplicative situation. To distinguish between these two sources, consider the logarithm of the odds ratio as the fundamental measure of individual response to exposure, a choice much better adapted to arithmetic averaging than the odds ratio itself. I shall demonstrate that far from being of limited utility, logistic models serve to rescue epidemiologic inference about individual level log odds ratios. In Greenland's notation, ru and rOl are personal risk probabilities under exposed or unexposed conditions, respectively. Holland (3) introduced the useful adjectives "factual" and "counterfactual" that we use below: the first describes ru when a person is exposed and rOi when unexposed, while the second describes rOl when the person is exposed and ru when unexposed. The term "counterfactual" suggests the unobservable nature of those risks. Now let /„ = log [ru/{\ - ru)] and ki = log [WO - rod] denote an individual's log-odds on disease under exposed and unexposed conditions, respectively, with individual risk log odds ratio lu - kiDirectly analogous to the interpretability property of risk differences we have the following.
Property. Absent confounding, the difference in average log-odds between an exposed and unexposed cohort is interpretable as the average of individual risk log odds ratios. (Equivalently, the ratio of geometric mean odds in an exposed cohort to that in an unexposed cohort is interpretable as the geometric mean of individual risk odds ratios). In particular, if individual log odds ratios are constant, the common value is measured by the difference in average log-odds. Absent confounding here means that the average counterfactual ki in the exposed cohort (of size N\) would equal the average factual ki in the unexposed cohort (of size No), i.e., (£i lo,)/Ni = (So lot/No. With this assumption, the identity follows as in the case of risk differences from (£, lu)/N, - (So lc,)/No = (Si /i,)/JVi - (Si k,)/N, = Si (A/ - lod/N,. Note that our nonconfoundmg assumption appears to be different from Greenland's: (Si ro,)/N, = (So r0,)/N0. These both follow, however, from a reasonable (albeit stronger) condition of nonconfounding: the distribution of counterfactual risks rOi in the exposed cohort is the same as the distribution of factual risks r
963
diseases using case-control and cohort studies. Int J Epidemiol 1984; 1387-93.
Ron Dewar Jack Siemiatycki Institut Armand-Frappier 531, boulevard des Prairies Ville de Laval, Quebec Canada H7V1B7
RE. "INTERPRETATION AND CHOICE OF EFFECT MEASURES IN EPIDEMIOLOGIC ANALYSES" Greenland (1,2) has argued that if one's objective is to study patterns of individual responses to exposure, then incidence proportions and densities are preferable to incidence odds. The deciding factor in this view seems to be that an observable difference in disease incidence proportions between an exposed and unexposed cohort has a simultaneous interpretation as an estimate of the average personal risk difference, assuming absence of confounding, whereas a disease incidence odds ratio does not have an analogous interpretation as an estimate of the average personal odds ratio. While I agree with this statement of fact, I do not share Greenland's larger view, that "logistic and log-linear models are useful [in epidemiologic inference] only insofar as they provide improved . . . estimates of incidence differences or ratios" (1, p. 761). Odds ratio non-collapsibility is one difficulty, of course, but another arises in the stochastic risk model by Greenland's use of arithmetic averages in a clearly multiplicative situation. To distinguish between these two sources, consider the logarithm of the odds ratio as the fundamental measure of individual response to exposure, a choice much better adapted to arithmetic averaging than the odds ratio itself. I shall demonstrate that far from being of limited utility, logistic models serve to rescue epidemiologic inference about individual level log odds ratios. In Greenland's notation, ru and rOl are personal risk probabilities under exposed or unexposed conditions, respectively. Holland (3) introduced the useful adjectives "factual" and "counterfactual" that we use below: the first describes ru when a person is exposed and rOi when unexposed, while the second describes rOl when the person is exposed and ru when unexposed. The term "counterfactual" suggests the unobservable nature of those risks. Now let /„ = log [ru/{\ - ru)] and ki = log [WO - rod] denote an individual's log-odds on disease under exposed and unexposed conditions, respectively, with individual risk log odds ratio lu - kiDirectly analogous to the interpretability property of risk differences we have the following.
Property. Absent confounding, the difference in average log-odds between an exposed and unexposed cohort is interpretable as the average of individual risk log odds ratios. (Equivalently, the ratio of geometric mean odds in an exposed cohort to that in an unexposed cohort is interpretable as the geometric mean of individual risk odds ratios). In particular, if individual log odds ratios are constant, the common value is measured by the difference in average log-odds. Absent confounding here means that the average counterfactual ki in the exposed cohort (of size N\) would equal the average factual ki in the unexposed cohort (of size No), i.e., (£i lo,)/Ni = (So lot/No. With this assumption, the identity follows as in the case of risk differences from (£, lu)/N, - (So lc,)/No = (Si /i,)/JVi - (Si k,)/N, = Si (A/ - lod/N,. Note that our nonconfoundmg assumption appears to be different from Greenland's: (Si ro,)/N, = (So r0,)/N0. These both follow, however, from a reasonable (albeit stronger) condition of nonconfounding: the distribution of counterfactual risks rOi in the exposed cohort is the same as the distribution of factual risks r
