Regression Methods for Estimating Attributable Risk in Population-based Case-Control Studies: A Comparison of Additive and Multiplicative Models

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1 American Journal of Epidemralogy Vol 133, No. 3 Copyright 1991 by The Johns Hopkins University School of Hygiene and Pubfc Health Printed m U.S.A. Al rights reserved Regression Methods for Estimating Attributable Risk in Population-based Case-Control Studies: A Comparison of Additive and Multiplicative Models Steven S. Coughlin, 1 Catharie C. Nass, 2-3 Linda W. Pickle, 14 Bruce Trock, 5 and Greta Bunin 3 A regression method that utilizes an additive model is proposed for the estimation of attributable risk in case-control studies carried out in defined populations. In contrast to previous multivariate procedures for the estimation of attributable risk, which have utilized logistic regression techniques to adjust for confounding factors, the model assumes an additive relation between the covariates included in the regression equation. As an empirical example, additive and logistic models were fitted to matched casecontrol data from a population-based study of childhood astrocytoma brain tumors. Although both models fitted the data well, the additive model provided a more satisfactory estimate of the risk attributable to multiple exposures, in the absence of significant additive interaction. In contrast to the results from the logistic model, the adjusted estimates of the risk attributable to each factor included in the additive model summed to the overall estimate for all of the factors considered jointly. Thus, the additive approach provides a useful alternative to existing procedures for the muttivariate estimation of attributable risk when the additive model is determined to be appropriate on the basis of goodness-of-fit. Am J Epidemiol 1991 ;133: biometry; birth weight; brain neoplasms; epidemtologic methods; logistic model; preventive medicine There has been increasing interest in the application of multivariate statistical techniques to the estimation of attributable risk (1, 2), which has also been referred to as the etiologic fraction (3) or attributable fraction (4, 5). Attributable risk, as discussed throughout this paper, is analogous to the concept of excess fraction, as recently defined by Greenland and Robins (5). The logistic regression model has been used by previous investigators to obtain estimates of attributable risks using both prospective and case-control data from defined populations (1, 2). Bruzzi et al. (2) have shown how this approach may be useful for estimating the attributable risk for an individual factor (or subset of factors) that is simultaneously adjusted for the risk attributable to the remaining factors included in the model (2), as detailed below. Thus, the potential public health impact of removing the exposure (or exposures) from the population may be estimated while adjusting for the effects of other confounding factors. Received for publication October 23, 1989, and in final torn August 3, Division of Biostatistics and Epidemiology, Department of Community and Family Medicine, Georgetown University School of Medicine, Washington, DC. 2 Department of Epidemiology, The Johns Hopkins School of Hygiene and Public Health, Baltimore, MD. 3 Division of Oncology, Children's Hospital of Phaadelphia, Philadelphia, PA. 4 Vincent T Lombard! Cancer Research Center, 305 Georgetown University Hospital, Washington, DC 5 Division of Cancer Control, Fox Chase Cancer Center, Philadelphia, PA. Reprint requests to: Dr. Steven S. Coughfin, Division of Biostatistics and Epidemiology, Kober-Cogan Hall, Room 404, Georgetown University School of Medicine, Washington. DC Dr. Coughlin is a current recipient of a First Independent Research Support and Transition Award (HL ) from the National Heart, Lung, and Blood Institute.

2 306 Coughlin et al. Because the logistic model assumes a multiplicative relation between the covariates included in the regression equation, adjusted estimates of attributable risks for individual factors will not add to the overall attributable risk for all of the factors considered jointly (2). An exception may occur when the exposure patterns for the cases and controls are disjoint (6), which rarely occurs in practice. Under a model of no interaction on a multiplicative scale, the complement of the summary attributable risk for all of the factors acting simultaneously is equal to the product of the complements of the adjusted risks attributable to each factor (2). Thus, when applying the logistic model, the overall attributable risk for all of the exposures combined is, in general, neither the sum nor the product of the adjusted attributable risks for the individual factors, and the exact relation may be difficult to predict (2). Furthermore, the sum of the adjusted attributable risks for each factor may exceed 100 percent, and the risk estimates may be difficult to interpret as a result. In the present report, an alternative approach to the estimation of attributable risk from case-control data is proposed which utilizes an additive model (7). This method allows for the estimation of adjusted attributable risks that are additive, which may be more satisfactory for many applications in public health and etiologic research (6, 8). An empirical example is provided in which additive and logistic models were fitted to matched case-control data from a population-based study of childhood astrocytoma brain tumors in the Greater Delaware Valley (9). STATISTICAL BACKGROUND The logistic regression model may be used in the estimation of attributable risk by obtaining estimates of the adjusted relative risks for various levels of each factor included in the model (1, 2). Although both prospective and case-control data may be utilized in this way, the present discussion will be limited to the latter. The adjusted relative odds derived from the regression coefficients provide reasonable estimates of the relative risks associated with each factor as long as the disease is rare in the population of interest (2). If the cases and controls have been matched on potentially confounding variables, the conditional logistic regression model may also be used in the same manner to estimate attributable risk (2, 10). A further requirement for the estimation of adjusted attributable risk is an estimate of the joint distribution of the exposures of interest among the population from which the cases and controls have been sampled (6, 11). In a community-wide case-control study, the frequencies of the exposures of interest among the controls may provide reasonable estimates, as long as they represent a random sample of all individuals in the population. If a matched design has been utilized, then estimates of the prevalence of the exposures of interest obtained from the controls may be biased (11). Bruzzi et al. (2) have demonstrated that, if the cases are a random sample of all cases from a defined population, adjusted attributable risk estimates may be obtained using the distribution of the exposures of interest among the cases alone, in addition to stratum-specific estimates of the relative risks. For each stratum j defined by the exposures of interest, a baseline strata./* may be defined which has the same levels of the confounding factor (or factors) as does stratum j but baseline levels of the factor (or factors) of interest (2). Although any stratum could be used as the referent, we will assume for simplicity thaty = 0 is the baseline stratum, where y = 0,..., J. If we define Nto be the total number of cases, «, to be the number of cases within stratum j, and Rj to be the relative risk for stratum j compared to the baseline stratum, then the following formula derived by Bruzzi et al. (2) may be used to estimate the risk attributable to the factor (or factors) of interest while controlling for the effects of one or more confounding factors: (l)

3 Regression Methods for Attributable Risk 307 The summary attributable risk for all of the factors considered jointly may be similarly estimated (2). The regression approach allows for the estimation of the adjusted attributable risk for a factor (or factors) even when the data within each stratum are relatively sparse. Furthermore, interaction terms may be included in the model to obtain a better fit, and the covariates need not be categorical (2). Specification of additive model Additive models have been proposed for the analysis of case-control data which assume an additive relation between the covariates included in the regression equation (7, 12-14). Breslow and Storer (7) have recently extended the work of previous authors to describe a family of relative risk functions for case-control comparisons which includes the additive model utilized in the present report. Under the additive model, the estimated relative risk R x associated with having specified values of the exposure variables (x u..., x K ), relative to not having been exposed to any of these factors, is given by: R(x) = 1 + (2) The regression coefficients are estimated using an iterative maximum likelihood approach that is analogous to a series of weighted least squares regression equations (7). The covariates included in the additive model may be coded so that the corresponding regression coefficients are positive, by selecting the level of each factor with the lowest risk as the baseline level (7). Once the regression coefficients have been obtained under the additive model, adjusted and overall attributable risks for the exposures of interest may be estimated using the equation derived by Bruzzi et al. (2) (equation 1). As shown in the Appendix, the adjusted estimates of the risk attributable to each factor included in the additive model sum to the overall estimate for all of the factors considered jointly, when no interaction terms are included in the model. Furthermore, the sum of the adjusted attributable risks for each factor may not exceed one (Appendix). These properties do not hold for the logistic model (2). EMPIRICAL EXAMPLE Methods In order to compare the utility of these techniques, additive and logistic models were fitted to data from a population-based case-control study of childhood astrocytoma brain tumors (9). The cases consisted of 96 children less than 10 years of age who were diagnosed with astrocytoma of the brain during in the 31-county area covered by the Greater Delaware Valley Pediatric Tumor Registry. General population controls matched to the cases on telephone exchange, race, and year of birth (plus or minus one or two years depending on the age of the respective case at the time of interview of the mother) were identified using a random digit dialing technique (15). Significant associations with maternal consumption of nitrite-processed lunchmeats during pregnancy and higher birth weight of the child were observed by Nass (9) in previous analysis of these data, and these exposures, which have also been implicated by other investigators (16, 17), were of interest in the present analysis of attributable risk. In stratified analysis, 95 percent confidence intervals for relative odds were estimated using Woolfs method (18). Unmatched estimates of the relative odds were obtained in some initial analyses because of the relatively small number of discordant matched pairs. An approximate chi-square test was used to test for interaction on an additive scale (18). Multivariate analysis was carried out using the PECAN computer program for the analysis of matched casecontrol data (19). Both conditional logistic and additive regression models were fitted to the childhood astrocytoma data (7, 10). The covariates of interest were coded as 0,1 binary variables with zero used as the reference category. Attributable risks for the exposures of interest, adjusting for the other

4 308 Coughlin et a). factor included in the regression equation, and overall attributable risks for both factors were then estimated using equation 1 (2). Results Males composed 59.4 percent (57 of 96) of the cases and 53.1 percent (51 of 96) of the controls (p > 0.05). The frequency of higher birth weight and maternal consumption of processed lunchmeat during pregnancy among the cases (n = 96) and controls (n = 96) is shown in table 1. A higher percentage of the cases than the controls weighed more than 3,000 g at the time of birth (86.5 percent vs percent). Maternal consumption of processed lunchmeat was also more frequent among the cases compared with the controls (79.2 percent vs percent). Little evidence of interaction on an additive scale was observed between higher birth weight and maternal consumption of processed lunchmeat (table 2). And, the point estimates of the unmatched relative odds suggested sub-multiplicative interaction between these two exposures. The expected relative odds associated with exposure to both factors is 7.4 under an additive model ( = 7.4) and 17.6 under a multiplicative model (4.0 x 4.4 = 17.6), using the strata-specific relative odds shown in table 2. A chi-square test for additive interaction was statistically nonsignificant (p > 0.05), although there was limited statistical power to detect any interaction because of the small sample size. Similarly, in multivariate modeling of the data, to test for interaction on a multiplicative scale, the interaction term between higher birth weight and maternal consumption of processed lunchmeat did not significantly improve the fit of the conditional logistic model already containing these two variables (p > 0.05). The multivariate results from the logistic model are shown in table 3. Statistically significant, independent associations were observed with both higher birth weight (p < 0.007) and maternal consumption of processed lunchmeat (p < 0.002). The point estimates of the adjusted relative odds for,- g CD 23 S

5 Regression Methods for Attributable Risk 309 these exposures (table 3) were somewhat smaller than the strata-specific estimates shown in table 2. When the additive model was fitted to the data, the associations were again found to be statistically significant (table 4). And, the point estimates of the relative odds agreed well with the strata-specific estimates (table 2). With respect to the goodness-of-fit of the models, the overall loglikelihood ratio statistics for the logistic and additive models containing both higher birth weight and processed lunchmeat were comparable (p < 0.01 in both instances). Finally, the adjusted attributable risks of childhood astrocytoma associated with higher birth weight and maternal consumption of processed lunchmeat obtained using the additive and logistic models are shown in table 5. The estimated risk attributable to higher birth weight, adjusting for maternal consumption of processed lunchmeat, was 41.6 percent and 55.8 percent when the additive and logistic models were used, respectively (table 5). The adjusted attributable risk for processed lunchmeat was 43.0 percent when the relative risk estimates from TABLE 2. Assessment of possible Interaction between higher birth weight and matemai consumption of processed lunchmeat* Birth weight >3,000g No Yes No Yes No No Yes Yes No. of cases No. of controls Chi-square test for Interaction on an addttive scale not significant (p > 0.05). t Fixed reference category. Relative odds 1.0t % confidence Interval TABLE 3. Beta coefficients, levels of significance, and relative odds from logistic model fitted to childhood astrocytoma data* Covariate Higher birth weight Processed lunctimeat 0 coefficient P valuer Relative odds * Cases and controls were matched on age and race. The covarlates were coded as 0,1 binary variables with zero used as the reference category. t From Iog-H<e8hood ratio test. TABLE 4. Beta coefficients, levels of significance, and relative odds from additive model fitted to childhood astrocytoma data* Covariate Higher birth weight Processed lunchmeat (9 coefficient P valuer Relative odds * Cases and controls were matched on age and race. The covariates were coded as 0,1 binary variables with zero used as the reference category. f Fromtog-likeihoodratio test. TABLE 5. Adjusted attributable risks of childhood astrocytoma from higher birth weight and maternal consumption of processed lunchmeat estimated using additive and logistic models Exposure Higher birth weight* Processed lunchmeat* Both exposures considered jointly ' Attributable risks are adjusted for the other exposure. Additive model Attributable risk Logistic model

6 310 Coughlin et a). the additive model were used and 52.1 percent when those obtained using the logistic model were used. The overall risk attributable to both exposures combined was 84.6 percent from the additive model, which is exactly the sum of the adjusted attributable risks for each exposure. In contrast, the sum of the adjusted attributable risks for higher birth weight and processed lunchmeat obtained from the logistic model is greater than percent and is considerably larger than the overall risk for both exposures obtained from the same model (table 5). Additional attributable risk estimates were obtained after reversing the coding of the covariates included in the models so that the highest level of each exposure was the baseline level. This change in the coding convention resulted in a reduction in the adjusted estimates of attributable risk obtained under the additive model. For example, the overall risk attributable to both higher birth weight and maternal consumption of lunchmeat was reduced from 84.6 percent, as shown in table 5, to 55.2 percent. The estimates obtained under the logistic model were unchanged. DISCUSSION The additive model described in this report provides a useful alternative to the logistic regression model in the estimation of attributable risk from case-control studies which are population-based. While both additive and multiplicative models may be used for the multivariate estimation of attributable risk, the additive approach has desirable properties which apply when the additive model is selected on the basis of goodness-of-fit. As observed with the astrocytoma data, the adjusted estimates of attributable risk obtained for each factor included in the model add to the summary estimate for both of the factors acting jointly, when only main effects are considered (table 5). Thus, the additive model may provide better estimates of the risk attributable to multiple exposures, which are more easily interpretable, when there is an absence of interaction on an additive scale between the exposures of interest. Of course, choices between alternative models should be principally based upon the fit of the respective models to the data. With interaction terms included in the regression equation, the additive model may also be useful for obtaining adjusted estimates of attributable risk in the presence of significant interaction on an additive scale between the exposures of interest. Both the additive and logistic regression methods for estimating attributable risk are also applicable to unmatched data. However, unconditional regression models should be fitted. A limitation of the additive model is that the conditional likelihood function is influenced by the baseline that is chosen for the covariates, and a change in the way the independent variables are coded may impact on the beta coefficients and the risk estimates (20). In the present analysis, the covariates were coded as 0,1 binary variables with zero used as the reference category, which is in accord with the recommendation by Breslow and Storer (7) that the level of each factor with the lowest risk be used as the baseline level. Although this ad hoc approach has been criticized (20), the additive model did provide reasonable point estimates of the relative odds (RO) for higher birth weight (RO = 4.7, p < 0.014) and maternal consumption of lunchmeat (RO = 5.6, p < 0.003) when the covariates were coded in this way. In examining the goodness-of-fit of the additive and logistic models to the childhood astrocytoma data, both models appeared to fit the data well (likelihood ratio chi square p value < 0.01). Because of the limited sample size, these goodness-of-fit tests did not provide strong evidence with which to choose between alternative models. Nonetheless, the results shown in table 2 suggested that an additive model which included only main effects was most appropriate, and the adjusted estimates of attributable risk obtained using this model were additive and therefore more easily interpreted (table 5). Furthermore, there was little evidence of a multiplicative relation between the exposures of interest (table 2). Thus, the additive

7 Regression Methods for Attributable Risk 311 approach provided a useful alternative to the multiplicative model in estimating attributable risk. An important caveat in the multivariate estimation of attributable risk is that unless all relevant factors are included in the model and the model has the correct parametric form, the estimates of the risk attributable to individual factors may be biased (11). In considering the childhood astrocytoma data, relatively little is known about the epidemiology of this tumor and there may be important exposures which were not considered in the analysis (9, 16, 17). The adjusted estimates of attributable risk for higher birth weight and maternal consumption of processed lunchmeat during pregnancy (table 5) may have been biased upward by a failure to adjust for (unknown) factors which are positively associated with these exposures. However, these are common exposures in this population (table 1), which largely accounts for the sizable estimates of attributable risk (table 5). In summary, both additive and logistic regression methods may be used to obtain adjusted estimates of attributable risk from population-based data. However, the proposed additive approach may be more advantageous for some applications in public health research (8, 11). In view of the multifactorial etiology of many diseases for which preventive efforts are currently directed (1 1), it is likely that multivariate techniques for the estimation of attributable risk will increasingly come into play. REFERENCES 1. Deubner DC, Wilkinson WE, Helms MJ, et al. Logistic model estimation of death attributable to risk factors for cardiovascular disease in Evans County, Georgia. Am J Epidemiol 1980;l 12: Bruzzi P, Green SB, Byar DP, et al. Estimating the population attributable risk for multiple risk factors using case-control data. Am J Epidemiol 1985;122:9O KJeinbaum DG, Kupper LL, Morgenstern H. Epidemiologic research: principles and quantitative methods. Belmont, CA: Lifetime Learning Publications, Ouellet BL, Romeder JM, Lance JM. Premature mortality attributable to smoking and hazardous drinking in Canada. Am J Epidemiol 1979; 109: Greenland S, Robins JM. Conceptual problems in the definition and interpretation of attributable fractions. Am J Epidemiol 1988; 128: Walter SD. Effects of interaction, confounding and observational error on attributable risk estimation. Am J Epidemiol 1983;117: Breslow NE, Storer BE. General relative risk functions for case-control studies. Am J Epidemiol 1985; 122: Rothman KJ, Greenland S, Walker AM. Concepts of interaction. Am J Epidemiol 1980;l 12: Nass CC. Parental occupational exposures and astrocytoma in children under ten years of age in the Greater Delaware Valley. Ph.D. dissertation. Baltimore: The Johns Hopkins University, Breslow NE, Day NE. Statistical methods in cancer research. Vol 1. The analysis of case-control studies. Lyon, France: International Agency For Research on Cancer, Walter SD. Prevention for multifactorial diseases. Am J Epidemiol 1980; 112: Dayal HH. Additive excess risk model for interaction in retrospective studies. J Chronic Dis 1980,33: Walker AM, Rothman KJ. Models of varying parametric form in case-referent studies. Am J Epidemiol 1982; 115: Thomas DC. General relative risk models for survival time and matched case-control analysis. Biometrics 1981;37: Ward EM, Kramer S, Meadows AT. The efficacy of random digit dialing in selecting matched controls for a case-control study of pediatric cancer. Am J Epidemiol 1984; 120: Gold E, Gordis L, Tonascia J, et al. Risk factors for brain tumors in children. Am J Epidemiol 1979;1O9:3O Preston-Martin S, Yu MC, Benton B, et al. N- nitroso compounds and childhood brain tumors: a case-control study. Cancer Res 1982;42: Schlesselman JJ. Case-control studies. Design, conduct, analysis. New York: Oxford University Press, Lubin JH. A computer program for the analysis of matched case-control studies. Comput Biomed Res 1981;14: Moolgavkar SH, Venzon DJ. Generalrelativerisk regression models for epidemiologic studies. Am J Epidemiol 1987; 126:

8 312 Coughlinetal. APPENDIX Let AR.,.2 represent the risk attributable to the presence of both factors 1 and 2, relative to the absence of both factors, controlling for the effects of one or more confounding factors. Let AR, represent the risk attributable to factor / (/ = 1,2), controlling for the other risk factor and any confounding factors. For simplicity, we assume that the risk factors of interest are binary. Also, let RIJ = 1 + Xfii + Xfi 2, where X t = 1 if factor 1 is present, 0 otherwise, and Xj = 1 if factor 2 is present, 0 otherwise. Theorem. AR U = AR, + AR 2 Proof. Under the independent additive model, the risk for persons in category ij relative to those in category i*j* is R,j/R rr. For example, Following text equation 1, R n _ 1 +0, i Ro\ I where pj = -A, the proportion of cases in stratum j. Extension to the adjustment of a second factor yields: AR, Under the additive model this becomes: Similarly, _ _ POO PlO POI, Pi 1 \_Roo/Roo\ R / R R / R \j AD 1 I j, Pii AR, = 1 - poo + Poi j AD 1 J^ J^ AR 2 = 1 - poo + Pio + POI Pii 1+02 j + _02_ = 2 - (poo + poi + Pio + Pi i) + Poo +. POI, Pll 1+0, ,. o , + 02

9 Regression Methods for Attributable Risk 313 Theorem. Under the independent additive model, 0 < AR, + AR 2 < 1. Proof. ARi + AR 2 = AR,, 2 as shown above. Thus, the conclusion follows from the definition of attributable risk. AR - 1 V El Assuming that factors 1 and 2 are risk factors for the disease implies that R o > 1 for each combination ij. Thus, -^ < p,j with equality only when R u = 1. Therefore, i,j=o Rij ij-o so that AR,, 2 = 1-2Jr = 0 if and only if R tj «1 for all ij. If any R u > 1, AR,, 2 = 1-2 ~ > 1-2 P,j = 0. Therefore, AR^ ^ 0 with equality if and only if R tj = 1 for all ij; that is, when neither factor 1 nor factor 2 is truly a risk factor for the disease. To show that AR U2 < 1, note that p,j > 0 for all ij. Therefore, > 0 and 2 2= 0. It follows directly that AR^ = 1 2TT '> ^tn where p :j = 0 for all ij. i quality only in the degenerate case