modeling other human carcinogens (e.g., arse- nic) (Lubin et al. 2008) and allows for the linear exposure–response in the usual linear RR model to be modified to exhibit various degrees of curvature. We also fit the Michaelis–Menten model of enzyme kinetics, in which the rate of reaction tails off as substrate concentration increases (Lehninger 1982). The model has been used rarely in epidemiology, but it is sim- ple to implement, because it merely involves the transformation of exposure and the estima- tion of one additional parameter to describe the exposure–response relationship. For the linear RR model, the model is RR = 1 + β1 × [cumexp/(cumexp + k)], where k represents a constant, analogous to the Km used in bio- chemistry for enzyme-substrate kinetics, and is estimated from the data. Another option, not often used in etio- logic research but used in studies of the effects of radiation (e.g., Little 2008), is not an RR model at all, but an excess absolute risk (EAR) model, also referred to as an addi- tive Poisson model. This model has the form Rexp = Rnonexp + β × cumexp. We fit this model following the presentation of Boshuizen and Feskens (2010), who present SAS code for the GENMOD procedure to fit this model via Poisson regression. Poisson regression requires categorized data for rates, unlike the Cox regression models used to fit the linear RR and log RR models here. We cross-categorized events and person-time by age, cumulative exposure (lagged 15 years), and calendar time for these analyses and compared the EAR model with the log RR model. We also used a scale-parameter statistic that adjusted for over- or underdispersion. Excess risk for environmental exposure to EtO according to each model was calculated using life table techniques accounting for all- cause mortality and applying an adaptation of the method described in the 1988 Biological Effects of Ionizing Radiation (BEIR) report (BEIR 1988). Because the exposure of inter- est was environmental (i.e., risk to the general public due to EtO in ambient air) and was assumed to begin at birth but with a 15-year lag, risk was effectively considered to begin at age 15 years. More details on the excess risk calculation can be found in Supplemental Material (doi:10.1289/ehp.1002521) and also in Appendix C of the 2006 draft U.S. EPA can- cer risk assessment for EtO (U.S. EPA 2006). Briefly, background mortality rates from U.S. vital statistics for 2000 (Centers for Disease Control and Prevention 2010) were stratified by 5-year age groups, and the probability of surviving each age interval was calculated. For incidence calculations, we calculated survival during the interval so that it was the product of the probability of surviving to the interval without a diagnosis of breast cancer and the Environmental Health Perspectives • volume 119 | number 6 | June 2011 df 1 1 10 2 1 1 1 10 2 1 2 p-Valuea 0.03 0.04 0.29 0.01 0.005 0.0030 0.0096 0.1249 0.0023 0.0007 0.0035 –2LL 1944.17 1944.68 1936.91 1940.48 1941.03 1942.27 1940.26 1933.94 1936.94 1937.49 1937.78 probability of not getting incident breast cancer during the interval. Surveillance, Epidemiology and End Results Program (SEER; National Cancer Institute 2010) rates for incident breast cancer for 1997–2001 were similarly strati- fied by 5-year age groups. Then, for each age interval, we calculated the cumulative prob- ability of getting incident breast cancer dur- ing the interval, given survival without breast cancer up to that interval. These age-specific probabilities of getting incident breast cancer were then summed across age groups to get the background lifetime (≤ age 85 years) risk of developing breast cancer (result: 0.147). We repeated this same procedure for the exposed group, except that the age-specific background incidence rates of breast cancer were multiplied by the rate ratio predicted by the exposure– response model for the cumulative exposure for that age group. The age-specific cumulative exposures were themselves derived by assuming a constant exposure intensity (set at a given level), which then accumulated daily with a 15-year lag (i.e., starting at age 15 years). Lifetime risk for the exposed population was again the sum of the conditional probabilities of getting breast cancer in each age-group inter- val, and excess risk was derived by subtracting the risk in the nonexposed population from the risk in the exposed (divided by 1 – lifetime risk in nonexposed). The EC01 (i.e., the estimated constant lifetime exposure level associated with a 1% excess lifetime risk of developing breast cancer) is determined by iteratively varying the exposure level until a lifetime excess risk of 0.01 is obtained. Similarly, a 95% one-sided lower confidence limit (LEC01) is estimated using the upper one-sided confidence limit of the rate ratio (β + 1.64 × standard error of β) and iterating until the exposure resulting in a 1% excess risk is found. As noted by Langholz and Richardson (2010), in linear RR models the profile likeli- hood may be less likely to be approximately Table 1. Log RR and linear models. Exposure Log RR models Log transformed Untransformed (linear) Categorical Two-piece linear spline Square root transformed Linear RR models Log transformed Untransformed (linear) Categorical Two-piece linear spline Square root transformed Linear exponential Abbreviations: df, degrees of freedom; –2LL, –2 log likelihood. ap-Value for chi square, the change in –2LL by the addition of exposure variable(s) to model. bAll models included three indicator variables for date of birth, one for family history, and one for parity. The –2LL for a model that included only the covariates date of birth, parity, and first-degree relative with breast cancer was 1948.93, and the –2LL for a model with no covariates was 1967.81. The AIC is derived as –2LL + 2 × (number of parameters estimated in model); a smaller AIC value indicates a better fit, adjusted for different number of parameters in the model. Risk estimation using epidemiologic data normal than in log-linear models, so that p-values based on the Wald statistic and the change in log likelihood may differ, and Wald- type confidence bounds may diverge from bounds derived from the profile likelihood. We found such divergence in our linear RR models and used bounds from the profile like- lihood to estimate the LEC01 for all models except the two-parameter linear-exponential model, where we used the delta method to derive the variance of the RR and the upper one-sided 95% confidence intervals (CIs). Results Table 1 gives the model fit results for the log RR and linear RR models, respectively. Overall, the linear RR models fit the data bet- ter than the log RR models, based on the cor- responding model likelihoods. Based on the Akaike information criterion (AIC; Akaike 1974), which is a measure of model fit, where a lower value signifies a better fit, the linear RR model using the two-piece linear spline fits better than all other models except the square- root model, which provided the best global fit among both log-linear and linear RR models. Results for Michaelis-Menten model are not shown. Graphically, this model looked almost identical to the square-root model, but the goodness of fit was not as good as the fit for the square-root model. The AIC was 1951.3; the best estimated k was about 13,000 ppm-days. The EAR (additive Poisson) model did not fit as well as the RR models. Because the Poisson model requires categorized data and has a different likelihood than do the Cox models used elsewhere here, it is not possible to directly compare the goodness of fit between the Poisson EAR model and the Cox RR models. Therefore, we compared the AIC from an EAR model with the AIC from a log RR Poisson model. The EAR model fit considerably less well than the log RR Poisson model (AIC 1918 AICb 1956.17 1956.68 1966.91 1954.48 1953.03 1954.27 1952.26 1963.94 1950.94 1949.49 1951.78 833