Steenland et al. vs. 1867; smaller is better). The scale parameter was close to 1 with both models. Consequently, we do not present quantitative risk estimates based on the EAR model. Figures 1 and 2 show the results for the log RR and linear RR models graphically. The shapes of the curves for corresponding log RR and linear RR models are similar. RR estimates 0 10,000 20,000 30,000 40,000 Cumulative exposure, lagged 15 years (ppm × days) Figure 1. Log RR models for breast cancer incidence and for EtO exposure. Log RR models as used here have the form log RR = β1(cumexp). Spline refers to a two-piece spline log RR model with a single knot at 5,800 ppm-days. Log square-root, log-log, and log-linear models refer to log RR models with exposure square root transformed, log transformed, or untransformed, respectively. Individual points indicate esti- mates from a model with exposure categorized into deciles among the exposed (reference = no exposure, points graphed at the midpoint of each exposure interval). Spline Log linear Log square root Log log Linear Linear exponential Spline Square root Linear log 2.0 R R 2.8 2.4 1.6 1.2 0.8 2.5 2.0 R R 1.5 1.0 0 10,000 20,000 30,000 40,000 Cumulative exposure, lagged 15 years (ppm × days) Figure 2. Linear RR models for breast cancer incidence and EtO exposure. Linear RR models as used here have the form RR = 1 + β1(cumexp). Spline refers to a two-piece spline linear RR model with a single knot at 5,800 ppm-days. Square root, linear-log, and linear models refer to the linear RR models with exposure square root transformed, log transformed, or untransformed, respectively. The linear-exponential model has the form RR = 1 + {β1(cumexp) × exp[β2(cumexp)]}. Individual points indicate estimates from a model with exposure categorized into deciles among the exposed (reference = no exposure; points graphed at the midpoint of each exposure interval). 834 ovlume 119 | number 6 | June 2011 • Environmental Health Perspectives based on the categorical models do not follow a monotonic trend with increasing exposure, but the overall pattern suggests increasing RRs with increasing exposure, with some attenua- tion at the highest exposure decile (i.e., a line fit through the other deciles would pass well above the highest decile). Note that although the cut points for forming decile categories were the same in Figures 1 and 2, the linear RR model gives different categorical results than the log-linear model. Modeling exposure as a simple continuous term appears to under- estimate the exposure–response relationship suggested by the categorical point estimates from corresponding log RR and linear RR models. Modeling log-transformed exposure produces log RR and linear RR model esti- mates that conform reasonably well to the categorical point estimates, but the exposure– response curve is very steep in the low-dose region. Estimates from the two-piece spline models appear to conform well to the categori- cal results and also suggest a pronounced expo- sure–response slope for exposures below the knot (< 5,600 ppm-days). The square-root log RR and linear RR models also fit the data well but generate a supralinear shape (although less pronounced than the log-transform model) in the low-dose region. Table 2 presents the resulting exposure levels that correspond to a 1% lifetime extra risk of breast cancer (EC01) based on the five different linear RR models, as well as the LEC01 based on using the upper one-sided 95% bound for the exposure–response coeffi- cient. The linear exposure model, with its low slope, had the highest EC01, whereas the log- transformed exposure model had the lowest, and the two-piece spline had an EC01 between the two. The EC01 for the two-piece spline resulted entirely from the first segment of the spline model, because the 1% excess life- time risk resulted from cumulative exposures < 5,800 ppm-days, that is, within the range of exposures corresponding to the first segment of the estimated exposure–response curve. Sensitivity analyses. We conducted several analyses to evaluate the sensitivity of results to different model specifications, including the use of a three-piece linear spline model with a second knot at 7,200 ppm-days, which was the optimal location based on a comparison of model fits. However, this model did not sig- nificantly improve the fit to the data over that of a two-piece linear spline model (data not shown). We also evaluated truncated linear RR models after excluding observations in the top 5% of the exposure distribution where the most extreme outliers in cumulative exposures occurred. The model of the untransformed (linear) exposure was the most affected by this exclusion, which resulted in a sharp increase in slope indicating its sensitivity to data in the high-exposure range (Figure 3). The EC01 for