the truncated data changed substantially for the linear model (from 0.040 to 0.020 ppm, a 50% decrease) but less for the log-transform model (from 0.0005 to 0.0006 ppm, a 20% increase) and for the two-piece model (from 0.0010 to 0.0011 ppm, a 10% increase); we observed no change for the linear-exponen- tial model or the square-root model (data not shown). In addition, we fit the linear RR model after excluding all observations with expo- sures above 5,800 ppm-days, the location of the knot used in the two-piece spline model, which eliminated the top 20% of exposures. The slope of the exposure–response curve for exposure modeled as an untransformed linear variable was similar (8% higher) to the slope of the lower piece of the two-piece spline model in Figure 2, but its variance was 34% higher (data not shown). This finding illus- trates that the slope of the estimated low-dose exposure–response relationship based on the two-piece spline model is not (as might be expected) the same as the slope that would be obtained by simply truncating the data at the preferred knot, because the two segments of the curve produced by the spline model covary, so that one influences the other. We believe it is generally preferable to use the complete data set to evaluate the low- dose region rather than some truncation of the data that would allow an untransformed linear model to fit well for two main reasons: to avoid some arbitrary point to truncate the data (in the two-piece model, the likelihood function provides an objective mechanism for knot selection), and to avoid information in the upper-dose range that can affect the slope and variance of the exposure–response relationship in the low-dose range. We also conducted analyses looking at a possible effect of dose rate (using the lin- ear RR model), which for some carcinogens modifies the effect of cumulative dose (e.g., Lubin et al. 2008). Specifically, we divided the observations into high- and low-dose rate groups defined as an average intensity Table 2. EC01 estimates for linear RR models. Exposure EC01 (ppm) LEC01a (ppm) Untransformed (linear) 0.0400 0.0165 Log transformed 0.00005 0.00002 Two-piece linear spline 0.0100 0.0039 Square root transformed 0.0016 0.0003 Linear exponential 0.0113 0.0058 aLower CI for EC01, using upper one-sided 95% con- fidence limit for exposure–response coefficient, as determined using profile likelihood for single parameter models and for the first piece of the two-piece spline model, applicable in the low-dose region of interest (below the knot, where the second parameter is 0). For the two-parameter linear-exponential model, we used the delta method to derive the variance of the RR and upper one-sided 95% CI using Wald-type variances for each parameter, given the added computational com- plexity of deriving profile-based bounds using the joint likelihood for both parameters. of exposure < 1 ppm and > 1 ppm, which is the current occupational standard and which divided the cases into approximately equal groups. Interaction terms between average intensity (dichotomous indicator term) and either linear (untransformed) exposure or the square root of exposure were not close to sta- tistical significance (p = 0.40 and 0.57, respec- tively). These results indicate that the effect of cumulative dose did not change significantly according to high- or low-dose rate. Discussion Our goal here has been to illustrate key issues in risk assessment related to the choice of a statistical model when risk attenuates at high exposures. We have chosen to rely on rela- tively simple, easily interpretable models that may be appropriate to risk assessors operating in a public health setting, without entering into a comprehensive discussion of statisti- cal modeling of exposure–response trends. Outside the realm of risk assessment, a wide variety of options are available for choos- ing the best model, but we argue that these choices are somewhat more limited if they are to conform to the needs of the risk assessor, especially if the goal is estimation of permis- sible environmental levels rather than occu- pational levels. Then the shape of the curve in the low-dose region is key. When there is a choice of different models that fit the data well, models that are approximately linear in the low-dose region are preferred, in part because they will not be highly sensitive to the relatively arbitrary choice of a point of Risk estimation using epidemiologic data departure for low-dose linear extrapolation. The point of departure is particularly sensi- tive to the shape of the curve in the low-dose region, whereas overall model fit pertains to the validity of the model across the entire exposure range. Occupational studies for a wide variety of chemicals, often used for risk assessment for both occupational and environmental settings, have frequently shown exposure–response relationships in which the risk of cancer pla- teaus or attenuates at high exposures (Stayner et al. 2003). A number of reasons have been advanced to explain such attenuation, includ- ing depletion of susceptible individuals in the population, saturation of biological pro- cesses, misclassification of exposure at high levels, and the healthy worker survivor effect. One might argue that the principal reason for such attenuation is increased misclassifi- cation and mismeasurement of exposure at high levels, in which case it might make more sense to attempt to correct the exposure–re- sponse data for measurement error than to model mismeasured data. There are several arguments against this approach. First, from a practical standpoint, it is rare that a gold standard of well-measured data is available to estimate the direction and amount of mis- measurement. Second, there is no reason a priori to assume that measurement error is the principal reason for observed attenuation, and it is not necessarily clear that attenuation will result from this error. Measurement error that is nondifferential and of the Berkson type (Armstrong 1998; as in occupational 2.5 R R 2.0 1.5 1.0 Linear Square root Spline Linear exponential Linear log Environmental Health Perspectives • volume 119 | number 6 | June 2011 0 10,000 20,000 30,000 Cumulative exposure, lagged 15 years (ppm × days) Figure 3. Linear RR models with top 5% of exposure eliminated (> 21,219 ppm-days). Models correspond to those described in Figure 2. 40,000 835