Steenland et al. studies where the mean job-specific exposure is assigned to all workers in that job) with constant error variance across exposure levels has been shown in RR models to result in a downward bias on the exposure–response rela- tionships in RR models (Armstrong 1990). However, others have shown in log-linear RR models that increased mismeasurement of high exposures versus low exposures does not necessarily lead to estimates of exposure– response relationships biased to the and can, in fact, lead to overestimates of expo- sure–response relationships (Steenland and Deddens 2000). Whether one might expect attenuation due to mismeasurement in linear RR models under this scenario is not clear. We argue that a two-piece spline model is a good candidate for risk assessment when there is risk attenuation at high exposure levels. We have used data on breast cancer incidence and EtO exposure, which exhibit attenuation, to illustrate the two-piece spline model; however, as noted above, there are many other examples where this approach would be suitable. We also argue that addi- tive models (linear RR) offer some advantages for risk assessors, namely, easy interpretabil- ity and a strictly linear form in the low-dose region of interest. When there is attenuation, the risk asses- sor has a number of choices. For example, if the exposure–response curve plateaus as exposure increases, the risk assessor may face a dilemma of whether to consider all the data or make some relatively arbitrary decision to exclude high-exposure data altogether from the exposure–response model to focus on the low-exposure region that is the most infor- mative and relevant for estimating low-dose risk (e.g., from environmental exposures). We have argued that the two-piece linear model provides one solution to this dilemma that avoids the need to make a subjective decision about what to define as high exposure and has the added benefit of a linear exposure– response curve in the low-exposure region. One potential disadvantage to the two- piece spline model can arise in choosing the knot based on the likelihood, particularly when data are sparse. Sometimes the best like- lihood for choosing the knot is not much bet- ter than another, that is, the profile likelihood is fairly flat or has several local maxima. This will also occur when the data are not sparse but the two-piece spline model does not pro- vide a good fit to the data. The variety of models illustrated here pro- vide a statistically reasonable fit to the overall breast cancer morbidity data considered, but they have very different shapes in the low- exposure region, resulting in very different EC01 estimates. The two-piece spline model yields estimates between the more extreme results obtained from the log-transformed 836 model (very supralinear in the low-exposure region) and the untransformed (linear in exposure) model (too sublinear in the low- exposure region). The square-root model also is somewhat supralinear in the low-dose region, resulting in a low EC01. An alternative to choosing the best of a series of models that fit the data reason- ably well is model averaging, in which the exposure–response results from different mod- els are averaged using some sort of weighting scheme. When these models have the same parametric form, a weighted average of the parameters can be calculated using weights determined from the likelihood in a frequen- tist setting or from Bayesian posterior prob- abilities for the parameters (assumed to be random variables). Schwartz et al. (2008) pro- vide an example from air pollution where a five-knot linear spline model was used with different lags on pollution exposure. A series of models was run in which the change in slope at each knot was constrained in differ- ent ways (e.g., a threshold model in which the slope of the first piece of the linear model was constrained to be 0). The parameters for slope in each piece were then averaged using as weights the Bayesian posterior probabilities, with noninformative priors or priors favor- ing a nonthreshold model over a threshold model. Wang et al. (2004) provide a good discussion of this same type of model averag- ing in a slightly different context. They show via simulation how Bayesian model averaging can outperform backward and forward selec- tion techniques to pick the best model from a large set of candidate predictors, then show how parameter estimates for a given set of pre- dictors from the best models can be averaged using posterior probabilities to allow inference about these parameters while reflecting the uncertainty of model selection. When the models have different paramet- ric forms, as in our case (e.g., square root– transformed model, two-piece model), model averaging might be done via calculation of dif- ferent EC01 values and taking a weighted aver- age across them using model likelihoods or Bayesian posterior probabilities. Noble et al. (2009) have used model averaging to calculate a benchmark concentration, analogous to an EC, in a study of lung function and coal dust, using as weights Bayesian posterior probabili- ties with noninformative priors, across linear, square root, and quadratic models, with the selection of a set of covariates, such as height and weight, from among six possible ones. The subset of best models used in the model averaging, out of all possible models, was chosen using a type of selection rule called Occam’s window (Noble et al. 2009), based on the posterior likelihood for the model. One difficulty which arises is deciding what the necessarily arbitrary cut point will be for excluding some range of poorly fitting models from the average. Model averaging makes more sense, in general, in situations where the results of dif- ferent models are similar, for example, using a set of best models chosen via Occam’s win- dow. Assuming a relatively broad inclusion rule, a model result that is an outlier on one side of the central tendency may have undue influence on the model average if the fit of the outlier model is reasonable (thus increasing the weight of its contribution to the average across all models) and if it is not balanced by an out- lier on the other side of the central tendency. In our data, we found that different models, all of which fit the data reasonably well, produced very different estimates of EC01. Although an AIC-weighted average of EC01 estimates from Table 2 yielded a model-average EC01 of 0.013, which was not far from the estimates based on the two-piece linear and linear- exponential models, this occurred because the two outliers (from the linear and log-trans- formed exposure models) were on opposite sides of 0.013—a situation that might not have occurred if we had overlooked one model form or another. Furthermore, models that are not linear in the low-dose region (which is approximately 0–1,000 ppm in our data) will lead to very different estimates of permissible exposure levels, depending on the somewhat arbitrary choice of EC01 or EC05 or EC001, for example, as the point of departure for low-dose extrapolation. In our case, limiting ourselves to models that were approximately linear in the low-dose region implies limiting ourselves to the linear exposure model, the two-piece lin- ear model, and perhaps the linear-exponential model. In our view the linear model is an out- lier that did not fit the data as well as the other two models, and we would not advise averag- ing these three models. In sum, from both a public health and risk assessment perspective, we believe it best to choose a best model based on a combination of model fit and a desire that the resulting model be linear or close to linear in the low-dose region. Ref Rencs Akaike H. 1974. 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