Steenland et al. such as models in which the exposure is log transformed or square root transformed (e.g., power models), are supralinear in the low-exposure region such that slope of the exposure–response curve is high at low expo- sures (see Rothman et al. 2008 for a discus- sion of transformation of exposure variables). This high slope in the low-exposure region of interest results in EC01s that are very low. Furthermore, without linearity or near linear- ity in the low-dose region, the slope of the linear low-exposure extrapolation used for risk assessment is very sensitive to the particular point of departure (e.g., LEC01 vs. LEC001) used for low-dose extrapolation. On the other hand, a model with a simple untransformed linear term for cumulative exposure may not fit the attenuating data as well and may under- estimate risk in the low-dose region, resulting in excessively large estimates of EC01. One alternative is to use a two-piece spline model, that is, a spline curve with one knot. Splines are piecewise polynomials of degree n, the simplest being a linear spline of degree 1. The point at which the pieces join is called a knot. Splines can be espe- cially useful in approximating the shapes of exposure–response curves, which may vary substantially over the range of exposure (see Rothman et al. 2008 for a discussion of the application of splines in the analysis of epide- miological data). In our approach, the data are used to fit log-linear or linear-risk models with two linear sections or pieces, which permit the exposure–response relation to have a higher slope in the lower exposure region than in the higher exposure region when it is attenu- ated. Linear spline models can be generalized in a straightforward manner to accommodate more complex exposure–response relations by using additional knots to generate a mul- tipiece linear spline model. In addition, qua- dratic and cubic splines may be used instead of linear ones. Quadratic and cubic splines with multiple knots are commonly used to fit exposure–response data in epidemiology (Eisen et al. 2004; Harrell et al. 1988; Steenland and Deddens 2004). These more complex spline models have disadvantages for risk assessment in that their shape is dependent on multiple knot choices, their parameters are not easily interpretable, and they are not linear in the low-dose region. However, such models may help determine an approximate shape for sub- sequent fitting of a parametric curve that may be more useful for risk assessment. Simpler polynomial or power models without sepa- rate pieces (i.e., that are not splines) do not require knot choice (Greenland 1995; Stayner et al. 1997), nor do linear-exponential mod- els that have been used in radiation research (Schneider and Walsh 2008). Another model of potential interest is the Michaelis-Menten model of enzyme kinetics, in 832 which the rate of reaction tails off as substrate concentration increases (Lehninger 1982). However, as previously noted, exposure– response curves based on these models may be supralinear in the low-dose region when the exposure–response attenuates at high exposures. Cohort data or nested case–control data, which are typically the basis of environmental or occupational risk assessment using human data, are usually fit using either survival analy- sis, or conditional logistic regression, where data are grouped in risk sets matched on time via Cox regression. Both linear RR and log- linear RR models for nested case–control data can be fit using common statistical programs such as SAS. Because these models are fit using the same likelihood (Cox’s partial like- lihood), their likelihoods can be compared (Langholz and Richardson 2010). Here we provide illustrations of the application of a variety of RR models using data from a cohort study of workers exposed to the sterilant gas ethylene oxide (EtO). Methods Briefly, we previously conducted a cohort analysis, with a focus on breast cancer, of 7,576 women exposed to EtO while sterilizing medi- cal supplies (see Steenland et al. 2003 for details of this study). Here we focus on the 5,138 women from that cohort who we were able to interview; of these women, 232 had been diag- nosed with breast cancer. Estimated daily expo- sures to EtO (average intensity) across different jobs and time periods ranged from 0.05 ppm to 77 ppm—the current U.S. Occupational Safety and Health Administration standard is 1 ppm (U.S. Department of Labor 2007), with an average duration of exposure of 11 years and a median of 7 years (range, 1–50 years). Cumulative exposure was calculated by mul- tiplying duration of exposure by intensity of exposure for each job or time period and sum- ming across different jobs and time periods, with a 15-year lag period such that exposure within 15 years of case diagnosis (or the corre- sponding date in noncases) was excluded from analyses. The 15-year lag was optimal (best fitting) in log-linear models (Steenland et al. 2003), and we used it again here for linear RR models for consistency. Models were fit using Cox regression, in which each case was compared with its matched risk set (with matching based on white or non- white race), cumulative exposure was time dependent, and the time variable was age. Analyses included the following potential con- founders: parity (continuous), date of birth, and family history of breast cancer in first-degree relatives (yes/no). In the linear RR models, one has the choice of combining additive terms for the main exposure with multiplicative terms for covariates, for example, RR = (1 + β1X1) × exp(β2X2) × exp(β3X3), where X2 and X3 represent covariates, or of using a strictly addi- tive model for both covariates and exposure (e.g., RR = 1 + β1X1 + β2X2 + β3X3). We present results for the strictly additive model, because an inspection of model likelihoods indi- cated that this type of linear RR model fit the data best. SAS NLP (version 9.1; SAS Institute Inc., Cary, NC) was used for analyses of both log RR models and linear RR models, as sug- gested by Langholz and Richardson (2010). As noted, the SAS procedure provides comparable model likelihoods for both types of models. We present a variety of log RR and linear RR models, as described below, including a categorical exposure model (deciles), a linear (untransformed) exposure model (e.g., RR = 1 + cumexp), a model using the log transforma- tion of exposure [(e.g., RR = 1 + log(cumexp)], a model using a square-root transforma- tion (e.g., RR = 1 + square root (cumexp), a two-piece spline model, a linear-exponential model (used only in the linear RR case), and a Michaelis–Menten model. The square-root transformation fits curves that attenuate at high exposures and was chosen to represent the class of polynomial models. We restricted ourselves to relatively simple statistical models and did not use any biologi- cally based models (e.g., two-stage models), which are not commonly used for risk assess- ment using human data. In categorical analyses, exposure was clas- sified into 11 groups, with the reference group including cases and noncases who had no EtO exposure > 15 years before the case was diag- nosed with breast cancer (62 breast cancers), and 10 groups with cumulative exposures > 0 with cut points selected such that an approxi- mately equal number of breast cancer cases (n = 17) occurred in each decile: 1–355, 356– 842, 843–1,361, 1,362–2,187, 2,188–3,772, 3,773–5,522, 5,523–7,891, 7,892–14,483, 14,484–25,112, and > 25,112 ppm-days. The linear spline model had the form (lin- ear RR version) of RR = 1 + β 1 × cumexp + β2z × maximum (0, cumexp – knot), where the value corresponding to maximum (0, cumexp – knot) is either the cumulative expo- sure value less the exposure level used to define the knot, or 0 if the cumulative exposure is less than the value of the knot. The slope of the first exposure–response segment based on the model is represented by β1, whereas the slope of the second segment is β1 – β2. Knots for the two-piece spline model were chosen to maxi- mize the log likelihood after trying knots at 100 ppm-day intervals. In our case, there was a clear optimal knot at 5,800 ppm-days (lin- ear RR model). [See Supplemental Material, Figure 1 (doi:10.1289/ehp.1002521).] The linear-exponential model had the form RR = 1 + (β1 × cumexp) × exp(β2× cumexp). This model has been used in radiation stud- ies (Schneider and Walsh 2008) as well as in ovlume 119 | number 6 | June 2011 • Environmental Health Perspectives