The results in Tables VIII and IX and Fig. 2 ( see Fig. 2A Fig. 2B Fig. 2C Fig. 2D show that the restriction of results to the low dose region has a stronger effect on (using the exponential relative risk model) than does dose adjustment. The graphical results in Fig. 2A and regression diagnostics (Appendix Fig. A2) further indicate that the main effects model with cumulative dose represented as a linear term in the exponent (see Eq. 3) does not provide an adequate description of the dose-response relation over the entire range of doses. This is the dose-response model that was used in previous studies of the X-10 only subcohort of Oak Ridge workers [22,23]. The results presented in Tables VI and VII are based on the ERR (see Eq. 4) which has been widely used in radiation epidemiology [4,24,25]. These two regression functions cannot be compared directly using likelihood ratio tests but can be compared indirectly using the values of the deviance and df for each model.
A more extensive analysis of all cancer mortality was conducted to further explain and clarify the differences between the ERR and exponential relative risk functions. Table XI contains the resulting summary statistics for seven relative risk functions for the all cancer mortality using unadjusted doses with a ten year lag for the X-10/Y-12 subcohort. This analysis is based on a main effects model with external rates for the baseline risk (see Appendix). The deviance for a Poisson regression model is an overall summary of the discrepancy between the fitted values and the data for each cell in the ADS and provides a measure of ''unexplained variation" similar to the residual sum of squares in least squares regression, i.e. smaller values indicate a ''better fit". The deviance for each relative risk function in column 1 is given in column 3, and these statistics can be used to calculate LRTs for nested models. The deviance for the null model of no dose-response relation (i.e. a constant relative risk of one) is 2020.05 (see line 1 of Table XI ). The value of the LRT statistic (3.55) for the null hypothesis of zero slope in the exponential model is obtained by subtracting the deviance on line 2 from that on line 1. The last line in Table XI is an unconstrained model, i.e. a relative risk parameter is estimated for each dose group, and can be used to construct a ``lack-of-fit" test for any of the other models in Table XI that impose constraints on the relative risk- dose relation . For example, the lack-of-fit test for the linear excess relative risk function (line 4 of Table XI ) with the unconstrained model as the alternative yields a LRT statistic of 2014.75 - 2012.08 = 2.67 with 8 df, which does not indicate lack-of-fit. The LRT for zero slope for the linear ERR model is 5.3 with 1 df. These results suggest that the ERR model provides a better description of the relative risk dose relation over the entire dose range than the linear exponential relative risk function.
Another less formal approach that can be used to identify ''good" models is the AIC (small values indicate a ''better fit"). The AIC values in column 4 of Table XI suggest that the best model among those considered is the linear ERR (see line 4) since it has the smallest AIC value. The fitted values for this linear ERR function are shown in Fig. 2A, where it appears to describe the relative risk over the entire dose range. The exponential relative risk estimates based on the low dose data are close to the linear ERR estimates in the low dose range. Results for a power law model for the ERR are given on line 6 of Table XI and in Fig. 2A. Both the power law model and the linear-quadratic model (line 5 Table XI ) can be used to construct likelihood ratio lack-of-fit tests for the linear ERR dose-response. These results further support the conclusion that the linear ERR model is a reasonable representation of the relative risk dose relation over the entire dose range. Fig. 2 also presents the results of fitting the exponential and linear ERR function to lung, digestive, and prostate cancer. Fig. 3 presents similar results for all cause mortality, diseases of the circulatory system, nonmalignant respiratory disease, and all external causes.