For each cancer site and for φ = 5, 6, ... 44, models were fitted to the 0 – 20 mSv male/female (M/F) data.
For the linear model only, I estimated ERR0.025,φ by fitting the model to the restricted subcohort of cells with mean weighted adjusted colon dose ≤ 0.5 mSv; ERR1,φ was obtained from fitting the 0 – 20 mSv data, as with all other modelling.
More precise optimal latencies φm, 95% CI's for and , and Goodness-of-Fit (GoF) statistics were determined for the nested models, including separate male and female results. The category model (M/F) was then fitted using the optimal latencies from the two-phase model.
Results are now outlined for specific cancers (M/F).
The linear model is never significant against the control model, but Figure 2 suggests two distinct latency periods: 5 ≤ φ ≤ 21, and 29 ≤ φ ≤ 41. Outside these periods, the nested models show no significant improvement on the control model. In both periods, the two-phase and transient models improve on the control model. The two-phase model significantly improves on the transient and linear models when 5 ≤ φ ≤ 21 and on the linear model when 29 ≤ φ ≤ 41. For the two-phase model, ERR0.025,φ and ERR1,φ are both positive when 5 ≤ φ ≤ 21, and negative when 29 ≤ φ ≤ 41 (Figure 3).
For the linear model (Table 1) φm = 6.66 is optimal in the region of ERR ≥ 0, but the model is not significant against the control model. For the transient model (Table 2) φm= 11.85 is optimal within the range for which ERR0.025,φ ≥ 0, but ERR is not significantly positive. The optimal latency for the two-phase model (Table 3) is φm = 11.89, when comparison with the control model has LRT2p-con = 21.190 (2 df), and comparisons with the transient and linear models have LRT2p-trans = 15.684 (1 df) and LRT2p-lin = 19.643 (1 df). Each LRT is highly significant. The two-phase model gives ERR0.025,φ = 0.391 with 95%CI (0.077, 0.857) while ERR1,φ = 0.459 (0.113, 0.942). When Dφ = Dmax = 0.012, ERR attains a local maximum of 0.539 (0.171, 1.037).
Applying the category model (Table 4) with φ = 11.89 gives LRTcat-con = 21.590 (3 df) and both β1 and β3 are significantly positive.
The models pass Goodness-of-Fit tests with Dev, but Pearson Chi-Square exceeds df due to one case in a cell with T = 0.4 p-y. The two-phase model is preferable to its nested alternatives by LRT comparisons, and clearly distinguishes the latency periods as regions of positive and negative ERR. At the optimal latency, R = 1 + σ/β= 262.62.
There are two distinct latency periods outside which the models are insignificant (Figure 4). When 10 ≤ φ ≤ 23 the two-phase model improves on the transient, linear, and control models, and ERR0.025,φ and ERR1,φ are negative in all three models (Figure 5). When 32 ≤ φ ≤ 41 the two-phase model improves on the transient, linear, and control models, and ERR0.025,φ and ERR1,φ are non-negative in all three models. When 34 ≤ φ ≤ 43 the transient and linear models each improve on the control model with positive ERR, and when 34 ≤ φ ≤ 38 the transient improves on the linear model.
The optimal latencies are 38.58 (linear), 36.90 (transient), and 36.90 if the two-phase model is optimised with ERR0.025,φ ≥ 0. At φm = 36.90 the two-phase model has LRT2p-con = 34.874 while LRT2p-trans = 16.755 and LRT2p-lin = 27.642, all highly significant comparisons. At the optimal latency, the two-phase model gives ERR0.025,φ = 1.099 (0.264, 2.374) while ERR1,φ = 1.428 (0.481, 2.954). The local maximum ERR occurs when Dφ = 0.013. Estimates from the transient and linear models are comparable.
Applying the category model with φ = 36.90 gives LRTcat-con = 33.844 and β1, β2, and β3 are all significantly positive. If "city" is included as a control, LRTcat-con = 34.939, β1 = 1.35 (0.72, 2.23), β2 = 0.73 (0.14, 1.60), β3 = 1.39 (0.70, 2.34).
The models pass both GoF tests. The two-phase model is preferable to its nested alternatives by LRT and gives wider periods of significant improvement on the control model, distinguished as regions of positive and negative ERR. At the optimal latency R = 204.11.
The linear model is never significant against the control model (Figure 6). The transient improves on the linear and control models when 5 ≤ φ ≤ 38. The two-phase improves on the linear and control models when 5 ≤ φ ≤ 21 and when 24 ≤ φ ≤ 38, and on the transient model when 32 ≤ φ ≤ 38. When 5 ≤ φ ≤ 21 the transient and linear models give positive ERR0.025,φ and ERR1,φ (Figure 7) as does the two-phase when 7 ≤ φ ≤ 21. When 22 ≤ φ ≤ 41 ERR0.025,φ and ERR1,φ are negative in the two-phase and transient models.
Optimal latencies are 16.90 (linear), 13.60 (transient), and 13.60 if the two-phase model is optimised with positive ERR0.025. At φm = 13.60 the transient model has LRTtrans-con = 15.701 while LRTtrans-lin = 14.394, both highly significant. At this latency the transient model gives ERR0.025,φ = 0.790 (0.195, 2.006) while ERR1,φ = 0.796 (0.189, 1.985). The two-phase model does not improve the transient and gives comparable ERR with somewhat wider CI's. It does improve on the linear model with LRT2p-lin = 14.734 and gives R = 86.39.
Applying the category model with φ = 13.60 gives LRTcat-con = 19.304 and β1, β2, and β3 are all significantly positive.
The 3 nested models pass both GoF tests. The transient model is preferable here.
None of the models give significant positive response for the M/F data (Figure 8, Figure 9). The transient model has occasional weak negative response.
The linear model is never significant against the control model (Figure 10). When 12 ≤ φ ≤ 16 the transient and two-phase models both improve on the control model, and ERR is negative in all three models (Figure 11). When 24 ≤ φ ≤ 33 ERR0.025,φ and ERR1,φ are positive in the two-phase and transient models. At the optimal latency φ = 26.91 (in the region of positive ERR) both the two-phase and transient models are weakly significant against the linear and control models. The 95%CI's do not exclude negative ERR.
Applying the category model with φ = 26.91 gives the weak result LRTcat-con = 8.077 with only β1 significantly positive.
When 22 ≤ φ ≤ 23 the two-phase model is significant against transient, linear, and control models (Figure 12) and ERR is negative in all models (Figure 13). When 39 ≤ φ ≤ 44 the transient model improves on the control model, and when 42 ≤ φ ≤ 43 the transient and two-phase improve on the linear and control models. When 37 ≤ φ ≤ 44 ERR is positive in all models. The optimal latency in all three nested models is φm = 43.86, at which all are significant against the control model and the transient and two-phase are significant against the linear model.
At φm = 43.86 the transient model has LRTtrans-con = 16.781 and LRTtrans-lin = 11.753, both highly significant, with ERR0.025,φ = 0.271 (0.070, 0.509) while ERR1,φ = 0.300 (0.034, 0.633). The two-phase model has LRT2p-con = 17.925, LRT2p-lin = 12.896, both highly significant, while LRT2p-trans = 1.143 is not significant. ERR0.025,φ = 0.269 (0.051, 0.538) while ERR1,φ = 0.272 (-0.044, 0.662). The linear model has LRTlin-con = 5.029 and ERR1,φ = 0.259 (0.030, 0.522). If applied to the 0 – 0.5 mSv data, the optimal latency is φm = 41.78 at which LRTlin-con = 5.562 and ERR0.025,φ = 0.175 (0.028, 0.344).
At φ = 43.86 the category model has LRTcat-con = 16.311 while β2 and β3 are significantly positive.
The models pass both GoF tests. The transient model is preferable here, for simplicity.
Comparison with LSS12
The ERR values found here by applying non-linear models to the 0 – 20 mSv subcohort and optimising latency are several orders of magnitude above those derived by extrapolating from results for a linear model applied with fixed 5 year lag to the entire dose range of A-bomb survivors, as in LSS12  whose Tables AII (Male/Female), AIII (Male) and AIV (Female) show ERR/Sv (organ dose). If a linear model is applied to a single dataset, ERR for d mSv will be (d/1000)(ERR/Sv). Extrapolations of this type are inherent in the ICRP estimate of the risks arising from low doses, which underpin its recommended annual dose limits.
Comparisons are shown in Table 5. Significant discrepancies occur at 10 mSv (ERR1) and 0.25 mSv (ERR0.025) with the two-phase model for stomach, liver, and lung; and with the transient model for liver and lung.
Estimates of ERR1 from the non-linear models are 2 to 3 orders of magnitude above extrapolations from LSS12; for ERR0.025 the discrepancy is 3 to 4 orders of magnitude.
As a first step towards understanding how dosimetry errors may affect these results, the models were fitted to portions of the 0 – 20 mSv data for the liver. Models were refined to control for "city", although this had little impact. There are 173 data cells with weighted adjusted colon dose = 0. If these cells are deleted from the 0 – 20 mSv subcohort and analysis is restricted to the remaining 2838 cells, fitting the two-phase and linear models for the liver (M/F) at φ = 36.90 years gives LRT2p-con = 37.639, LRT2p-lin = 30.095, and ERR1 = 1.478 (0.52, 3.04). Thus the results for the liver are not caused by any special features of the zero-dose cells.
Alternatively, choose 0 = x0 < x1 < x2 < ... < x9 < x10 = 20 mSv. Define Si as the set of data cells in the 0 – 20 mSv subcohort for which the weighted adjusted colon dose does not fall in the interval [xi-1, xi]. The xi may be chosen so that the Si have roughly equal p-y of observation. Then, fitting the two-phase and linear models for the liver (M/F) with φ = 36.9 to the reduced datasets Si gives LRT2p-lin and ERR1 values as shown in Table 6. LRT2p-lin values are ≥ 18.94 (1 df) and ERR1 is fairly stable, varying from 1.21 to 1.74.
The DS86 dosimetry used here and in LSS12 was re-investigated by a Joint Working Group using physical measurements, resulting in a new dosimetry system DS02 . A dataset using DS02 and enabling comparison with DS86 was released by RERF last year and can be downloaded as DS02can.dat from the RERF website . However, the cancer mortality fields in this dataset only show deaths from solid cancers (combined), liquid cancers, and leukaemia. Investigation of the stomach, liver, and lung as individual sites is not possible from this public dataset. Furthermore, the stratification of the new dataset differs from that in LSS12 and it is not possible to simply read off the DS02 values for cells in the LSS12 data.
Nonetheless for those cells in the DS02 dataset which have a DS86 dose, the weighted adjusted DS02 colon dose using neutron RBE = 10, shown as "cola02w10", can be compared with the corresponding DS86 dose variable shown as "cola86w10".
Only 11 cells, with 13.83 p-y and no solid cancer deaths, have cola02w10 < 0.005 Sv and cola86w10 ≥ 0.005 Sv. Only 5 cells, with 11.24 p-y and no solid cancer deaths, have cola02w10 ≥ 0.005 Sv and cola86w10 < 0.005 Sv. For cells with cola86w10 ≥ 0.005 Sv the ratio θ = cola02w10/cola86w10 varies from 0.216 to 1.925 with mean 1.08 and standard deviation 0.082. There are only 7 cells, with 10.47 p-y and no solid cancer deaths, for which θ < 0.5. Likewise 92 cells, with 195.84 p-y and 3 solid cancer deaths, have θ > 1.5. In the subcohort with 0.005 Sv ≤ cola86w10 ≤ 0.02 Sv, θ varies from 0.216 to 1.27 with mean 1.026 and standard deviation 0.077. There are 15 cells with 78.5 p-y and 2 solid cancer deaths, for which θ < 0.8, and 9 cells with 9.88 p-y and 1 solid cancer death, for which θ > 1.2
Thus if DS02 is an accurate estimate of the flash dose, there is virtually no misclassification of DS86 between the categories "above 5 mSv" and "below 5 mSv", and above 5 mSv the DS86 dose is a reasonable estimate, though 8% below DS02 on average.
The scope in this paper for misclassification of the dose is therefore reduced if doses below 5 mSv are taken as baseline and doses from 5 to 20 mSv are taken as a single category which is then analysed with latency. While that approach is too crude to detect non-linearity, it gives very similar results for the liver to those found with the linear model.
As shown in Table 1 the optimal latency for the liver (M/F) using the linear model is φ = 38.58, for which comparison with the control model has LRTlin-con = 11.21, β = ERR1 = 0.69 (0.25, 1.26). Now define E = 1 if Time-Since-Exposure ≥ 38.58 and 5 mSv ≤ colon dose ≤ 20 mSv, E = 0 otherwise. As Time-Since-Exposure is known, errors in E can only arise if doses below 5 mSv were misclasssified as above 5 mSv, or vice versa. Fitting the model defined by ERR = γE to the liver (M/F) data for the 0 – 20 mSv dose range and comparing with the control model (γ = 0) gives LRT = 11.04, γ = 0.67 (0.24, 1.21). Thus the results for the linear model can be reproduced with a two-category model using a cutpoint of 5 mSv, and these categories are almost identical whether defined by DS86 or DS02.
As a further test, the liver data were modelled in the extended DS86 dose range 5 mSv – 500 mSv, where DS86 and DS02 are in reasonable agreement. The two-phase model is a significant improvement on the linear model at latency φ = 36.9 years, with LRT2p-lin = 10.37 (1 df) and ERR1 = 0.74 (0.10, 1.79).
Similar results for the liver are obtained from the 0 – 500 mSv dose range. The two-phase model is a significant improvement on the linear model at latency φ = 36.9 years, with LRT2p-lin = 16.86 (1 df) and ERR1 = 0.76 (0.15, 1.66).
For comparison, the results for the liver obtained from the 0 – 20 mSv dose range at latency φ = 36.9 years, are LRT2p-lin = 29.67 (1 df) and ERR1 = 1.46 (0.50, 3.00). Similar results when "city" is omitted from the model are shown in Table 3.
The DS02 public dataset does not show the liver but does allow modelling of all-solid cancers. The subcohort S defined by cola02w10 ≤ 0.02 and Time-Since-Exposure ≤ 45.39 years (the maximum value in the 0 – 20 mSv subcohort of LSS12 data) has 1682335.39 p-y and 4363 cases, roughly comparable to the LSS12 subcohort analysed with DS86 in this paper. Models were defined as previously, but using cola02w10 in place of the DS86 weighted adjusted colon dose. None of the cells in S contain Nagasaki factory workers. Controls for "city" and "ground distance category" (proximal or distal) were introduced, though they had little impact. If the linear model is fitted to S with latency φ = 43.29 years then LRTlin-con = 3.79 (1 df). For the two-phase model LRT2p-con = 10.49 (2 df) and LRT2p-lin = 6.70 (1 df), again showing non-linearity. ERR1 = 0.223 (-0.07, 0.58). For the DS86 results, Table 3 shows ERR1 = 0.272 (-0.04, 0.66) at latency 43.86 years.
The estimates of ERR1 for all-solid cancers are similar whether derived from the 0 – 20 mSv subcohort of LSS12, or the comparable 0 – 20 mSv subcohort of the DS02 data.
The DS02 data contains an additional 10 years of follow-up. Non-linearity is still present in the extended cohort. Define the subcohort T by cola02w10 ≤ 0.02 without restricting Time-Since-Exposure. At latency 43.29 years the linear model is indistinguishable from the control model (LRT = 0.47) but the two-phase model has LRT2p-con = 9.49 (2 df) and LRT2p-lin = 9.02 (1 df). ERR1 = 0.08 (-0.07, 0.26).