Dose–responses from multi-model inference for the non-cancer disease mortality of atomic bomb survivors

The non-cancer mortality data for cerebrovascular disease (CVD) and cardiovascular diseases from Report 13 on the atomic bomb survivors published by the Radiation Effects Research Foundation were analysed to investigate the dose–response for the influence of radiation on these detrimental health effects. Various parametric and categorical models (such as linear-no-threshold (LNT) and a number of threshold and step models) were analysed with a statistical selection protocol that rated the model description of the data. Instead of applying the usual approach of identifying one preferred model for each data set, a set of plausible models was applied, and a sub-set of non-nested models was identified that all fitted the data about equally well. Subsequently, this sub-set of non-nested models was used to perform multi-model inference (MMI), an innovative method of mathematically combining different models to allow risk estimates to be based on several plausible dose–response models rather than just relying on a single model of choice. This procedure thereby produces more reliable risk estimates based on a more comprehensive appraisal of model uncertainties. For CVD, MMI yielded a weak dose–response (with a risk estimate of about one-third of the LNT model) below a step at 0.6 Gy and a stronger dose–response at higher doses. The calculated risk estimates are consistent with zero risk below this threshold-dose. For mortalities related to cardiovascular diseases, an LNT-type dose–response was found with risk estimates consistent with zero risk below 2.2 Gy based on 90% confidence intervals. The MMI approach described here resolves a dilemma in practical radiation protection when one is forced to select between models with profoundly different dose–responses for risk estimates. Electronic supplementary material The online version of this article (doi:10.1007/s00411-012-0410-4) contains supplementary material, which is available to authorized users.


Preston baseline model
The Preston baseline model is as follows: These naming conventions are taken from R13models.log. Age at exposure is denoted by e, age attained by a. Model parameters in Eq. (A1) are italicised. Parameters cmH, cfH, cmN, and cfN represent constant factors (cmH and cfH, for example, are constants related to males and females in Hiroshima), parameter e30l70 describes variations of the hazard with multiplicative effects of age attained and age at exposure, while parameters e30m and e30f describe variations of the hazard with age at exposure for males and females, respectively. Parameters e30qspm and e30qspf mark 3 the dependence on a quadratic spline function with age knot at e = 30 years for males and females, respectively. The Preston baseline model therefore uses 29 model parameters including the four age knots in e30qsp, e50qsp, lage40qsp, and lage70qsp.

Streamlined baseline models
One common (i.e. joint) baseline model was used for the cerebrovascular disease (CVD) data. Our streamlined baseline model for CVD contains 21 statistically significant baseline parameters -8 parameters less than Preston's baseline model h 0 from Eq. (A1): e30sqm = e30qspm = e50qspm = e50qspf = l70f = 0 which made the three age knots related to e30qspm, e50qspm, and e50qspf obsolete. In other words: for the fit of the CVD data one single joint baseline model was used, namely Eq. (A1) with e30sqm = e30qspm = e50qspm = e50qspf = l70f = 0. In addition, it was found that the model fit significantly improved when the age knots at 40 and 70 years in lage40qsp and lage70qsp, respectively, were allowed to be free. Note that for reasons of clarity the related adjustable parameters are denoted by l40agem, l40agef, l70agem, and l70agef although best estimates different than 40 or 70 were found in the related model fits (Table S1). The fit also significantly improved when the age at exposure knot at 30 years in e30qsp was allowed to be free; the related model parameter is denoted by e30agef.
For cardiovascular diseases, we proceeded analogously as for CVD. Each of the 29 parameters of the Preston baseline model was tested for its significance resulting in a streamlined baseline model with 14 model parameters less than the Preston baseline model: e30qspm = e30qspf = e50qspm = e50qspf = l70m = l70sqf = l70qspm = 0 with the five age knots related to e30qspm, e30qspf, e50qspm, e50qspf, and l70qspm obsolete. In other words: for the fit of the data for cardiovascular diseases one single joint baseline model was used, namely Eq. (A1) with e30qspm = e30qspf = e50qspm = e50qspf = l70m = l70sqf = l70qspm = 0. Furthermore, it was found that for cardiovascular diseases there was no statistically significant city effect: instead of the four baseline parameters cmH, cfH, cmN, and cfN applied for CVD (Table S1) only two remain for cardiovascular diseases: cm and cf (Table S2). The streamlined baseline model for cardiovascular diseases therefore has 15 (29 -12 -2) model parameters (see Table S2 in the Online Resource). In addition, it was found that the model fit significantly improved when the age knots at 40 and 70 in lage40qsp and lage70qsp, respectively, were allowed to be free (Table S2).

Dose-effect modifiers
Three dose-effect modifiers were implemented into the various risk models. For an excess relative risk (ERR) model, the following form was is any of the dose-responses from Fig. 1. Here, dem 1 , dem 2 , and dem 3 are three adjustable parameters related to the three dose-effect modifiers sex, age at exposure, and age attained. The naming conventions for e30 and lage70 are provided after Eq. (A1).
When fitting the mortality data for cardiovascular diseases we found for the EAR-LNT model and the EAR-quadratic model that age was a statistically significant dose effect modifier with the related parameter dem3 = 5.1 (Table S2). That gives the factor exp(5.1 × ln(a/70)). At an attained age of a = 56 years, the mean age attained of all individuals registered within the data set for cardiovascular diseases, we therefore have h = h 0 + 0.29 × ear(D). For a = 70 years we simply have h = h 0 + ear(D). Consequently, for lower ages the EAR (shown in Fig. 3 for a = 70 years) calculated with the EAR-LNT model is strongly decreased. This explains the seemingly inconsistent shape of the EAR for the EAR-LNT model in For all other model fits no significant effect modifiers were found (Tables S1 and S2).

Poisson regression
The MECAN software (Kaiser 2010) uses the maximum likelihood method to estimate the values of the adjustable model parameters by fitting the model to the data. Because maximizing the likelihood is equivalent to minimizing the −ln(Likelihood), the latter problem, which is numerically better tractable, is solved in MECAN to find the best model solution. For grouped person-year data such as the grouped LSS data, the likelihood corresponding to a Poisson model is used: where n i is the observed number of cases (i.e. the number of fatal CVDs or cardiovascular diseases) in group i and Λ i is the calculated (expected) number of cases in group i. The deviance is defined as dev := −2 × ln(MaxLikelihood).

ERR and EAR calculated from different models
For CVD, we found that two different ERR models are preferable (Table 1). The general form of an ERR model is where RR is the relative risk. Consequently, when we calculate the EAR from an ERR model we get For CVD, the baseline model, h 0 , depends on city and sex via model parameters cmH, cfH, cmN, and cfN (Table S1). Therefore, the EAR-values for CVD (Table 2) also depend on city and sex: they are only valid for males from Hiroshima, as stated in the Results section of the main text. When the ERR is calculated from an ERR model, then only the shape of the dose-responses related to the excess risk from radiation enters the risk estimate. Fig. 2 is therefore valid for males and females from both cities.
For cardiovascular diseases, it was found that three different EAR models are preferable ( found that h 0 is dependent on sex (but not city) via model parameters cm and cf (Table S2). Therefore, the ERR-values for cardiovascular diseases 6 ( Table 3) also depend on sex: they are only valid for males. Related to Fig. 3 it can be said that the EAR calculated from an EAR model only depends on the shapes of the preferable dose-responses for the excess risk from radiation. Therefore, Fig. 3 is valid for males and females from both cities. Table 2 In the previous section it has been derived that for CVD the EAR can be calculated from an ERR model as follows: EAR = h 0 × ERR. The cityaveraged EAR-values for males, <EAR> m , can be calculated as follows:

Derivation of city-averaged EAR-values to be used in
Here, w H = PY H /PY tot is the number of person years in Hiroshima divided by the total number of person years in the data set; w N = PY N /PY tot is the number of person years in Nagasaki divided by the total number of person years in the data set, and h 0,mH is the streamlined baseline hazard for males in Hiroshima, i.e. Eq. (A1) with the numerical values for the model parameters taken from Table S1 using e30sqm = e30qspm = e50qspm = e50qspf = l70f = 0 (refer to section 2 above) and with cfH = cmN = cfN = 0. An analogous definition holds for h 0,mN . For the ERR-LNT model we therefore have h 0,mH = exp{−9.57 + 0.44 × e30 × lage70 + 0.504 × e30 + 2.07 × e30 + 0.522 × e30sq − 0.581 × e30qsp − 11.32 × lage70 − 14.3 × lage70sq − 5.6 × lage70sq + 16.33 × lage40qsp + 8.9 × lage40qsp − 120 × lage70qsp − 134 × lage70qsp} = exp{−9.57} × exp{rest} where rest stands for all other terms in the exponential function. 7 The right hand side of Eq. (A2) can be expanded as follows: Considering Eq. (A2) and using EAR = h 0 × ERR within Eq. (A4), one obtains The term EAR mH stands for the EAR-values for males in Hiroshima as given in  Table 2 the city-averaged EAR-values (for example: for the MMI the EAR-value at 1 Gy for males in Hiroshima with an age attained of 70 years, exposed at an age of 30 years, is 6.6 per 10 4 PY; therefore, the city-averaged EAR-value for males is 6.6 × 1.1 = 7.3 per 10 4 PY. This risk prediction holds for males in Hiroshima and Nagasaki).

Neglection of categorical and Gompertz models for MMI
It is noted that the categorical model (#11 in Fig. 1) is a non-nested model like those summarized in Table 1. It was, however, not used for MMI because of its very small contributions to the AIC-weights. The categorical model fit to the data for CVD yielded ∆AIC = 9.85 (with dev = 3565.9 and Npar = 28). Using Eq.
The fit of the data for cardiovascular diseases with the Gompertz function led to p 1 = 0.035; therefore this model was not used for MMI either.

Table S1
Results from fitting the final three non-nested models to the joint mortality data for CVD in males and females: Best estimates and Wald-type standard errors (in parentheses) for the model parameters of the three preferable model fits for CVD. The numbers in brackets refer to the eleven dose-responses depicted in Fig. 1 in the main text. Parameters 1 to 21 are the baseline parameters, parameters 22 and 23 are radiation related parameters, i.e. they refer to the risk models #1, #2, and #6 depicted in Fig. 1

Table S3
Final deviances for CVD in males analyzed with ERR-step model: Final deviances obtained by forward calculations using the ERR-step model (with D th = 0.62 Gy) in combination with the mortality data for CVD in males. For the calculations the best estimates from Table S1 were applied. Data were grouped into four dose categories and five age categories. As a comparison, the deviances are also shown for Preston's ERR-LNT model.