Methodological improvements to meta-analysis of low dose rate studies and derivation of Dose and Dose-rate Effectiveness Factors

Abstract

Epidemiological studies of cancer rates associated with external and internal exposure to ionizing radiation have been subject to extensive reviews by various scientific bodies. It has long been assumed that radiation-induced cancer risks at low doses or low dose rates are lower (per unit dose) than those at higher doses and dose rates. Based on a mixture of experimental and epidemiologic evidence the International Commission on Radiological Protection recommended the use of a Dose and Dose-Rate Effectiveness Factor for purposes of radiological protection to reduce solid cancer risks obtained from moderate-to-high acute dose studies (e.g. those derived from the Japanese atomic bomb survivors) when applied to low dose or low dose-rate exposures. In the last few years there have been a number of attempts at assessing the effect of extrapolation of dose rate via direct comparison of observed risks in low dose rate occupational studies and appropriately age/sex adjusted analyses of the Japanese atomic bomb survivors. The usual approach is to consider the ratio of the excess relative risks in the two studies. This can be estimated using standard meta-analysis with inverse weighting of ratios of relative risks using variances derived via the delta method. In this paper certain potential statistical problems in the ratio of estimated excess relative risks for low dose rate studies to the excess relative risk in the Japanese atomic bomb survivors are discussed, specifically the absence of a well-defined mean and the theoretically unbounded variance of this ratio. A slightly different method of meta-analysis for estimating uncertainties of these ratios is proposed, motivated by Fieller’s theorem, which leads to slightly different central estimates and confidence intervals for the dose rate effectiveness factor. However, given the uncertainties in the data, the differences in mean values and uncertainties from the dose rate effectiveness factor estimated using delta-method-based meta-analysis are not substantial, generally less than 70%.

Introduction

Epidemiological studies of cancer rates associated with external and internal exposure to ionizing radiation have been subject to extensive reviews by various scientific bodies (Committee to Assess Health Risks from Exposure to Low Levels of Ionizing Radiation 2006, International Commission on Radiological Protection (ICRP) 2007, United Nations Scientific Committee on the Effects of Atomic Radiation (UNSCEAR) 2008). In particular, the United Nations Scientific Committee on the Effects of Atomic Radiation (UNSCEAR) 2006 Report assessed cancer incidence (Preston, Ron et al. 2007) and mortality data (Preston, Pierce et al. 2004) relating to the Japanese atomic bomb survivor Life Span Study (LSS) cohort using the (then current) Dosimetry System 2002 (DS02) dose estimates, as well as many other studies of persons exposed occupationally, environmentally and medically (therapeutically or diagnostically) (United Nations Scientific Committee on the Effects of Atomic Radiation (UNSCEAR) 2008).

It has long been assumed that radiation-induced cancer risks at low doses or low dose rates are lower (per unit dose) than those at higher doses and dose rates (International Commission on Radiological Protection (ICRP) 2007, United Nations Scientific Committee on the Effects of Atomic Radiation (UNSCEAR) 2008). Based on a mixture of experimental and epidemiologic evidence the International Commission on Radiological Protection (ICRP) recommended that for radiological protection purposes a Dose and Dose-Rate Effectiveness Factor (DDREF) of 2 be applied to reduce solid cancer risks obtained from moderate-to-high acute dose studies (e.g. those derived from the Japanese atomic bomb survivors) to those relating to low dose or low dose-rate exposures (International Commission on Radiological Protection (ICRP) 2007). Recently the ICRP Task Group 91 (TG 91) has re-examined the evidence for such ameliorating effects of lower dose or lower dose-rate exposures. Determining the effect of curvature in the dose response and its impact on low dose effects is separate from the effect that may be produced by amelioration of dose rate. The so-called dose-rate extrapolation factor (DREF) is the ratio of dose response slopes (i.e. excess risk per unit dose) at high dose rate to those at low dose rate, so that a DREF > 1 would imply that high dose rate risks per unit dose are greater than those at low dose rate, while a DREF < 1 would imply that high dose rate risks are less than those at low dose rate. The so-called low dose extrapolation factor (LDEF) is the ratio of the high dose slope to the low dose slope (e.g. as derived via fitting a linear-quadratic model) fitted to a specific dose range. LDEF is used to derive the degree of over- (if LDEF > 1) or under-estimation (if LDEF < 1) of low dose risk by linear extrapolation from effects at higher doses. There has been much recent work associated with the work of ICRP TG91 that assesses LDEF and DREF in experimental radiobiologic data (Haley, Paunesku et al. 2015, Tran and Little 2017, Zander, Paunesku et al. 2020) and also deriving estimates of LDEF from epidemiologic studies (Little, Pawel et al. 2020). The measure of DDREF used by ICRP to some extent combines these two different quantities, DREF and LDEF.

In the last few years there have been a number of attempts at assessing the effect of extrapolation of dose rate (i.e., DREF) via direct comparison of risks in low dose rate occupational studies and appropriately age/sex adjusted analyses of the Japanese atomic bomb survivors. In particular this method has been employed in the studies of Jacob et al (Jacob, Rühm et al. 2009), Shore et al (Shore, Walsh et al. 2017) and Kocher et al (Kocher, Apostoaei et al. 2018). As is discussed in the present paper there are certain statistical aspects that require attention in the ratio of estimated excess relative risks (ERR) for low dose rate studies, $ER{R}_{LDR}$, to the ERR in the Japanese atomic bomb survivors, $ER{R}_{LSS}$, that is employed in the studies of Jacob et al (Jacob, Rühm et al. 2009), Shore et al (Shore, Walsh et al. 2017) and Hoel (Hoel 2018). Kocher et al (Kocher, Apostoaei et al. 2018) estimate the inverse ratio, the ratio of estimated ERR for the LSS, $ER{R}_{LSS}$, to that of the low dose rate studies, $ER{R}_{LDR}$, and as is discussed below this analysis does not involve the particular aspects of the other three studies, because of truncation of the distribution of the ratio. Jacob et al (Jacob, Rühm et al. 2009), Shore et al (Shore, Walsh et al. 2017) and Hoel (Hoel 2018) therefore estimate 1/DREF, whereas Kocher et al (Kocher, Apostoaei et al. 2018) estimate DREF. While the approach of Kocher et al (Kocher, Apostoaei et al. 2018) more directly estimates the quantity of interest, a simple inversion of the estimates of Jacob et al (Jacob, Rühm et al. 2009), Shore et al (Shore, Walsh et al. 2017) and Hoel (Hoel 2018) also yields DREF. Related to the issues of ratio estimates in these studies there are also issues in combining multiple such sets of ratios in a meta-analysis.

As noted in the Supplement the fundamental issue is that the ratio $X/Y$ of two non-trivial (i.e. with non-zero variance) normal random variables has unbounded variance $(\mathrm{var}[X/Y]=\infty )$; indeed, as is shown there, it also fails to be absolutely integrable $(E[\left|X/Y\right|]=\infty )$ , so that the mean is also undefined. Since the standard types of meta-analysis rely on the variance of the estimators to optimally weight the estimates (DerSimonian and Laird 1986, Viechtbauer 2005) these methods will break down. [It is a consequence of the Gauss-Markov theorem that inverse variance weighting is known to be best linear unbiased (BLUE), and for this reason this weighting scheme is often but not always used; for example Kocher et al (Kocher, Apostoaei et al. 2018) used a non-standard scheme to weight individual studies, which was controversial (Kocher, Apostoaei et al. 2019, Wakeford, Azizova et al. 2019).] However confidence intervals (CIs) can still be constructed and will often, although not invariably, be finite (Fieller 1940, Fieller 1954). The ratio method could be saved by assuming the error distributions, in particular those for the denominator random variable, were taken from probability distributions that were in some sense bounded away from 0. Alternatively, the problem can be circumvented by truncation of the DREF distribution, as was done by Kocher et al (Kocher, Apostoaei et al. 2018) by removing all values <0.2 or >20. Arguably Kocher et al (Kocher, Apostoaei et al. 2018) are imposing too stringent conditions on the DREF in doing this, since in many cases the denominator random variable (i.e., the risk in the respective low dose rate study) can clearly not be assumed to be bounded away from 0 in the way that would be implied by this (for example the CI may include 0). Arguably the restriction of range used by Kocher et al (Kocher, Apostoaei et al. 2018) is arbitrary (although possibly not unreasonable), and the results obtained will likely depend on the precise limits used. The apparent lack of problems that could arise from taking even a large number of Monte Carlo simulations when one does not bound the distributions in this way, for example when using Normal errors may be misleading, as it would be expected (from the theoretical results given in the Supplement) that as the number of Monte Carlo samples increases then the sample variances and the mean of the absolute values will become increasingly large, becoming unbounded as $n\to \infty$.

There are additional issues relating to the correlations that may arise in the denominators of the two or more different ERR ratios, $ER{R}_{LDR,i}/ER{R}_{LSS,i}$, because the denominators may be based on the same, or at least overlapping (if mortality and incidence are being considered) versions of the LSS. As such the standard BLUE or 1-step random effects estimate (DerSimonian and Laird 1986) will not have the usual indicated distributions. In principle this problem could be overcome by taking account of the correlations between component ERR ratios, $ER{R}_{LDR,i}/ER{R}_{LSS,i}$, although this does not appear to have been done by Jacob et al (Jacob, Rühm et al. 2009), Shore et al (Shore, Walsh et al. 2017), Hoel (Hoel 2018) or Kocher et al (Kocher, Apostoaei et al. 2018). It is judged here that there may be a fairly simple solution to both of these issues, motivated by Fieller’s theorem (Fieller 1940, Fieller 1954) and related findings of Chakravarti (Chakravarti 1971). In the present paper the calculations needed are outlined, and a few comparisons of the results of this new method with the standard meta-analysis based on the delta method are provided.

Methods

Full details of the methods used here are given in an electronic supplement. Assume that there is a series of low dose rate studies, and for each study the $ER{R}_{LDR,i}$ is derived, together with some estimate of the associated variance, ${\sigma }_{LDR,i}{}^2$, and the $ER{R}_{LSS,i}$ is also derived, together with some estimate of the associated variance, ${\sigma }_{LSS,i}{}^2$, in the matched or adjusted LSS data. [In many cases one only has the 95% CI for $ER{R}_{LDR,i}$, say, $[ER{R}_{LDR,i,0.025},ER{R}_{LDR,i,0.975}]$, and likewise for $ER{R}_{LSS,i}$, say $[ER{R}_{LSS,i,0.025},ER{R}_{LSS,i,0.975}]$ in which case approximate variance estimates can be derived via ${\sigma }_{LDR,i}{}^2={((ER{R}_{LDR,i,0.975}-ER{R}_{LDR,i,0.025})/2N{(0,1)}_{0.975})}^2{\sigma }_{LSS,i}{}^2$ and ${\sigma }_{LSS,i}{}^2={((ER{R}_{LSS,i,0.975}-ER{R}_{LSS,i,0.025})/2N{(0,1)}_{0.975})}^2$, where $N{(0,1)}_{0.975}\approx 1.96$ is the 97.5% centile point of the standard normal $N(0,1)$ random variable, with mean 0, variance 1.]. The inverse-variance weighted ERR for the low dose rate studies can then be computed:

$$ER{R}_{LDR,tot}=\sum _{i=1}^k{w}_{LDR,i}ER{R}_{LDR,i}=\frac{\sum _{i=1}^k(1/{\sigma }_{LDR,i}{}^2)ER{R}_{LDR,i}}{\sum _{i=1}^k(1/{\sigma }_{LDR,i}{}^2)}$$ (1)

and similarly for the LSS:

$$ER{R}_{LSS,tot}=\sum _{i=1}^k{w}_{LSS,i}ER{R}_{LSS,i}=\frac{\sum _{i=1}^k(1/{\sigma }_{LSS,i}{}^2)ER{R}_{LSS,i}}{\sum _{i=1}^k(1/{\sigma }_{LSS,i}{}^2)}$$ (2)

This leads to the ratio estimate of 1/DREF, specifically:

$$ER{R}_{LDR,tot}/ER{R}_{LSS,tot}$$ (3)

From Eqs. (1) and (2) the aggregate measures of variance for the low dose rate studies can be computed:

$$Va{r}_{LDR,tot}=\frac{1}{\sum _{i=1}^k(1/{\sigma }_{LDR,i}{}^2)}$$ (4)

and for the LSS:

$$Va{r}_{LSS,tot}=\frac{1}{\sum _{i=1}^k(1/{\sigma }_{LSS,i}{}^2)}$$ (5)

However, more refined estimates of the covariance of the LSS variance could also be derived, taking account of the correlations between particular estimates. In particular the aggregate variance could be readily estimated using bootstrap methods. Having derived these variance estimates one can slightly adapt the uncorrelated sample formula (S5) from the Supplement to obtain the 100(1−δ)% CI for the ratio estimate $ER{R}_{LDR,tot}/ER{R}_{LSS,tot}$ as:

$$\frac{1}{1-\mathrm{\Phi }}\left[\frac{ER{R}_{LDR,tot}}{ER{R}_{LSS,tot}}\pm \frac{N{(0,1)}_{1-\delta /2}}{ER{R}_{LSS,tot}}\sqrt{\frac{ER{R}_{LDR,tot}{}^2}{ER{R}_{LSS,tot}{}^2}Va{r}_{LSS,tot}+Va{r}_{LDR,tot}(1-\mathrm{\Phi })}\right]$$ (6)

where:

$$\mathrm{\Phi }=\frac{Va{r}_{LSS,tot}N{(0,1)}_{1-\delta /2}{}^2}{ER{R}_{LSS,tot}{}^2}$$ (7)

and where $N{(0,1)}_{1-\delta /2}$ is the $1-\delta /2$ centile of a $N(0,1)$ distribution with mean 0 and variance 1; so for example for $\delta =0.05$ (i.e., for 95% CI) $\mathrm{\Phi }=\frac{Va{r}_{LSS,tot}{1.96}^2}{ER{R}_{LSS,tot}{}^2}$. Eqs. (6) and (7) yield confidence intervals $[C{I}_{LDR/LSS,Fieller,\delta /2},C{I}_{LDR/LSS,Fieller,1-\delta /2}]$.

Meta-analysis using the delta method would estimate the aggregate estimate of $Y/X$ via:

$$\begin{gathered} ER{R}_{LDR/LSS}=\frac{\sum _{i=1}^k{w}_{LDR/LSS,i}ER{R}_{LDR,i}/ER{R}_{LSS,i}}{\sum _{i=1}^k{w}_{LDR/LSS,i}} \\ =\frac{\sum _{i=1}^k(1/Var[ER{R}_{LDR,i}/ER{R}_{LSS,i}])ER{R}_{LDR,i}/ER{R}_{LSS,i}}{\sum _{i=1}^k(1/Var[ER{R}_{LDR,i}/ER{R}_{LSS,i}])} \end{gathered}$$ (8)

It should be noted that the statistic $ER{R}_{LDR/LSS}$ will generally (for k > 1) be different from the ratio which use of Fieller’s theorem and associated derivations suggests, as given by Eq. (3). The variance associated with $ER{R}_{LDR/LSS}$ is formally:

$$Var[ER{R}_{LDR/LSS}]=\frac{1}{\sum _{i=1}^k(1/Var[ER{R}_{LDR,i}/ER{R}_{LSS,i}])}$$ (9)

The variances of the ratios via the delta method can be estimated as:

$$Var[ER{R}_{LDR,i}/ER{R}_{LSS,i}]=\frac{Var[ER{R}_{LDR,i}]}{ER{R}_{LSS,i}{}^2}+\frac{Var[ER{R}_{LSS,i}]ER{R}_{LDR,i}{}^2}{ER{R}_{LSS,i}{}^4}$$ (10)

Finally the 95% CI on $ER{R}_{LDR/LSS}$ is estimated via:

$$ER{R}_{LDR/LSS}\pm 1.96Var{[ER{R}_{LDR/LSS}]}^{0.5}$$ (11)

resulting in confidence intervals $[C{I}_{LDR/LSS,delta,0.025},C{I}_{LDR/LSS,delta,0.975}]$. The difference made by calculation of central estimates and CI using either Eqs. (3) and (6) or Eqs. (8) and (11) are illustrated with a few examples, using data taken from Table 2 of the paper of Shore et al (Shore, Walsh et al. 2017). In all cases variances of the ERR were derived from the 90% or 95% CI (${\mathit{ERR}}_{\mathit{low}},{\mathit{ERR}}_{\mathit{high}}$) given by Shore et al (Shore, Walsh et al. 2017) by taking $Var={[(ER{R}_{high}-ER{R}_{low})/(2*N{(0,1)}_{0.975})]}^2$ if the CI were 95%, or: $Var={[(ER{R}_{high}-ER{R}_{low})/(2*N{(0,1)}_{0.95})]}^2$ if the CI were 90%. It should be noted that there are slight deviations from the central estimates of the ERR ratio from those given by Shore et al (Shore, Walsh et al. 2017), resulting from the 2 decimal place rounding in the published estimates. We are particularly interested in the percentage change in the width of the lower and upper confidence intervals:

$$100\left(\frac{[ER{R}_{LDR,i}/ER{R}_{LSS,i}-C{I}_{LDR/LSS,Fieller,0.025}]}{[ER{R}_{LDR,i}/ER{R}_{LSS,i}-C{I}_{LDR/LSS,delta,0.025}]}-1\right)$$ (12)

$$100\left(\frac{[C{I}_{LDR/LSS,Fieller,0.975}-ER{R}_{LDR,i}/ER{R}_{LSS,i}]}{[C{I}_{LDR/LSS,delta,0.975}-ER{R}_{LDR,i}/ER{R}_{LSS,i}]}-1\right)$$ (13)

and for meta-analyses the related changes also in the mean.

Table 2.

Meta-analysis of $ER{R}_{LDR}/ER{R}_{LSS}$ using delta-method based confidence intervals (CIs) and Fieller-based CIs (using data taken in part from Shore et al (Shore, Walsh et al. 2017))

Studies used	Delta method	Fieller method	% difference of mean [Fieller vs delta]	% difference in length of lower CI / % difference in length of upper CI [Fieller vs delta method]
	$ER{R}_{LDR}/ER{R}_{LSS}$ (+95% CI)	$ER{R}_{LDR}/ER{R}_{LSS}$ (+95% CI)
All studies (using all Mayak data)	0.33 (0.13, 0.53)	0.43 (0.19, 0.67)	29	19 / 22
All mortality studies (using all Mayak data)	0.35 (0.14, 0.57)	0.45 (0.20, 0.71)	28	17 / 20
All studies (excluding Mayak data)	0.54 (0.09, 0.99)	0.91 (0.28, 1.56)	69	41 / 44
All mortality studies (excluding Mayak data)	1.12 (0.40, 1.84)	1.25 (0.49, 2.04)	12	6 / 9

Results

Table 1 shows that the CIs generated by use of Fieller’s theorem have generally wider CIs, with the upper CI in particular often (but not always) somewhat wider than that given by the delta method, in some cases up to 40% wider. For example the estimate of $ER{R}_{LDR}/ER{R}_{LSS}$ implied by the delta method for the INWORKS study is 1.34 (95% CI 0.22, 2.47), whereas that implied by the Fieller method is 1.34 (0.30, 2.70), so that the length of the interval between the upper 95% CI and the central estimate is 21% larger. The estimate of $ER{R}_{LDR}/ER{R}_{LSS}$ implied by the delta method for the Chernobyl liquidator study is 2.52 (95% CI −0.44, 5.48), whereas that implied by the Fieller method is 2.52 (95% CI −0.19, 6.64), so that the length of the interval between the upper 95% CI and the central estimate is 39% larger (Table 1). Table 2 shows that the same thing is true of the ratios of ERR derived via meta-analysis, and as predicted in the Methods section the mean also shifts. In some cases the length of the interval between the upper 95% CI and the central estimate is up to 45% wider. For example the estimate of $ER{R}_{LDR}/ER{R}_{LSS}$ implied by the delta method for all studies is 0.33 (95% CI 0.13, 0.53), whereas that implied by the Fieller method is 0.43 (0.19, 0.67) (Table 2), so that the mean is 29% larger and the length of the interval between the upper 95% CI and the central estimate is 22% bigger. The estimate of $ER{R}_{LDR}/ER{R}_{LSS}$ implied by the delta method for all studies excluding the Mayak data is 0.54 (95% CI 0.09,0.99), whereas that implied by the Fieller method is 0.91 (0.28, 1.56) (Table 2), so that the mean is 69% larger and the length of the interval between the upper 95% CI and the central estimate is 44% larger.

Table 1.

Delta-method based confidence intervals (CIs) and Fieller-based CIs of ratio of $ER{R}_{LDR}/ER{R}_{LSS}$ (using data taken in part from Table 2 of Shore et al (Shore, Walsh et al. 2017))

	Delta method	Fieller method	% difference in length of lower CI / % difference in length of upper CI [Fieller vs delta method]
Occupational studies, mortality	$ER{R}_{LDR}/ER{R}_{LSS}$ (+95% CI)	$ER{R}_{LDR}/ER{R}_{LSS}$ (+95% CI)
INWORKS	1.34 (0.22, 2.47)	1.34 (0.30, 2.70)	−7 / 21
Japanese nuclear workers	0.43 (−3.39, 4.26)	0.43 (−4.14, 5.52)	20 / 33
Chernobyl liquidators	2.52 (−0.44, 5.48)	2.52 (−0.19, 6.64)	−8 / 39
Mayak (all workers)	0.28 (0.05, 0.50)	0.28 (0.07, 0.55)	−7 / 18
Mayak (little plutonium exposure)	0.47 (−0.09, 1.02)	0.47 (−0.07, 1.10)	−3 / 14
Rocketdyne	−0.87 (−8.48, 6.74)	−0.87 (−8.97, 7.05)	6 / 4
Uranium millers	0.72 (−6.86, 8.31)	0.72 (−7.19, 8.79)	4 / 6
US nuclear power plant workers	1.13 (−6.27, 8.54)	1.13 (−6.80, 9.56)	7 / 14
Canadian nuclear workers	−3.33 (−13.42, 6.75)	−3.33 (−15.47, 7.14)	20 / 4
Port Hope	0.44 (−2.74, 3.63)	0.44 (−2.86, 3.84)	4 / 7
Sweden nuclear workers	−1.71 (−26.49, 23.08)	−1.71 (−29.40, 25.25)	12 / 9
German nuclear power plant workers	−3.06 (−12.48, 6.36)	−3.06 (−14.58, 6.76)	22 / 4
Rocky Flats	−5.82 (−15.32, 3.68)	−5.82 (−18.70, 3.39)	36 / −3
Belgium nuclear workers	−1.69 (−24.95, 21.58)	−1.69 (−28.04, 23.85)	13 / 10
Finland nuclear workers	543.75 (−2797.78, 3885.28)	543.75 (−3058.22, 4417.03)	8 / 16
Spain nuclear workers	3.00 (−42.23, 48.23)	3.00 (−47.03, 54.60)	11 / 14
Australia nuclear workers	35.26 (−161.21, 231.73)	35.26 (−171.99, 255.32)	5 / 12
Slovakia nuclear workers	23.75 (−96.85, 144.35)	23.75 (−108.16, 172.04)	9 / 23
Environmental studies, mortality
Techa river	1.15 (−0.03, 2.33)	1.15 (−0.01, 2.40)	−2 / 6
Yangjiang HBRA	0.39 (−4.62, 5.40)	0.39 (−4.82, 5.66)	4 / 5
Incidence data only
Kerala HBRA	−0.38 (−1.92, 1.15)	−0.38 (−2.06, 1.20)	9 / 3
Taiwan dwellings	0.24 (−0.39, 0.87)	0.24 (−0.39, 0.91)	1 / 7
Korea NPP	3.68 (−6.17, 13.53)	3.68 (−6.38, 15.09)	2 / 16

Discussion

For the present study, a variant method has been developed for estimating the ratio and associated CIs of ERRs from low dose rate studies to those from the LSS (or other high dose rate studies) and results are compared with those predicted by the more standard delta method for the studies considered by Shore et al (Shore, Walsh et al. 2017). The results obtained imply that the CIs generated using the methods suggested by Fieller’s theorem are generally (but not invariably) somewhat wider than those predicted using the delta method, with the upper CIs in particular often (but not always) somewhat wider than those given by the delta method. When the Fieller method is applied to meta-analysis of results from several studies, the same is also true, and the mean can also shift. However, given the uncertainties in the data the differences in mean values and uncertainties are not substantial, generally <70% (Tables 1, 2).

As noted in the Supplement the fundamental issue is that the ratio of two random variables $X/Y$ in which Y has non-trivial density in a neighbourhood of 0 will have unbounded variance, indeed fails to be absolutely integrable, so that the mean is also undefined. In particular, it should be noted that if for a particular case the CIs in the LSS were “narrow” (in some sense), it would not guarantee that the ratio $ER{R}_{LDR}/ER{R}_{LSS}$ would not misbehave, and the fact that a blow up of the variance (or mean) was not observed when using 1000 Monte Carlo samples of $ER{R}_{LDR}$ and $ER{R}_{LSS}$, as Jacob et al (Jacob, Rühm et al. 2009) did, does not prove that this will not happen. Indeed, it is clear from the paper of Jacob et al (Jacob, Rühm et al. 2009) that because the authors appear to be simulating numerator and denominator from two sampled normal distributions that the issue identified in the Supplement will certainly apply to estimates of variance derived from the Monte Carlo samples, were the samples to be made big enough.

Other solutions to the issues laid out here could be envisaged, for example via restating the issue as one of regression, treating the LSS estimates as the independent variables, and regarding the corresponding estimates from LDR studies as the dependent variables. The inverse of the DREF is the slope of the resulting regression, if the intercept were to be fixed at 0. The correlation in the uncertainties in the LSS estimates could be treated using well established techniques for adjusting for covariate measurement error, taking account of shared error, which would be considerable, possibly using techniques recently developed by Stram et al (Stram, Preston et al. 2015, Zhang, Preston et al. 2017). Even without taking account of correlations, treating the LSS estimates as taken from a population with overall variance ${\sigma }^2$ (for example as given by the variance over all endpoints of all estimates for ERR in a given age/sex subgroup in the LSS) and measured with classical error with variance ${\omega }^2$ (for example as given by the error in the particular estimate), which to a first approximation is likely be the case, then the standard regression calibration adjustments for classical error could be applied, implying that the slopes should be corrected by a factor $1+{\omega }^2/{\sigma }^2$ (Thomas, Stram et al. 1993, Carroll, Ruppert et al. 2006). If one were only interested in testing the equality of two sets of ERR, for example to test whether the DDREF was 1, another obvious approach is to consider not the ratio of the estimates but the difference, in the way that was done by Hamra et al (Hamra, MacLehose et al. 2014) in the context of two different carcinogenic exposures, but readily applied to the present situation. Arguably the difference in means may be a more natural measure. However, it would be substantially different between different cancer sites, and would also not be immediately applicable to the current system of radiological protection in the way that DREF is.

The method applied here fundamentally assumes normality of the component random variables, the $ER{R}_{LDR,i}$ and $ER{R}_{LSS,i}$. Often this may not be the case, in particular if these variables have CIs that are not symmetric about the central estimate. One solution in that case would be to first approximately symmetrise them via an appropriate transformation, $ER{R}_{LDR,i}{}^T=f(ER{R}_{LDR,i})$ and $ER{R}_{LDR,i}{}^T=f(ER{R}_{LSS,i})$. Having computed CI for the ratio $ER{R}_{LDR,i}{}^T/ER{R}_{LSS,i}{}^T$ the functional inverse of the transformation $f^{-1}(.)$ would then be applied to derive CI for the ratio $ER{R}_{LDR,i}/ER{R}_{LSS,i}$ on the original scale. The function $f(.)$ would have to be chosen so that $f(xy)=f(x)f(y)$; the only such continuous functions are the power transformations $f(x)=x^p$.

As noted in the Introduction, the issue of DREF is just one part of the ICRP concept of DDREF (International Commission on Radiological Protection (ICRP) 2007). The present results (Table 2) are broadly consistent with those of Shore et al (Shore, Walsh et al. 2017) in implying that a DREF of 2 is consistent with the epidemiologic data. However, this is only the case if the Mayak data is included in the meta-analysis; if the Mayak data is excluded a DREF of about 1 is suggested (Table 2). Again these results mirror the findings of Shore et al (Shore, Walsh et al. 2017). The other major part of DDREF is LDEF, relating to the extrapolation of high dose to low dose. The marked estimates of dose-response curvature for many, although not all, malignant endpoints in the analysis conducted by Little et al (Little, Pawel et al. 2020) of the latest LSS mortality data suggest that there is a substantial ameliorating effect of low-dose compared with high-dose exposure for these endpoints, implying that the LDEF is greater than 1. There is much (quite old) experimental data yielding information on LDEF, and the somewhat related idea of DREF which is the focus of the paper (Rühm, Woloschak et al. 2015). The findings on LDEF of Little et al (Little, Pawel et al. 2020) are consistent with this older body of data (Rühm, Woloschak et al. 2015), also moderately consistent with results of recent re-analysis of various large bodies of experimental animal data (Haley, Paunesku et al. 2015, Tran and Little 2017, Zander, Paunesku et al. 2020). The LDEF of about 2 found by Little et al (Little, Pawel et al. 2020) for most malignant endpoints is consistent with the DDREF of 2 adopted by ICRP (International Commission on Radiological Protection (ICRP) 2007). The findings of the present paper and those of Shore et al (Shore, Walsh et al. 2017) and Hoel (Hoel 2018) suggest that there may also be a DREF of 2; however, this is critically dependent on the Mayak data, so that if this dataset is removed from the analysis the ratio of ERRs in low dose rate studies and LSS are consistent with a DREF of 1 (Table 2).

Although not the focus of the present paper, the same meta-analytic technique used to estimate DREF or its inverse and associated uncertainties could be applied to estimate relative biological effectiveness (RBE), although to the best of our knowledge this has never been done.

Availability of data and material

The paper has no data, so this is judged not applicable.