Analysis of cataract in relationship to occupational radiation dose accounting for dosimetric uncertainties in a cohort of US radiologic technologists

Abstract

Cataract is one of the major morbidities in the US population and it has long been appreciated that high and acutely delivered radiation doses of 1 Gy or more can induce cataract. Some more recent studies, in particular of the US Radiologic Technologists, have suggested that cataract may be induced by much lower chronically delivered doses of ionizing radiation. It is well recognized that dosimetric measurement error can alter substantially the shape of radiation dose-response relationship and hence the derived study risk estimates, and can also inflate the variance of the estimates. In the present study we evaluate the impact of uncertainties in eye-lens absorbed doses on the estimated risk of cataract in the US Radiologic Technologists Monte Carlo Dosimetry System, using both absolute and relative risk models. Among 11,345 cases we show that the inflation in the standard error for the excess relative risk (ERR) is generally modest, at most about 20% of the unadjusted standard error, depending on the model used for the baseline risk. The largest adjustment results from use of relative risk models, so that the ERR/Gy and its 95% confidence intervals (CI) change from 1.085 (0.645, 1.525) to 1.085 (0.558, 1.612) after adjustment. However, the inflation in the standard error of the excess absolute risk (EAR) coefficient is generally minimal, at most about 0.04% of the standard error.

Introduction

Cataract is one of the major morbidities in the US population, with overall prevalence of ~24% above age 40 years but increasing to ~71% above age 75 (1). High radiation doses of 1 Gy or more can induce cataract (2), and accumulating evidence from follow-up studies of the Japanese atomic bomb survivors (3-5) and worker populations (6-12) suggests that cataracts may be induced by cumulative lower doses, of the order of 100-250 mGy. A recent study of cataract in the US radiologic technologists (USRT) demonstrated a significant cataract risk associated with cumulative ionizing radiation exposure below 100 mGy (13, 14). Low and moderate dose radiation risk is of general public health concern. Use of diagnostic computed tomography (CT) in the US is substantial with about 18% of visits to the emergency room associated with use of CT (15). Among persons receiving one or more CTs it is estimated that about 15% will receive cumulative effective radiation doses of 100 mGy or more (16).

The problem of allowing for uncertainties in radiation dose assessments when estimating dose-response relationships has recently been the subject of much research (17). Typically errors, that is to say the difference between the true and assigned values, are assumed to be of one of two types, classical or Berkson. Classical errors, in which the measured (estimated) doses are assumed to be distributed around the true dose, with the errors independent of true dose, generally result in a downward bias of the dose-response parameter (17). Berkson errors, in which the true dose is randomly distributed around a measured or computed dose estimate, with the errors independent of measured dose, do not result in biased estimates of the dose-response parameter for linear models, although for non-linear models that is not the case (17). Biases can also result when errors in a given variable (whether of Berkson or classical form) are correlated with errors in other variables (17, 18). Both types of error can be either multiplicative or additive with respect to the underlying true dose (for classical error) or measured or computed dose (for Berkson error). In practice, the errors associated with estimation of doses are often a mixture of classical and Berkson form and each type of dose uncertainty can include both a shared component, common to all individuals within a group, and an unshared part, unique to an individual within a cohort (19). When exposure estimates in a cohort are constructed using complicated physical and biological models, uncertainties in the dosimetry system can become very complex, with uncertainties of some parameters affecting a large group of study participants simultaneously (20, 21). Zhang et al (21) outlined a novel method to estimate the impact of shared error on parameter uncertainties, and applied it to simulated data from a Monte Carlo dosimetry system (MCDS) developed for the Mayak nuclear worker data. The MCDS provides both dose estimates and associated uncertainty information, in contrast to traditional dosimetry systems. A MCDS has also been developed for the USRT cohort (22); however, until now the only analysis making full use of the entirety of the Monte Carlo dose simulations (rather than simply the mean of the sample) in the USRT has been one of chromosome aberrations (23) and analysis of hemopoietic malignancies (24). The errors in the USRT MCDS are largely of Berkson form, arising largely from a mixture of shared and unshared assignment errors in the exposure score function. However, the exposure score also incorporates variables with classical error, fundamentally derived from the classical errors in the film badge doses. (We treat these matters at greater length in the Discussion.)

The objective of the present study is to evaluate the impact of uncertainties in the eye-lens absorbed doses on the risk of cataract in the 2014 USRT MCDS (22). We shall examine the degree to which the confidence intervals in radiation-associated excess absolute and excess relative risk are modified by dose uncertainties. The models fitted will be of Poisson excess relative risk (ERR) form, in contrast to the previous analysis which fitted Poisson excess absolute risk (EAR) models to this cohort (14); however, we shall also fit Poisson EAR models. We shall use a person year table that is disaggregated by person, in contrast to the version of the person year table used previously (14), which was summed over persons in the cohort. The Poisson ERR models that will be fitted are more similar in terms of the risk metric to the Cox proportional hazards models (25) fitted to this cohort by Little et al (13), although the statistical form of this model is quite different from this paper and the previous one (14). It should be emphasized that the main point of the paper is to explore the degree of inflation of confidence intervals that results from shared dose errors. Although the central estimates of risk may be different from those derived previously, because of the use of slightly different models and a disaggregated person year table, the method we shall use to account for dose uncertainties does not affect these risk estimates. We shall explore the sensitivity of the calculations to choice of baseline (zero dose) model.

Data and Methods

Overview

The USRT study population, cohort follow-up, and dosimetry methods have been described elsewhere (13, 22, 26, 27) (see also www.radtechstudy.nci.nih.gov). Four questionnaires (administered during 1983–1989, 1994–1998, 2003–2005, and 2012–2014) collected information on health outcomes (including self-reports of cataract in all but the first questionnaire), work history and other variables. Since data on cataract were only elicited in the second through fourth questionnaires, we only included responders to the second or third questionnaire and at least one subsequent questionnaire for follow-up of these ocular endpoints. Exclusions are as previously used (13) yielding a total of 67,246 technologists eligible for study, from which a further 3894 technologists were excluded with trivial follow-up (with date of entry the same as date of exit), resulting in an analysis cohort of 63,352 technologists (14). Follow-up began at the first survey in which lack of cataract was reported, and ended at the earlier of the date the last questionnaire answered, or the date of the first assessed cataract. We also censored follow-up after a diagnosis of cancer (other than non-melanoma skin cancer [NMSC]) because of the potential for radiotherapy that the subject might receive. Additional details are given in our earlier papers (13, 14).

Occupational Dosimetry

A historical dose reconstruction was undertaken to estimate annual radiation absorbed doses to the eye lens from occupational exposure for each radiologic technologist (13, 14, 22). The 2014 occupational dosimetry estimation system (22) has a stochastic design to model shared and unshared uncertainties, and to account for missing and uncertain dose-related parameters. Using that system, we produced 1000 simulations from the distribution of actual (true) occupational eye-lens dose for all study subjects (22). Additional details have been given elsewhere (22). As previously detailed, doses were estimated up to the earlier of death or the end of 1997 (13, 14).

Covariates

Data for the a priori-specified potential confounding variables were obtained from the first questionnaire responded to. Questionnaire-derived variables selected a priori as adjusting covariates because of their known association with cataract prevalence (28-30) included sex, racial/ethnic group, birth year, diabetes, body mass index (BMI), smoking status (current smoker/ex-smoker/never smoker, numbers of cigarettes/day, age stopped smoking), and cumulative ambient ultraviolet B (UVB) radiant exposure. The metric for UVB radiant exposure has been described elsewhere (31). Because UVB radiation can only be estimated for the subset of persons who provided lifetime residential history in the third questionnaire, an indicator for UVB missing data was also included in the baseline risk model. Those other variables that were used in the risk models (racial group, baseline diabetes status, baseline BMI, baseline smoking status, baseline cigarettes/day smoked, baseline age stopped smoking) and for which there was missing information there was also a level of each variable to code for being missing or unknown, as indicated in Appendix A Table A1. The distribution of the covariates (including missing values) is given in Table 1 of each of Little et al (13, 14).

Statistical methods

An ungrouped person year table was generated for the dataset using the variables listed in Appendix Table A1. Risks for cataract were assessed using a model in which the number of cases in cell ($i$, $j$) for individual $i=1,\dots ,n$ [n=number of persons] and age/calendar year of follow-up interval of observation $j=1,\dots ,J_i$ [$J_i$ being the number of age/calendar year follow-up intervals for individual $i$], was assumed to have a Poisson distribution with mean:

$${\alpha }_{ij}=h_{ij}[{({\beta }_p)}_{p=0}^M,{(C_{ijp})}_{p=1}^M,X_{ij}]t_{ij}$$ (1)

where $t_{ij}$ is the number of person years spent in cell ($i$, $j$), $X_{ij}$ is the associated true five-year lagged cumulative radiation dose to the eye lens, and we use a linear relative risk formulation for $h_{ij}[{({\beta }_p)}_{p=0}^M,{(C_{ijp})}_{p=1}^M,X_{ij}]$:

$$h_{ij}[{({\beta }_p)}_{p=0}^M,{(C_{ijp})}_{p=1}^M,X_{ij}]=h_{ij,0}[{({\beta }_p)}_{p=1}^M,{(C_{ijp})}_{p=1}^M][1+{\beta }_0{X}_{ij}]=\text{exp}\left[\sum _{p=1}^M{C}_{ijp}{\beta }_p\right][1+{\beta }_0{X}_{ij}]$$ (2)

The cumulative doses $X_{ij}$ are summed for each individual $i$ over all ages/calendar periods up to the age at risk minus the lag period (5 years). We also assumed an excess absolute risk formulation in which:

$$h_{ij}[{({\beta }_p)}_{p=0}^M,{(C_{ijp})}_{p=1}^M,X_{ij}]=h_{ij,0}[{({\beta }_p)}_{p=1}^M,{(C_{ijp})}_{p=1}^M]+{\beta }_0{X}_{ij}=\text{exp}\left[\sum _{p=1}^M{C}_{ijp}{\beta }_p\right]+{\beta }_0{X}_{ij}$$ (3)

Here ${(C_{ijp})}_{p=1}^M$ are some explanatory variables (e.g., age, sex, diabetes status, cumulative UVB radiant exposure) for the specific cell in the ungrouped person-year table, and ${({\beta }_p)}_{p=0}^M$ are some parameters to be determined. The initial baseline (zero dose) model, of the form:

$$h_{ij,0}[{({\beta }_p)}_{p=1}^M,{(C_{ijp})}_{p=1}^M]=\text{exp}\left[\sum _{p=1}^M{C}_{ijp}{\beta }_p\right]$$ (4)

was that used in a previous analysis of this dataset (14), in which variables (as given in Appendix A Table A1) were selected by a forward stepwise method, using a p-value for selection of p=0.05 (32). However, the inverse condition number (33) of the associated Fisher information matrix for this model was below the machine precision (~2.2 x 10⁻¹⁶), so effectively numerically non-invertible. We therefore simplified the model in various ways to make the information matrix numerically invertible, using only the log-linear and log-quadratic terms in ln[age], calendar year, birth year and BMI in the function $h_{ij,0}[.]$, rather than the higher order polynomial expansion employed previously. We successively dropped terms from the baseline model of Little et al (14) to derive a set of models that had an acceptable condition number – that is to say Model C. We further simplified this baseline model to get models (B, A) with successively higher inverse condition numbers. Three candidate models were chosen, as detailed in Appendix A Table A2. Further details on the adjustments implied by the assumed error structure to the regression coefficients are given in Appendix A. For model fitting purposes the true lagged doses in each stratum, $X_{ij}$, were replaced by the mean lagged doses from the Monte Carlo dosimetry, $Z_{ij}=E[X_{ij}]$. The model was fitted via Poisson maximum likelihood (32), using Epicure (34). The computation of the adjusted parameter covariance estimates, which made full use of the dose distributions, was done using R (35). As shown in Appendix A, the elements of the asymptotic covariance matrix associated with the parameters ${({\beta }_p)}_{p=0}^M$ are ultimately a function of the covariances of the simulated true doses from the MCDS. This is subject to the approximations inherent in use of the first step of the first order Fisher scoring algorithm (32), or equivalently the Newton algorithm. We make no assumptions about the Gaussian nature of the true dose distributions, which would of course be nonsense – these dose distributions are highly skewed. However, in a large dataset like USRT these approximations made by using the first step in the Fisher algorithm are likely to be good ones. Having evaluated for the specified model the adjusted (asymptotic) parameter covariance, $\text{cov}[{({\widehat{\beta }}_p)}_{p=0}^M]$ we estimated the adjusted 95% confidence intervals (CI) via ${\widehat{\beta }}_0\pm N(0.975\mid 0,1){(\text{cov}{[{({\widehat{\beta }}_p)}_{p=0}^M]}_{00})}^{0.5}$ where $\text{cov}{[{({\widehat{\beta }}_p)}_{p=0}^M]}_{00}$ is the first diagonal entry in the estimated covariance matrix and $N(0.975\mid 0,1)\approx 1.96$ is the 97.5% point of the standard Normal distribution. This is in contrast to the profile likelihood-based confidence intervals used in the previous analyses (13, 14). The R and Epicure code used is attached.

Results

Table 1 provides a summary of key details of the cohort, in particular detailing the large number (11,345) of cataract cases in the cohort. The mean follow-up in the cohort is 13.1 years (=832,491/63,352). The mean cumulative 5-year lagged eye-lens absorbed dose at the end of follow-up is very low, about 50 mGy, but spanning a considerable range, from 0 to 1.51 Gy (Table 1).

Table 1.

Details of the subset of the US Radiologic Technologist cohort employed in the present analysis

Persons	63,352
Person years	832,491
Birth year range (mean)	1910.9 - 1966.5 (1949.8)
Cumulative (end of follow-up) eye-lens dose (Gy) (mean)	0.0-1.514 (0.050)
Age at start of follow-up range (years) (mean)	31.797 - 84.860 (46.175)
Age at end of follow-up range (years) (mean)	34.923 - 99.926 (61.045)
Number of self-reported cataract cases	11,345

The values of the parameter estimates, standard error estimates (both corrected and uncorrected), and confidence intervals are shown in Table 2. All three baseline (zero dose) models used (details given in Appendix A Table A2) have inverse condition numbers at least four orders of magnitude above the machine precision (Appendix A Table A3). Baseline model A, the simplest model of the three, has the highest inverse condition numbers, 3.7 x 10⁻⁸ for the absolute risk model and 7.0 x 10⁻⁸ for the relative risk model (Appendix A Table A3), and arguably most trust should therefore be attached to the numerical results of this pair of models. Table 2 indicates that the inflation in the standard errors for the excess absolute risk (EAR/ Gy) is almost negligible, no more 0.04% of the respective standard error, whichever of three baseline models (A, B, C) is chosen. However, the inflation in the excess relative risk (ERR) coefficient standard error (ERR / Gy) is more substantial, although still modest. For the simplest model (model A) the 95% CI about the central estimate of ERR of 1.085 Gy⁻¹, changed from (0.645, 1.525) unadjusted for dose uncertainties to (0.558, 1.612) for the uncertainty-adjusted model; this represents inflation of the standard errors by about 19.9% (Table 2).

Table 2.

Estimates of relative and absolute risk model dose parameters, as a function of the baseline (zero dose) risk model used (models A, B, C given in Appendix A Table A2).

	Models of excess relative risk (ERR)
Baseline model	Parameter Estimate (ERR /Gy)	Uncorrected Standard Error Estimate (ERR /Gy)	Uncorrected 95% CI (ERR /Gy)	Corrected Standard Error Estimate (ERR /Gy)	Corrected 95% CI (ERR /Gy)	Difference between corrected and uncorrected standard error (ERR /Gy)	% difference adjusted vs unadjusted standard error
A	1.085	0.224	(0.645, 1.525)	0.269	(0.558, 1.612)	0.045	19.864
B	1.249	0.234	(0.790, 1.708)	0.234	(0.790, 1.708)	0.000011	0.005
C	0.949	0.223	(0.512, 1.385)	0.223	(0.511, 1.386)	0.000268	0.120
	Models of excess absolute risk (EAR)
	Parameter Estimate (EAR /104 PY Gy)	Uncorrected Standard Error Estimate (EAR /104 PY Gy)	Uncorrected 95% CI (EAR /104 PY Gy)	Corrected Standard Error Estimate (EAR /104 PY Gy)	Corrected 95% CI (EAR /104 PY Gy)	Difference between corrected and uncorrected standard error (EAR /104 PY Gy)	% difference adjusted vs unadjusted standard error
A	197.5	23.615	(151.215, 243.785)	23.615	(151.215, 243.785)	0.000318	0.001
B	97.75	27.925	(43.018, 152.482)	27.925	(43.017, 152.483)	0.000320	0.001
C	137.1	27.822	(82.569, 191.631)	27.832	(82.550, 191.650)	0.009868	0.035

Discussion

We have shown that uncertainties in the true eye-lens absorbed doses do not substantially affect confidence intervals in cataract radiogenic risk, whether for absolute or relative risk models, and in the case of the absolute risk models the modification is essentially negligible. These results may be due to the fact that there is little sharing in uncertainty across the whole cohort for a given ensemble of doses (i.e. replication from the MCDS); that is the variance of the shared random effect $\text{var}[{\epsilon }_{SM}]\approx 0$, as we discuss below. The absence of such shared error will result in little augmentation of the standard error, as shown by expressions (A8), (A9) and (A9’) in Appendix A.

Analysis of a small subset (using 1000 randomly chosen individuals and 100 randomly chosen Monte Carlo dose replications) of the dose data using a number of log-linear mixed models with various sets of explanatory (fixed-effect) covariates suggest in all cases that the replication random effect, which accounts for the shared dose error across individuals, is highly significant (p<0.001) but nevertheless small, so that the overall magnitude of the shared random effect variance ($\text{var}[{\epsilon }_{SM}]\approx 0.024$) is about two orders of magnitude less than the unshared error variance ($\text{var}[{\epsilon }_{i,NS}]\approx 1.49$) on a log scale. This is also suggested by graphical analyses conducted by Simon et al. (22), although these do no more than suggest that the overall magnitude of shared error is modest. As such, all this suggests that shared error is perhaps not very important in this dataset, reinforced by the results of our analysis.

Strengths of the present study include the large population with low (mostly <100 mGy) cumulative protracted radiation doses and prospective cohort design, with prospectively collected information on many relevant risk factors (e.g., cigarette smoking, diabetes status, BMI, UVB radiant exposure). There is a comprehensive occupational MCDS with estimated absorbed eye-lens doses (13). Although a substantial proportion of the estimated cumulative occupational dose is derived from questionnaires (22), the dosimetry has been extensively validated, in particular via chromosome aberrations evaluated with fluorescence in situ hybridization (FISH) (23).

Our study had several limitations, particularly the self-report and lack of clinical validation of cataracts. However, the population of radiologic technologists reported here is medically literate, so that self-reports of cataract and other medical conditions such as diabetes should be reasonably reliable. All analyses adjust for age, which should largely eliminate the major risk factor for tendency to mis-recall diagnostic information. However, recall bias in reporting some of these conditions cannot be ruled out. A related weakness is lack of information on cataract subtype. As with many occupational studies, cohort members had to survive to answer the second questionnaire and be free of cataract at that point. Such selection will not necessarily bias our analysis, since everyone had to survive to answer a questionnaire, and all risk was assessed conditional on that. Follow-up was censored at the date of the last informative questionnaire answered; as previously discussed (13, 14) it is plausible that such censoring was non-informative for cataract. The main conclusions of the paper regarding the impact of dose uncertainty on the estimated risk of cataract may depend on the modeling assumptions included in the 2014 MCDS, regarding the shared/unshared sources of uncertainty and, more importantly, the assumed magnitude of dose uncertainties. Because of changes in personnel in the USRT study team, we have no easy way of assessing the impact of these.

As outlined in the Statistical Methods and Appendix A, the approximations inherent in the adjusted calculations of covariance, making use of the Fisher scoring method (32) (or equivalently the Newton algorithm), are likely good ones for a dataset of this size. Alternative methods could be envisaged to account for dose uncertainties, in particular Bayesian inference based on Markov Chain Monte Carlo (MCMC) algorithms or Monte Carlo maximum likelihood (MCML). In a dataset of the present size (~70,000 subjects) a standard Bayesian analysis using the Metropolis-Hastings algorithm would be very time consuming, taking probably weeks or months per run in order to guarantee parameter convergence, although Hamiltonian Monte Carlo methods (36) could potentially considerably speed this up. MCML is also likely to be computationally challenging, although not to the degree of a Bayesian MCMC analysis. However, MCML would also necessitate writing code in a high level compiled language such as Fortran or C++. The tradeoff would naturally be different for a much smaller dataset, for example of about a tenth the size of the present one, where the accuracy of the approximations in the Fisher scoring algorithm (or Newton algorithm) would be lower, and Bayesian MCMC computations relatively quicker.

Indeed, there have been previous analyses of the effects of dosimetric error in the USRT, in particular analysis of chromosome translocations in relation to occupational ionizing radiation dose, in which regression-calibration was compared with two full-likelihood methods, Bayesian MCMC and MCML (23). There was little difference between the results of these three methods, in particular little inflation in confidence intervals introduced by the full-likelihood methods, which would in principle take account of shared uncertainties also (23). All of these results suggest that shared error is a generally pretty small component of dose uncertainties in this cohort. Very similar results have been observed in analysis of hemopoietic neoplasm mortality in the USRT using methods similar to those of the present paper for a relative risk model (24); it is very likely that the results obtained here and for chromosome aberration and hemopoietic neoplasms apply to the effects of occupational radiation exposure in relation to all other disease endpoints in this cohort. More generally, correction for dose uncertainties via regression calibration, Bayesian MCMC and MCML methods in a number of radiation exposed datasets yields in almost all cases remarkably similar results (37-39), very likely a result of the modest magnitude of dosimetric errors, as in the present cohort.

The degree of shared and unshared dosimetric error is not homogenous among members of the USRT cohort, and will differ depending on the particular parameters that determine an individual’s dose, in particular the presence of film badge data, and the extent to which these may be the same for another individual, as we now detail. As discussed by Simon et al (22) there are a number of sources of exposure data for this cohort, in particular 921,134 individual film badge records for 79,959 cohort members, as well as data on work history and protection practices from three self-administered questionnaire surveys and a number of literature-based surveys of mean workplace exposure. The historical data is reconstructed from surveys over each of three distinct employment periods, pre-1940, 1940-1949, 1950-1955, while for the period 1956-1997 the film badge records are used; the methods used are described in more detail elsewhere (22). The Monte Carlo dosimetry assigns doses with a log-normal distribution, with geometric mean (GM) and geometric standard deviation (GSD) depending on whether or not a film-badge dose is present. When present the dose assigned has GM based on a weighted combination of the population dose for that year and the individual, with a constant GSD = 1.2 (22). When a film-badge record is not present the GM depends on the estimated mean dose for the year multiplied by variables determined by the exposure score category, which is based on the calendar-period specific historical assessments discussed above; likewise the GSD is a function of the exposure score category. The exposure score is given by a regression model fitted to the post-1956 individual dose records taking account of questionnaire-derived weekly numbers of various types of diagnostic radiographic procedures, and some other occupational variables (22). The extent to which individuals share parameters in the exposure score induces a variable degree of shared uncertainty particularly in those components of dose received before 1956. The MCDS therefore generates ensembles of plausible realizations of “true” doses with a fundamentally Berkson error structure, as is made clear by Table 6 in Simon et al (22). As Simon et al (22) note: “The USRT dosimetry system uses simulation to provide multiple values of annual badge dose for each study participant taking into account individual dosimetric uncertainties as well as sources of shared errors. The goal in designing the system was to provide annual badge dose realizations that represent a sample from the actual (true) badge dose distribution given what is known about the uncertainties in the time-dependent population dose distribution, the individual’s work patterns and practices and individual badge dose measurements (when available).” This implies, in particular, that the sample mean of a large number of cumulative (for an individual up to a given year) eye-dose simulations, for a given individual and follow-up interval [often, as here, used as the assigned eye-dose (13, 14)], will approximately equal the true mean of the underlying actual (true) cumulative eye-dose distribution for that individual and follow-up interval. However, the variables that make up the exposure score function are measured with what is often classical error, derived ultimately from the classical error in the film badge doses.

The method used here only adjusts the confidence intervals for the component of shared error in the dose estimates. There are other types of errors, in particular of misclassification of outcome, the effect of which can be to bias the outcome in either direction. For example, were some non-radiogenic outcome misreported as cataract the effect expected would be to bias the dose response downward, whereas if a radiogenic endpoint with greater sensitivity than cataract were to be diagnosed as cataract then the dose response would be biased upwards; however, as cataract is among the most radiogenic endpoints (40, 41), this second possibility can be largely discounted. There have been a few studies of such diagnostic misclassification in radiation studies, in particular analysis of possible misclassification of cancer and non-cancer outcomes in the Japanese atomic bomb survivors (42); it was concluded that if misclassification of cancer as non-cancer was independent of dose such misclassification was unlikely to effect the non-cancer dose response, although a small amount of dose-dependent misclassification could result in a pronounced upward bias in the non-cancer dose response (42). Variations in completeness of ascertainment are unlikely to bias dose response if independent of dose, although if dependent on dose there is the potential for substantial bias. It would be expected that self-reported outcome data such as cataract in the present cohort would be more subject to variation in degree of reporting, although it is unlikely that, once adjusted for age and calendar period, that it would be related to eye-lens dose. Confounding and other forms of bias have the potential to alter the dose response trend in either direction, and cannot be easily quantified. Some attempts have been made at accounting for such biases in a meta-analysis of power-frequency magnetic fields and childhood leukemia (43).

The present study yields estimates of excess relative risk per Gy of between 0.949 (95% CI 0.511, 1.386) and 1.249 (95% CI 0.790, 1.708) for cataract incidence (Table 2). These are 40-80% greater than the estimate, fitted by means of a Cox proportional hazards model, to the same data of excess hazard ratio per Gy of 0.69 (95% CI 0.27, 1.16) (13). If one uses the identical background model employed in a previous paper (14) to fit a linear excess relative risk model the resulting excess relative risk per Gy is 0.67 (95% CI 0.26, 1.13); the previous paper (14) used the same aggregated version of the person year table, summed over persons, as that in the present paper. However, the disaggregated nature of the person year table would be expected to yield different model fits, even if the fitted models were the same, which they are not. Indeed, fitting the same form of baseline EAR models as previously (14) to the disaggregated person year table employed here yields an estimated radiation-associated EAR of 183.5 /10⁴ PY Gy, different from that found previously, 94.21 /10⁴ PY Gy (14), as well as all the estimates (197.5 (model A), 97.75 (model B), 137.1 (model C), Table 2) presented here using simplified forms of this background model. The Cox model used by Little et al (13) makes use of continuous measures of dose, whereas the person year table used in the present paper and the previous analysis (14) used person-year weighted average dose and of other variables (e.g., age, calendar year). It has been suggested by Loomis et al (44), comparing results of a Poisson regression analysis using a finely disaggregated person year table, one based on a coarser stratification and one using a Cox proportional hazards model that the use of coarsely aggregated estimates of exposure can introduce substantial bias in trend risk estimates compared with the other two models. The present analysis and the comparison with the Cox model analysis (13) suggests that the degree of aggregation in our dataset is sufficiently fine that the degree of bias thereby introduced is modest.

The method employed here has been used in a simulation study closely modeled on the actual Monte Carlo dosimetry system employed in the cohort of Mayak nuclear workers (21). In the model which assumed a large shared dose error component, somewhat similar to what is thought to be the case, the standard error of the relative risk trend increased by a factor of 2 over the unadjusted estimate. Although the analysis of the real data has yet to be done, this suggests that in this cohort shared error has considerable effect. This is also clearly the case in a dataset of thyroid nodules among person exposed to atmospheric nuclear tests in Kazakhstan (45). Hoffmann et al (46) presented results of a number of simulation studies based closely on lung cancer in relation to radon exposure in the French uranium miners cohort, which suggested that within-individual shared error has the potential to cause appreciable attenuation in the dose response trend when using Cox or excess hazard ratio survival models; the degree of attenuation was much less if only unshared error (of Berkson or classical type) was assumed. Hoffmann et al (47) also analyzed lung cancer mortality in relation to radon exposure in this miner cohort by means of a Bayesian model that accounted for by a mixture (depending on time period of exposure) of unshared Berkson and classical error. To the best of our knowledge no analysis taking account of shared error has been done in any other radiation-exposed dataset, although there are quite a few examples of analyses adjusting for unshared classical error, many of them in the Japanese atomic bomb survivor Life Span Study, using both classical likelihood-based regression calibration methods (48-52) and Bayesian techniques, in particular by Little et al (39, 53, 54), an approach used also in a report of the United Nations Scientific Committee on the Effects of Atomic Radiation (55). These methods generally predict modest adjustments, generally of the order of 10-20%.

Appendix A. Details on statistical methods

We assume that there are true five-year lagged cumulative doses $X=(X_{ij})$ for individuals $i$ and age/calendar year of follow-up interval $j$ [see Appendix A Table A1], but we only have an estimate of the means of the joint distribution of these $Z=(Z_{ij})=(E[X_{ij}])$. The number of years spent by individual $i$ in follow-up interval $j$ is $t_{ij}$, with $i=1,\dots ,N$, $j=1,\dots ,J_i$. Let $D_{ij}$ denote the number of events in cell ($i$, $j$). So for a given individual, $D_{ij}=0$ for $1\le j\le J_i$ and $D_{i{J}_i}=0$ or 1. The person year table is ungrouped to the extent that each individual contributes a number of individual records to the person year table i.e. the cells defined by particular levels of the stratifying variables (age, time etc) are not summed over the individuals in the cohort. The person years in each stratifying cell $t_{ij}$ can, unlike $D_{ij}$, take many values, because of the fact that the temporal stratifying timepoints are not spaced by single units of time/age (see Appendix A Table A1), also by the nature of a Lexis diagram. We further assume that $D_{ij}\sim Poisson[h_{ij}{t}_{ij}]$ where $h_{ij}=h_{ij}({({\beta }_p)}_{p=0}^M,{(C_{ijp})}_{p=1}^M,(X_{ij}))$ is the hazard function (instantaneous event rate), a function of the true doses. Here ${(C_{ijp})}_{p=1}^M$ are some explanatory variables (e.g., age, sex, diabetes status, cumulative UVB radiant exposure) for the specific cell in the ungrouped person-year table, and ${({\beta }_p)}_{p=0}^M$ are some parameters to be determined. Then the Poisson likelihood (which as per Zhang et al (21) is the same, up to a multiplicative constant, as that derived from the exponential process) is given by:

$$L\left[{({\beta }_p)}_{p=0}^M\right]=\prod _{i=1}^N\left(\prod _{j=1}^{J_i}{[h_{ij}{t}_{ij}]}^{D_{ij}}∕D_{ij}!\right)\text{exp}\left(-\sum _{j=1}^{J_i}{h}_{ij}{t}_{ij}\right)=\prod _{i=1}^N{[h_{i{J}_i}{t}_{i{J}_i}]}^{D_{i{J}_i}}\text{exp}\left(-\sum _{j=1}^{J_i}{h}_{ij}{t}_{ij}\right)$$ (A1)

so that the log-likelihood is given by:

$$l=C+\sum _{i=1}^N\sum _{j=1}^{J_j}[D_{ij}\text{ln}[h_{ij}]-h_{ij}{t}_{ij}]$$ (A2)

where the constant $C={\sum }_{i=1}^N{\sum }_{j=1}^{J_i}{D}_{ij}\text{ln}[t_{ij}]$ is not a function of any model parameters. [Note that the $D_{ij}!$ after the first equality in (A1) are all 1, as $D_{ij}=0,1$ and so these terms drop out.] The score vector is given by:

$$S{({({\beta }_p)}_{p=0}^M)}_k=\frac{\partial l}{\partial {\beta }_k}=\sum _{i=1}^N\sum _{j=1}^{J_i}{D}_{ij}\frac{1}{h_{ij}}\frac{\partial h_{ij}}{\partial {\beta }_k}-\frac{\partial h_{ij}}{\partial {\beta }_k}{t}_{ij}$$ (A3)

and the Fisher information, $I$, is given by:

$$\begin{gathered} {(I)}_{km}=-E\left[\frac{{\partial }^{2}l}{\partial {\beta }_k{\beta }_m}\right]=E\left[\frac{\partial l}{\partial {\beta }_k}\frac{\partial l}{\partial {\beta }_m}\right] \\ =\sum _{i=1}^N\sum _{j=1}^{J_i}-E[D_{ij}]\frac{1}{h_{ij}}\frac{{\partial }^2{h}_{ij}}{\partial {\beta }_k\partial {\beta }_m}+E[D_{ij}]\frac{1}{h{{}_{ij}}^2}\frac{\partial h_{ij}}{\partial {\beta }_k}\frac{\partial h_{ij}}{\partial {\beta }_m}+\frac{{\partial }^2{h}_{ij}}{\partial {\beta }_k\partial {\beta }_m}{t}_{ij} \\ =\sum _{i=1}^N\sum _{j=1}^{J_i}\frac{t_{ij}}{h_{ij}}\frac{\partial h_{ij}}{\partial {\beta }_k}\frac{\partial h_{ij}}{\partial {\beta }_m}=\sum _{i=1}^N\sum _{j=1}^{J_i}{h}_{ij}{t}_{ij}\left[\frac{1}{h_{ij}}\frac{\partial h_{ij}}{\partial {\beta }_k}\right]\left[\frac{1}{h_{ij}}\frac{\partial h_{ij}}{\partial {\beta }_m}\right] \\ =Q^{k}diag[h_{ij}{t}_{ij}]{[Q^m]}^T=Q^{k}diag[E[D_{ij}]]{[Q^m]}^T \end{gathered}$$ (A4)

where $Q^k=(Q_{ij}^k)=\left(\frac{1}{h_{ij}}\frac{\partial h_{ij}}{\partial {\beta }_k}\right)=\left(\frac{\partial \text{ln}[E[D_{ij}]]}{\partial {\beta }_k}\right)$ is to be understood as a row vector with the $T={\sum }_{i=1}^N{J}_i$ indicated elements strung out, and similarly for $Q_m$, making use of the fact that $E[D_{ij}]=t_{ij}{h}_{ij}$. [In general we use the notation $diag[x_1,x_2,\dots ,x_N]$ to mean an $N\mathrm{x}N$ matrix with 0s in the off-diagonal entries and $x_1,x_2,\dots ,x_N$ in the diagonal entries, so that here $diag[h_{ij}{t}_{ij}]$ is to be understood as the $T\mathrm{x}T$ matrix with 0 off-diagonal entries and the $h_{ij}{t}_{ij}$ along the diagonal in the same order as the indexing for $Q^k$.] We can also rewrite the score vector (A3) as:

$$S{({({\beta }_p)}_{p=0}^M)}_k=\sum _{i=1}^N\sum _{j=1}^{J_i}{D}_{ij}{Q}_{ij}^k-E[D_{ij}]Q_{ij}^k$$ (A3)

By the standard variance decomposition formula (a consequence of the tower property of conditional expectations), extended trivially to covariances:

$$\begin{gathered} \text{cov}[(D_{ij}),(D_{kl})\mid Z]=E[\text{cov}[(D_{ij}),(D_{kl})\mid Z,X]\mid Z]+ \\ \text{cov}[E[(D_{ij})\mid Z,X],E[(D_{kl})\mid Z,X]\mid Z] \end{gathered}$$ (A5)

Assume now that:

$$h_{ij}=h_{ij,0}[1+{\beta }_0{X}_{ij}]$$ (A6)

Then from (A5) and the fact that $D_{ij}\sim Poisson[t_{ij}{h}_{ij}]=Poisson\left[t_{ij}{h}_{ij,0}[1+{\beta }_0{X}_{ij}]\right]$, then by standard properties of Poisson distributions:

$$\begin{gathered} E[D_{ij}\mid Z,X]=t_{ij}{h}_{ij,0}[1+{\beta }_0{X}_{ij}] \\ \text{cov}[D_{ij},D_{kl}\mid Z,X]=\text{var}[D_{ij}\mid Z,X]1_{(i,j)=(k,l)}=E[D_{ij}\mid Z,X]1_{(i,j)=(k,l)} \end{gathered}$$ (A7)

we have that:

$$\begin{gathered} \text{cov}{[(D_{ij}),(D_{kl})\mid Z]}_{ij,kl}=E[1_{(i,j)=(k,l)}E(D_{ij}\mid Z,X)\mid Z]+\text{cov}[E[(D_{ij})\mid Z,X],E[(D_{kl})\mid Z,X]\mid Z] \\ =1_{(i,j)=(k,l)}E(D_{ij}\mid Z)+\text{cov}[t_{ij}{h}_{ij,0}[1+{\beta }_0{X}_{ij}],t_{kl}{h}_{kl,0}[1+{\beta }_0{X}_{kl}]\mid Z] \\ =1_{(i,j)=(k,l)}E(D_{ij}\mid Z)+\beta {{}_0}^2{t}_{ij}{t}_{kl}{h}_{ij,0}{h}_{kl,0}\text{cov}[X_{ij},X_{kl}\mid Z] \end{gathered}$$ (A8)

so that from (A3’) and (A4) we have that:

$$\begin{gathered} \text{cov}[(S_k),(S_l)\mid Z]=\phantom{[}\text{cov}[\sum _{i=1}^N\sum _{j=1}^{J_i}{Q}_{ij}^k(D_{ij}-E[D_{ij}]),\sum _{i^{\prime }=1}^N\sum _{j^{\prime }=1}^{J_{i^{\prime }}}{Q}_{i^{\prime }{j}^{\prime }}^l(D_{i^{\prime }{j}^{\prime }}-E[D_{i^{\prime }{j}^{\prime }}])\mid Z] \\ =\sum _{i,i^{\prime }=1}^N\sum _{j=1}^{J_i}\sum _{j^{\prime }=1}^{J_{i^{\prime }}}{Q}_{ij}^k{Q}_{i^{\prime }{j}^{\prime }}^l\text{cov}[D_{ij},D_{i^{\prime }{j}^{\prime }}\mid Z] \\ =\sum _{i,i^{\prime }=1}^N\sum _{j=1}^{J_i}\sum _{j^{\prime }=1}^{J_{i^{\prime }}}{Q}_{ij}^k{Q}_{i^{\prime }{j}^{\prime }}^l[1_{(i,j)=(i^{\prime },j^{\prime })}E(D_{ij}\mid Z)+\beta {{}_0}^2{t}_{ij}{t}_{i^{\prime }{j}^{\prime }}{h}_{ij,0}{h}_{i^{\prime },j^{\prime },0}\text{cov}[X_{ij},X_{i^{\prime }{j}^{\prime }}\mid Z]] \\ ={(I)}_{kl}+\beta {{}_0}^2\sum _{i,i^{\prime }=1}^N\sum _{j=1}^{J_i}\sum _{j^{\prime }=1}^{J_{i^{\prime }}}{Q}_{ij}^k{Q}_{i^{\prime }{j}^{\prime }}^l{t}_{ij}{t}_{i^{\prime }{j}^{\prime }}{h}_{ij,0}{h}_{i^{\prime }{j}^{\prime },0}\text{cov}[X_{ij},X_{i^{\prime }{j}^{\prime }}\mid Z] \end{gathered}$$ (A9)

From the Fisher scoring algorithm (32) (alternatively Newton’s algorithm) we have that, approximately (considering only the first step in the algorithm), the maximum likelihood estimate $\widehat{\beta }={({\widehat{\beta }}_p)}_{p=0}^M$ as a function of true lagged dose is:

$${({\widehat{\beta }}_p)}_{p=0}^M-{({\beta }_p)}_{p=0}^M\approx I^{-1}S({(\beta )}_{p=0}^M)$$ (A10)

and since the Fisher information $I$ is constant we have that approximately:

$$\text{cov}[{({\widehat{\beta }}_p)}_{p=0}^M]\approx I^{-1}\text{cov}[S({(\beta )}_{p=0}^M)]I^{-1}$$ (A11)

where $\text{cov}[S({(\beta )}_{p=0}^M)]$ is the covariance of the score statistic. We can plug in an evaluation of $\text{cov}[S({(\beta )}_{p=0}^M)]$ from (A9) to evaluate this, using the empirical covariances from the MCDS to estimate each $\text{cov}[X_{ij},X_{i^{\prime }{j}^{\prime }}\mid Z]$.

We may alternatively assume an absolute risk formulation as a function of lagged dose:

$$h_{ij}=h_{ij,0}+{\beta }_0{X}_{ij}$$ (A6’)

Again from (A5) we have, using similar intermediate derivations to (A7):

$$\text{cov}{[(D_{ij}),(D_{kl})\mid Z]}_{ij,kl}=1_{(i,j)=(k,l)}E(D_{ij}\mid Z)+t_{ij}{t}_{kl}\beta {{}_0}^2\text{cov}[X_{ij},X_{kl}\mid Z]$$ (A8’)

so that from (A3’) we have that:

$$\begin{gathered} \text{cov}[(S_k),(S_l)\mid Z]=\text{cov}\phantom{[}[\sum _{i=1}^N\sum _{j=1}^{J_i}{Q}_{ij}^k(D_{ij}-E[D_{ij}]),\sum _{i^{\prime }=1}^N\sum _{j^{\prime }=1}^{J_{i^{\prime }}}{Q}_{i^{\prime }{j}^{\prime }}^l(D_{i^{\prime }{j}^{\prime }}-E[D_{i^{\prime },j^{\prime }}])\mid Z] \\ =\sum _{i,i^{\prime }=1}^N\sum _{j=1}^{J_i}\sum _{j^{\prime }=1}^{J_{i^{\prime }}}{Q}_{ij}^k{Q}_{i^{\prime }{j}^{\prime }}^l\text{cov}[D_{ij},D_{i^{\prime }{j}^{\prime }}\mid Z] \\ =\sum _{i,i^{\prime }=1}^N\sum _{j=1}^{J_i}\sum _{j^{\prime }=1}^{J_{i^{\prime }}}{Q}_{ij}^k{Q}_{i^{\prime }{j}^{\prime }}^l[1_{(i,j)=(i^{\prime },j^{\prime })}E(D_{ij}\mid Z)+t_{ij}{t}_{i^{\prime }{j}^{\prime }}{\beta }_0^2\text{cov}[X_{ij},X_{i^{\prime },j^{\prime }}\mid Z]] \\ ={(I)}_{kl}+\sum _{i,i^{\prime }=1}^N\sum _{j=1}^{J_i}\sum _{j^{\prime }=1}^{J_{i^{\prime }}}{Q}_{ij}^k{Q}_{i^{\prime }{j}^{\prime }}^l{t}_{ij}{t}_{i^{\prime }{j}^{\prime }}{\beta }_0^2\text{cov}[X_{ij},X_{i^{\prime },j^{\prime }}\mid Z] \end{gathered}$$ (A9’)

which can again be plugged into (A11), again using the empirical covariances from the MCDS to estimate each $\text{cov}[X_{ij},X_{i^{\prime }{j}^{\prime }}\mid Z]$ to thereby evaluate the covariance on the model parameters.

Having evaluated for the specified model the adjusted (asymptotic) parameter covariance, $\text{cov}[{({\widehat{\beta }}_p)}_{p=0}^M]$ we estimated the adjusted 95% confidence intervals via ${\widehat{\beta }}_0\pm N(0.975\mid 0,1){(\text{cov}{[{({\widehat{\beta }}_p)}_{p=0}^M]}_{00})}^{0.5}$ where $\text{cov}{[{({\widehat{\beta }}_p)}_{p=0}^M]}_{00}$ is the first diagonal entry in the estimated covariance matrix and $N(0.975\mid 0,1)\approx 1.96)$ is the 97.5% point of the standard Normal distribution.

Table A1. Variables used to stratify ungrouped person year table.

Variable	Levels
Age (years)	0 / 5 / 10 / … / 95 / 100 / ∞
Calendar year	1990-1-1 / 1995-1-1 / 2000-1-1 / 2005-1-1/ 2010-1-1 / ∞
Birth year	1800 / 1900 / 1910 / 1920 / 1930 / 1940 / 1950 / 1960 / ∞
Cumulative occupational eye-lens absorbed dose (Gy)	0 / 0.001 / 0.002 / 0.005 / 0.01 / 0.02 / 0.05 / 0.1 / 0.2 / 0.5 / 1.0 / 2.0 / ∞
Cumulative UVB radiant exposure (MJ cm2)	0 / 0.01 / 0.02 / 0.03 / 0.05 / 0.075 / 0.1 / 0.2 / ∞
Sex	male / female
Racial group	white / black / Asian+Pacific islander / American Indian / other / unknown
Body mass index (BMI) (kg/m2)	missing / 0 / 18.5 / 25.0 / 30 .0 / ∞
Baseline diabetes	no / yes / unknown
Baseline smoking status	unknown / never smoker / ex-smoker / current smoker
Baseline number of cigarettes/day smoked	unknown / 0 / 10 / 20 / 30 / 40 / 50 / 60 / ∞
Baseline age last smoked	unknown / 0 / 20 / 30 / 40 / 50 / 60 / 70 / ∞

Table A2. Model parameters used to model baseline (zero dose) rate in log-linear models (2) and (3).

Unless otherwise indicated all were analyzed as time-varying variables.

No.	Variable name
Model A
1	sex at baseline
2	diabetes at baseline
3	current smoking status at baseline [current smoker, ex-smoker vs non smoker/unknown]
4	smoking age stopped at baseline
5	smoking quantity (cigarettes / day) at baseline
6	UVB cumulative radiant exposure (MJ cm−2) (including indicator for UVB exposure being missing)
7	ln[age]
8	calendar year
9	birth year at baseline
10	body mass index known at baseline (Y/N)
11	body mass index (kg m−2) at baseline (if known)
Model B
1	sex at baseline
2	diabetes at baseline
3	current smoking status at baseline [current smoker, ex-smoker vs non smoker/unknown]
4	smoking age stopped at baseline
5	smoking quantity (cigarettes / day) at baseline
6	UVB cumulative radiant exposure (MJ cm−2) (including indicator for UVB exposure being missing)
7	ln[age]
8	(ln[age])2
9	calendar year
10	(calendar year)2
11	birth year at baseline
12	body mass index known at baseline (Y/N)
13	body mass index (kg m−2) at baseline (if known)
14	(body mass index at baseline (if known))2
Model C
1	sex at baseline
2	diabetes at baseline
3	current smoking status at baseline [current smoker, ex-smoker vs non smoker/unknown]
4	smoking age stopped at baseline
5	smoking quantity (cigarettes / day) at baseline
6	UVB cumulative radiant exposure (MJ cm−2) (including indicator for UVB exposure being missing)
7	ln[age]
8	(ln[age])2
9	calendar year
10	(calendar year)2
11	birth year at baseline
12	(birth year at baseline)2
13	body mass index known at baseline (Y/N)
14	body mass index (kg m−2) at baseline (if known)
15	(body mass index (kg m−2) at baseline (if known))2

Table A3. Inverse condition numbers for the baseline (zero dose) models and risk type (relative risk, absolute risk) used.

The baseline models used are as described in Appendix A Table A2.

Baseline (zero dose) model	Type of radiation risk model	Inverse condition number
A	Relative risk	6.96 x 10−8
A	Absolute risk	3.73 x 10−8
B	Relative risk	4.32 x 10−11
B	Absolute risk	3.24 x 10−11
C	Relative risk	8.47 x 10−12
C	Absolute risk	6.00 x 10−12