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The effect of censoring on cancer risk estimates based on the Canadian National Dose Registry of occupational radiation exposure

Journal of Exposure Science & Environmental Epidemiology volume 15, pages 398–406 (2005)Cite this article

Abstract

Cohort studies represent an important epidemiological tool for exploring the potential adverse health effects of low-dose exposure to ionizing radiation in the workplace. Analyses of data from the National Dose Registry of Canada have suggested that occupational radiation exposure leads to increased risk of several specific types of cancer, as well as increased overall risk of cancer. An important aspect of such studies is the censoring in recorded exposures induced by dosimetry detection limits. Such a censoring effect can lead to significant underestimation of cumulative doses which, in turn, can result in overestimation of the excess cancer risk associated with occupational radiation exposure. In this article, we present analytic results, supported by a simulation study, on the magnitude of overestimation of risk based on the additive relative risk model used in the analysis of the NDR data that can occur due to censoring. Our results indicate that overestimation of risk is modest, being less than 20% in all situations considered here. Because censoring also results in ovestimation of the precision of the risk estimates, the significance levels of Wald-type statistical tests for increased risk based on the ratio of the estimate to its standard error are virtually unaffected by censoring. These results suggest that although the application of the additive excess relative risk model in the presence of censoring may lead to some overestimation of risk, the model does not lead to invalid conclusions regarding the association between occupational radiation exposure and cancer risk based on data from the NDR.

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Acknowledgements

We are grateful to Drs. Willem Sont and Pat Ashmore for their helpful comments on this paper, and to the referees, whose comments greatly strengthened our manuscript. D. Krewski is the NSERC/SSHRC/McLaughlin Chair in Population Health Risk Assessment at the University of Ottawa. This work was supported in part by grant CDC 5 R01 OH07864-01 from the National Institute of Occupational Health Sciences and by a grant from the Canadian Institutes of Health Research.

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Authors and Affiliations

  1. R. Samuel McLaughlin Centre for Population Health Risk Assessment, Ottawa, ON, Canada

    Hwashin Shin, Timothy Ramsay & Daniel Krewski

  2. Healthy Environments and Consumer Safety Branch, Health Canada, Ottawa, K1A 0K9, ON, Canada

    Jan M Zielinski

  3. Department of Epidemiology and Community Medicine, Faculty of Medicine, University of Ottawa, K1N 6N5, ON, Canada

    Jan M Zielinski

Authors
  1. Hwashin Shin
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  2. Timothy Ramsay
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  3. Daniel Krewski
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  4. Jan M Zielinski
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Corresponding author

Correspondence to Timothy Ramsay.

Appendix A. Proof of asymptotic result

Appendix A. Proof of asymptotic result

Notation

Each of the observed cases of cancer used to fit the ERR model falls into one cell of a two-way table. Each of the C cells is defined by a 5-year time period and a 5-year age group; any individual who was in a given age group during a given time period contributes person-years to that cell. We assume that the subjects are partitioned into S strata (S1, …, Ss) defined by the occupational group, and that the individual dose measurements of individuals in a given stratum are independently and identically distributed. We will use the subscripts i, j, k, and l to index cells, people, individual measured doses, and strata, respectively.

We suppose that our sample consists of dose records for N individuals, with the jth subject's measurements constituting a time series of exposure measurements for that individual. If the jth individual contributes person-years to the ith cell, his/her cumulative uncensored and censored doses upon leaving that cell will be denoted by Dij and Cij, respectively. The number of person-years contributed by subject j to cell i will be denoted by tij, and the total number of dose measurements obtained for subject j before he/she leaves cell i will be denoted by mij. We assume that for some non-negative constant c (the minimum detection limit), the jth individual's kth censored dosimeter measurement cjk is defined by

Define Φi to be the set of individuals who contribute person-years to the ith cell and ni=∣Φi∣ to be the number of individuals in Φi. For example, the expression Φi={3,4,7} means that the third, fourth, and seventh subjects (and no others) contribute person-years to the first cell, so that n1=3. Define ti and D̄i to be the total number of person-years and the weighted average of the cumulative dose (weighted by the number of person-years tij) in the ith cell, respectively. Define yi to be the observed number of lung cancer cases in the ith cell and λi to be the baseline lung cancer rate (cases per person-year for the population of interest) for the ith cell. Although we used historical values for λi in our study, λi can also be estimated as part of the model-fitting procedure. The ERR model then implies that, for some constant β, the expected number of cancer cases for cell i is given by the expression

The parameter β measures the ERR due to radiation exposure. Suppose that the jth subject contributes tij person-years to the ith cell, and has an average cumulative dose of Dij in cell i. Then the weighted average D̄i is estimated by

We assume that the probability that a given individual is in stratum l is p(l). Furthermore, we assume that for a subject in the lth stratum, either all of that subject's exposure measurements are zero, or that all of that subject's exposure measurements are independently and identically distributed (iid) random variables with density function fl. The probability that a subject's dose distribution is identically zero is pl(0). Furthermore, assume both that the stratum memberships of different individuals are independent and that the number of person-years contributed to a given cell by different individuals are iid. Given that an individual is in stratum l, the expected value of his/her censored reading is a constant multiple of the expected value of an uncensored dose measurement:

where the constant αl is defined as

Fitting the ERR model

The iterative Gauss–Newton algorithm used to fit the excess relative risk model can be very compactly described in matrix notation. Define the vectors

The fitting algorithm proceeds as follows.

Initialization step:

  1. 1)

    Choose initial estimate β̂0.

  2. 2)

    Define the C × C diagonal matrix Ŵ0 with ith diagonal entry equal to [tiλi(1 + β̂0D̄i)]−1.

mth Iterative step:

  1. 1)

    Compute updated estimate β̂m = β̂m−1 + (VtWm − 1V)−1 VtWm−1[Y − W− 1m − 11].

  2. 2)

    Update the C × C diagonal matrix Ŵm so that ith diagonal entry equals [tiλi(1 + β̂mD̄i)]−1

Termination step:

Stop when ∣∣β̂m − β̂m − 1∣∣ < ɛ

Output:

β̂:= β̂m, Ŵ:=Ŵm

Regardless of the starting value β̂0, β̂m converges to the maximum likelihood estimate of β and (VtŴV)−1/2 converges to the standard error of β̂.

Effect of censoring on β̂

Let β̂ and β̂(c) denote, respectively, the estimates of β fit using uncensored and censored dose measurements. Asymptotically, the t-statistics β̂/SE(β̂) and β̂(c)/SE(β̂(c)) for testing H0:β=0 are equal. The proof of this fact involves two steps. First, we will show that the average cumulative censored dose for the ith cell, C̄i, approaches αD̄i, for some constant α that is independent of i. This will be true when ni=∣Φi∣, the number of people contributing to cell i, approaches infinity. Next, we show that this fact implies that censoring does not affect the value of the t-statistic.

We now show that censoring results in the average dose DÌ„i for the ith cell being multiplied by a constant which is independent of i. The average cumulative dose DÌ„i for the ith cell is calculated using the expression

Let t̄i = E[tij], m̄i = E[mij], and d̄l be the expected value of one measurement for a subject with nonzero dose in the lth stratum. The independence of tij and djk implies that

as ni → ∞. A parallel argument for the convergence of C̄i shows that the average cumulative dose computed from the censored measurements converges in probability to that computed from the uncensored dose as ni → ∞, with the constant α defined by

Finally, we show that, at least asymptotically, censoring does not change the value of the t-statistic for testing the significance of β̂. At the mth iterative step, the updated estimate β̂m is

The estimate for β̂(c) is obtained from this expression by substituting β̂(cm−1 for β̂m−1 and C̄i for D̄i. If we choose both the starting values β̂0 and β̂0(c to be 0, Eq. (1) implies that β̂1(c) converges in probability to β̂1/α when ni → ∞. By induction on m, we see that β̂m(c) converges in probability to β̂m/α for all m and, therefore, that

Similarly, one can show that

It follows that

as ni → ∞.

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Shin, H., Ramsay, T., Krewski, D. et al. The effect of censoring on cancer risk estimates based on the Canadian National Dose Registry of occupational radiation exposure. J Expo Sci Environ Epidemiol 15, 398–406 (2005). https://doi.org/10.1038/sj.jea.7500416

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