Abstract
In this paper, the influence of measurement errors in exposure doses in a regression model with binary response is studied. Recently, it has been recognized that uncertainty in exposure dose is characterized by errors of two types: classical additive errors and Berkson multiplicative errors. The combination of classical additive and Berkson multiplicative errors has not been considered in the literature previously. In a simulation study based on data from radio-epidemiological research of thyroid cancer in Ukraine caused by the Chornobyl accident, it is shown that ignoring measurement errors in doses leads to overestimation of background prevalence and underestimation of excess relative risk. In the work, several methods to reduce these biases are proposed. They are new regression calibration, an additive version of efficient SIMEX, and novel corrected score methods.
Keywords: Berkson measurement error, Chornobyl, Classical measurement error, Corrected scores, Dose-response, Radiation epidemiology, Regression calibration, SIMEX
1. Introduction
As a result of the 1986 Chornobyl accident, significant territory of Ukraine, Russia, and Belarus were under radioactive contamination and the inhabitants of that territories suffered from radioactive exposure.
Even 5–6 years after the accident, an inflation of the incidence of thyroid cancer cases was observed for children and adolescents who lived in the territories where the estimated thyroid exposure doses were quite high, see Likhtarev, Sobolev and others (1995b), Jacob and other (2006), and Buglova and others (1996).
In fact, the growth of thyroid cancer prevalence for children and adolescents caused by internal irradiation from Chornobyl fallouts turned out to be the main (if not the unique) statistically reliable effect of the Chornobyl accident. Consequently this effect was of great interest for radiation epidemiologists all over the world, leading to a series of studies in Ukraine, Belarus and Russia, see Likhtarov, Kovgan, Vavilov, Chepurny, Ron and others (2006), Kopecky and other (2006), and Zablotska and other (2011).
However, interpretation of the results for most of the radiation epidemiological studies was based on risk estimation methods which do not take into account the presence of significant uncertainties in doses. One of the consequences of the assumption about the absence of errors in doses can be that the risk estimates are biased and the dose-response curve is distorted. The reasons for risk estimates distortions are not only systematic but also due to random errors in the dose estimates. In radiation epidemiology, various attempts have been made to construct statistical methods for analyzing not only uncertainty in the effect of the dose but also uncertainty in the dose itself, see Mallick and other (2002), Carroll and other (2006), Lyon and other (2006), Kopecky and other (2006), Li and other (2007), Hofer (2008), Kukush and other (2011), and Likhtarov, Kovgan, Masiuk and others (2014). The literature now recognizes that dose measurements are inevitably affected by errors of either classical or Berkson type, or a combination of the two, see Mallick and other (2002). Unfortunately, the most popular computer package in radiation epidemiology, Epicure (Preston and other, 1993) does not account for dose uncertainty.
Previous attempts at dose–response estimation while accounting for uncertainties in doses have almost exclusively treated the dose uncertainties as multiplicative in structure. However, in the Chornobyl accident, recent detailed analyses of radioactivity registration mechanisms have shown that classical errors in thyroid exposure doses that were reconstructed in Likhtarov, Kovgan, Vavilov, Chepurny, Bouville and others (2005), and Likhtarov, Kovgan, Masiuk and others (2014) are of additive rather than multiplicative type, see Likhtarov, Masiuk and others (2013). In addition, Likhtarov, Masiuk and others (2013) show that thyroid radioactivity registration errors have a Poisson distribution. Because in most cases the intensity of measurements was quite high (Likhtarev, Prohl and others, 1993; Likhtarev, Goulko and others, 1995a), the exposure dose measurement errors can be regarded normally distributed, although heteroscedastic, see Likhtarov, Masiuk and others (2013).
The aim of the present paper is to study radiation risk estimates and methods of risk estimation in models with additive measurement errors and multiplicative Berkson errors in exposure doses. In Section 2, we present the measurement error model and the risk model. In Section 3, we note that standard methods perform poorly in our context, and we develop three new methods: (a) a novel version of Corrected Scores, (b) a new version of Regression Calibration, and (c) a new version of efficient SIMEX (see Cook and Stefanski, 1994; Carroll and other, 2006; Kukush and other, 2011). Section 4 presents results of simulation studies, while Section 5 has concluding remarks. Technical details are given in Appendices of supplementary material available at Biostatistics online.
2. Models
2.1. Model of dose with classical additive and Berkson multiplicative errors
In May and June 1986, >150 000 measurements of thyroid radioactivity were made among inhabitants of the northern part of Ukraine, which suffered from the most intensive radionuclide fallouts, including 115 000 measurements among children and adolescents aged 0–18 years (Likhtarev, Prohl and others, 1993; Likhtarev, Goulko and others, 1995a). Further, the measurements will be denoted 
. In what follows, a superscript “mes” refers to measured versions of the true variables, and a superscript “tr” refers to the true variables. Here 
denotes an individual.
As shown in Likhtarov, Kovgan, Vavilov, Chepurny, Bouville and others (2005) and Likhtarov, Kovgan, Masiuk and others (2014), the measured individual thyroid dose for the 
th person can be written as
 |
(2.1) |
where 
is the measured thyroid mass, 
is a factor that is obtained from the ecological model of radioactivity transition, and 
is the measured radioactivity in the thyroid.
Ecological coefficient 
includes the error of Berkson type, see Likhtarov, Kovgan, Masiuk and others (2014). Denote the factor with Berkson error 
, so that (2.1) becomes
 |
(2.2) |
The true dose is decomposed as
 |
Here the relation between 
and 
includes multiplicative Berkson error of the form 
, where 
and 
, where 
is known. The variables 
and 
are stochastically independent, for details, see Eq. (8) in Kukush and other (2011). The empirical distribution of 
and its characteristics (expectation, variance, etc.) can be obtained by the Monte-Carlo procedure described in Likhtarov, Kovgan, Masiuk and others (2014).
As shown in Likhtarov, Masiuk and others (2013), radioactivity measurements of the thyroid are now known to have additive error, so that 
, the measured thyroid radioactivity, is
 |
(2.3) |
where the 
are independent standard normal variables, the value 
is known and 
are independent random variables.
Plug (2.3) into (2.2) and set 
. We get
 |
(2.4) |
The random variables 
, 
, and 
are jointly independent, although we allow correlation between 
and 
. Define 
, then (2.4) takes a form
 |
(2.5) |
 |
(2.6) |
Actually, (2.5) and (2.6) are a model of dose observations with additive classical and multiplicative Berkson errors. It is straightforward to see that 
.
2.2. Prevalence model
In order to model cases of cancer for a fixed time interval, we use a model of rare events with binary response variable 
, where 
in the case of thyroid cancer and 
in the absence of disease. Define 
to be background prevalence intensity, i.e., in the absence of dose, and define 
. Then define
 |
(2.7) |
where EAR is excess absolute risk. Then the conditional distribution of 
given the exposure dose is defined by
 |
(2.8) |
The observed sample consists of couples 
, for 
. The parameters 
and 
(or, in other parameterization, 
and EAR), are positive and to be estimated.
3. Methods
3.1. Existing methods
Common methods include (a) the naïve estimator, which is maximum likelihood estimator not accounting for measurement errors in doses; (b) parametric and linear regression calibration as defined in Appendix A of supplementary material available at Biostatistics online; and (c) the ordinary SIMEX method (Cook and Stefanski, 1994; Carroll and other, 2006). The simulation results show that methods (a) and (b) yield estimates with significant bias, see Appendix A of supplementary material available at Biostatistics online. This can be explained by specific structure of the data problem, where we have a kind of mixture of lognormal and normal variables. The ordinary SIMEX has larger bias compared with the efficient SIMEX, see Kukush and other (2011). Instead, we developed three new methods described in Sections 3.2–3.5.
3.2. Corrected Score estimator
Within the Corrected Score method, we adjust the unbiased estimating function to measurement errors (Carroll and other, 2006, Section 7.4). Introduce the estimating function 
as a solution to the deconvolution problem
 |
where 
is an unbiased estimating functions, see Appendix B of supplementary material available at Biostatistics online; 
is a product of a matrix and a vector
 |
(3.1) |
The explicit expression for 
is
 |
(3.2) |
A consistent estimator of 
is a solution 
to an unbiased estimating equation, namely a solution to
 |
(3.3) |
Equation (3.3) is linear in 
and 
, and therefore, it can be solved efficiently.
In Appendix B of supplementary material available at Biostatistics online, we establish the asymptotic normality of 
, and construct a data-based covariance matrix estimator.
3.3. New regression calibration handling Berkson error
As mentioned in Section 3.1, the conventional parametric regression calibration has quite poor behavior in our simulation studies. In this section, we develop an approximation to regression calibration that has much more satisfactory behavior.
The idea is to treat the additive normal error in dose (2.5) as if it was multiplicative log-normal error, but with approximately the same conditional variance of 
given 
. Denote the log-normal error by 
. Equating the variance of the multiplicative error 
to the relative variance 
and replacing the unknown 
with a feasible value 
, we obtain
 |
This yields
 |
Then calibration is performed in the same manner as described in Kukush and other (2011), namely
 |
Here the estimators of 
and 
are taken from Likhtarov, Masiuk and others (2013), namely
 |
(3.4) |
 |
(3.5) |
where
 |
After preliminary calibration of doses, the maximum likelihood method described in Masiuk and other (2013) is used for accounting for Berkson error, see Appendix C of supplementary material available at Biostatistics online.
3.4. Efficient SIMEX
As a prerequisite to classical SIMEX method, assume that we can evaluate an estimator 
in the model without measurement errors (e.g., the maximum likelihood estimator).
Classical SIMEX algorithm is described in Carroll and other (2006, Section 5). It consists of the following steps:
Select a “large” number
and a finite set of non-negative numbers 
.
For all
and all 
, generate normal random variables 
, where 
comes from (2.5).
- For all
and 
, evaluate the naive estimator for perturbed data
and evaluate averaged estimate
Extrapolate
to point 
and assign 
.
In Kukush and other (2011), the “efficient SIMEX estimator” of the risk parameters of the model with multiplicative error was derived as an alternative to the ordinary SIMEX. It differed in the way that 
is perturbed only if 
. Here we develop this idea in the model with additive errors.
Setting tuning parameters. Select a “large” number
and a finite set of non-negative numbers 
. We use 
and 
in our numerical work.
- Simulation. For all
and all 
such that 
, generate normal random variables
As an optional refinement, generate them such that
.
-
Estimation. For all
and 
, solve the system of equations in 
and 
(3.6)
The perturbed dose
(3.7) 
can be negative, and significant negative doses break down the naïve estimator. Therefore, we use the censored perturbed doses given by 
. Denote the solution as 
, 
.
For
average 
and 
in 
:
-
Extrapolation. Extrapolate numerically the functions
and 
to 
. In extrapolation, we approximate 
and 
with quadratic polynomial. Such a choice of extrapolant function is the simplest one, and it allows to express the estimates explicitly through 
and 
, see Kukush and other (2011).
The values
and 
are the efficient SIMEX estimates of 
and EAR.
3.5. Efficient SIMEX handling Berkson error
In this section, we introduce the SIMEX estimator which uses variances of both classical and Berkson errors. We start with unbiased estimating equation in the model with Berkson error only, see Appendix C.2 of supplementary material available at Biostatistics online. Assume for the moment that 
are known. Denote the conditional probability
 |
The following equations are unbiased:
 |
that is, with true parameters substituted, expectations of the left-hand and right-hand sides of these equations coincide.
With (2.6), the expression for 
is
 |
(3.8) |
where expectation is taken for nonrandom 
and lognormal 
, 
, see Likhtarov, Masiuk and others (2013). In generic case 
, 
, 
, and the integral in (3.8) is taken from 
to 
. In other case, we integrate over the interval where the numerator 
is positive.
Now, consider the model with both Berkson and classical errors. In SIMEX method, perturbed measured doses are substituted for true doses. Therefore, substitute 
for 
:
 |
Change the right-hand side of the second equation to 
. This is equivalent to formal adding to the latter equation the unbiased equation 
; the unbiasedness holds true because
 |
This simplification is done in order to avoid perturbations of doses for non-cases 
. We get
 |
(3.9) |
 |
(3.10) |
The efficient SIMEX estimator is defined similarly to the one in Section 3.4. We just replace equations (3.6) and (3.7) with (3.9) and (3.10).
For significant perturbations, the modified dose 
may be negative, which may break down the estimation procedure. Therefore, the negative doses are changed to zeros, i.e., 
is used instead of 
.
4. Simulation study
4.1. Simulation setup
In order to simulate exposure doses, we used a real subpopulation of children and adolescents under 18, consisting of 
13 000 persons from the settlements of Zhytomyr, Kyiv, and Chernihiv, which had direct measurements of thyroid activity in May–June 1986. Exposure doses for this subpopulation were constructed via the framework of the Ukrainian-American project on thyroid cancer prevalence in Ukraine after the Chornobyl accident; see Likhtarov, Kovgan, Masiuk and others (2014).
Parameters of the absolute risk model (2.7) for the observation period from 1991 to 2001 were given by values close to ones obtained in epidemiological studies of thyroid cancer in Ukraine, see Jacob and other (2006) and Likhtarov, Kovgan, Vavilov, Chepurny, Ron and others (2006), namely
 |
(4.1) |
In our simulation study, 1000 different data sets were simulated for different levels of classical (
) and Berkson (
) uncertainty. The classical error level was defined as the constant value 
varied from 0.2 to 1. The Berkson error level was set in such a way that geometric standard deviation of 
given 
, 
, took on the values 1 (no error), 1.5, 2, 3, 5, and 8. All the listed values are realistic.
Simulation study is performed in four steps:
Initial doses
are taken from the real thyroid doses of children and adolescents internally exposed to 
I in 1986, see Figure 1.
True dose values are generated for the cohort by using
and taking into account the uncertainty levels 
given in the first column of Tables 1 and 2, see (2.6).
Using the data from Step (2), as well as the model in equations (2.7) and (2.8), with the parameter values
and EAR in (4.1), a disease vector is generated.
Initial doses
were perturbed, and thus, the measured doses 
were generated according to equation (2.5), with the error standard deviation 
, where 
enters the second column of Tables 1 and 2. As a result, we obtain an observation model with classical additive and Berkson multiplicative errors in doses.
Based on the measured doses
, the information of measurement errors 
and 
, as well as the disease vector generated in Step (3), the parameter values 
and EAR are estimated by three methods.
Fig. 1.
Histogram of 
.
Table 1.
Estimates of baseline incidence rate (medians over 1000 simulations and standard deviations)
Estimates of  by different methods |
|||||||||
|---|---|---|---|---|---|---|---|---|---|
| Error |
Naïve |
New calibrate handling Berkson error |
Corrected Score |
Efficient SIMEX handling Berkson error |
|||||
 |
 |
Median | SD | Median | SD | Median | SD | Median | SD |
| 1 no error | 0 | 1.95 | (0.53) | 1.95 | (0.53) | 1.93 | (0.97) | 1.94 | (0.53) |
| 0.2 | 1.99 | (0.54) | 2.00 | (0.55) | 1.95 | (1.01) | 1.94 | (0.55) | |
| 0.4 | 2.20 | (0.56) | 2.11 | (0.57) | 1.98 | (1.11) | 1.93 | (0.75) | |
| 0.6 | 2.57 | (0.59) | 2.46 | (0.58) | 1.93 | (1.28) | 2.52 | (1.11) | |
| 0.8 | 2.91 | (0.61) | 2.79 | (0.59) | 1.90 | (1.49) | 3.52 | (1.39) | |
| 1 | 3.15 | (0.62) | 3.05 | (0.60) | 1.98 | (1.80) | 4.49 | (1.59) | |
| 1.5 | 0 | 1.96 | (0.55) | 1.95 | (0.55) | 1.97 | (0.97) | 1.95 | (0.55) |
| 0.2 | 2.01 | (0.56) | 2.01 | (0.58) | 1.97 | (1.01) | 1.95 | (0.56) | |
| 0.4 | 2.20 | (0.58) | 2.14 | (0.60) | 1.97 | (1.12) | 1.93 | (0.73) | |
| 0.6 | 2.57 | (0.58) | 2.46 | (0.59) | 2.00 | (1.25) | 2.40 | (1.10) | |
| 0.8 | 2.89 | (0.59) | 2.79 | (0.59) | 2.04 | (1.48) | 3.50 | (1.44) | |
| 1 | 3.11 | (0.60) | 3.06 | (0.58) | 2.06 | (1.79) | 4.43 | (1.59) | |
| 2 | 0 | 1.95 | (0.54) | 1.94 | (0.55) | 2.05 | (0.94) | 1.93 | (0.54) |
| 0.2 | 2.00 | (0.54) | 1.99 | (0.56) | 2.05 | (1.02) | 1.94 | (0.54) | |
| 0.4 | 2.21 | (0.54) | 2.14 | (0.59) | 2.05 | (1.10) | 1.95 | (0.71) | |
| 0.6 | 2.57 | (0.58) | 2.47 | (0.59) | 2.07 | (1.24) | 2.40 | (1.13) | |
| 0.8 | 2.90 | (0.58) | 2.80 | (0.60) | 2.06 | (1.42) | 3.43 | (1.38) | |
| 1 | 3.13 | (0.59) | 3.04 | (0.58) | 2.11 | (1.67) | 4.39 | (1.56) | |
| 3 | 0 | 2.00 | (0.56) | 1.97 | (0.57) | 2.18 | (0.94) | 1.95 | (0.56) |
| 0.2 | 2.04 | (0.56) | 2.02 | (0.57) | 2.20 | (0.94) | 1.95 | (0.56) | |
| 0.4 | 2.23 | (0.56) | 2.15 | (0.60) | 2.19 | (1.03) | 1.94 | (0.74) | |
| 0.6 | 2.60 | (0.57) | 2.42 | (0.58) | 2.20 | (1.18) | 2.43 | (1.14) | |
| 0.8 | 2.89 | (0.59) | 2.82 | (0.61) | 2.24 | (1.34) | 3.46 | (1.34) | |
| 1 | 3.12 | (0.59) | 3.06 | (0.60) | 2.23 | (1.62) | 4.38 | (1.52) | |
| 5 | 0 | 2.12 | (0.55) | 2.03 | (0.57) | 2.37 | (0.83) | 1.94 | (0.55) |
| 0.2 | 2.17 | (0.56) | 2.02 | (0.59) | 2.38 | (0.88) | 1.95 | (0.56) | |
| 0.4 | 2.34 | (0.57) | 2.12 | (0.59) | 2.39 | (0.96) | 1.95 | (0.73) | |
| 0.6 | 2.65 | (0.57) | 2.44 | (0.58) | 2.38 | (1.08) | 2.40 | (1.07) | |
| 0.8 | 2.94 | (0.57) | 2.75 | (0.57) | 2.39 | (1.21) | 3.40 | (1.37) | |
| 1 | 3.14 | (0.57) | 2.99 | (0.57) | 2.44 | (1.46) | 4.24 | (1.46) | |
| 8 | 0 | 2.23 | (0.56) | 2.07 | (0.56) | 2.59 | (0.76) | 1.92 | (0.57) |
| 0.2 | 2.27 | (0.56) | 2.04 | (0.58) | 2.58 | (0.78) | 1.93 | (0.56) | |
| 0.4 | 2.43 | (0.57) | 2.13 | (0.58) | 2.59 | (0.85) | 1.92 | (0.76) | |
| 0.6 | 2.71 | (0.57) | 2.45 | (0.57) | 2.59 | (0.91) | 2.39 | (1.08) | |
| 0.8 | 2.96 | (0.57) | 2.71 | (0.56) | 2.61 | (1.03) | 3.25 | (1.38) | |
| 1 | 3.13 | (0.57) | 2.93 | (0.55) | 2.64 | (1.26) | 4.04 | (1.40) | |
True value 
.
Table 2.
Estimates of absolute excess risk (medians over 1000 simulations and standard deviations)
Estimates of  by different methods |
|||||||||
|---|---|---|---|---|---|---|---|---|---|
| Error |
Naïve |
New calibrate handling Berkson error |
Corrected Score |
Efficient SIMEX handling Berkson error |
|||||
 |
 |
Median | SD | Median | SD | Median | SD | Median | SD |
| 1 no error | 0 | 4.98 | (1.03) | 5.01 | (1.03) | 5.03 | (1.58) | 4.99 | (1.04) |
| 0.2 | 4.93 | (1.02) | 5.05 | (1.02) | 4.97 | (1.64) | 5.00 | (1.03) | |
| 0.4 | 4.63 | (1.01) | 4.88 | (1.03) | 4.93 | (1.76) | 5.04 | (1.27) | |
| 0.6 | 4.01 | (0.92) | 4.07 | (0.87) | 4.98 | (2.03) | 4.07 | (1.65) | |
| 0.8 | 3.39 | (0.84) | 3.31 | (0.80) | 5.02 | (2.42) | 2.36 | (1.90) | |
| 1 | 2.91 | (0.73) | 2.80 | (0.72) | 5.02 | (2.91) | 0.84 | (2.09) | |
| 1.5 | 0 | 4.98 | (1.04) | 4.99 | (1.04) | 5.00 | (1.59) | 4.99 | (1.04) |
| 0.2 | 4.90 | (1.02) | 5.07 | (0.99) | 4.94 | (1.65) | 4.99 | (1.04) | |
| 0.4 | 4.57 | (0.98) | 4.90 | (0.99) | 4.93 | (1.77) | 5.04 | (1.31) | |
| 0.6 | 3.98 | (0.88) | 4.05 | (0.92) | 4.94 | (2.03) | 4.17 | (1.73) | |
| 0.8 | 3.38 | (0.83) | 3.34 | (0.82) | 4.94 | (2.42) | 2.46 | (1.96) | |
| 1 | 2.91 | (0.76) | 2.83 | (0.71) | 4.90 | (2.86) | 0.88 | (2.13) | |
| 2 | 0 | 4.90 | (1.06) | 5.04 | (1.06) | 4.84 | (1.59) | 5.00 | (1.07) |
| 0.2 | 4.84 | (1.02) | 5.05 | (1.04) | 4.84 | (1.68) | 5.01 | (1.06) | |
| 0.4 | 4.52 | (0.99) | 4.89 | (1.05) | 4.78 | (1.85) | 5.04 | (1.34) | |
| 0.6 | 3.92 | (0.89) | 4.03 | (0.92) | 4.82 | (2.07) | 4.13 | (1.85) | |
| 0.8 | 3.33 | (0.85) | 3.31 | (0.79) | 4.80 | (2.35) | 2.57 | (2.02) | |
| 1 | 2.87 | (0.77) | 2.80 | (0.70) | 4.79 | (2.73) | 0.98 | (2.06) | |
| 3 | 0 | 4.75 | (1.02) | 5.15 | (1.01) | 4.54 | (1.57) | 5.03 | (1.08) |
| 0.2 | 4.68 | (1.01) | 5.08 | (1.07) | 4.51 | (1.64) | 5.02 | (1.09) | |
| 0.4 | 4.36 | (0.97) | 4.89 | (1.07) | 4.48 | (1.79) | 5.03 | (1.37) | |
| 0.6 | 3.75 | (0.90) | 4.02 | (1.34) | 4.44 | (1.96) | 4.13 | (1.90) | |
| 0.8 | 3.19 | (0.84) | 3.30 | (0.82) | 4.40 | (2.26) | 2.37 | (2.15) | |
| 1 | 2.74 | (0.76) | 2.78 | (0.73) | 4.42 | (2.71) | 0.90 | (2.18) | |
| 5 | 0 | 4.30 | (0.99) | 5.07 | (0.99) | 3.79 | (1.42) | 5.00 | (1.22) |
| 0.2 | 4.24 | (0.98) | 5.13 | (1.25) | 3.80 | (1.49) | 4.98 | (1.23) | |
| 0.4 | 3.94 | (0.96) | 4.99 | (1.26) | 3.77 | (1.57) | 4.96 | (1.49) | |
| 0.6 | 3.37 | (0.90) | 4.04 | (1.06) | 3.73 | (1.71) | 4.07 | (2.09) | |
| 0.8 | 2.85 | (0.81) | 3.32 | (0.94) | 3.72 | (1.95) | 2.28 | (2.38) | |
| 1 | 2.50 | (0.74) | 2.80 | (0.83) | 3.71 | (2.38) | 0.79 | (2.26) | |
| 8 | 0 | 3.64 | (0.90) | 5.15 | (1.15) | 2.99 | (1.25) | 4.98 | (1.43) |
| 0.2 | 3.59 | (0.89) | 5.19 | (1.48) | 2.96 | (1.29) | 5.00 | (1.44) | |
| 0.4 | 3.30 | (0.86) | 4.98 | (1.41) | 2.89 | (1.37) | 4.94 | (1.79) | |
| 0.6 | 2.88 | (0.79) | 4.02 | (1.22) | 2.87 | (1.55) | 3.96 | (2.21) | |
| 0.8 | 2.41 | (0.75) | 3.30 | (1.08) | 2.89 | (1.72) | 2.23 | (2.51) | |
| 1 | 2.08 | (0.67) | 2.78 | (1.95) | 2.84 | (2.04) | 0.65 | (2.51) | |
True value 
.
Steps (1) to (5) are repeated 1000 times and the median values of the estimated risk coefficients as well as standard deviations are presented in Tables 1 and 2.
Sometimes measured doses 
can take negative values as a result of large errors in the additive error model (2.5). In such cases, negative doses were replaced by a small positive number, except for the Corrected Score estimator, because the Corrected Score method can handle negative doses.
For each of the various values of 
, the averaged number of cases over 1000 realizations was 68, with corresponding frequency of thyroid cancer disease 0.51%.
4.2. Results and discussion
Estimation of absolute risk parameters was performed by the naïve method, the Corrected Score method presented in Section 3.2 that takes into account only classical error, and also by the new regression calibration method and the efficient SIMEX method described in Sections 3.3 and 3.5, respectively. The latter two methods take into account both classical and Berkson errors. Because in our case the distribution of data set 
is strictly positive and its logarithm is approximately symmetric (see Figure 1), in our simulation any parametric method assumes a log-normal distribution of 
.
The medians of the estimates of the baseline incidence rates and the standard deviations (SD) of the estimates are given in Table 1, while the medians of the estimates of the excess absolute risk and the standard deviations of the estimates are given in Table 2. In Appendix D of supplementary material available at Biostatistics online, we display 95% deviance intervals computed based on the obtained empirical distribution for risk parameters estimators with truncation of 2.5% quantiles from both sides, and hence an interval estimate for risk parameters.
4.2.1. Naïve estimator.
The simulation results showed that the naïve method underestimates EAR and overestimates background prevalence intensity. The risk estimates have larger bias for larger measurement errors in doses. For 
, EAR is underestimated twice. The level of uncertainty 
for additive measurement errors in doses corresponds to the geometric standard deviation equal 
for multiplicative errors. Comparison with results from Kukush and other (2011) shows reasonable consistency. It is worth mentioning that for 
and for 
, the bias of the background prevalence and the bias of EAR do not exceed 5%. Thus, for small level of uncertainty, the naïve method gives quite satisfactory results, as expected.
Nevertheless the effect of Berkson error on the results of risk analysis is much smaller. If 
, then the effect is negligible. When 
is increasing up to 3 and more, then the bias of the estimate is more essential and should be taken into account.
4.2.2. Regression calibration and efficient SIMEX.
Though parametric regression calibration defined in Likhtarov, Masiuk and others (2013) takes into account the shape of the distribution of 
, the estimates computed by this method are considerably biased, with underestimated background prevalence intensity and overestimated of EAR (the results are shown in Appendix A of supplementary material available at Biostatistics online). This is unexpected effect compared with simulation results from Kukush and other (2011), where for multiplicative measurement errors in doses, the parametric estimates were quite acceptable. It looks like the reason for this is the structure of the normal measurement errors 
and the log-normal distribution of 
, but we have no definite explanation.
Estimates obtained by the new regression calibration are much more stable and less biased compared with the ones obtained by other methods of regression calibration, and are quite satisfactory when the classical error in dose is not too large, in particular for 
. However, when 
, there is considerable bias.
Estimates of absolute risk parameters obtained by efficient SIMEX method fit the model values only for small classical errors. The estimates are satisfactory (that is bias does not exceed 10%) if 
. However, when 
, there is considerable bias.
These methods can handle quite large Berkson errors.
4.2.3. Corrected Score method.
The Corrected Score estimator is the least biased of all ones presented in this paper. For the error-level 
, the maximal absolute bias for EAR and for 
does not exceed 5%. Of course, the Corrected Score estimator has the widest deviance intervals, reflecting the well-known phenomenon that bias correction typically leads to increased variability of estimates.
Using this estimator, only classical error in the factor 
(see (2.3)) was taken into account. This leads to bias for large Berkson errors.
4.2.4. Influence of Berkson error.
For moderate levels 
, the effect of Berkson error on ultimate estimates is insignificant. But if 
increases to 3 and more, then the influence of Berkson error is indeed significant and should be taken into account. Simulation showed that in the naïve estimates the Berkson error, as well as the classical error but to a smaller extent, leads to underestimation of EAR and overestimation of 
.
5. Conclusions
There are classical additive errors and Berkson multiplicative errors in exposure doses in the linear model for rare events. That is a fact that requires a new statistical methodology. To solve this problem, we have developed new methods of regression calibration, corrected scores, and efficient SIMEX that are appropriate for the actual dose uncertainties. We performed simulations based on real data from epidemiological studies. The thyroid absorbed doses were taken from the results of Ukrainian–American project involving the Chornobyl accident, and cases were modeled based on the underlying risk model. The true absolute risk parameters were chosen to be typical for the epidemiological studies in this important context. Estimators of the parameters were constructed by the naïve method (that is without taking into account dose measurement errors) with the package EPICURE and also by the methods mentioned above.
We showed that the naïve estimator has significant bias. The bias increases as the classical or Berkson error variance increases. The efficient SIMEX and new regression calibration approaches improve the estimators, but mainly for moderate classical uncertainty levels such as 
. They give quite good result for significant Berkson error. The new Corrected Score estimator has little bias for small Berkson errors. However, this estimator has the largest deviance intervals, and it does not take Berkson error into account.
In general, methods of radiation risk estimation in cases of the classical additive dose error work more poorly than in case of the classical multiplicative error (Kukush and other, 2011). At first glance the reason is as follows: the size 
of underlying cohort in the latter paper is larger, namely 
around 70 000 persons vs. 
around 13 000 persons in the present paper. However, additional simulations showed that in this case artificial enlargement of the sample size does not significantly improve the risk estimates. Therefore, we believe that this phenomenon has to do with the combination of normal dose errors 
and lognormally distributed random variables 
. This assertion is confirmed by other investigations we have done but that are not reported in the present paper.
Choosing among the methods, other than the naïve estimate which is clearly unacceptable, is difficult. However, for a concrete radiation risk estimation problem, it is reasonable to perform a preliminary simulation study. Such a simulation will make it possible, for a given dose distribution and prevalence level, to analyze the behavior of estimates obtained by various methods and also the influence of nuisance parameters on the model, such as effect modifiers and confounders, see Health Risks from Exposure to Low Levels of Ionizing Radiation (2006).
Funding
The research was supported by the Ukrainian Radiation Protection Institute. Carroll's research was supported by a grant from the National Cancer Institute (U01-CA057030).
Supplementary material
Supplementary material is available at http://biostatistics.oxfordjournals.org.
Acknowledgments
Conflict of Interest: None declared.
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