6.3.1. Setup and Methods
The simulation results are presented in –.
Estimates of baseline incidence rate for the first scenario.
Estimates of excess absolute risk for the second scenario.
The estimation of the absolute risk parameters was performed by several methods:
- Naïve: The ordinary maximum likelihood method in which the dose estimates are taken to be error-free.
- Parametric Full ML: Full maximum likelihood where the error in the measured activity was taken into account by means of the integral convolution over the distribution of true thyroid doses (Carroll et al., 2006; Masiuk et al., 2008), assuming that f Qtr/Mmes has a lognormal distribution. The error in the thyroid mass was handled by relation (13).
- Nonparametric Full ML: The full maximum likelihood method described above, but without assumption about the distribution of f Qtr/Mmes. The latter distribution is approximated according to formula (12).
- Parametric Regression Calibration: Parametric regression calibration as described in Section 4.1.
- Nonparametric Regression Calibration: Nonparametric calibration as described in Section 4.2.
- Ordinary SIMEX: The SIMEX method described in Section 5 with refinement (A.3).
- Efficient SIMEX: Our computationally efficient modification of the SIMEX method presented in Section 5.
Each estimate was computed for 100 different realizations of doses and cases. Then the corresponding estimates were averaged.
For methods 1, 4 and 5, the confidence intervals were not calculated due to their well-known statistical inconsistency in the presence of dose errors, and hence we used the deviance interval (DI), calculated based on the 2.5th and the 97.5th percentiles of the estimates over 100 simulations.
For the estimators 2, 3, 6 and 7 the confidence intervals were constructed based on the asymptotic covariance matrices. The method applied to construct the confidence intervals for both SIMEX estimators is given in Appendix A.3
6.3.4. Nonparametric Regression Calibration and Maximum Likelihood
Nonparametric regression calibration and nonparametric full maximum likelihood methods seem to be more adequate, because they estimate the dose distribution in a nonparametric way. As a result those methods show quite modest bias even for the nonsmooth underlying distribution in Scenario 1.
However, in Scenario 2, things might be expected to be different because of the large right tail of true doses, for which the regression calibration approximation might break down. Nonparametric full maximum likelihood behaves quite well in this case, while nonparametric regression calibration has some bias, perhaps because the nonlinear effects of the underlying logistic type model are revealed.
6.3.6. Numerical Comparisons
For a particular comparison, we consider the case that GSDQ = 3 and GSDM = 1.5.
In Scenario 1, the %-bias in λ0 for Naïve, Parametric ML, Nonparametric ML, Parametric regression calibration, Nonparametric regression calibration, Ordinary SIMEX and Efficient SIMEX were 95%, 31%, 2%, 35%, 1%, 9% and 1%, respectively. The corresponding biases for Scenario 2 were 102%, 16%, 4%, 6%, 9%, 2% and 2%, respectively.
In Scenario 1, the %-bias in Excess Absolute Risk for Naïve, Parametric ML, Nonparametric ML, Parametric regression calibration, Nonparametric regression calibration, Ordinary SIMEX and computationally Efficient SIMEX were 70%, 28%, 0.1%, 13%, 1%, 53% and 0.2%, respectively. The corresponding biases for Scenario 2 were 50%, 11%, 2%, 5%, 3%, 11% and 2%, respectively.
In terms of variability, we ignore the Naïve estimate because of its severe bias. We considered the length of the 95% CI as a measure of standard deviation, and define the variance efficiency of a method compared to the computationally efficient SIMEX method as the square of the ratio of efficient SIMEX interval length to the method’s interval length. For the methods the CI were not computed, the deviance intervals were used.
In Scenario 1, the variance efficiency in λ0 for the Parametric ML, Nonparametric ML, Parametric regression calibration, Nonparametric regression calibration and Ordinary SIMEX compared to computationally Efficient SIMEX were 100%, 71%, 59%, 118% and 73%, respectively. The corresponding efficiencies for Scenario 2 were 121%, 150%, 100%, 125% and 79%, respectively.
In Scenario 1, the variance efficiency in Excess Absolute Risk for the Parametric ML, Nonparametric ML, Parametric regression calibration, Nonparametric regression calibration and Ordinary SIMEX compared to computationally Efficient SIMEX were 125%, 94%, 158%, 83% and 155%, respectively. The corresponding efficiencies for Scenario 2 were 125%, 81%, 99%, 107% and 137%, respectively.