and . Then var[log(LAR)] may be estimated by
where V is the estimated variance-covariance matrix of , which is available as a component of the output from the computer program used to estimate the risk models. The standard error of the log of estimated LAR is the square root of the estimate of this variance. A 95% confidence interval for log (LAR) is obtained as the estimate of log (LAR) plus and minus 1.96 times the standard error, and the confidence interval for LAR is obtained by taking the antilogarithm of these end points.
The LAR based on absolute risk transport is
The issues and computations involve only slight modifications of what has been described above. For scenarios that involve a weighted average of different ages at exposure and for relative and absolute risk models for leukemia, which involve quadratic-in-dose terms and different modifiers including interactions, the computations differ but the ideas behind the delta method calculations are the same as above.
The confidence intervals in Tables 12-5A and 12-5B for risks of cancer incidence and mortality at specific sites were based on the same procedure as above, but without accounting for the uncertainty in γ and η, since, with a few exceptions, these quantities were fixed at their values estimated from all solid cancers combined (although the values of γ and η used in site-specific models were compatible with data for each site, the fixed values cannot be considered unbiased estimates of the correct values). For most sites, uncertainty in the estimated coefficient of dose (β) is quite large and is expected to dominate the uncertainty in the estimated LAR.
A single estimate of LAR is obtained from estimates based on ERR and EAR transport models as a combination on the log scale: log (LAR) = [p (log (LARERR) + (1 − p) log (LAREAR)], where LARERR and LAREAR are the estimates based on ERR and EAR transport, respectively, and p is a number between 0 and 1, reflecting the relative strength of belief in the two transport models. For most cancers, a value of .7 was taken for p. Exceptions were lung cancer, where p = .3, and thyroid cancer, where only an ERR model developed from data on Caucasian women was available. A further adjustment to the single estimate of LAR, due to the presumed curvature in the dose-response, is obtained by dividing this combined estimate by the presumed DDREF. A value of 1.5 was used for DDREF, which is an estimate of the median of the Bayesian posterior probability distribution for DDREF, as discussed in the chapter.
The uncertainty analysis here arrives at an approximate variance for log (LAR), emanating from the individual variances in LARERR and LAREAR (sampling variability from the LSS risk model estimation, as discussed above), p (uncertainty in the knowledge of whether absolute risk or excess risk is transportable from Japanese A-bomb survivors to the U.S. population), and DDREF (uncertainty in estimating dose-response curvature from animal studies and uncertainty with which the animal curvature applies to humans).
To accomplish this, the model above is written more formally as depending on four sets of unknown quantities: θR, the parameters in the relevant ERR model; θA, the parameters in the EAR model; IR, an indicator variable that takes on the value 1 if the ERR model is the correct one for transport and 0 if the EAR model is the correct one; and θDDREF, the unknown DDREF. The LAR associated with an acute radiation dose D at age e may be written as
where LARR(e, D; θR) and LARA(e, D; θA) are the LARs based on EAR and ERR transport, prior to DDREF adjustment, and θDDREF is the correct DDREF value. Notice that if the ERR model is the correct one for transport, then IR is 0 and the LAR expression above reduces to LARA(e, D; θA / θDDREF. Similarly, if the relative risk model is the correct one for transport, then the LAR expression reduces to the excess relative risk LAR with DDREF adjustment.
The estimated LAR can be expressed by the same formula, but with the known parameters replaced by their estimators: , where and are parameter estimates for the ERR and EAR models; is the (subjective) probability that the relative risk model is the correct one for transport; and is the (subjective) estimate of DDREF. Every quantity with a “hat” on it is an uncertain estimator and has a variance associated with it. The variance in the estimated LAR, consequently, is that which is propagated by the variances of these estimators.
Statistically, it is best to consider this propagation on the log scale:
With the simplifying approximation that the “hats” can be dropped from and in the middle term and the assumption that the uncertainties due to risk model estimation, subjective assessment of DDREF, and subjective as-