The models for dependence on dose are generally incorporated into risk models by assuming that the excess risk functions are proportional to *f*(*d*), where the multiplicative constant (in dose) depends on *a*, *e*, and *s*.

In general, cancer rates vary considerably as functions of attained age, and there is strong evidence indicating that cancer risks associated with radiation exposure also vary as functions of attained age and age at exposure. For example, it has been observed that after instantaneous exposure to radiation, leukemia and bone cancer rates rise for a short period of time (≈ years) and then decrease to baseline rates over a longer period of time (≈ years). In contrast, the available evidence suggests, and it is generally believed, that rates for most other cancers increase after exposure to radiation and possibly remain at elevated levels at all ages.

Models for the dependence of risk on variables such as age at exposure, attained age, and time since exposure are often empirical and are justified more by epidemiologic and statistical principles than by radiobiological theory. A useful class of models that includes the modifying effects on radiation dose-response of attained age, age at exposure, and gender has the form

for EAR models, and

for ERR models, where *g*(*a*, *e*, *s*) is a function of attained age, age at exposure, and gender. Because time since exposure is equal to the difference *t* = *a* − *e*, this class of models includes models defined as functions of time since exposure. Often *g* depends on *e* and *t* via exponential and power functions.

For example, the committee’s preferred model for solid cancer uses

where is *e* − 30 years if *e* is less than 30, and 0 if *e* is greater than or equal to 30; and γ, η, and θ are unknown parameters, which must be estimated from the data.

Models describe the mathematical form of a risk function, but the parameters in the model must be estimated from data. For example, a linear dose model presupposes that risk increases linearly with dose but the slope of the line, which measures the increase in risk for a unit increase in dose, must be estimated from data. Similarly, models for the effect of modifying factors depend on parameters that must be estimated from data. The most common method of fitting risk model data (*i.e.*, estimating the unknown parameters in the model) is the method of maximum likelihood reference. Given a model for the probability density of the observed data, a likelihood is obtained by evaluating the density at the observed data. The likelihood is a function of the data and the unknown parameters in the probability density model. The parameters are estimated by those values in the parameter space (the set of all allowable parameter values) that maximize the likelihood for the given data values.

There are several approaches for the numerical calculations of likelihood analysis. Estimation based on grouped data using a Poisson form of the likelihood (Clayton and Hills 1993) has been used for the analyses of atomic bomb survivors and other major epidemiologic studies of radiation health risks.

This analysis is facilitated by forming a table so that individuals contributing information to each cell of the table have equal, or approximately equal, background rates. In particular, the table is formed by the cross-classification of individuals into categories of age at exposure, time period, exposure dose, and all other variables that appear in the model. The key summary variables required for each cell are the total person-years (PY) of observation in the cell, the number of new cases of cancer, the mean dose, the mean age at exposure, and the mean age or mean time since exposure.

For an RR model, the contribution to the likelihood from the data in each cell of the table has the same form as a Poisson likelihood (thus permitting well-understood and straightforward computations), with the mean equal to the product of PY; a parameter for the common, cell-specific background rate; and the RR 1 + *fg*, where *f* and *g* are functions of dose and of age, age at exposure, and sex, described previously.

The full likelihood is the product of the cell-specific Poisson likelihoods. Numerical optimization is required to maximize the likelihood, and statistical inference generally is based on large-sample approximations for maximum likelihood estimation.

The models developed as described above can be used to estimate both lifetime risks and probabilities of causation, both of which are discussed below. Following this, several limitations in the use of these models, which lead to uncertainties in estimated risks, are discussed. Further discussion of uncertainties and the committee’s approach to quantifying them can be found in Chapter 12.

To calculate the lifetime risk for a particular age at exposure and a particular gender, one essentially follows a sub-