showing that EAR(*t*) describes the additive increase in incidence rate associated with exposure. For example, if the EAR is constant, EAR(*t*) = *b*, then the effect of exposure is to increase the incidence rate by the constant amount *b* for all time periods. Note that *b* = 0 corresponds to the case of no association.

A second common measure of discrepancy is the *relative risk* (RR), defined as

Rearranging terms shows that

so that RR(*t*) describes the multiplicative increase in incidence rate associated with exposure. When the RR is constant, RR(*t*) = *r,* the effect of exposure is to alter incidence rate by the factor *r*. If exposure increases risk, then *r* > 1; if exposure decreases risk, then *r* < 1, and *r* = 1 corresponds to the case of no association. The *excess relative risk* ERR(*t*) is

The ERR of the exposed and unexposed incidence rates are related via the equation

The previous section defined the fundamental quantities used in risk estimation: risks, rates, EAR, RR, and ERR, and established their relevance to the study of environmental carcinogens. These measures enable the study of differences in disease occurrence in relationship to time, by studying either EAR(*t*) or ERR(*t*) between unexposed and exposed groups. For most carcinogens, exposure is not a simple dichotomy (unexposed, exposed) but occurs on a continuum. That is, the exposure or dose *d* can vary from no exposure (*d* = 0) upward. In such cases the relationship between risk—or EAR(*t*) or ERR(*t*)—and dose is of fundamental importance. For all carcinogens it is generally agreed that sufficiently large doses increase the risk of cancer. By definition there is no increase in risk in the absence of exposure (*d* = 0). That is, when *d* = 0, both EAR(*t*) = 0 and ERR(*t*) = 0. Thus, for many carcinogens the only open or unresolved issue is the dependence of risk on small or low doses. Low-dose ranges are often the most relevant in terms of numbers of exposed individuals. They are also the most difficult ranges for which to obtain unequivocal evidence of increased risk. These difficulties result from the fact that small increases in risk associated with low levels of exposure are difficult to detect (using statistical methods) in the presence of background risks.

The difficulties can be seen by considering the estimates of risk from the longitudinal follow-up study described in “Rates, Risks, and Probability Models.” For a time period *L _{j}*, let

A key objective of this report is the calculation of quantitative estimates of human health risks (*e.g.*, cancer) associated with exposure to ionizing radiation for specific subpopulations defined by stratification on variables such as sex, age, exposure profile, and smoking history. In theory, such estimates could be derived by identifying a large group of individuals having common exposure profiles within each stratum and following the groups over a long period of time. As described above, the proportion of individuals in each group who develop cancer in specific time periods provides the desired estimates of risk. However, this approach is not feasible because sufficient data are not available. At low levels of exposure, cancer risks associated with exposure are small relative to baseline or background risks. The increases in observed cancer rates associated with exposure are small relative to the natural random fluctuations in baseline cancer rates. Thus, very large groups of individuals would have to be followed for very long periods of time to provide sufficiently precise estimates of risk associated with exposure. Consequently, direct estimates of risk are not possible for stratified subpopulations. The alternative is to use mathematical models for risk as functions of dose and stratifying variables such as sex and age.

Model-based estimation provides a feasible alternative to direct estimation. Model-based estimates efficiently exploit the information in the available data and provide a means of deriving estimates for strata and dose profile combinations for which data are sparse. This is accomplished by exploiting assumptions about the functional form of a risk model. Of course, the validity of estimates derived from models depends on the appropriateness of the model; thus model choice is important. The accepted approach in radiation epi-