Cancer Risk Modeling with Radiation and Smoking

Regression models to describe cancer risks included a description of the rates for unexposed (zero dose) nonsmokers (baseline rate) with additional terms for radiation and smoking effects. We described the joint effects of radiation and smoking in various ways, including additive and multiplicative ERR models and additive excess rate models (EAR). Ignoring smoking, the ERR model was:
$$BK{G}_{ALL}\left(1+ER{R}_{rad}\right),$$
where $BK{G}_{ALL}$ represents baseline rates for those not exposed to radiation (i.e., unexposed), and $ER{R}_{rad}$ was the excess relative risk for radiation exposure.
The multiplicative ERR model for the joint effect of radiation and smoking was:
$$BK{G}_{NS}\left(1+ER{R}_{smk}\right)\left(1+ER{R}_{rad}\right),$$
in which $BK{G}_{NS}$ was the baseline rate for unexposed nonsmokers, $ER{R}_{smk}$ was the excess relative risk for smoking, and $ER{R}_{rad}$ was the excess relative risk for radiation. In this model, $ER{R}_{rad}$ described the radiation-associated proportional increase in rates relative to unexposed people with the same smoking history. If smoking was not an effect modifier (that is, $ER{R}_{rad}$ did not depend on smoking history), this increase was independent of smoking history.
The additive ERR model of the joint effect of radiation and smoking was:
$$BK{G}_{NS}\left(1+ER{R}_{smk}+ER{R}_{rad}\right).$$
In this model, $ER{R}_{rad}$ describes the radiation-associated proportional increase in rates relative to the risk for unexposed, nonsmokers.
An additive excess rate (or EAR) model for the joint effect of radiation and smoking on cancer rates was:
$$BK{G}_{NS}+EA{R}_{smk}+EA{R}_{rad},$$
where $EA{R}_{smk}$ and $EA{R}_{rad}$ described the smoking and radiation effects in terms of rate differences.
What follows are details of the model forms used for the baseline, ERR and EAR terms considered in these analyses.