of reduced risk. An independently acting background cause (or aggregation of causes not including A or B) can be allowed for as yet another independently operating factor, so that if Pr[D | A,B] is the probability of D in the presence of both exposures, simple independent action would require: Pr[D¯|A,B]Pr[D¯|A¯,B¯] = Pr[D¯|A,B¯]Pr[D¯|A¯,B]Pr[D¯|A¯,B¯]Pr[D¯|A¯,B¯], where the overbar indicates nonoccurrence of the exposure or outcome. The idea again is that the probability of escaping the causes associated with A, with B, and the causes associated with the background factors is the product of the three separate escape probabilities if A and B act independently of each other and of all other unmeasured causal factors (background). Taking logarithms yields a model that is additive in the log-complement, adjusted by the background: {ln(1−Pr[D|A,B])−ln(1−Pr[D|A¯,B¯])}={ln(1−Pr[D|A,B¯])−ln(1−Pr[D|A¯,B¯])}+{ln(1−Pr[D|A¯,B])−ln(1−Pr[D|A¯,B¯])},, Suppose D is rare, with risk near to 0. Then, because mathematically -ln(1-x) converges to x as x goes to 0, this condition is approximately the same as: (Pr[D|A,B]−Pr[D|A¯,B¯])=(Pr[D|A,B¯]−Pr[D|A¯,B¯])+(Pr[D|A¯,B]−Pr[D|A¯,B¯]), that is, additivity on the absolute risk scale. Even if D is not rare, if one is looking at hazards across instantaneous time the event is rare in each small time interval. Thus if λAB(t) is the instantaneous hazard rate at time t
