Now note that under independence of A and B in the source population, the ratio in the denominator is 1 and what is left is the numerator ratio, i.e. the odds ratio for A versus B among cases only. The remarkable feature of this design is that its efficiency for detecting departures from multiplicative joint effects is as good as that of a study with an infinitely large control group, because in effect the controls are only serving to estimate that denominator “1” with increasing precision. However, the user should beware, because the limitations are drastic for the case-only design: even when the primary assumption holds, main effects cannot be studied at all and only multiplicative interactions can be assessed. Often a genetic and an environmental factor will not occur independently in the population, violating the method’s primary assumption, because attrition of risk factors can be jointly selective. These constraints handicap case-only inference related to joint effects.
