If genetic variant j is not a valid IV, in particular because it has a direct effect on the outcome (αj≠0), then we have βj=β+αjγj . The ratio estimate based on genetic variant j in an infinite sample will equal the true causal effect β plus an error term αjγj . In the same way, the TSLS and IVW estimates will tend towards: β+∑j=1JγjσYj−2αj∑j=1Jγj2σYj−2=β+Bias(α,γ). This implies that the TSLS estimate is consistent when the assumption IV3 is true and all the αj parameters are zero. It is also consistent if the pleiotropic effects happen to cancel out, such that the bias term is equal to zero.24 Although this will not be universally plausible, we explore the condition that the correlation between the genetic associations with the exposure (the γj parameters) and the direct effects of the genetic variants on the outcome (the αj parameters) is zero. We refer to the condition that the distributions of these parameters are independent as InSIDE (Instrument Strength Independent of Direct Effect). It can be viewed as a weaker version of the exclusion restriction assumption.
