All statistical methods make assumptions (and those assumptions are inevitably always violated), but the extent to which they are violated and the impact of that violation on estimates is often unclear.

Our original method for non-linear Mendelian randomization (residual-stratified method) made a strong and unrealistic assumption that the effect of genetic variants on the exposure is constant for all individuals in the dataset.

While we assessed sensitivity to this assumption in the original methods paper, violations of this assumption in practice are stronger than we assessed, and realistic violations of the assumption can lead to substantial bias in practice.

We developed a second method (doubly-ranked method) which makes a strictly weaker assumption that the ordering of individuals' exposure values would be the same if their genetic instrument were fixed to take any value (rank preserving assumption).

This is a strictly weaker assumption than the constant genetic effect assumption, in that it allows the magnitude of the genetic effect on the exposure to vary, but it still requires some degree of homogeneity in the genetic effect on the exposure.

However, this assumption can also be violated. Enter Ang's manuscript! Ang shows that if we can model the heterogeneity in the genetic effect on the exposure, then we can correct for this heterogeneity in the doubly-ranked method.

For instance, genetic associations with 25(OH)D levels (a biomarker of vitamin D status) are larger in the summer and smaller in the winter, and genetic associations with several traits differ between men and women, and with socioeconomic markers.

If we subtract the GxE interaction from the exposure, then we can stratify on this corrected exposure value. This correction is only necessary in the stratification step; the estimation can proceed using the uncorrected exposure values.

This makes a substantial difference to estimates for LDL-cholesterol, and a detectable but much smaller difference to estimates for BMI and vitamin D. The obvious limitation is this only holds for GxE interactions we can measure and account for.