  Standard Mendelian randomization analysis can produce biased results if the
genetic variant defining the instrumental variable (IV) is confounded and/or
has a horizontal pleiotropic effect on the outcome of interest not mediated by
the treatment. We provide novel identification conditions for the causal effect
of a treatment in presence of unmeasured confounding, by leveraging an invalid
IV for which both the IV independence and exclusion restriction assumptions may
be violated. The proposed Mendelian Randomization Mixed-Scale Treatment Effect
Robust Identification (MR MiSTERI) approach relies on (i) an assumption that
the treatment effect does not vary with the invalid IV on the additive scale;
and (ii) that the selection bias due to confounding does not vary with the
invalid IV on the odds ratio scale; and (iii) that the residual variance for
the outcome is heteroscedastic and thus varies with the invalid IV. We formally
establish that their conjunction can identify a causal effect even with an
invalid IV subject to pleiotropy. MiSTERI is shown to be particularly
advantageous in presence of pervasive heterogeneity of pleiotropic effects on
additive scale, a setting in which two recently proposed robust estimation
methods MR GxE and MR GENIUS can be severely biased. In order to incorporate
multiple, possibly correlated and weak IVs, a common challenge in MR studies,
we develop a MAny Weak Invalid Instruments (MR MaWII MiSTERI) approach for
strengthened identification and improved accuracy MaWII MiSTERI is shown to be
robust to horizontal pleiotropy, violation of IV independence assumption and
weak IV bias. Both simulation studies and real data analysis results
demonstrate the robustness of the proposed MR MiSTERI methods.
