Table 1. Columns left to right show: Statistical formulae for GMTE(1), GMTE(0), RGMTE, MR and CAT estimates; Sufficient assumptions each one relies upon to consistently estimate the GMTE (or zero in the case of the GMTE(0) estimate); Estimate-specific confounder test statistics; generic R code to obtain each estimate.

For the GMTE(0) estimate, $T_{CAT}=\widehat{E}\left[G\right|T=1]T$, for the GMTE(0) estimate T⁻ = 1 − T, T*⁻ = T⁻ G, for the RGMTE estimate T* = TG and ${\widehat{T}}^{*}=\widehat{E}\left[T^{*}\right|G]$. Note that the GMTE(0) estimate does not directly target the GMTE, but rather zero under the PG assumptions.

Estimate	Statistical Formula	Sufficient Assumptions	Confounder Test	Fit in R
CAT
${\widehat{\beta }}_{CAT}\left(Y\right)$	$\frac{\widehat{E}\left[Y\right|T=1]-\widehat{E}\left[Y\right|T=0]}{\widehat{E}\left[G\right|T=1]}$	{PG1 ∪ PG3} ∩ NUC ∩ Hom	${\widehat{\beta }}_{CAT}\left(G\right)=0,{\widehat{\beta }}_{CAT}\left(Z\right)=0$	Coef. of TCAT in Y ∼ TCAT + Z
GMTE(0)
${\widehat{\beta }}_{GMTE\left(0\right)}\left(Y\right)$	$\widehat{E}\left[Y\right|T=0,G=1]-\widehat{E}\left[Y\right|T=0,G=0]$	PG3 ∩ {(PG1 ∩ PG2) ∪ NUC}	${\widehat{\beta }}_{GMTE\left(0\right)}\left(Z\right)=0$	Coef. of T*− in Y ∼ T− + T*− + Z
GMTE(1)
${\widehat{\beta }}_{GMTE\left(1\right)}\left(Y\right)$	$\widehat{E}\left[Y\right|T=1,G=1]-\widehat{E}\left[Y\right|T=1,G=0]$	PG3 ∩ {(PG1 ∩ PG2) ∪ NUC}	${\widehat{\beta }}_{GMTE\left(1\right)}\left(Z\right)=0$	Coef. of T* in Y ∼ T + T* + Z
RGMTE
${\widehat{\beta }}_{RGMTE}\left(Y\right)$	${\widehat{\beta }}_{GMTE\left(1\right)}\left(Y\right)-{\widehat{\beta }}_{GMTE\left(0\right)}\left(Y\right)$	PG1∪NUC	${\widehat{\beta }}_{RGMTE}\left(Z\right)=0$	Coef. of T* in $Y\sim T+T^{*}+{\widehat{T}}^{*}+Z$
MR
${\widehat{\beta }}_{MR}\left(Y\right)$	$\frac{\widehat{E}\left[Y\right|G=1]-\widehat{E}\left[Y\right|G=0]}{\widehat{E}\left[T^{*}\right|G=1]-\widehat{E}\left[T^{*}\right|G=0]}$	{PG1 ∪ Hom} ∩ {PG2 ∪ NUC} ∩ PG3	${\widehat{\beta }}_{MR}\left(Z\right)=0$	Coef. of ${\widehat{T}}^{*}$ in $Y\sim {\widehat{T}}^{*}+Z$