Table 1.

Description and parameter formulisation across each simulation scenario

Simulation Scenario	Description	Parameter values
RCT: 10% dropout†	A randomized controlled	ϕ = 0
	trial with treatment	α0 s. t. P (A0 = 1) = 0.5
	randomly allocated to 50%	${\mathrm{\pi }}_{\mathrm{i},{\mathrm{A}}_1}=\theta a_{0i}$
	of the population at baseline,	θ = 0.9
	with 10% treatment dropout.	αY s. t. P (Y = 1) = 0.2
		${\beta }_{A_0}={\beta }_{A_1}=\log \left(0.5\right)$
		${\beta }_{X_0}={\beta }_{X_1}=\log \left(1.5\right)$
Observational: 50%	An observational study	ϕ =  log (2)
treated	where 50% of the population	θ =  log (2)
	have treatment.	αj : j = 0, 1 s. t. P (Aj = 1) = 0.5
		αY s. t. P (Y = 1) = 0.2
		${\beta }_{A_0}={\beta }_{A_1}=\log \left(0.5\right)$
		${\beta }_{X_0}={\beta }_{X_1}=\log \left(1.5\right)$
Observational: 20%	An observational study	ϕ =  log (2)
treated	where 20% of the population	θ =  log (2)
	have treatment.	αj : j = 0, 1 s. t. P (Aj = 1) = 0.2
		αY s. t. P (Y = 1) = 0.2
		${\beta }_{A_0}={\beta }_{A_1}=\log \left(0.5\right)$
		${\beta }_{X_0}={\beta }_{X_1}=\log \left(1.5\right)$

^† : results from across a range of percentage dropouts (values of θ) gave similar results as those for the RCT: 10% dropout scenario and so are omitted. They are available on request.