Table 1 Variables included in the simulation study, generating models, and notation

Number of simulated data sets

N _sim

Sample size each data set

N _obs

Treatment allocated

Z _i  = 0/1 for standard of care/intervention

Treatment received

X _i  = 0/1 for standard of care/intervention

Baseline hazard function for time until CVD event

\( {h}_0\left({t}_c\right)={\gamma}_c{\lambda}_c^{\gamma_c}{t}_c^{\gamma_c-1},\kern0.5em {\lambda}_c=36,\kern0.62em {\gamma}_c=1.2 \)

Baseline hazard function for time mortality

\( {h}_0\left({t}_m\right)={\gamma}_m{\lambda}_m^{\gamma_m},\kern0.5em {\lambda}_m=55,\kern0.5em {\gamma}_c=1.2 \)

Individual random effects

ε _i  ~ N(0, σ ²), σ = 0.7

Intervention effect

β = − 0.32

Individual hazard function for time until CVD event

\( h\left({t}_{i c}\right) = {h}_0\left({t}_c\right)\ {e}^{\left(\beta {X}_i+{\varepsilon}_i\right)} \)

Individual hazard function for time until mortality

\( h\ \left({t}_{i m}\right)={h}_0\left({t}_m\right)\ {e}^{\left({\varepsilon}_i\right)} \)

Probability of a patient refusing intervention if offered and average patient refusal

\( \begin{array}{l}{p}_i, p={\varSigma}_i{p}_i/{N}_{obs}\\ {}\end{array} \)

Probability of clinician refusing to offer the intervention to the patient and average clinician refusal

\( \begin{array}{l}{q}_i, q={\varSigma}_i{q}_i/{N}_{obs}\\ {}\end{array} \)

Trial follow-up time in years

T ^max = 3

Censoring indicator

C _i  = I(T _ic  ≥ min(T _im , T _max  ))

Observed outcome

Y _i  = min(T _ic , T _im , T _max )

Observed trial data

{Y _i , C _i , Z _i , X _i }