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Table 1 Description of all variables used in simulation

From: Cohort Multiple Randomised Controlled Trials (cmRCT) design: efficient but biased? A simulation study to evaluate the feasibility of the Cluster cmRCT design

Number of patients in cohort, control arm and intervention arm

N, Ncon, Nint

Number of clusters in trial

K

Size of each cluster

J = 620

Treatment allocated to kth cluster

Zk = 0/1 for control/intervention

Treatment received by ith individual from kth cluster

Xik = 0/1 for control/intervention

Time until CVD event for ith individual from the kth cluster

\( {\mathrm{T}}_{\mathrm{ik}}^{\mathrm{c}}\sim \mathrm{Weibull}\left({\upgamma}_{\mathrm{c}},{\uplambda}_{\mathrm{c}}{\mathrm{e}}^{-\left(\upbeta {\mathrm{X}}_{\mathrm{ik}}+{\upvarepsilon}_{\mathrm{ik}}+{\mathrm{U}}_{\mathrm{k}}\right)/{\upgamma}_{\mathrm{c}}}\right) \)

Time until mortality (censoring distribution) for the ith individual from the kth cluster

\( {\mathrm{T}}_{\mathrm{ik}}^{\mathrm{m}}\sim \mathrm{Weibull}\left({\upgamma}_{\mathrm{m}},{\uplambda}_{\mathrm{m}}{\mathrm{e}}^{-\left({\upvarepsilon}_{\mathrm{ik}}+{\mathrm{U}}_{\mathrm{k}}\right)\kern0.1em /\kern0.1em {\upgamma}_{\mathrm{m}}}\right) \)

Common baseline hazard function for time until CVD event

\( {\mathrm{h}}^{\mathrm{c}}\left(\mathrm{t}\right)={\upgamma}_{\mathrm{c}}{\mathrm{t}}^{\upgamma_{\mathrm{c}}-1}/\kern0.1em {\uplambda_{\mathrm{c}}}^{\upgamma_{\mathrm{c}}},\kern1em {\upgamma}_{\mathrm{c}}=1.2,{\uplambda}_{\mathrm{c}}=36 \)

Common baseline hazard function for time until mortality

\( {\mathrm{h}}^{\mathrm{m}}\left(\mathrm{t}\right)={\upgamma}_{\mathrm{m}}{\mathrm{t}}^{\upgamma_{\mathrm{m}}-1}/{\uplambda_{\mathrm{m}}}^{\upgamma_{\mathrm{m}}},\kern0.5em {\upgamma}_{\mathrm{m}}=1.2,{\uplambda}_{\mathrm{m}}=55 \)

Individual hazard function for time until CVD event

\( {\mathrm{h}}_{\mathrm{ik}}^{\mathrm{c}}\left(\mathrm{t}\right)={\mathrm{h}}^{\mathrm{c}}\left(\mathrm{t}\right){\mathrm{e}}^{\left({\upvarepsilon}_{\mathrm{ik}}+{\mathrm{U}}_{\mathrm{k}}+\upbeta {\mathrm{X}}_{\mathrm{ik}}\right)} \)

Individual hazard function for time until mortality

\( {\mathrm{h}}_{\mathrm{ik}}^{\mathrm{m}}\left(\mathrm{t}\right)={\mathrm{h}}^{\mathrm{m}}\left(\mathrm{t}\right){\mathrm{e}}^{\left({\upvarepsilon}_{\mathrm{ik}}+{\mathrm{U}}_{\mathrm{k}}\right)} \)

Individual level random effects

ɛ ik N(0, σ ɛ 2)

Cluster level random effects

U k N(0, σ u 2)

Intervention effect

β = − 0.32

Ten year risk of a CVD event

rik = P(T cik  < 10| Xik = 0, εik, Uk)

Individual and average probability of patient refusing treatment

\( {p}_{ik},\;p={\displaystyle {\sum}_{i,k}\frac{p_{ik}}{N}} \)

Individual and average probability of clinician refusing to offer treatment

\( {q}_{ik},\kern0.5em q={\displaystyle {\sum}_{i,k}\frac{q_{ik}}{N}} \)

Correlation between patient refusal probability and patient risk

ρ p

Correlation between clinician refusal probability and patient risk

ρ q

Censoring indicator

C ik  = I(T ik c ≥ min(T ik m, T max ))

Trial follow up time

T max  = 3

Random variable observed for each patient

Y ik  = min(T ik c, T ik m, T max )