Table 1 Research questions investigated and description of applied missingness models and prior structures for δ_ik

A1

pattern-mixture

identical

intervention-specific

MAR on average

\( {\delta}_{ik}={\delta}_{t_{ik}} \), \( {\delta}_{t_{ik}}\sim N\left(0,1\right) \)

MAR on average is compared with extreme scenarios (on average). Intervention 1 is the reference of the network.

MME on average

\( {\delta}_{ik}={\delta}_{t_{ik}} \), \( {\delta}_{t_{ik}}\sim N\left(\mathit{\ln}(2),1\right) \)

MMNE on average

\( {\delta}_{ik}={\delta}_{t_{ik}} \), \( {\delta}_{t_{ik}}\sim N\left(-\mathit{\ln}(2),1\right) \)

BC on average

\( {\delta}_{ik}={\delta}_{t_{ik}} \), \( {\delta}_{t_{ik}}\sim N\left(\mathit{\ln}(2),1\right) \),  t_ik ≠ 1

\( {\delta}_{ik}={\delta}_{t_{ik}} \), \( {\delta}_{t_{ik}}\sim N\left(-\mathit{\ln}(2),1\right) \),  t_ik = 1

WC on average

\( {\delta}_{ik}={\delta}_{t_{ik}} \), \( {\delta}_{t_{ik}}\sim N\left(-\mathit{\ln}(2),1\right) \),  t_ik ≠ 1

\( {\delta}_{ik}={\delta}_{t_{ik}} \), \( {\delta}_{t_{ik}}\sim N\left(\mathit{\ln}(2),1\right) \),  t_ik = 1

A2

pattern-mixture

identical

intervention-specific

MAR fixed

\( {\delta}_{ik}={\delta}_{t_{ik}} \), \( {\delta}_{t_{ik}}=0 \)

‘On average’ scenarios are compared with corresponding ‘fixed’ scenarios. Intervention 1 is the reference of the network.

MME fixed

\( {\delta}_{ik}={\delta}_{t_{ik}} \), \( {\delta}_{t_{ik}}=\mathit{\ln}(2) \)

MMNE fixed

\( {\delta}_{ik}={\delta}_{t_{ik}} \), \( {\delta}_{t_{ik}}=-\mathit{\ln}(2) \)

BC fixed

\( {\delta}_{ik}={\delta}_{t_{ik}} \), \( {\delta}_{t_{ik}}=\mathit{\ln}(2) \),  t_ik ≠ 1

\( {\delta}_{ik}={\delta}_{t_{ik}} \), \( {\delta}_{t_{ik}}=-\mathit{\ln}(2) \),  t_ik = 1

WC fixed

\( {\delta}_{ik}={\delta}_{t_{ik}} \), \( {\delta}_{t_{ik}}=-\mathit{\ln}(2) \),  t_ik ≠ 1

\( {\delta}_{ik}={\delta}_{t_{ik}} \), \( {\delta}_{t_{ik}}=\mathit{\ln}(2) \),  t_ik = 1

B1

pattern-mixture

identical

common-within-network

MAR on average

δ_ik = δ, δ~N(0, 1)

Identical is compared with hierarchical for each structural assumption.

trial-specific

δ_ik = δ_i, δ_i~N(0, 1)

intervention-specific

\( {\delta}_{ik}={\delta}_{t_{ik}} \), \( {\delta}_{t_{ik}}\sim N\left(0,1\right) \)

hierarchical

common-within-network

MAR on average

δ_ik~N(Δ, σ²), Δ~N(0, 1), σ~U(0, 1)

trial-specific

\( {\delta}_{ik}\sim N\left({\varDelta}_i,{\sigma}_i^2\right) \), Δ_i~N(0, 1), σ_i~U(0, 1)

intervention-specific

\( {\delta}_{ik}\sim N\left({\varDelta}_{t_{ik}},{\sigma}_{t_{ik}}^2\right) \), \( {\varDelta}_{t_{ik}}\sim N\left(0,1\right) \), \( {\sigma}_{t_{ik}}\sim U\left(0,1\right) \)

B2a

pattern-mixture

identical

as in analysis B1

MAR on average

as in analysis B1

Structural assumptions are compared with each other.

B2b

pattern-mixture

hierarchical

as in analysis B1

MAR on average

as in analysis B1

Structural assumptions are compared with each other.

B3

pattern-mixture

identical

intervention-specific

MAR on average

\( {\delta}_{ik}={\delta}_{t_{ik}} \), \( {\delta}_{t_{ik}}\sim N\left(0,1\right) \)

Pattern-mixture model is compared with selection model for each structural assumption.

selection

C1

pattern-mixture

identical

intervention-specific

MAR on average

\( {\delta}_{ik}={\delta}_{t_{ik}} \), \( {\delta}_{t_{ik}}\sim N\left(0,1\right) \) – moderate

\( {\delta}_{ik}={\delta}_{t_{ik}} \), \( {\delta}_{t_{ik}}\sim N\left(\mathrm{0,0.25}\right) \) – liberal

\( {\delta}_{ik}={\delta}_{t_{ik}} \), \( {\delta}_{t_{ik}}\sim N\left(0,4\right) \) – conservative

Moderate prior variance for δ_ik is compared with liberal and conservative prior variance.