Table 2 Models for the analysis of pnRCTs

Model 1

Linear regression (ignore clustering)

y_i = β₀ + θt_i + ϵ_i

    • \( {\epsilon}_i\sim N\left(0,{\sigma}_{\epsilon}^2\right) \) the individual level variation

No

Model 2

Fully clustered (impose clustering)

y_ij = β₀ + θt_ij + u_j + ϵ_ij

    • \( {u}_j\sim N\left(0,{\sigma}_u^2\right) \) a random effects term representing between cluster        variation

    • \( {\epsilon}_{ij}\sim N\left(0,{\sigma}_{\epsilon}^2\right) \) the individual level variation

No

Model 3

Partially nested homoscedastic

y_ij = β₀ + θt_ij + u_jt_ij + ϵ_ij

    • \( {u}_j\sim N\left(0,{\sigma}_u^2\right) \) a random effects term representing between-cluster variation        in clustered arm

    • \( {\epsilon}_{ij}\sim N\left(0,{\sigma}_{\epsilon}^2\right) \) the individual level variation

No

Model 4

Partially nested heteroscedastic

y_ij = β₀ + θt_ij + u_jt_ij + r_ij(1 − t_ij) + ϵ_ijt_ij

    • \( {u}_j\sim N\left(0,{\sigma}_u^2\right) \) a random effects term representing between cluster-variation        in clustered arm

    • \( {r}_{ij}\sim N\left(0,{\sigma}_r^2\right) \) the individual level variation in the non-clustered control arm.

    • \( {\epsilon}_{ij}\sim N\left(0,{\sigma}_{\epsilon}^2\right) \) the individual level variation in the clustered arm

Yes