Table 2 Data generation models for simulations under each scenario

Standard

log(μ_ij)=α+θ×X_ij+b_iDescription: No confounding, early adoption, or effect modification

None.

1

Confounding

log(μ_ij)=α+θ×X_ij+β_ij+b_iDescription: At each time period j, \(n^{\prime }_{j}\) clusters randomly exposed to event inducing confounding for remainder of study period; \(n^{\prime }_{j}\sim Binomial(N'_{j},1/N)\), where \(n^{\prime }_{j}\) is total number of clusters unexposed to event prior to time period j.

β_ij∼unif[−1,0] if cluster i exposed during time period j; 0 otherwise.

2.1

β_ij∼unif[0,1] if cluster i exposed during time period j; 0 otherwise.

2.2

Early adoption

\(\log (\mu _{ij}) = \alpha +\theta _{ij}\times 1_{\{X_{ij}=0\}} + \theta \times X_{ij} + b_{i}\)Description: At each time period j, \(n^{*}_{j}\) control clusters prematurely adopt intervention components; \(n^{*}_{j}\sim Binomial(N^{*}_{j},\frac {N-N^{*}_{j}+1}{2\times N})\), where \(n^{*}_{j}\) is the number of control clusters not receiving the intervention prior to time period j.

θ_ij∼unif[θ,0] if control cluster i prematurely adopts intervention at time period j; 0 otherwise.

3

Confounding + Early adoption (or Effect modification)

\(\log (\mu _{ij}) = \alpha +\beta _{ij}+\theta _{ij}\times 1_{\{X_{ij}=0\}} + \theta \times X_{ij} + b_{i}\)Description: At each time period j, \(n^{\prime }_{j}\) clusters are randomly exposed to confounding events and \(n_{j}^{*}\) control clusters prematurely adopt intervention components, where \(n^{\prime }_{j}\) and \(n_{j}^{*}\) are defined above. Control clusters may be exposed to both confounding factors and early adoption. Data generation model for effect modification is similar.

β_ij∼unif[−1,0] if cluster i exposed to confounding event during time period j; 0 otherwise. θ_ij is defined as above.

4.1

β_ij∼unif[0,1] if cluster i exposed to confounding event during time period j; 0 otherwise. θ_ij is defined as above.

4.2