Table 1.

Data generation models for simulations under each scenario.

Scenario	Data generating model and scenario description	Impact on outcome	Index
Standard	log(µij) = α + θ × Xij + bi Description: No confounding, early adoption, or effect modification	None.	1
Confounding	log(µij) = α + θ × Xij + βij + bi Description: At each time period j, ${n^{\prime }}_j$ clusters randomly exposed to event inducing confounding for remainder of study period; ${n^{\prime }}_j\sim Binomial\left({N^{\prime }}_j,1/N\right)$, where ${N^{\prime }}_j$ is total number of clusters unexposed to event prior to time period j.	βij ∼ unif [−1, 0] if cluster i exposed during time period j; 0 otherwise.	2.1
		βij ∼ unif [0, 1] if cluster i exposed during time period j; 0 otherwise.	2.2
Early adoption	$\text{log}\left({\mu }_{ij}\right)=\alpha +{\theta }_{ij}\times 1_{\left\{X_{ij}=0\right\}}+\theta \times X_{ij}+b_i$ Description: At each time period j, $n_j^{*}$ control clusters prematurely adopt intervention components; $n_j^{*}\sim Binomial\left(N_j^{*},\frac{N-N_j^{*}+1}{2\times N}\right)$, where $N_j^{*}$ is the number of control clusters not receiving the intervention prior to time period j.	θij ∼ unif [θ, 0] if control cluster i prematurely adopts intervention at time period j; 0 otherwise.	3
Confounding + Early adoption (or Effect modification)	$\text{log}\left({\mu }_{ij}\right)=\alpha +{\beta }_{ij}+{\theta }_{ij}\times 1_{\left\{X_{ij}=0\right\}}+\theta \times X_{ij}+b_i$ 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 $n^{\prime }$ 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.	βij ∼ unif [−1, 0] if cluster i exposed to confounding event during time period j; 0 otherwise. θij is defined as above.	4.1
		βij ∼ unif [0, 1] if cluster i exposed to confounding event during time period j; 0 otherwise. θij is defined as above.	4.2

Data is simulated under 4 general scenarios. The data generating model for each simulation scenario is displayed in the second column. Here µ_ij is the expected rate of opioid overdose deaths in cluster i during time period j, θ is the intervention effect and is set to log(0.6), and X_ij is an indicator of whether cluster i is scheduled to receive intervention during time period j and is based on the SWD represented by Figure 1. The fixed intercept α is set to −10 and the random intercept b_i is simulated from a N (0, 0.30) distribution. A description of the selection process for exposure to confounding events or early adoption is provided in the second column (below the data generating model). The impact of confounding factors and/or early adoption on the outcome is detailed in the third column. In scenarios 2 and 4, we allow confounding factors to have either a positive impact on the outcome (scenarios 2.1 and 4.1) or a negative impact on the outcome (scenarios 2.2 and 4.2).