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Table 2 Data generation models for simulations under each scenario

From: Accounting for confounding by time, early intervention adoption, and time-varying effect modification in the design and analysis of stepped-wedge designs: application to a proposed study design to reduce opioid-related mortality

Scenario Data generating model and scenario description Impact on outcome Index
Standard log(μij)=α+θ×Xij+biDescription: No confounding, early adoption, or effect modification None. 1
Confounding log(μij)=α+θ×Xij+βij+biDescription: 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. βijunif[−1,0] if cluster i exposed during time period j; 0 otherwise. 2.1
   βijunif[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. θijunif[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. βijunif[−1,0] if cluster i exposed to confounding event during time period j; 0 otherwise. θij is defined as above. 4.1
   βijunif[0,1] if cluster i exposed to confounding event during time period j; 0 otherwise. θij is defined as above. 4.2
  1. Data is simulated under 4 general scenarios. The data generating model for each simulation scenario is displayed in the second column. Here μij 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 Xij is an indicator of whether cluster i is scheduled to receive intervention during time period j and is based on the SWD represented by Fig. 1. The fixed intercept α is set to -10 and the random intercept bi 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)