Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
The main outcome is a binary indicator for any rearrest by T* months. Using PA DOC's 2025 facility release counts as the planning benchmark, a one-year rollout yields approximately 10,331 release events before exclusions. Under the planned 40/30/30 allocation, the expected group sizes are about 4,132 control, 3,099 T1, and 3,099 T2. At a benchmark control-group rearrest rate of 0.35, the binary-outcome standard deviation is sqrt(0.35 x 0.65) = 0.477. Under the exact two-sample difference-in-proportions calculation with alpha = 0.05, two-sided, and 80 percent power, the minimum detectable effect is 3.1 percentage points for T1 versus Control and T2 versus Control, and 3.4 percentage points for T1 versus T2. Under a conservative 75 percent delivery scenario, the treatment-versus-control MDE rises to about 4.2 to 4.4 percentage points. As a clustering sensitivity, simulation-based power calculations that allow facility-level ICC up to 0.01 yield MDEs of about 3.3 percentage points for treatment versus control and about 3.5 percentage points for T1 versus T2, reflecting individual randomization within facility and analysis models with facility fixed effects.