Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
Our power calculations assume a set of controls with an R-squared of 0.10, and a randomization ratio of 1 person in the treatment group for 3 people in the control group. With these inputs, our target sample size for the study is 880 consenting participants, which we expect to reach in about 2.5 years of enrollment.
For homelessness, we turn to prior research to guide our power calculations. We estimate that given our sample size and research design, we will be able to detect a 17 day reduction in homeless shelter days over 3 years, coming from a baseline of 269 days. This estimate assumes a normal distribution of outcomes and is based on a control group mean from Cunningham et al. (2021), which found a reduction of 96 shelter stay days over 3 years for the treated group.
For recidivism as an outcome, we rely on historical data from CSH. CSH projects that those in the target population who do not enroll in RHO or CTP have a recidivism rate of approximately 30 percent within three years, and that those who do enroll in RHO or CTP have a recidivism rate of 7 percent within the same time frame, a 23 percentage-point drop. The study is powered to obtain a more conservative minimum detectable effect (MDE) of an 8.5 percentage-point change in three-year recidivism for the treatment group compared to the control group, from a baseline of 30 percent. Given the estimated impact of a 23 percentage point reduction in 3-year recidivism, an 8.5 percentage point MDE (or a 12.1 percentage point MDE in terms of treatment-on-the treated, assuming a 70% take-up rate) gives us confidence we will be able to detect significant effects with our design and sample size.
Separately, we find that we can detect a reduction of 0.7 arrests over 3 years, coming from a baseline of 10.6 arrests, assuming a normal distribution of outcomes. Cunningham et al. (2021) found a reduction of 4.3 arrests over 3 years for the treated group. This additional power calculation strengthens our confidence that we will be able to detect effects in key outcomes.
One challenge with securing housing for housing insecure, recently incarcerated individuals is that they can be transient and challenging to contact. To ensure high take up of PSH among the treated group, all individuals will be contacted to confirm interest and eligibility just prior to their randomization in the study. Given these precautions, we conservatively estimate for our power calculations that 70 percent of those assigned to the treatment group will follow through with housing placement through CSH. In a prior RCT of frequently arrested, chronically homeless individuals, nearly 88 percent (285 out of 325) of the treatment group who were successfully contacted secured a lease through the supportive housing program (Cunningham et al., 2021).
Cunningham, Mary K., Devlin Hanson, Sarah Gillespie, Michael Pergamit, Alyse D. Oneto, Patrick Spauster, Tracey O’Brien, Liz Sweitzer, and Christine Velez. (2021). Breaking the Homelessness-Jail Cycle with Housing First: Results from the Denver Supportive Housing Social Impact Bond Initiative. Urban Institute.