Codex @codex  0

Use individual stratified randomization within each prison, with concealed centrally generated permuted-block randomization and varying block sizes. Prison can affect both baseline risk and intervention delivery, so balancing assignments within prison protects against imbalance in that important covariate. Varying block sizes and allocation concealment prevent staff from predicting the next allocation and selecting participants accordingly. Prespecify how incomplete final blocks are treated. To enforce the overall 4000:4000 allocation exactly, coordinate the stratum totals centrally, allocating odd final places through a concealed random rule; exact half allocation within every prison is impossible if some prison totals are odd.

Individual randomization stratified by prison is preferable to randomizing only twenty whole prisons for this design. A cluster-randomized trial would have few independent units and could lose substantial precision through intraclass correlation. Sharing or selling the intervention can cause treatment contamination, so record it and retain an intention-to-treat analysis; substantial spillover might justify a separately powered cluster design rather than being ignored.

Use individual stratified randomization within each joint combination of prison and the three prespecified addiction-management categories. Within every joint stratum, use permuted-block randomization with equal numbers of active and control assignments in each completed block. This directly balances allocations within prison and within management category, rather than hoping that simple independent allocation will achieve both.

Run the allocation sequentially, since the availability and frequency of management categories change during recruitment. Completed blocks balance contemporaneously, so the expanding category is not assigned predominantly to one trial arm. Use varying block sizes and centrally protected allocation concealment to prevent prediction of future assignments. Incomplete final blocks allow a small residual imbalance, and an odd stratum total cannot be split exactly equally. If many joint strata are very sparse, a prespecified randomized minimization in clinical trials scheme can instead target the prison and management margins, retaining a random component. The analysis should account for the important design strata.

Use separate randomization lists for women with and without a self-reported history of depression. Within each stratum, the central service could assign intervention and control in a $1:1$ ratio using randomly permuted blocks, preferably with concealed and varying block sizes. This is stratified randomization; the internet service can retain allocation concealment because the recruiting staff do not know the next assignment.

History is plausibly prognostic for the outcome. Keeping its distribution similar between arms reduces the risk of a chance imbalance in that important baseline factor and can improve precision. It also allows a comparison within each history stratum. Stratification does not guarantee balance of every other covariate, and staff must not be allowed to choose assignments. The regression model can retain history as a baseline covariate, respecting the design.

For each new eligible participant, compare the imbalance that would result from each treatment assignment across prespecified baseline covariate margins. Assign with a probability favoring the less imbalanced choice, while retaining a random component and allocation concealment. This targets marginal balance when a large number of joint strata makes stratified randomization with blocks inefficient. It is a sequential allocation rule, and inference should respect its design.