Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 46 / 2 / b / iii / Solution

An ethical objection concerns the proposed saving of control-prison staff costs: the trial is explicitly based on individual informed consent, and institutional allocation does not justify omitting control participants' information, consent or appropriate outcome follow-up. These responsibilities remain in both arms. A cluster-randomized trial can sometimes be ethically defensible, but changing the randomized unit alone does not make those participant protections unnecessary.

The statistical objection is loss of independent information and potential prison-level imbalance. Outcomes within a prison may share baseline risks, management practices and release conditions, yielding positive intraclass correlation. For equal cluster size $m$, the usual exchangeable approximation gives design effect $1+(m-1)\rho$, so the individual-randomization calculation understates the sample required when $\rho >0$. There are also fewer independent randomized units, and prison-specific adoption of management practices can be unbalanced across arms. Thus randomizing prisons generally loses precision for this individually deliverable intervention, unless compensating design features and additional clusters are supplied.