Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 41 / 1 / a / Solution

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.