Grizzle (1982) describes Relative Efficiency (RE) as the relative change in variance when using stratification versus simple randomisation. Researchers may consider a threshold where the RE gets low enough that stratified randomisation is imperative (as the variance associated with the parameter of interest gets too large with simple randomisation).
Consider the same analysis as proposed in Grizzle (1982) but where the prognostic factor is sex with a population prevalence of 0.5 (i.e. using the dichotomy X=1 if a patient is female, 0 otherwise).
(a) In this scenario given, what does the RE tell us about randomisation?
(b) You believe it’s plausible that for a trial with sample size n = 20 with equal sized treatment groups to have an imbalance where there are 6 females in one treatment arm and 3 females in the other treatment arm. What is the relative efficiency in this scenario?
(c) You decide to prepare for a scenario where the balance is worse than that in (b) and where g = 1 − h, with n1 = n2 = 10. What is the minimum value for g such that the relative efficiency remains above 0.8?
(d) You are later told that a much larger study is needed and the allocation ratio of treatment 1 to treatment 2 must be 2:1. What is the RE for n = n1 + n2 = 60, g = 0.4 and h = 0.6?
(e) For the scenario in (d), would you recommend the use of stratified or simple randomisation?