# When using stratified randomization in a RCT, how many envelopes per stratum should be prepared?

If one considers a hypothetical RCT, say a total sample size of 300, separated into three strata based on surgical procedure (open surgery, closed surgery, or combined procedure). Blocking in groups of 2 will be done within each stratum.

When generating the randomization sequence, obviously one cannot know ahead of time how many patients will be randomized into each stratum.

So, how many randomization envelopes should be made for each stratum? Based on previous data, one would expect about 33/33/33 for each group, but if only 100 envelopes are made for each group, this provides no flexibility in the case that the historical data is wrong.

Should one:

• play it safe and do 300 per stratum (and have a lot left over after the trial is done), or
• guess the required number based on previous data, with a buffering factor built-in?

Any tips would be appreciated!

There are no consequences to having too many random assignments left over at the end, and pretty large consequences to having your randomization scheme fail mid-assignment for a trial. Given the cost of an envelop and a piece of paper, I'd vote on the side of safety and make 300 of each.

If you're trying to trim the budget a bit, I'd go with 150 or so, based on whuber's answer, and be in the "absurdly unlikely" category.

• Thanks - in line with what I thought might be the best idea. – pmgjones Sep 12 '11 at 1:47

The stratum populations have a multinomial distribution. 300 is large enough that the Normal approximation will be accurate, implying the population in any given stratum has a Normal$(100, \sqrt{\frac{1}{3}\frac{2}{3}300})$ distribution. Although all three counts are correlated, it's still a good approximation to assume they are not and apply a Bonferroni correction: that is, to be assured of a $1-\alpha$ chance of not exhausting the envelopes for any stratum, find the upper $1-\alpha/3$ quantile of this distribution. (For $\alpha=.01$, for example, this quantile equals 122.2, which we round up to 123) Create this many envelopes for each stratum.

If you mistrust these approximations, it's simple to simulate many experiments. In 1,000,000 trials, 10,064 had a stratum with 123 or more patients, indicating that 122 envelopes would have been sufficient 98.99+% of the time.

You probably don't even want to run a 1/100 risk. If you prepare 128 envelopes, the risk of running out is only 1/1000; with 133 envelopes, it drops to 1/10,000. Simulation of 4,000,000 experiments bears out this expectation.

• Thank you. Useful answer. For experienced trialists, though, I still wonder what the "rule of thumb" is for stratified RCTs whose historical probabilities are not as firmly known. What if I could only say "I expect 40-60% to be open, 20-40% closed, and the rest combined"? Any disadvantages of creating more envelopes than necessary, as long as blocked randomization is used? – pmgjones Sep 11 '11 at 23:34
• @propofol I don't understand the role of "historical probabilities" here. My understanding is that the experiment is under your control and that you are randomizing using a random number generator; therefore, the probabilities are determined, not estimated. Perhaps you could edit your question to elaborate on exactly how randomization is being used in this experiment. – whuber Sep 12 '11 at 15:16