# an alternative allocation method for small RCTs?

stratification and minimization are two randomization options to achieve balance in terms of covariates or baselines in small RCTs. in recent years, researchers seem to have favoured minimization, particularly for sequential allocation designs with a high number of balancing variables. in short, the method involves the choice of some imbalance criterion and then sequentially allocates each new individual to the study arm leading to the smallest new criterion value. this allocation can be done purely deterministically or involving some element of chance.

our question is if the approach could be simplified when the assignment to study arms takes place only after inclusion of all participants is completed. our idea is to use simple randomization without any constraints to generate a large number, say N = 1000, of complete and fully random allocation schemes. in the next step we would identify the, say, n = 100 schemes with the smallest imbalances (using a similar criterion as for minimization) or, alternatively, all schemes with a criterion value below some prespecified cut-off value. finally, we would choose one of these remaining schemes at random.

the whole process would be carried out by a third person not involved in the study intervention or the collection of study outcomes. the investigators would only receive the last resulting allocation scheme from the person responsible for allocation.

the rationale behind our idea is simplicity and that we would like not to sacrifice too much randomness for balance.

has anyone heard of such an allocation strategy before? what do you think of it? are there any considerations concerning bias? (in some way, we would just be rejecting allocation schemes as long as we don't like them because of intolerably low balance. on the other hand, minimization or even stratification contain similar aspects...) and what about the implications for the statistical analysis? would you still adjust for the covariates using covariance analysis? any other thoughts?

thank you very much for any feedback!

• This proposed method is reminiscent of a simple study I described at stats.stackexchange.com/a/157646/919 in which I selected one sample from a set of random samples according to a simple "innocent" criterion. I think your method might be subject to the same criticisms as the one I examined: namely, it would have a bias that is unknown and potentially huge. How would you cope with that problem? – whuber Nov 27 '18 at 17:07
• thank you for your comment, whuber. i don't exactly understand what kind of bias you'd be afraid of; bias related to the choice of the random seed? but the final draw from the restricted set of allocation schemes would be done by some third party who would have to worry about what seed they use. anyway, in the meantime i found out that "my" suggestion is well known and accepted under several names, with "covariate constraint randomisation" being one of the most frequently used. – schotti Jun 26 '19 at 12:21
• The bias arises because your procedure for selecting 100 out of 1000 possible samples results in a set that are a fortiori non-random. They are designed to be unusually "balanced," but there is no assurance that balanced samples are representative of the population in all ways that are important to the analysis. A bias is possible, and therefore has to be suspected, but there's no way to assess or quantify that bias. Moreover, your procedure explicitly violates the assumptions of any standard probability model used to test hypotheses, measure confidence, and construct prediction intervals. – whuber Jun 26 '19 at 14:19
• i see no reason why the selected 100 samples should not be random. of course, not all 1000 samples would enjoy the same probability to be chosen as the final draw; that probability would be greater for balanced samples. but just having a greater probability doesn't make something non-random, right? i tend to agree with the reminder of your comment, whuber, but isn't designing "unusually balanced" samples the very idea of all "balanced allocation" methods? i can not see how "my" method would deserve more scepticism in that respect than e.g. common stratified randomisation. – schotti Jun 27 '19 at 21:12
• If you want to treat this as a probability sample that's great--but you need to know the probabilities in order to make reasonable estimates from the data! Your method is not stratified. The sceptic has the upper hand on you, because you have the duty to demonstrate your procedure is valid, not the other way around. – whuber Jun 27 '19 at 21:15