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In blind experiments subjects are randomly assigned to one of groups. The most commonly used solution is to use (equivalent of) a coin toss, with the same probability to be assigned to each group.

I have read about two modifications of a procedure of random assignment in blind tests:

  • fair / just coin, where probability of assignment to group A is proportional to the number of subjects assigned to other group (group B) divided by number of all assigned subjects.

    This solution is meant to have more equal distribution of subjects to groups (as truly random coin can result in one group to be over-represented).

    It is known as covariate-minimization randomization.

  • result-based probability, where probability of assignment to given group is proportional to how well this group performs in experiment (e.g. in drug test).

    It is known as response-adaptive randomization, or outcome-adaptive randomization.

How based on results of such not fully random assignment method can one recover (calculate) the same estimates and error bounds as for random sampling?

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Here's a paper that may be helpful regarding your second point: Power and bias in adaptively randomized clinical trials. When looked at from the Bayesian standpoint, you simply compute posterior probabilities. The randomization mechanism doesn't matter. If you want to get frequentist operating characteristics, you can run simulations.

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