# Randomization in Experiment Design

I have seen two ways to conduct randomization in an experiment. I am confused about the difference between them and which one is the correct one.

Assume we have an even number of subjects (say 20).

Method 1. let each subject flip a coin, if it is Head, then go to group A, otherwise, go to group B. This procedure stops until one group has 10 people, then the rest subjects who haven't flip the coin, all go to the other group.

This method will guarantee that each group will have exactly the same number of people. Think about the extreme case, when n = 2

Method 2. let all subjects flip a coin, if it is Head, then go to group A, otherwise, go to group B.

This method will very likely result in the case that group A and B do not have same number of subjects.

Which method is the correct randomization in experiment? And why the other method is wrong?

My hunch is that the Method 2 is correct, but I don't know what's wrong with Method 1. Especially, if n=2 (for theoretical purpose), then I would favor Method 1.

My idea is the following: in order to claim causality in the end, we have to make sure that each subject has same probability of being assigned to group A and B. The Method 2 can guarantee this. However, the situation for Method 1 is tricky. Namely, before the first guy has flipped the coin in Method 1, it is indeed that all subjects have same probability of being assigned to Group A and B. However, once the first guy has flipped the coin and is assigned to one group, say Group A, then the last guy's chance of being assigned to Group A is less than the chance of being assigned to Group B.

Both are correct under the usual assumption of independent sampling, because, as you note, each subject has in prospect an equal chance of ending up in each condition. What I would recommend in practice is neither of these. Instead, randomly assign the two conditions within each pair of subjects. [By this I mean randomly decide (with equal probability) for either (a) subject 1 to get treatment 1 and subject 2 to get treatment 2 or (b) subject 1 to get treatment 2 and subject 2 to get treatment 1. Then do the same thing for subjects 3 and 4, subjects 5 and 6, etc.] This will give you cell sizes that are close to equal throughout the run of the study, which is convenient if you want to look at the data before the study's done, or you don't have a predetermined sample size.

• Could you please elaborate it a little? What do you mean by consecutive pair of subjects? Say I have 20 subjects in total, then using what mechanism, you make 10 pairs from these 20 subjects? Then for each pair, you flip a coin, to determine treatment A or treatment B? Aug 1, 2016 at 13:02
• Sorry, I worded that poorly. I've edited it. Aug 1, 2016 at 14:17
• By the way, you answered several questions I have posted before. I really like your way of thinking these real life probability questions. Could you please recommend me some books or may materials written by you, e.g., your blog. I am self studying experimental design and seeking for good textbooks. Thanks Aug 1, 2016 at 19:54
• Correct me if I am wrong. So I first make pairs (1,2),(3,4),...,(19,20). Then within each pair, say (1,2), I flip a coin. If it is Head, then subject 1 goes with treatment A, subject 2 goes with treatment B; if it is Tail, then subject 1 goes with treatment B, subject 2 goes with treatment A. And I do this procedure for all the rest pairs. But is this the same as Method 1? Namely, you eventually end up with exactly same number of subjects in treatment A versus treatment B if the total number of subjects is even Aug 1, 2016 at 20:15
• Thank you! If you like my answers, you should accept them (by clicking the check mark under the voting arrows). More generally, you should accept an answer for every question you've asked, so long as you got a satisfactory answer. Feel free to check out my website ( arfer.net ), which has some relevant research papers and essays, and also this trio of posts I did on a My Little Pony-themed subreddit, but right now, most of my writing on data analysis is on Cross Validated. Aug 1, 2016 at 22:46