# Permuted Block Randomization

Statistics Definitions > Permuted Block Randomization

## What is Permuted Block Randomization

Permuted block randomization is a way to randomly allocate a participant to a treatment group, while maintaining a balance across treatment groups. Each “block” has a specified number of randomly ordered treatment assignments.

For example, let’s you had treatment groups A and B, and you plan to enroll 10 new patients per week. Your first 3 blocks might look like this:

• Week 1: BAAABABBAB.
• Week 2: ABABBABABA.
• Week 3: AAABBBAABB.

Note that each block has 5 As and five Bs, maintaining a balance of the two despite the random order.

If your experiment involves a number that isn’t divisible by the block size, then your treatment groups may not have the exact same amounts. For example, if you only enrolled 28 patients in this particular permuted block randomization scheme, you would have two fewer Bs than As:

• Week 1: BAAABABBAB.
• Week 2: ABABBABABA.
• Week 3: AAABBBAA.

For large trials, a small imbalance usually doesn’t make a big difference, but this is something to take into consideration for smaller trials.

## Assigning Blocks

Two basic methods for assigning blocks are random number generation and permutations.

1. Random Number Generation.

Step 1: Randomly generate a number for each treatment assignment. For example, if you had a block with treatments AABB, you might get:

• A = 4
• A = 88
• B = 9
• B = 17

Step 2: Rank the generated numbers from highest to lowest:

• A = 88
• B = 17
• B = 9
• A = 4

This gives you your first block, ABBA.

Step 3: Repeat the process to assign a new block.

2. Permutations

Step 1: Write a list of all permutations for the block size. In other words, write out a list of all possible variations. The formula is b! / ((b/2)! (b/2!)) where b is the block size and ! is a factorial.
For a block of size 4, you would have six possible arrangements:
4! ((4/2)! (4/2)!) = 4! / (2)! x (2)! = 4 x 3 x 2 x 1 / ((2 x 1) x (2 x 1)) = 24 / 4 = 6.
The arrangements are:

• AABB
• ABAB
• BAAB
• BABA
• BBAA
• ABBA

Step 2: Randomly choose one arrangement for each block.

## Issues

If a researcher is made aware of the block sizes, they may be able to figure out some of the treatment groups. for example, if the researcher knows that the block size is 10, and 9 patients have been assigned to five treatment groups and one control group, the last patient would be assigned to the control group. The researcher can then decide to replace that patient with another in order to stack the odds for the trial. A way to avoid this is to add a further level of randomization: making the blocks random sizes.

For the same reason, you should never use a block size of two: once the first treatment is known, the second treatment group is also known.