It’s A Math, Math World (Block Design)

Blocking is a method to create balance in treatment assignments over time as recruitment progresses. Randomization, by its very nature, can create treatment groups of unequal sizes.

First, we have to define a treatment assignment ratio (allocation ratio) as the ratio of the number of persons in one treatment group relative to another.  For example:

A ratio of 1:1 indicates a treatment and control group of equal sizes

A ratio 1:1:1:1:5 indicates 4 test treatments and 1 control treatment and there are 5 times as many control subjects as in any of the test treatment groups. (Note: The control group is always listed last).

The block size must be a multiple of the sum of the digits of the allocation ratio.


Three treatments A, B and C

Allocation Ratio is 1:1:2 (¼ allocated to A, ¼ allocated to B and ½ allocated to C)

Sum of digits = 1+1+2 = 4

Then block sizes must be multiple of 4. Within each block, we randomly assign the treatments according to the allocation ratio.

There is one down side of using the same size block of 4 in this case. Once the first 3 people had been assigned treatments in that block, we would know the treatment assignment for the 4th person.  Thus, the mask can be broken on the study this way.            

However, by varying the block sizes (sometimes 4, sometimes 8, etc), we can prevent this from happening.

Methods of Varying Block Sizes:

1) Alternating

Ex:  1:1 allocation with treatments A and B


1st Block size 4

2nd Block size 6

3rd Block size 4


This approach is the easiest but the pattern can be figured out. The treatment assignment of the 4th, 10th, 14th, 20th, etc, persons are determined by previous assignments.

2) Random Varying


Where “?” = either 4 or 6

A two step process:

First, select a random number to determine block size

If ui < 0.5, then block i has 4 subjects

If ui > 0.5, then block i has 6 subjects

Second, once block size has been determined, select the random ordering of blocks


Nurse will not know if 10th subject will be a “B” or “A” because he/she does not know what the first and second block sizes are.

Next time, we will look at stratification to reduce variation in our clinical trial data.

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One Response to “It’s A Math, Math World (Block Design)”

  • LUISA says:

    Hi, how are you? How long has this blog been around? I have been searching for this kind of information for the past week and a half. Greetings from Brighton and Hove