## 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.**

**Example:**

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 4^{th} 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

4-6-4-6-4-6-4-6-…

1^{st} Block size 4

2^{nd} Block size 6

3^{rd} Block size 4

etc.

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

**2) Random Varying**

4-4-6-4-6-?-?-?

Where “?” = either 4 or 6

**A two step process:**

**First, select a random number to determine block size**

If u_{i} < 0.5, then block *i* has 4 subjects

If u_{i} > 0.5, then block *i* has 6 subjects

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

Ex. AABABAABA

Nurse will not know if 10^{th} 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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