Minimization Algorithm to Achieve Treatment Balance across Strata in Stratified Randomization

In randomized controlled clinical trials, we usually need to consider if there are any prognostic factors that may have influence on the primary outcome measure. The prognostic factors usually include demographic information (gender, age), severity of the disease (mild, moderate, and severe), use of concomitant medication (patients on background therapy versus patients not on background therapy), genetic sub-type of the disease, biomarker measures at baseline…

Dealing with the prognostic factors can be at the statistical analysis stage through the sub-group analyses to investigate the impact of the prognostic factors on the treatment outcome.

However, if there are factors known to have impact on the treatment outcome, it is better to consider them in the planning stage through the stratified randomization. The prognostic factors are considered as stratified factors. In stratified randomization, the random treatment assignment is actually performed within each strata. I had an old blog article to explain the stratified randomization “Stratified randomization to achieve the balance of treatment assignment within each strata”

When we plan for the stratified randomization, the number of stratification factors and number of levels or categories for each stratification factor need to be considered. The total number of strata is the multiplication of the number of levels of stratification factors.

Suppose we just have one stratification factor with two levels (age group: 'less than 65 years old' versus 'greater than and equal to 65 years old'), the total number of strata = 1 x 2 = 2. The randomization (specifically the block randomization) will be implemented within each stratum of 'less than 65 years old' or 'greater than and equal to 65 years old'.

Suppose we have two stratification factors: age group with two levels ('less than 65 years old' versus 'greater than and equal to 65 years old') and disease severity with three levels (mild, moderate, and severe), the total number of strata = 2 x 3 =6. The randomization (specifically the block randomization) will be implemented within each of the following 6 strata (combinations):

• less than 65 years old and mild severity
• less than 65 years old and moderate severity
• less than 65 years old and severe severity
• greater than and equal to 65 years old and mild severity
• greater than and equal to 65 years old and moderate severity
• greater than and equal to 65 years old and severe severity

The question arises when more and more stratification factors are added to the list. How many stratification factors can we have in a study while still maintaining the overall balance in treatment assignment across all of these strata?

The number of strata can increase rapidly. Suppose we have three stratification factors:

• Age group with two levels ('less than 65 years old' versus 'greater and equal to 65 years old')
• Disease severity with three levels (mild, moderate, and severe)
• Geographic region with four categories (Asia, Europe, North America, and South America)

The total number of strata = 2 x 3 x 4=24. The randomization (specifically the block randomization) will be implemented within each of the 24 strata (combinations):

• less than 65 years old, mild severity, Asia
• less than 65 years old, moderate severity, Asia
• less than 65 years old, severe severity, Asia
• ……

While there is not rule how many stratification factors or strata can be used in a clinical trial, I usually stick with a number to keep the number of stratification factors no more than two or total number of strata no more than 6.

With the limited total sample size, if there are too many stratification factors or too many strata, it can actually cause the imbalance in treatment assignment on the study level. Even though the treatment assignment is intended to be balanced within each stratum by implementing the block randomization, there will be always incomplete blocks across all strata, which causes the imbalance in treatment assignment on the study level.

One approach to address this issue is an approach called ‘minimization’ algorithm. Minimization algorithm is one of the adaptive randomization approaches. It is used in the situation where there is a limited or small overall sample size, but many critical stratification factors.

Here are a couple of screen shots for illustrating how the minimization works.

The minimization requires the calculation after each subject is randomized. It is almost not possible to implement the minimization with the manual process. Luckily, with IRT (Interactive Response Technology) including IVRS (interactive voice response system) and IWRS (interactive web response system), the minimization can be implemented through programming. The algorithm for minimization is built into the IRT system and the calculation after each subject randomization is automatically calculated by the system. The IRT vendors such as conduit, sovuda, Datatrak et al can all do the implementation of the minimization algorithm.

References: 

• Scott NW et al (2002) The method of minimization for allocation to clinical trials. a review
• Levy and Blood (NESUG 17) Minimization: Testing A Dynamic Randomization Algorithm
• O'Callaghan (2014) OxMaR: Open Source Free Software for Online Minimization and Randomization for Clinical Trials
• Saghaei (2011) An Overview of Randomization  and Minimization Programs for Randomized Clinical Trials
• Altman and Bland (2005) Treatment Allocation by Minimisation
• Online course: Treatment Allocation and Randomization