## Strategies for allocating treatments to patients in a clinical trial

## Introduction

Within a typical clinical trial researchers want to compare the health outcomes obtained from one of more treatments. In an ideal study the patients enrolled within a clinical study are identical except for the treatment that they are assigned. In real life there are any number of factors that differ between patients (such as their age, diet, etc), these factors are known as confounders. Because of the presence of confounders clinical researchers need methods for allocating treatments to patients in such a way that there is no significant difference between treatment groups (eg. each group has the same distribution of patient ages).

The simplest study design is one where when each new patient is enrolled in the study the patient is assigned to either a drug treatment or a placebo group depending upon the toss of a coin. This approach is known as simple randomization. There are many advantages in using randomization within a clinical trial. On average randomization ensures that there is no significant difference between treatment groups (a patient within one age category is just as likely to receive the treatment or the placebo). Randomization provides this balance on both the confounding variables that the research team were aware of and have measured, as well as those variables that were unknown. Without randomization medical staff might knowingly or unknowingly decide to allocate particular treatments to particular patients (hence biasing a study).

The problem with simple randomization is that it does not guarantee a balance between treatment groups. There is no guarantee that simple randomization will produce the same number of patients within each group, or that the patient groups are similar with respect to confounding variables such as the age. This balance is obtained by simple randomization on average, but is rarely obtained within any specific trial. To address these issues with simple randomization, more advanced methods are generally used for treatment allocation including blocking, stratification, and minimization. These methods also use randomization, but perform randomization in such a way as to obtain balanced treatment groups. This tutorial article will address each of these treatment allocation methods in turn, will explain how these methods work, and the advantages of using these techniques within a clinical trial.

## Blocking

Blocking methods are used to ensure that there is an equal number of patients within each treatment group. A simple design might involve two treatment groups (a drug treatment and a placebo). Patients are then recruited into the study two patients at a time (a block of two). Within each block one of the patients will randomly be assigned to receive the drug treatment, and one will be assigned to receive the placebo. This simple study design uses randomization to obtain similar treatment groups (with respect to the confounding variables), while also ensuring that as the study proceeds that we always have the same number of patients within the drug treatment and placebo groups.

Blocking can also be used when we have more than two treatment groups within our study (eg. when comparing three treatments we might use a block of three patients, where within each block we assign one patient to each of the three treatment groups). We can also use blocks of a larger size than the number of treatment groups. For a study with two treatments we could use a block size of two, however this places some restrictions on the number of ways for randomly allocating treatments to patients. Alternatively we could use a block size of four where within each block two patients are assigned to the drug treatment and two patients are assigned to the placebo. For a study with two treatments we could use a block size that is any multiple of two, larger blocks allow a greater level of randomization but also allow a larger possibility that we might end up with treatment groups of different sizes at the end of the study.

## Stratification

Blocking is used to obtain treatment groups of similar sizes, to obtain treatment groups with similar characteristics (eg similar age distributions) we might choose to use stratification. In stratification we divide our patient population into categories (eg. we might use age categories of 0-20 years, 21-40 years, and over 40 years). Within each age category we then divide treatments to patients. Within each age category we might then commonly use a method such blocking in order to ensure that there are the same number of patients that receive each drug treatment within each age category (eg. the same number of young people will receive the drug treatment and the placebo). It should be noted that if stratification is used within a study design then treatment should not be allocated to patients using simple randomization within each category, in such a case the use of simple randomization with stratification has no advantages over simple randomization without stratification. With stratification we might also choose to use more than one stratification variable (eg. we might have patient groups of young females, young males, older females, etc). In addition some studies make use of more than one measurement site (eg. hospitals), in this case we might also choose to have a different strata for each site.

When using stratification we might also ask how many strata should we use. The basic principle is that if we use too many strata then we will have very few patients within each strata (and hence little chance of having the same number of patients for each treatment group within each strata). If we have too few strata then we are not making full use of the benefits of stratification (our treatment groups will be less likely to have similar characteristics such as age distribution). Hence we should choose the number of strata that provides a balance between these two extremes.

With a stratified design we might also choose how many patients we want to enroll within each strata. In some designs we might choose to have the same number of patients within each strata. In other designs we know that the variability within some strata is larger than in others, hence we might choose to enroll more patients within those strata where there is more variability in patient characteristics.

## Adaptive allocation

The study designs considered above make use of randomization and stratification. An alternative study design uses an approach known as adaptive allocation. For these designs the probability of treatment assignment is not constant but is determined by the current balance between the treatment groups. Two common methods of adaptive allocation are urn designs and minimization.

A simple method used to first introduce the concept of probability is to consider an urn containing one white and one black ball. Within a clinical trial we might then select one ball at random from the urn for each new patient. If the white ball is selected then the patient is given treatment A, and if the black ball is selected then the patient is given treatment B. This is the approach used in simple randomization, however we might consider a slight alternative to this approach in order to improve the balance between the two treatment groups. In this case we count the number of patients that have already received treatment A (n_{A}) and those that have received treatment B (n_{B}). The probability that a new patient will receive treatment A and treatment B is then given by:

P(A) = n_{B} / (n_{A} + n_{B})

P(B) = n_{A} / (n_{A} + n_{B})

Using this alternative approach means that if we already have a large number of patients who have received treatment A then the next patient has a large probability of receiving treatment B. We could then use such an urn approach, with a different “urn” for each strata in a stratified design (as described previously).

Minimization is an alternative method of adaptive allocation. Within this study design we define a measure (a mathematical formula) that describes the current level of imbalance between two patient groups. This measure of imbalance can describe both an imbalance in the number of patients within each treatment group as well as the imbalance in patient characteristics (eg. the age distribution within each treatment group). With each new patient in our study we need to determine whether to give this patient treatment A or treatment B. To determine this treatment allocation we will evaluate (a) our measure of imbalance if the patient is allocated treatment A, and (b) our measure of imbalance if the patient is allocated treatment B. We will then allocate the treatment that produced the smallest measure of imbalance. If the two measures are the same then the treatment is allocated at random. The advantage of minimization is that it allows a tighter control over the balance between the groups compared to a blocked strata design, but it also means that the imbalance measures need to be able to be computed as each new patient enters the study.

## Unequal group sizes

For many studies we might choose to have the same number of patients within each patient group. In other studies we might choose to have a different number of patients within each group. This is commonly done if we would expect that the variation in health outcomes in larger in one group compared to the other, or if one treatment has a significantly larger financial cost than the other. In the former case we might calculate the ratio between the size of each patient group as:

N_{1}/N_{2} = variance(group 1) / variance(group 2)

In the latter case we might calculate the size ratio as:

N_{1}/N_{2} = square root (cost of treatment 2 / cost of treatment 1)

Once we have decided the desired ratio between the group sizes we can incorporate this information into our blocking, stratified, urn or minimization designs. Within a standard blocked design we might have the same number of people receiving each treatment within each block, in the case of unequal group sizes we can have a different number of patients receiving each treatment within each block (eg. within a block 70% of patients might receive treatment 1). Within a stratified design we might choose to have more patients within those strata where we would expect a larger variance in patient outcomes. Within a standard urn design we start with an equal number of “balls” for treatment 1 and treatment 2, for unequal group sizes we will start with a different number of balls (eg. 70% of the balls might represent treatment 1). In the case of minimization we might design our measure of imbalance so that the desired group sizes (eg. 70% for treatment 1) produces the smallest level of imbalance.

## Conclusion

A good study design allocates treatments to patients in such a way that a desired number of patients are allocated to each treatment group (eg. 50% of patients to treatment 1, and 50% of patients to treatment 2). At the same time we want to try and ensure that the distribution of patients is the same within each patient group (eg. the same number of younger patients are allocated to treatment 1 and treatment 2). In order to achieve this balance between treatment groups a number of designs can be used. These include blocked, stratified, urn and minimization techniques. These methods vary in terms of how tightly they control the patient characteristics (such as age) and in terms of how much of the study design can be specified before the study (compared to adjustments that need to be made with each new patient).

These techniques are vital as an imbalance between treatment groups can seriously undermine the validity of the results from any clinical trial.

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