Minimization

Note

This page tries to summarize the details of how minimization works as a method of treatment allocation. It mostly just summarises the original Pocock and Simon article found here

Minimization balances treatment assignments across multiple prognostic factors simultaneously. Let’s start with some notation - we’re running a trial and we have:

• \(N\) different treatments to assign participants to.

• \(M\) prognostic factors that we want to balance (all categorical 1).

  • Each factor has \(n_i\) levels.

At any point during the trial, \(x_{ijk}\) is the number of participants assigned to treatment \(k\), who have level \(j\) of factor \(i\).

A new patient coming into the trial has a factor level for each of the \(M\) factors \(r_1, \dots, r_M\). If that patient is assigned to treatment \(k\), there will be new values \(x_{ijl}^k\):

\[x_{i r_i k}^k = x_{ijk} + 1\]

\[x_{ijl}^k = x_{ijl} \text{ for } l \neq k, j \neq r_i\]

i.e. we add 1 to all the counts \(x_{ijk}\) that match the treatment \(k\) and the factor levels the participant has, and all the other counts are unchanged.

Calculating imbalance

Imbalance within each factor

We start by calculating the imbalance within each factor. We choose a function \(D(\{ z \})\) that measures the variation in the participant counts \(\{ z \}\) (you can choose different functions like \(D = \text{range}\)). For each treatment \(k\) that a new participant could be assigned to, we can get the resulting counts that would be produced by assigning to \(k\), and calculate:

\[d_{ik} = D( \{ x_{i r_i l}^k \}_{l = 1}^N)\]

i.e. the imbalance in treatment numbers across all treatments, for all participants with level \(r_i\) on factor \(i\).

Note

Only participants who match the new participant’s factor level \(r_i\) matter for the calculation of imbalance for factor \(i\), participants with other factor levels don’t affect the imbalance calculation

Total imbalance

Once we have the imbalance within each factor, we combine them to produce a total imbalance score. We choose a function \(G\) and calculate

\[G_k = G(d_{1k}, \cdots, d_{Mk})\]

that combines the \(d_{ik}\) for all the \(M\) factors (an obvious choice for \(G\) is just the sum). Then \(G_k\) is the total imbalance you would have, if the new patient is assigned to treatment \(k\). We calculate \(G_k\) for all the \(N\) treatments.

Assigning probabilities

You can then rank the treatments in order of increasing \(G_k\) values (increasing = larger amount of imbalance), assigning ascending ranks \((1), (2), \dots, (N)\).

Then pick some way of assigning decreasing probabilities of treatment assignment \(p_k\) to these, i.e.

\[p_1 \geq p_2 \geq \cdots \geq p_N\]

(\(p_k\) could be a function of \(G_k\), the amount of imbalance, but this may be unnecessarily complex, and in practice just using the ranks should produce good balance).

Additional details

Unequal treatment allocation

Han et al. (2009) discussed how minimization could be adapted to trials with unequal treatment allocation ratios.

The original minimization method can be adapted by simply dividing the count of participants in each arm (within each factor) by that arm’s allocation ratio - e.g. if the allocation ratio is 1:2 and the current counts for female participants are 11 and 20, then we would divide and get 11 and 10 - and then calculate the factor imbalance score from these values.

However this method is not perfect, as it results in a bias: slightly more participants are allocated to arms with low allocation ratios than desired, and slightly less to arms with high allocation ratios. Han et al. (2009) also propose a “biased coin minimization” method that solves this issue.

Biased coin minimization

In the biased coin minimization method, the treatment that would produce least imbalance is assigned with a probability given by:

\[p^H_{(i)} = 1 - \frac{ \sum_{k \neq i}^N r_{(k)} }{ \sum_{k \neq 1}^N r_{(k)} } \left( 1 - p^H_{(1)} \right)\]

where \(p^H_{(1)}\) is the probability that the treatment with the lowest allocation ratio will be assigned, when it produces the lowest imbalance.

The remaining probability is divided between treatments according to the formula:

\[p^L_{(i), H = (j)} = \frac{ r_{(i)} }{ \sum_{k \neq j}^N r_{(k)} } \left( 1 - p^H_{(j)} \right)\]

Footnotes

1

Continuous factors can be accounted for in some extensions of the minimization method, but they are not implemented in smallerize