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The Randomization Process

  • Chapter
Fundamentals of Clinical Trials

Abstract

The randomized controlled clinical trial is the standard by which all trials are judged. In the simplest case, randomization is a process by which each participant has the same chance of being assigned to either intervention or control. An example would be the toss of a coin, in which heads indicates intervention group and tails indicates control group. Even in the more complex randomization strategies, the element of chance underlies the allocation process. Of course, neither trial participant nor investigator should know what the assignment will be before the participant’s decision to enter the study. Otherwise, the benefits of randomization can be lost. The role that randomization plays in clinical trials has been discussed in Chap. 5 as well as by numerous authors [1–12]. While not all accept that randomization is essential [10, 11], most agree it is the best method for achieving comparability between study groups, and the most appropriate basis for statistical inference [1, 3].

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Author information

Authors and Affiliations

  1. North Bethesda, MD, USA

    Lawrence M. Friedman

  2. Division of Public Health Sciences, Wake Forest School of Medicine, Winston-Salem, NC, USA

    Curt D. Furberg

  3. Department Biostatistics and Medical Informatics, University of Wisconsin, Madison, WI, USA

    David L. DeMets

  4. Department of Biostatistics, Wake Forest School of Medicine, Winston-Salem, NC, USA

    David M. Reboussin

  5. Department of Medicine, Duke University, Durham, NC, USA

    Christopher B. Granger

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Appendix: Adaptive Randomization Algorithm

Appendix: Adaptive Randomization Algorithm

Adaptive randomization can be used for more than two intervention groups, but for the sake of simplicity only two will be used here. In order to describe this procedure in more detail, a minimum amount of notation needs to be defined. First, let

x ik = the number of participants already assigned intervention k

(k = 1, 2) who have the same level of prognostic factor i

(i = 1, 2, … , f) as the new participant.

and define

$$ \begin{array}{l}{x}_{ik}^t={x}_{ik}\mathrm{if}t\ne k\\ {}={x}_{ik}+1\mathrm{if}t=k\end{array} $$

The xtik represents the change in balance of allocation if the new participant is assigned intervention t. Finally, let

B(t) = function of the xtik’s, which measures the “lack of balance” over all prognostic factors if the next participant is assigned intervention t.

Many possible definitions of B(t) can be identified. As an illustrative example, let

$$ B(t)={\displaystyle {\sum}_{i=1}^f{w}_i\ Range\ \Big({x}_{i1}^t,{x}_{i2}^t}\Big) $$

where wi = the relative importance of factor i to the other factors and the range is the absolute difference between the largest and smallest values of xti1 and xti2.

The value of B(t) is determined for each intervention (t = 1 and t = 2). The intervention with the smaller B(t) is preferred, because allocation of the participant to that intervention will cause the least imbalance. The participant is assigned, with probability p > 1/2, to the intervention with the smaller score, B(1) or B(2). The participant is assigned, with probability (1 − p), to the intervention with the larger score. These probabilities introduce the random component into the allocation scheme. Note that if p = 1 and, therefore, 1 − p = 0, the allocation procedure is deterministic (no chance or random aspect) and has been referred to by the term “minimization” [51, 59].

As a simple example of the adaptive stratification method, suppose there are two groups and two prognostic factors to control. The first factor has two levels and the second factor has three levels. Assume that 50 participants have already been randomized and the following table summarizes the results (Table 6.A1).

Table 6.A1 Fifty randomized participants by group and level of factor (xik’s)a
Full size table

In addition, the function B(t) as defined above will be used with the range of the xik’s as the measure of imbalance, where w1 = 3 and w2 = 2; that is, the first factor is 1.5 times as important as the second as a prognostic factor. Finally, suppose p = 2/3 and 1 − p = 1/3.

If the next participant to be randomized has the first level of the first factor and the third level of the second factor, then this corresponds to the first and fifth columns in the table. The task is to determine B(1) and B(2) for this participant as shown below.

  1. 1.

    Determine B(1)

    1. (a)

      Factor 1, Level 1

       

      K

      x1k

      x11k

      Range (x111, x112)

      Group

      1

      16

      17

      \( \left|17\hbox{--} 14\right| \) = 3

      2

      14

      14

      		 
    2. (b)

      Factor 2, Level 3

       

      K

      x2k

      x12k

      Range (x121, x122)

      Group

      1

      4

      5

      \( \left|5\hbox{--} 6\right| \) = 1

      2

      6

      6

      		 

      Using the formula given, B(1) is computed as 3 × 3 + 2 × 1 = 11.

  2. 2.

    Determine B(2)

    1. (a)

      Factor 1, Level 1

       

      K

      x1k

      x21k

      Range (x211, x212)

      Group

      1

      16

      16

      \( \left|16\hbox{--} 15\right| \) = 1

      2

      14

      15

      		 
    2. (b)

      Factor 2, Level 3

       

      K

      x2k

      x11k

      Range (x121, x122)

      Group

      1

      4

      4

      \( \left|4\hbox{--}7\right| \) = 3

      2

      6

      7

      		 

      Then B(2) is computed as 3 × 1 + 2 × 3 = 9.

  3. 3.

    Now rank B(1) and B(2) from smaller to larger and assign with probability p the group with the smaller B(t).

    t

    B(t)

    Probability of assigning t

    2

    B(2) = 9

    p = 2/3

    1

    B(1) = 11

    1 − p = 1/3

    Thus, this participant is randomized to Group 2 with probability 2/3 and to Group 1 with probability 1/3. Note that if minimization were used (p = 1), the assignment would be Group 2.

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Friedman, L.M., Furberg, C.D., DeMets, D.L., Reboussin, D.M., Granger, C.B. (2015). The Randomization Process. In: Fundamentals of Clinical Trials. Springer, Cham. https://doi.org/10.1007/978-3-319-18539-2_6

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