# 11.2: Randomization schemes for individual participants

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## 2.1 Unrestricted randomization

Simple random allocation of individuals between the different intervention groups is carried out most conveniently by using a computer. For example, in Microsoft Excel, the instruction ‘= RANDBETWEEN(1,3)’ will produce a random number between 1 and 3, i.e. each of the numbers 1, 2, or 3 has an equal chance of being generated. The equivalent of tossing a coin is = RANDBETWEEN(1,2). Some calculators also have a key which generates a random number on the display (usually a decimal number between 0 and 1, so that, for example, the equivalent of coin tossing would be to allocate a number less than 0.5000 as ‘heads’ and a number 0.5000 or greater as ‘tails’).

In large trials, it is common for a centralized randomization system to be used. When an investigator has decided that a participant meets the entry criteria for a trial, and the participant has given informed consent to be randomized to one of the trial interventions, the investigator telephones, or sends a text, to a central office to give the identification details for the participant, and the office then tells, or texts, the investigator to which intervention the participant has been randomly assigned or, in the case of a double-blind trial, the code for the intervention that should be administered to the participant. Systems are now commonly used whereby this process has been automated and does not require an individual to answer the telephone in the central office or for a similar automated procedure to be followed over the Internet. The advantage of this method of intervention assignment is that there is no way in which the investigator can influence the randomization procedure, and if, for example, the investigator decides not to allocate an intervention to a participant after knowing the random assignment, there is a central record of this.

For investigators who cannot set up access to a procedure for remote randomization, a frequently used alternative procedure is for a set of opaque, sealed, and numbered envelopes to be prepared, containing the intervention allocations (or possibly even the actual interventions if these are, for example, drugs). The envelopes are opened in numerical sequence, as each new person is entered into the trial. Entry criteria must be checked and eligibility satisfied before an envelope is opened, in order to exclude the possibility that the decision to accept a subject into the trial is influenced by the knowledge of the group to which he or she would be allocated. For large trials, the use of envelopes may be too cumbersome. Coding systems and alternative procedures appropriate for use in the case of ‘double-blind’ designs are discussed in Section 5.

Where the study product (for example, drug, vaccine) package is individually numbered and labelled (and randomization has been done before the numbering and labelling and where there is an indistinguishable placebo or control intervention), randomization may simply be achieved by registering each new recruit and assigning them the number on the product package.

In some circumstances, it may be better to design the randomization system, such that it is completely transparent to participants that a random allocation process is being used. A trial may be more acceptable if the trial population is involved in the randomization procedure. For example, in a trial in Ghana, the allocation of insecticide-impregnated bed-nets was randomized, such that, in some communities, all households received a bed-net immediately and, in other communities, the distribution of nets was deferred until a later time (Binka et al., 1996). At a public meeting involving all of the trial communities, the name of each community was written on a slip of paper. All the slips were put in a bucket, and a child was asked to draw some of the slips from the bucket to determine which communities received the bed-nets first. By using this procedure, it was apparent that the allocation was random and that no favouritism was operating. The fairness of the procedure was demonstrated to the population by the fact that, by chance, the community in which the area chief resided was not selected for early bed-net allocation (much to the surprise of the population)! (Fred Binka, personal communication.)

Unrestricted randomization is often employed in large trials, as it is likely that any imbalance between the intervention groups with respect to risk factors for the occurrence of the outcomes of interest will tend to even out. Furthermore, it is possible to adjust for any residual imbalance during the analysis of the study without important loss of statistical power.

## 2.2 Restricted randomization

Although an unrestricted randomization procedure should lead to approximately equal numbers of participants in each group, this is not guaranteed. For example, there is more than a 5% chance that, if 20 participants are allocated to one of two groups at random, six or fewer may be allocated to one group, and 14 or more to the other. A better balance is achieved by using a ‘restricted randomization’ procedure, also called ‘blocked randomization’ or ‘randomization with balance’. This procedure ensures equal numbers in each group, after there have been a fixed number of allocations. For example, the allocation procedure might be designed in blocks of ten, such that, in every ten allocations, five are to one group and five to the other. The total number of intervention groups must be a multiple of the size of the blocks.

In order to minimize the possibility that an allocation can be deduced from previous allocations, the block size should not be too small (in particular, it should not be two!), and, if possible, it should not be known to the investigator responsible for the administration of the interventions. Indeed, as far as possible, those giving the interventions should not be aware that blocking has been carried out, or, if the block size is a fixed number, the person giving the intervention would know in advance what the intervention allocation of the last individual or group in the block would be. Another safeguard is to use several different block sizes for allocating interventions in a trial. For example, in a trial with two arms, the block size might be varied, at random, between eight, ten, and 12.

Two different procedures for carrying out restricted randomization are described in Sections 2.2.1 and 2.2.2, one appropriate for small block sizes and the other appropriate for larger block sizes, say eight or more.

### 2.2.1 Small block sizes

If two interventions, say A and B, are to be allocated using a block size of, say four, it is possible to list all the different possible combinations of the allocations that will yield two As and two Bs. This is illustrated in Table 11.1. A number is allocated to each combination, and a random number is chosen to select a particular allocation.

The selection of each random number (between 1 and 6) generates four intervention allocations. Thus, if the random numbers 4, 5, and 1 are generated, these yield a list of twelve intervention allocations (to be assigned to participants in sequence) (Table 11.2).

### 2.2.2 Larger block sizes

Listing all possible combinations of allocations within a block becomes unmanageable, as the block size increases. For example, with a block size of ten, there are 252 different possible combinations, each yielding five participants in each of two intervention groups A and B. An alternative approach is necessary therefore. Suppose the block size is to be 12 and six allocations are to be made to group A and six to group B. Random numbers between 1 and 12 are generated, until six different numbers in that range have been generated (numbers that duplicate a previous one are ignored). Algorithms are easily available on the Internet to generate such random numbers. (For example, at <http://www.random.org/integers>, it is straightforward to generate X random integers between Y and Z where the user inserts values for X, Y, and Z.) Thus, we might request six random numbers between 1 and 12 and obtain 1, 2, 4, 7, 11, and 12. Then, the first, second, fourth, seventh, eleventh, and twelfth participants within the block are allocated to one of the interventions, say A, and the other participants to B. The complete sequence for the block of 12 is shown in Table 11.3.

A similar procedure, with a different set of random numbers, is used to allocate interventions in the next block (i.e. 13 to 24), and so on.

In general, it is better to choose block sizes which are not too large, in order to reduce the risk of a long sequence of individuals being allocated to the same intervention. A maximum block size of 12 is suggested.

Table 11.1 Example of allocation rule for a block size of four, with two intervention groups A and B

Allocation | Corresponding random number |
---|---|

AABB | 1 |

BBAA | 2 |

ABAB | 3 |

BABA | 4 |

ABBA | 5 |

BAAB | 6 |

Table 11.2 Example of random allocation to two groups using a block size of four

Block number | 1 | 2 | 3 |
---|---|---|---|

Random number | 4 | 5 | 1 |

Allocation sequence | BABA | ABBA | AABB |

## 2.3 Stratified randomization

If different subgroups of participants, say males and females, have different background rates of disease, it may be desirable to design the allocation procedure such that the interventions are equally divided in each subgroup. This may be achieved though ‘stratified’ randomization. The population is stratified, for example, by sex or by age group, and the allocation of the interventions is carried out separately in each stratum.Stratification may be based on more than one factor. For example, there may be a separate allocation of interventions in each of a number of different age–sex groups. The greater the number of strata, the more complex the organization of the randomization is; in general, the number of strata should be kept small. Separate randomization lists will have to be maintained for each stratum. This may be achieved by using different sets of coloured envelopes, packages, or sticky labels for each stratum.Stratified randomization should be considered if it is known that there are large differences in disease risk between different groups of individuals in a trial (or in response to treatment in the case of a therapeutic trial) and if it is possible to place individuals in strata corresponding to different levels of risk prior to entry to the trial. The objective of stratification is to try to include in each stratum those at similar risk of disease (or response to treatment) and to randomize between interventions separately within each stratum. In multicentre trials, randomization is often stratified on study site.

Table 11.3 Example of random allocation to two groups using a block size of 12

Participant | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

Intervention | A | A | B | A | B | B | A | B | B | B | A | A |