Chapter 6 Randomization and allocation, blinding and placebos
6.1 Methods of allocation
There exist several methods to allocate patients to the different treatment groups in a clinical trial.
6.1.1 Simple randomization
Using simple randomization for allocation of \(2n\) patients to two treatment groups, A and B, the number of patients \(N_A\) in group A has a binomial distribution: \[ N_A \sim \mathop{\mathrm{Bin}}(2n, 1/2), \] which is the same for \(N_B\). However, one is fixed by the other: \(N_B = 2n - N_A\).
The distribution of the larger group size \(N_{\max}=\max(N_A,N_B)\) can then be derived as
\[ \Pr(N_{\max}=r) = \left\{ \begin{array}{rl} 2^{-2n} {2n \choose n} & \mbox{ for } r=n \\[.25cm] 2^{1-2n} {2n \choose r} & \mbox{ for } r=n+1,\ldots, 2n. \end{array} \right. \]
Proof. Given a specific \(N_A\), \(N_{\max}\) takes the following value:
| \(N_A\) | \(0\) | \(1\) | \(\dots\) | \(n - 1\) | \(n\) | \(n + 1\) | \(\dots\) | \(2n - 1\) | \(2n\) |
| \(N_B\) | \(2n\) | \(2n - 1\) | \(\dots\) | \(n + 1\) | \(n\) | \(n - 1\) | \(\dots\) | \(1\) | \(0\) |
| \(N_{\max}\) | \(2n\) | \(2n - 1\) | \(\dots\) | \(n + 1\) | \(n\) | \(n + 1\) | \(\dots\) | \(2n - 1\) | \(2n\) |
Equal sample size for both groups, \(r = n\), is a special case as it appears only once: \[\begin{equation*} \Pr(N_{\max} = n) = \Pr(N_A = n) = \binom{2n}{n} \left(\frac{1}{2}\right)^{2n} = 2^{-2n} \binom{2n}{n}. \end{equation*}\]
For \(r = n + 1, \dots, 2n\), the larger group can be either A or B, leading to: \[\begin{eqnarray*} \Pr(N_{\max} = r) & = & \Pr(N_A = r) + \Pr(N_A = 2n - r) \\ & = & \binom{2n}{r} \left(\frac{1}{2}\right)^{2n} + \binom{2n}{2n - r} \left(\frac{1}{2}\right)^{2n} \\ & \overset{\text{symm.}}{=} & 2 \cdot \binom{2n}{r} \left(\frac{1}{2}\right)^{2n} = 2^{1 -2n} \binom{2n}{r}. \end{eqnarray*}\]
For example, for \(n=15\), \(\Pr(N_{\max} \geq 20) = 0.10\). So, there is a substantial chance that the two groups will end up with markedly differing sizes, as illustrated by Figure 6.1. Unequal group sizes lead to a loss in power. For example, for Cohen’s \(d=1\), total sample size \(2n = n_A+n_B=30\) and \(\alpha=0.05\) the loss in power is illustrated in Figure 6.2.
Figure 6.1: Probability for unequal group sizes with simple randomization and a total sample size of \(2n=30\).
Figure 6.2: Loss in power for unequal group sizes in a scenario with \(d=1\), \(n_1+n_2=30\) and \(\alpha=0.05\).
6.1.2 Block randomization
The problem of unbalanced group sizes can be solved by using a form of restricted randomization with so-called random permuted blocks (RPBs). For example, with blocks of length 4, there are the following six different sequences of length 4 that comprise two As and two Bs:
| 1 | A | A | B | B |
| 2 | A | B | B | A |
| 3 | A | B | A | B |
| 4 | B | B | A | A |
| 5 | B | A | A | B |
| 6 | B | A | B | A |
In this case, the randomization selects randomly a block for every group of four patients. Block randomization using RPBs of length 4 ensures that group sizes never differ by more than 2. After every fourth patient, the two treatment groups must have the same size.
RPBs can also be used with other block lengths or for more than two groups, e.g. for a block length of 6 and three groups, there are 90 different sequences:| 1 | A | A | B | B | C | C |
| 2 | A | B | B | A | C | C |
| 3 | A | B | A | B | C | C |
| \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) |
| 90 | C | C | B | B | A | A |
The following R code computes the different sequences for RPBs of length 4:
library(randomizeR)
## bc: length of block
## K: number of treatment groups
obj4 <- pbrPar(bc = 4, K = 2)
getAllSeq(obj4)##
## Object of class "pbrSeq"
##
## design = PBR(4)
## bc = 4
## N = 4
## groups = A B
##
## The first 3 of 6 sequences of M:
##
## 1 B B A A
## 2 B A B A
## 3 A B B A
## ...Example 6.1 An RCT was conducted to examine the use of cold atmosperic plasma as a method to treat diabetic foot ulcers (Mirpour et al. 2020). The Methods section states that the patients were randomly assiged to the two treatment groups using block randomization with mixing block sizes of \(4\).
In some instances, the knowledge of the block length and of previous treatments allows the next treatment to be predicted. This may cause selection bias. RPBs with random block length have been suggested to avoid the potential for selection bias.
The following R code creates the randomization list for RBPs with random
block length of 2 and 4:
##
## Object of class "rRpbrSeq"
##
## design = RPBR(2,4)
## rb = 2 4
## filledBlock = FALSE
## seed = 1678498339
## N = 100
## K = 2
## ratio = 1 1
## groups = A B
##
## RandomizationSeqs BlockConst
## B A B A A B A B ... 4 4 2 2 4 ...## [1] "B" "A" "B" "A" "A" "B" "A" "B" "B" "A" "A" "B" "B"
## [14] "A" "A" "B" "A" "A" "B" "B" "B" "A" "B" "A" "B" "A"
## [27] "B" "A" "B" "A" "B" "A" "B" "A" "B" "A" "B" "A" "A"
## [40] "B" "A" "B" "A" "B" "B" "A" "A" "B" "B" "A" "A" "B"
## [53] "B" "B" "A" "A" "B" "B" "A" "A" "B" "A" "B" "B" "A"
## [66] "A" "A" "B" "B" "A" "B" "A" "B" "B" "A" "A" "A" "B"
## [79] "B" "A" "A" "B" "A" "B" "B" "A" "B" "A" "A" "B" "B"
## [92] "A" "A" "A" "B" "B" "A" "B" "B" "B"6.1.3 Unequal randomization
Unequal allocation can provide greater experience of a new treatment and may even encourage recruitment in certain trials. If the imbalance is no greater than 2:1, the loss in power is small (e.g. from 90% to 86% for \(\alpha=5\%\)). The treatment allocation sequences could be built by randomly selecting from the 15 blocks of length 6, comprising 4 As and 2 Bs.
The following R code computes the 2:1 allocation sequences
for a 2:1 allocation ratio using RPBs of length 6 for two groups:
library(randomizeR)
## bc: length of each block
obj <- pbrPar(bc=6, K = 2, ratio=c(2,1))
genSeq(obj, r=3, seed=123)##
## Object of class "rPbrSeq"
##
## design = PBR(6)
## seed = 123
## N = 6
## ratio = 2 1
## groups = A B
## bc = 6
##
## The sequences M:
##
## 1 A B A A B A
## 2 B A A B A A
## 3 A B B A A A6.1.4 Stratification
Stratification is useful when there is imbalance with respect to prognostic factors. Suppose we wish to compare a new treatment \(T\) with placebo \(P\) to see if it improves a certain continuous outcome. An RCT is conducted with \(n\) patients in each group, all younger than 16 years of age. However, there is an important binary prognostic factor (e.g. age) which has level \(A\) (child, < 12 years) in a proportion of the eligible patients, otherwise it has level \(B\) (adolescent, \(\geq\) 12 years). Suppose that there is no treatment effect, but the mean outcomes \(\mu_A\) and \(\mu_B\) for type \(A\) and \(B\) differ: \(\mu_A \neq \mu_B\). Let us assume that we have conducted one particular trial with \(n\) patients per treatment group and
- \(n_T\) patients of type \(A\) (children) in group \(T\) and
- \(n_P\) patients of type \(A\) in group \(P\).
The expected outcome difference between group \(T\) and \(P\) then is
\[(n_T-n_P)(\mu_A-\mu_B)/n.\]
Although there is no treatment effect, this will be non-zero if \(n_T \neq n_P\). This bias is called allocation bias (see 4.2.2). Balance (\(n_T = n_P\)) can only be guaranteed on average across all possible trials, but not for our specific trial. Hence, the trial design should ensure that \(n_T \approx n_P\).
Stratification aims to control the imbalance between groups not with respect to their size, but with respect to their composition. The idea is to use different RPBs for different strata defined by relevant prognostic factors. For example, if age is an important prognostic factor, one may use an RPB design for children and another one for adults. The number of children (and of adults) receiving each treatment will then be very similar.
6.1.5 Minimization
If there are many prognostic factors, using a separate RPB for each possible combination of these factors becomes impractical. For example, if there are 5 binary prognostic factors, there will be already \(2^5=32\) strata. Minimization aims to balance the groups with respect to each factor, but not for each combination of the factors. The minimization algorithm is:
- Suppose a new patient enters the trial with certain values \(x_1, \ldots, x_J\) of relevant prognostic factors.
- The difference \(D_j\) of numbers of patients allocated to group \(A\) and patients allocated to group \(B\) is computed for the observed value \(x_j\) of each prognostic factor \(j=1,\ldots,J\).
- Compute the total difference \(D = \sum_j D_j\) and proceed as follows (with some \(p>1/2\)):
\[ \mbox{If} \left\{ \begin{array}{c} D = 0 \\ D < 0 \\ D > 0 \end{array} \right\} \begin{array}{c} \mbox{then allocate the new patient} \\ \mbox{to group A with probability} \end{array} \left\{ \begin{array}{c} 0.5 \\ p \\ 1-p \end{array} \right\}. \]
6.2 Blinding and placebos
6.2.1 Single and double blindness
Blinding is used to avoid assessment bias. We distinguish between
Single-blind trial: the patient is unaware of the treatment being given, and
Double-blind trial: neither the doctor nor the patient knows what treatment is being given. Moreover, blinded outcome assessment is often possible even if the treating doctors cannot be blinded. In some trials the statistical analysis* is also carried out blinded, because of subjective elements in any statistical analysis, e.g.
Should I transform the outcome variable?
Should I compute risk differences, risk ratios or odds ratios for binary outcomes?
Should I include a covariate in the analysis?
However, a precise statistical analysis plan helps.
Example 6.1 (continued) In the RCT from Mirpour et al. (2020), the patient were blinded, and the data were collected by a trained physician and nurse who were blinded to the randomization method and the treatment assignment. Moreover, the data were analyzed by a blinded investigator to the study groups.
6.2.2 Placebos
Placebos are treatments that look similar to the true treatment but contain no active ingredient. They have essentially two roles:
Placebos facilitate blindness in RCTs: A treatment versus no treatment at all comparison cannot be blinded, but treatment versus placebo can often be blinded.
Placebos help to control the placebo effect.
The placebo effect occurs if a patient exhibits a response to treatment, even though the treatment has no active component and cannot be having a direct effect. If only untreated controls (no treatment at all) are used, then we would not be able to tell whether an observed effect is due to treatment or to the placebo effect. Therefore a placebo control group is usually selected to control the placebo effect. In some trials, both untreated and placebo control groups are used in addition to the usual intervention group to directly assess the size of the placebo effect.
6.2.3 The double-dummy technique
Placebos can also help in the comparison of two active treatments. Suppose a blue tablet is to be compared to a red tablet. Color cannot be changed, but the drug companies can produce placebo versions of the blue and red tablet, respectively. The double-dummy technique then gives both a blue and red tablet to each patient, in one group with blue active treatment and red placebo and vice versa in the other group. For example, this method is useful to blind a comparison of a treatment given as a tablet with an intravenous treatment.
6.3 Additional references
The design of experiments is discussed in M. Bland (2015) (Chapter 2). Allocation methods, assessment, blinding and placebos are discussed in J. N. S. Matthews (2006) (Chapters 4–5). Studies where the methods from this chapter are used in practice are for example Berlin et al. (2014), Van den Aardweg et al. (2011), Ballard et al. (2005).