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"It has long been an axiom of mine that the little things are infinitely the most important" (Sherlock Holmes)  ### Global randomization

#### Simple global randomization

Units could theoretically be allocated to each treatment group by simply tossing a coin. This, however, is not recommended as it cannot subsequently be verified. The best way is to use random number tables, or a sequence or random numbers generated by a computer programme.  We could for example assign even digits to Group #1 and odd digits to Group #2. Let the first ten numbers in a random number table be 8, 7, 9, 2, 5, 6, 9, 1, 7 & 4 Hence we allocate the first 10 individuals to treatments A1, A2, A2, A1, A2, A1, A2, A2, A2 & A1. This randomization list would give us four individuals in Group #1 and six in Group #2.

If there is a large difference in the number of units in each group, the randomization process can be repeated once or twice until similar group sizes are achieved. However, this approach is open to bias - for example if the investigator wants particular units assigned to particular treatment, he can re-randomize until he gets the allocation he wants. To avoid this possibility, the criteria for repeating the randomization should be set before starting the process (for example, it should be repeated if groups differ in size by more than ten individuals).

Since one of the reasons for randomization is to prevent selection bias, the random allocation sequence should always be concealed to the individuals in charge of enrolment and treatment allocation. Otherwise, knowledge of the upcoming allocation would permit selective assignment.  #### Restricted global randomization

Where the total number of units is fixed and known at the start of the experiment, there are several ways to ensure equality of treatment group sizes. One way is to initially assign treatments to units completely at random, but then once the required number of units for one treatment has been met, only the other treatments are assigned. This process continues until each treatment has been assigned to the same number of units. Alternatively one can allocate a number to each unit and then randomly select numbers, allocating the units sequentially to each treatment group.

Where units are recruited sequentially into a trial, other methods are required. With random permuted blocks blocks are constructed comprising all possible permutations of equal numbers of allocations to each treatment for a given block size. Randomly chosen blocks of allocations are then combined to give the desired total sample size. Let's take an example for blocks of four individuals and two treatments:

 Each possible permutation for equal numbers per treatment is allocated a block number: A1A1A2A2 = 1 ; A1A2A1A2 = 2 ; A1A2A2A1 = 3 ; A2A2A1A1 = 4 ; A2A1A2A1 = 5 ; A2A2A1A1 = 6 Blocks are then chosen randomly using a random number table. The first six numbers in the table are 8, 7, 9, 2, 5, & 6. We ignore numbers above 6 so we select blocks 2, 5 & 6. Hence we allocate the first 12 individuals to A1A2A1A2,  A2A1A2A1, &  A2A2A1A1. This randomization list would give us 6 individuals in each group.

In a trial without stratification block sizes should be reasonably large so as to reduce predictability. (e.g. >15)

A quite different approach is to use the biased coin method. Here if the numbers previously allocated to each group are the same, the next allocation is made completely at random. If more have previously been allocated to one group than the other, then the probability of assigning to the smaller group is made greater than 0.5. For small trials (<50 individuals) a probability of 0.67 is most appropriate, whilst for larger trials a probability of 0.6 is suitable. Let's take an example of a small trial. Again we generate a sequence of random numbers between 1 and 9:  If numbers previously allocated to each group are the same, we assign even digits to group #1 and odd to group #2. If numbers previously allocated to each group are different, we assign digits 1-3 to the larger group and digits 4-9 to the smaller group - in other words the probability of being allocated to the smaller group is 0.67 rather than 0.5. The first ten numbers in our random number table are 8, 7, 9, 2, 5, 6, 9, 1, 7 &4 Hence the first 10 individuals are allocated to:   A1, A2, A2, A2, A1, A1, A2, A2, A1 & A1. This randomization list would give us 5 individuals in each group.

### Stratified Randomization

#### Randomized complete block design

Here experimental units (often plots) that are similar are grouped together into blocks containing the same number of units as the number of treatment levels. In the diagram we have three treatment levels, each with one replicate in four blocks. Randomization is carried out separately in each block to the three plots numbered (1)-(3).

 Randomized block design with three treatments Blocks I 1 2 3 II 1 2 3 III 1 2 3 IV 1 2 3

In this example we have used a ranking method to assign treatments to the plots.

 Randomized block design with three treatments Blocks I A1 A2 A3 II A2 A3 A1 III A3 A1 A2 IV A2 A1 A3

Random numbers are assigned to each treatment within each block. The rank of those numbers within each block then designates the plot. We have demonstrated this procedure in the table below:

 Block Treatment : A1 A2 A3 I Random number 006 432 731 Rank (= plot no.) 1 2 3 II Random number 561 111 345 Rank (= plot no.) 3 1 2 III Random number 327 423 045 Rank (= plot no.) 2 3 1 IV Random number 567 389 659 Rank (= plot no.) 2 1 3

#### Latin square design

The first step is to construct a Latin Square for the number of treatment levels you wish to compare. You can make up your own square (as we have done below), or you could select a standard square from one of those given in some statistical books . Whichever you do, you will still have to randomize rows and columns before using it.

1. First list your four treatment levels in random order, say A3, A2, A1, A4. Then list them again starting from the second in the list, then from the third, and finally the fourth, so giving you the full square.

 A3 A2 A1 A4 A2 A1 A4 A3 A1 A4 A3 A2 A4 A3 A2 A1

2. Then randomize the columns using random number tables.

 A1 A3 A4 A2 A4 A2 A3 A1 A3 A1 A2 A4 A2 A4 A1 A3

3. Then randomize the rows in the design.

 A4 A2 A3 A1 A2 A4 A1 A3 A3 A1 A2 A4 A1 A3 A4 A2

After randomizing, check that there is still only one of each treatment in each row and column. Note that randomization does not prevent bias for individual replicates - you could for example end up randomly allocating the rows and columns to their original positions. Randomization can only prevent systematic bias. The Latin Square is now ready for use.

Although you can analyse the results of just one Latin square, it is always better to replicate the square several times. This is especially true for the smaller 3×3 and 4×4 squares.  #### Clinical trials

Random permuted blocks within strata

This is the most frequently used method to allocate treatments in clinical trials. Each individual is first allocated to his or her 'stratum'.

 stratum 1: age < 5 and male stratum 2: age > 5 and male stratum 3: age < 5 and female stratum 4: age > 5 and female

Say you were looking at the effect of iron supplementation on the incidence of infectious illness in children. Other trials indicate that both gender and age of children have an important effect on the outcome so you stratify by age and gender. If you create two levels for each factor, you end up with four strata. If the choice of a dividing line is somewhat arbitrary, then the choice of split should be made so as to make the number in each stratum similar.

A separate randomization list is then prepared for each stratum, using random permuted blocks as described above in order to get approximately equal numbers of each treatment level within each stratum.

 As before each possible permutation for equal numbers per treatment is allocated a block number: A1A1A2A2 = 1 ; A1A2A1A2 = 2 ; A1A2A2A1 = 3 ; A2A2A1A1 = 4 ; A2A1A2A1 = 5 ; A2A2A1A1 = 6 Blocks are selected randomly using a random number table. For the four different strata we obtained the block sequences: 5,3,4; 6,3,3; 3,5,3; 6,1,4; Hence the first 12 individuals in each stratum are allocated as below: Stratum 1: A2A1A2A1 A1A2A2A1 A2A2A1A1 Stratum 2: A2A2A1A1 A1A2A2A1 A1A2A2A1 Stratum 3: A1A2A2A1 A2A1A2A1 A1A2A2A1 Stratum 4: A2A2A1A1 A1A1A2A2 A2A2A1A1 This randomization list would give us 24 individuals in each group.

Minimization

We will base our example on a randomized controlled trial by Welschen et al. (2004) one of the examples for this More Information page. The aim was to evaluate the effectiveness of an intervention aiming at reducing antibiotic prescription rates for respiratory tract symptoms in primary care. Either intervention or non-intervention was allocated to each of the twelve experimental units which were groups of general practitioners known as peer review groups. There were three major confounding factors which it was thought would affect outcome, namely the volume of antibiotic prescribing, whether they were working in a rural or urban area, and the number of practitioners per group. Minimization could (and probably should ) have been used to balance the groups for the confounding factors.

Using minimization, one allocates the first unit a treatment (A1 or A2) at random. For each subsequent unit one then determines which treatment would lead to a better balance of confounding factors between the groups. To see how this is done, the table below shows the result after allocating seven units using the minimization process. The characteristics of each of those units as regards the confounding factors is shown below - notice that each of the seven units is entered three times, once for each confounding factor:

The next unit to be allocated () has low volume antibiotic prescribing, is in an urban area and has many general practitioners.

 Outcome after 7 units were allocated Properties of next unitto beallocated Factor Level Number in each treatment A1 A2 Volume antibioticprescribing Low 1 1 ⇐ High 2 3 Rural/urban Rural 3 3 ⇐ Urban 1 0 Few/many Few 2 1 ⇐ Many 2 2

To find out which treatment we are going to allocate, we add up the totals of units already allocated with the same characteristics as the new recruit for each treatment - for A1 the sum is 4 (=1+1+2), whilst for A2 the sum is 3 (=1+0+2). We therefore allocate the eighth unit to the smaller of the two groups (treatment A2), and update the table with the characteristics of that unit - as shown below.

 Outcome after 8 units were allocated Properties of last unitto beallocated Factor Level Number in each treatment A1 A2 Volume antibioticprescribing Low 1 2 ⇐ High 2 3 Rural/urban Rural 3 3 ⇐ Urban 1 1 Few/many Few 2 1 ⇐ Many 2 3

Clearly with this method a randomization list cannot be prepared in advance - instead the table is updated as each unit is recruited to the study and the appropriate allocation made. There are also many variations on this basic theme. Probably the commonest is to allocate the treatment with the lowest sum to the next unit with a probability less than 1, but still greater than 0.5 (say 0.8) in order to introduce a greater element of chance into the process.  ### Other designs

Randomization for factorial experiments and repeated measures designs is implemented in much the same way as for the methods we have already covered. Factorial experiments are carried out using the factorial set of treatments in one of the main designs above - namely completely randomized, randomized block or Latin square.