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Randomized experimentsOn this page: Simple global randomization Restricted global randomization Stratified Randomization Latin square design Clinical trials Other designs
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.
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:
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:
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).
In this example we have used a ranking method to assign treatments to the plots.
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:
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
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
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.
This is the most frequently used method to allocate treatments in clinical trials. Each individual is first allocated to his or her 'stratum'.
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.
We will base our example on a randomized controlled trial by Welschen et al.
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.
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.
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.
If there are multiple treatment levels and/or the design is stratified, analysis of
Methods for analysis of repeated measures experiments differ somewhat. Crossover trials for binary outcomes can be analyzed using matched pairs odds