StATS: BMJ letter (October 11, 2001)

Here's a letter I submitted to the web letters to the editor for the British Medical Journal in response to an article about randomisation (BMJ 1999 Sep 11;319(7211):703-704).

A simple approach for randomisation.

I'm glad that the British Medical Journal is devoting space to pragmatic aspects of research such as the article "How to randomise" by Douglas G. Altman and J. Martin Bland. I want to supplement their fine article by outlining an alternative approach for randomisation. This approach is simple and intuitive, and you can apply it in a variety of research settings.

In essence, you can randomise by following these three steps:

  1. arrange your treatments in a single column in a systematic order,
  2. create a second column of random numbers, and
  3. sort by these random numbers.

This approach has intuitive appeal. Sorting on a set of random numbers produces a random ordering.

Let's look at a simple example. Suppose we want to allocate three treatments (T1, T2, and T3) randomly to a total of twelve subjects.

First, arrange your treatments in a systematic order. This is what your data would look like:

T1
T2
T3
T1
T2
T3
T1
T2
T3
T1
T2
T3

Next, generate a second column of random numbers. You can find a random number generating function in most software. In Microsoft Excel, for example, you would use the RAND() function. The table below shows what you would get. Actually, you will get something different. If it weren't different, your numbers wouldn't be random, would they?

T1 0.02338
T2 0.00018
T3 0.50797
T1 0.03322
T2 0.35942
T3 0.23288
T1 0.59740
T2 0.63826
T3 0.20776
T1 0.47897
T2 0.90778
T3 0.41530

Finally, you sort this table by the column of random numbers. Your treatments now have a random and unpredictable order.

T2 0.00018
T1 0.02338
T1 0.03322
T3 0.20776
T3 0.23288
T2 0.35942
T3 0.41530
T1 0.47897
T3 0.50797
T1 0.59740
T2 0.63826
T2 0.90778

That was easy, wasn't it? With this approach, you don't have to worry about throwing out some of the random numbers, even when the number of treatments doesn't divide easily into 10 or 100.

You can set up a randomisation like this in five minutes using a simple spreadsheet. Spreadsheets have the annoying feature of recalculating a new set of random numbers after any operation, including a sort, but this is only a cosmetic problem.

The approach I have outlined extends readily to more complex situations like block or stratified random sampling. Let's look at the same research design, and let's randomize separately within blocks of size six.

First, you need to lay out your treatments and blocks in a systematic fashion. Then add a column of random numbers.

T1 B1 0.4280
T2 B1 0.7577
T3 B1 0.0912
T1 B1 0.3344
T2 B1 0.4102
T3 B1 0.5281
T1 B2 0.2790
T2 B2 0.8477
T3 B2 0.0850
T1 B2 0.3631
T2 B2 0.4929
T3 B2 0.0537

To restrict your randomisation to within individual blocks, apply a multiple column sort, specifying the block column first and then the random number column.

T3 B1 0.0912
T1 B1 0.3344
T2 B1 0.4102
T1 B1 0.4280
T3 B1 0.5281
T2 B1 0.7577
T3 B2 0.0537
T3 B2 0.0850
T1 B2 0.2790
T1 B2 0.3631
T2 B2 0.4929
T2 B2 0.8477

This is much simpler than the approach suggested by Altman and Bland; you don't have to list the 90 possible orderings of three treatments within a block of six.

You can use this same approach to randomise a stratified study or a crossover study. Just lay out your design systematically, add random numbers, and apply a multiple column sort.

Here is another adaptation of this approach that allows you to make a random selection from a small population of patients:

  1. list your patients in a systematic or arbitrary order,
  2. generate a random number for each patient,
  3. sort by the random numbers, and
  4. select your patients from the top of the sorted list.

Let's look at an example where you have 13 patients and you need to randomly select five. A table after sorting by the random numbers might look like this:

P02 0.09299
P05 0.16621
P11 0.39768
P09 0.48882
P01 0.49291
P06 0.51756
P13 0.52436
P04 0.56682
P07 0.58734
P10 0.79923
P12 0.93658
P08 0.96063
P03 0.96669

You would select patients 2, 5, 11, 9, and 1.

You can even use this approach to generate a randomisation light bulb joke: "take to in How many a screw does statisticians it lightbulb?"

In summary, if you want to randomise, attach a column of random numbers. Sorting by these random numbers will produce an unpredictable ordering of your treatments. This approach extends easily to a variety of research settings.

Steve Simon, Research Biostatistician, Children's Mercy Hospital, Kansas City MO, USA

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