• If the experimental units are not used simultaneously, you can random-
ize the order in which they are used.
• If the experimental units are not used at the same location, you can
randomize the locations at which they are used.
• If you use more than one measuring instrument for determining re-
sponse, you can randomize which units are measured on which instru- ments.
When we anticipate that one of these might cause a change in the response, we can often design that into the experiment (for example, by using blocking; see Chapter 13). Thus I try to design for the known problems, and randomize everything else.
One tale of woe Example 2.1
I once evaluated data from a study that was examining cadmium and other metal concentrations in soils around a commercial incinerator. The issue was whether the concentrations were higher in soils near the incinerator. They had eight sites selected (matched for soil type) around the incinerator, and took ten random soil samples at each site.
The samples were all sent to a commercial lab for analysis. The analysis was long and expensive, so they could only do about ten samples a day. Yes indeed, there was almost a perfect match of sites and analysis days. Sev- eral elements, including cadmium, were only present in trace concentrations, concentrations that were so low that instrument calibration, which was done daily, was crucial. When the data came back from the lab, we had a very good idea of the variability of their calibrations, and essentially no idea of how the sites differed.
The lab was informed that all the trace analyses, including cadmium, would be redone, all on one day, in a random order that we specified. Fortu- nately I was not a party to the question of who picked up the $75,000 tab for reanalysis.
2.3
Performing a Randomization
Once we decide to use randomization, there is still the problem of actually doing it. Randomizations usually consist of choosing a random order for
a set of objects (for example, doing analyses in random order) or choosing Random orders and random subsets
random subsets of a set of objects (for example, choosing a subset of units for treatment A). Thus we need methods for putting objects into random orders
18 Randomization and Design
and choosing random subsets. When the sample sizes for the subsets are fixed and known (as they usually are), we will be able to choose random subsets by first choosing random orders.
Randomization methods can be either physical or numerical. Physical randomization is achieved via an actual physical act that is believed to pro- duce random results with known properties. Examples of physical random- ization are coin tosses, card draws from shuffled decks, rolls of a die, and
Physical
randomization tickets in a hat. I say “believed to produce random results with known prop-
erties” because cards can be poorly shuffled, tickets in the hat can be poorly mixed, and skilled magicians can toss coins that come up heads every time. Large scale embarrassments due to faulty physical randomization include poor mixing of Selective Service draft induction numbers during World War II (see Mosteller, Rourke, and Thomas 1970). It is important to make sure that any physical randomization that you use is done well.
Physical generation of random orders is most easily done with cards or tickets in a hat. We must order N objects. We take N cards or tickets,
numbered1 through N , and mix them well. The first object is then given the
Physical random
order number of the first card or ticket drawn, and so on. The objects are then sorted
so that their assigned numbers are in increasing order. With good mixing, all orders of the objects are equally likely.
Once we have a random order, random subsets are easy. Suppose that the N objects are to be broken into g subsets with sizes n1, . . ., ng, with
n1+ · · · + ng = N . For example, eight students are to be grouped into one
Physical random subsets from random orders
group of four and two groups of two. First arrange the objects in random order. Once the objects are in random order, assign the first n1 objects to
group one, the nextn2objects to group two, and so on. If our eight students
were randomly ordered 3, 1, 6, 8, 5, 7, 2, 4, then our three groups would be (3, 1, 6, 8), (5, 7), and (2, 4).
Numerical randomization uses numbers taken from a table of “random” numbers or generated by a “random” number generator in computer software.
Numerical
randomization For example, Appendix Table D.1 contains random digits. We use the table
or a generator to produce a random ordering for our N objects, and then
proceed as for physical randomization if we need random subsets.
We get the random order by obtaining a random number for each object, and then sorting the objects so that the random numbers are in increasing order. Start arbitrarily in the table and read numbers of the required size sequentially from the table. If any number is a repeat of an earlier number, replace the repeat by the next number in the list so that you getN different
numbers. For example, suppose that we need 5 numbers and that the random
Numerical
random order numbers in the table are (4, 3, 7, 4, 6, 7, 2, 1, 9, . . .). Then our 5 selected