Comparing a Set of Treatments in Blocks

13.7 Comparing a Set of Treatments in Blocks

In Section 13.2, we discussed the idea of blocking, that is, isolating sets of experi- mental units that are reasonably homogeneous and randomly assigning treatments to these units. This is an extension of the “pairing” concept discussed in Chapters

9 and 10, and it is done to reduce experimental error, since the units in a block have more common characteristics than units in different blocks. The reader should not view blocks as a second factor, although this is a tempting way of visualizing the design. In fact, the main factor (treatments) still carries the major thrust of the experiment. Experimental units are still the source of error, just as in the completely randomized design. We merely treat sets of these units more systematically when blocking is accomplished. In this way, we say there are restrictions in randomization. Before we turn to a discussion of blocking, let us look at two examples of a completely randomized design. The first example is a chemical experiment designed to determine if there is a difference in mean reaction yield among four catalysts. Samples of materials to be tested are drawn from the same batches of raw materials, while other conditions, such as temperature and concentration of reactants, are held constant. In this case, the time of day for the experimental runs might represent the experimental units, and if the experimenter believed that there could possibly be a slight time effect, he or she would randomize the assignment of the catalysts to the runs to counteract the possible trend. As a second example of such a design, consider an experiment to compare four methods

13.8 Randomized Complete Block Designs 533 of measuring a particular physical property of a fluid substance. Suppose the

sampling process is destructive; that is, once a sample of the substance has been measured by one method, it cannot be measured again by any of the other methods. If it is decided that five measurements are to be taken for each method, then 20 samples of the material are selected from a large batch at random and are used in the experiment to compare the four measuring methods. The experimental units are the randomly selected samples. Any variation from sample to sample will

appear in the error variation, as measured by s 2 in the analysis.

What Is the Purpose of Blocking?

If the variation due to heterogeneity in experimental units is so large that the sensitivity with which treatment differences are detected is reduced due to an

inflated value of s 2 , a better plan might be to “block off” variation due to these units and thus reduce the extraneous variation to that accounted for by smaller or more homogeneous blocks. For example, suppose that in the previous catalyst illustration it is known a priori that there definitely is a significant day-to-day effect on the yield and that we can measure the yield for four catalysts on a given day. Rather than assign the four catalysts to the 20 test runs completely at random, we choose, say, five days and run each of the four catalysts on each day, randomly assigning the catalysts to the runs within days. In this way, the day- to-day variation is removed from the analysis, and consequently the experimental error, which still includes any time trend within days, more accurately represents chance variation. Each day is referred to as a block.

The most straightforward of the randomized block designs is one in which we randomly assign each treatment once to every block. Such an experimental layout is called a randomized complete block (RCB) design, each block constituting

a single replication of the treatments.