Blocking and Randomization

Blocking and randomization are used in planning of experiments, in order to increase the precision of the outcome and ensure the validity of the inference. Blocking is used to reduce errors. A block is a portion of the experimental material that is expected to be more homogeneous than the whole aggregate. For example, if the experiment is designed to test the effect of polyester coating of electronic circuits on their current output, the variability between circuits could be considerably bigger than the effect of the coating on the current output. In order to reduce this component of variance, one can block by circuit. Each circuit will be tested under two treatments: no-coating and coating. We first test the current output of a circuit without coating. Later we coat the circuit, and test again. Such a comparison of before and after a treatment, of the same units, is called paired-comparison.

Another example of blocking is the boy’s shoes examples of Box et al. (2005). Two kinds of shoe soles’ materials are to be tested by fixing the soles on $n$ pairs of boys’ shoes, and measuring the amount of wear of the soles after a period of actively wearing the shoes. Since there is high variability between activity of boys, if $m$ pairs will be with soles of one type and the rest of the other, it will not be clear whether any difference that might be observed in the degree of wearout is due to differences between the characteristics of the sole material or to the differences between the boys. By blocking by pair of shoes, we can reduce much of the variability. Each pair of shoes is assigned the two types of soles. The comparison within each block is free of the variability between boys. Furthermore, since boys use their right or left foot differently, one should assign the type of soles to the left or right shoes at random. Thus, the treatments (two types of soles) are assigned within each block at random.

Other examples of blocks could be machines, shifts of production, days of the week, operators, etc.

Generally, if there are $t$ treatments to compare, and $b$ blocks, and if all $t$ treatments can be performed within a single block, we assign all the $t$ treatments to each block. The order of applying the treatments within each block should be randomized. Such a design is called a randomized complete block design. We will see later how a proper analysis of the yield can validly test for the effects of the treatments.

If not all treatments can be applied within each block it is desirable to assign treatments to blocks in some balanced fashion. Such designs, to be discussed later, are called balanced incomplete block designs (BIBD).

Randomization within each block is important also to validate the assumption that the error components in the statistical model are independent. This assumption may not be valid if treatments are not assigned at random to the experimental units within each block.