In many experiments where the investigator is comparing a set of treatments there is the possibility of one or more sources of variability in the experimental measurements that can be accounted for during the design stage of the experimentation. For example we might be investigating four different pieces of machinery using say two different operators, who would be expected to display different degrees of competence with the equipment. Or we might not be able to run all of the experimental combinations in one session so we would want to take into account systematic differences that are due to experiments in the various sessions.
The least complicated scenario is where we would have a single (nuisance) factor that we want to control for in the experiment. The statistical model used to describe the data collected in such an experiment could be written in the form:
where there are v treatments in b blocks and the number of units in each block does not have to be the same and is denoted using the k subscript.
In a complete block design all treatments occur the same number of times in every block, usually one replicate of all treatments per block. There will be situations where the number of treatments is too large for all of them to be included in every block of the design. In these situations an incomplete block design would be used for running an experiment.
A special type of design is the balanced incomplete block design (BIBD), where the v treatments are investigated by allocating them to b blocks of equal size k. We have that k is less than t and b and k are chosen so that b * k is a multiple of v. All of the treatments occur exactly r times in the design and every pair of treatments occur together in lambda of the b blocks.
Two-way analysis of variance (ANOVA) is used to analyse data collected from an experiment using a block design, as discussed elsewhere in this post.