In the statistical theory of the design of experiments, blocking is the arranging of experimental units in groups (blocks) that are similar to one another. Blocking can be used to tackle the problem of pseudoreplication.
Blocking reduces unexplained variability. Its principle lies in the fact that variability which cannot be overcome (e.g. needing two batches of raw material to produce 1 container of a chemical) is confounded or aliased with a(n) (higher/highest order) interaction to eliminate its influence on the end product. High order interactions are usually of the least importance (think of the fact that temperature of a reactor or the batch of raw materials is more important than the combination of the two - this is especially true when more (3, 4, ...) factors are present); thus it is preferable to confound this variability with the higher interaction.
In the statistical theory of the design of experiments, blocking is the arranging of experimental units in groups (blocks) that are similar to one another. Typically, a blocking factor is a source of variability that is not of primary interest to the experimenter. An example of a blocking factor might be the sex of a patient; by blocking on sex, this source of variability is controlled for, thus leading to greater accuracy.
In Probability Theory the blocks method consists of splitting a sample into blocks (groups) separated by smaller subblocks so that the blocks can be considered almost independent. The blocks method helps proving limit theorems in the case of dependent random variables.
The blocks method was introduced by S. Bernstein:^{[1]} The method was successfully applied in the theory of sums of dependent random variables and in extreme value theory.^{[2]}^{[3]}^{[4]}
When we can control nuisance factors, an important technique known as blocking can be used to reduce or eliminate the contribution to experimental error contributed by nuisance factors. The basic concept is to create homogeneous blocks in which the nuisance factors are held constant and the factor of interest is allowed to vary. Within blocks, it is possible to assess the effect of different levels of the factor of interest without having to worry about variations due to changes of the block factors, which are accounted for in the analysis.
A nuisance factor is used as a blocking factor if every level of the primary factor occurs the same number of times with each level of the nuisance factor. The analysis of the experiment will focus on the effect of varying levels of the primary factor within each block of the experiment.
The general rule is:
Blocking is used to remove the effects of a few of the most important nuisance variables. Randomization is then used to reduce the contaminating effects of the remaining nuisance variables. For important nuisance variables, blocking will yield higher significance in the variables of interest than randomizing.
One useful way to look at a randomized block experiment is to consider it as a collection of completely randomized experiments, each run within one of the blocks of the total experiment.
Name of Design | Number of Factors k | Number of Runs n |
---|---|---|
2-factor RBD | 2 | L_{1} * L_{2} |
3-factor RBD | 3 | L_{1} * L_{2} * L_{3} |
4-factor RBD | 4 | L_{1} * L_{2} * L_{3} * L_{4} |
k-factor RBD | k | L_{1} * L_{2} * * L_{k} |
with
Suppose engineers at a semiconductor manufacturing facility want to test whether different wafer implant material dosages have a significant effect on resistivity measurements after a diffusion process taking place in a furnace. They have four different dosages they want to try and enough experimental wafers from the same lot to run three wafers at each of the dosages.
The nuisance factor they are concerned with is "furnace run" since it is known that each furnace run differs from the last and impacts many process parameters.
An ideal way to run this experiment would be to run all the 4x3=12 wafers in the same furnace run. That would eliminate the nuisance furnace factor completely. However, regular production wafers have furnace priority, and only a few experimental wafers are allowed into any furnace run at the same time.
A non-blocked way to run this experiment would be to run each of the twelve experimental wafers, in random order, one per furnace run. That would increase the experimental error of each resistivity measurement by the run-to-run furnace variability and make it more difficult to study the effects of the different dosages. The blocked way to run this experiment, assuming you can convince manufacturing to let you put four experimental wafers in a furnace run, would be to put four wafers with different dosages in each of three furnace runs. The only randomization would be choosing which of the three wafers with dosage 1 would go into furnace run 1, and similarly for the wafers with dosages 2, 3 and 4.
Let X_{1} be dosage "level" and X_{2} be the blocking factor furnace run. Then the experiment can be described as follows:
Before randomization, the design trials look like:
X_{1} | X_{2} |
---|---|
1 | 1 |
1 | 2 |
1 | 3 |
2 | 1 |
2 | 2 |
2 | 3 |
3 | 1 |
3 | 2 |
3 | 3 |
4 | 1 |
4 | 2 |
4 | 3 |
An alternate way of summarizing the design trials would be to use a 4x3 matrix whose 4 rows are the levels of the treatment X_{1} and whose columns are the 3 levels of the blocking variable X_{2}. The cells in the matrix have indices that match the X_{1}, X_{2} combinations above.
Treatment | Block 1 | Block 2 | Block 3 |
---|---|---|---|
1 | 1 | 1 | 1 |
2 | 1 | 1 | 1 |
3 | 1 | 1 | 1 |
4 | 1 | 1 | 1 |
By extension, note that the trials for any K-factor randomized block design are simply the cell indices of a k dimensional matrix.
The model for a randomized block design with one nuisance variable is
where
The theoretical basis of blocking is the following mathematical result^{[citation needed]}. Given random variables, X and Y
The difference between the treatment and the control can thus be given minimum variance (i.e. maximum precision) by maximising the covariance (or the correlation) between X and Y.