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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Blocking (statistics) | 2/3 | https://en.wikipedia.org/wiki/Blocking_(statistics) | reference | science, encyclopedia | 2026-05-05T09:49:20.919707+00:00 | kb-cron |
In our previous diet pills example, a blocking factor could be the sex of a patient. We could put individuals into one of two blocks (male or female). And within each of the two blocks, we can randomly assign the patients to either the diet pill (treatment) or placebo pill (control). By blocking on sex, this source of variability is controlled, therefore, leading to greater interpretation of how the diet pills affect weight loss.
=== Definition of blocking factors === 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.
=== Block a few of the most important nuisance factors === The general rule is:
"Block what you can; randomize what you cannot." 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.
== Implementation == Implementing blocking in experimental design involves a series of steps to effectively control for extraneous variables and enhance the precision of treatment effect estimates.
=== Identify nuisance variables === Identify potential factors that are not the primary focus of the study but could introduce variability.
=== Select appropriate blocking factors === Carefully choose blocking factors based on their relevance to the study as well as their potential to confound the primary factors of interest.
=== Define block sizes === There are consequences to partitioning a certain sized experiment into a certain number of blocks as the number of blocks determines the number of confounded effects.
=== Assign treatments to blocks === You may choose to randomly assign experimental units to treatment conditions within each block which may help ensure that any unaccounted for variability is spread evenly across treatment groups. However, depending on how you assign treatments to blocks, you may obtain a different number of confounded effects. Therefore, the number of as well as which specific effects get confounded can be chosen which means that assigning treatments to blocks is superior over random assignment.
=== Replication === By running a different design for each replicate, where a different effect gets confounded each time, the interaction effects are partially confounded instead of completely sacrificing one single effect. Replication enhances the reliability of results and allows for a more robust assessment of treatment effects.
== Example ==
=== Table === 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.
with
L1 = number of levels (settings) of factor 1 L2 = number of levels (settings) of factor 2 L3 = number of levels (settings) of factor 3 L4 = number of levels (settings) of factor 4
⋮
{\displaystyle \vdots }
Lk = number of levels (settings) of factor k
=== Example === 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.
==== Description of the experiment ==== Let X1 be dosage "level" and X2 be the blocking factor furnace run. Then the experiment can be described as follows:
k = 2 factors (1 primary factor X1 and 1 blocking factor X2) L1 = 4 levels of factor X1 L2 = 3 levels of factor X2 n = 1 replication per cell N = L1 * L2 = 4 * 3 = 12 runs Before randomization, the design trials look like:
==== Matrix representation ==== An alternate way of summarizing the design trials would be to use a 4x3 matrix whose 4 rows are the levels of the treatment X1 and whose columns are the 3 levels of the blocking variable X2. The cells in the matrix have indices that match the X1, X2 combinations above.
By extension, note that the trials for any K-factor randomized block design are simply the cell indices of a k dimensional matrix.
=== Model === The model for a randomized block design with one nuisance variable is
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{\displaystyle Y_{ij}=\mu +T_{i}+B_{j}+\mathrm {random\ error} }
where