Blocking (statistics)

In the statistical theory of the design of experiments, blocking is the arranging of experimental units that are similar to one another in groups (blocks) based on one or more variables. These variables are chosen carefully to minimize the effect of their variability on the observed outcomes. There are different ways that blocking can be implemented, resulting in different confounding effects. However, the different methods share the same purpose: to control variability introduced by specific factors that could influence the outcome of an experiment. The roots of blocking originated from the statistician, Ronald Fisher, following his development of ANOVA.

In the statistical theory of the design of experiments, blocking is the arranging of experimental units that are similar to one another in groups (blocks) based on one or more variables. These variables are chosen carefully to minimize the effect of their variability on the observed outcomes. There are different ways that blocking can be implemented, resulting in different confounding effects. However, the different methods share the same purpose: to control variability introduced by specific factors that could influence the outcome of an experiment. The roots of blocking originated from the statistician, Ronald Fisher, following his development of ANOVA. ===Examples=== Male and female: An experiment is designed to test a new drug on patients. There are two levels of the treatment, drug, and placebo, administered to male and female patients in a double blind trial. The sex of the patient is a blocking factor accounting for treatment variability between males and females. This reduces sources of variability and thus leads to greater precision. Elevation: An experiment is designed to test the effects of a new pesticide on a specific patch of grass. The grass area contains a major elevation change and thus consists of two distinct regions – 'high elevation' and 'low elevation'. A treatment group (the new pesticide) and a placebo group are applied to both the high elevation and low elevation areas of grass. In this instance the researcher is blocking the elevation factor which may account for variability in the pesticide's application. Intervention: Suppose a process is invented that intends to make the soles of shoes last longer, and a plan is formed to conduct a field trial. Given a group of n volunteers, one possible design would be to give n/2 of them shoes with the new soles and n/2 of them shoes with the ordinary soles, randomizing the assignment of the two kinds of soles. This type of experiment is a completely randomized design. Both groups are then asked to use their shoes for a period of time, and then measure the degree of wear of the soles. This is a workable experimental design, but purely from the point of view of statistical accuracy (ignoring any other factors), a better design would be to give each person one regular sole and one new sole, randomly assigning the two types to the left and right shoe of each volunteer. Such a design is called a "randomized complete block design." This design will be more sensitive than the first, because each person is acting as his/her own control and thus the control group is more closely matched to the treatment group block design ===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. ===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 L 1 = number of levels (settings) of factor 1 L 2 = number of levels (settings) of factor 2 L 3 = number of levels (settings) of factor 3 L 4 = number of levels (settings) of factor 4 \vdots L k = 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 X 1 be dosage "level" and X 2 be the blocking factor furnace run. Then the experiment can be described as follows: k = 2 factors (1 primary factor X 1 and 1 blocking factor X 2 ) L 1 = 4 levels of factor X 1 L 2 = 3 levels of factor X 2 n = 1 replication per cell N = L 1 * L 2 = 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 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. 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 Y_{ij} = \mu + T_i + B_j + \mathrm{random\ error} where Y ij is any observation for which X 1 = i and X 2 = j X 1 is the primary factor X 2 is the blocking factor μ is the general location parameter (i.e., the mean) T i is the effect for being in treatment i (of factor X 1 ) B j is the effect for being in block j (of factor X 2 ) ===Estimates=== Estimate for μ : \overline{Y} = the average of all the data Estimate for T i : \overline{Y}_{i\cdot} - \overline{Y} with \overline{Y}_{i\cdot} = average of all Y for which X 1 = i. Estimate for B j : \overline{Y}_{\cdot j} - \overline{Y} with \overline{Y}_{\cdot j} = average of all Y for which X 2 = j. ===Generalizations=== Generalized randomized block designs (GRBD) allow tests of block–treatment interaction, and has exactly one blocking factor like the RCBD. Latin squares (and other row–column designs) have two blocking factors that are believed to have no interaction. Latin hypercube sampling Graeco-Latin squares Hyper-Graeco-Latin square designs ==See also== Algebraic statistics Block design Combinatorial design Generalized randomized block design Glossary of experimental design Optimal design Paired difference test Dependent and independent variables Blockmodeling Paired data Block bootstrap Controlling for a variable ==References== ==Bibliography== Pre-publication chapters are available on-line.

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