Randomized block design

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.

Blocking to “remove” the effect of nuisance factors

For randomized block designs, there is one factor or variable that is of primary interest. However, there are also several other nuisance factors.

Nuisance factors are those that may affect the measured result, but are not of primary interest. For example, in applying a treatment, nuisance factors might be the specific operator who prepared the treatment, the time of day the experiment was run, and the room temperature. All experiments have nuisance factors. The experimenter will typically need to spend some time deciding which nuisance factors are important enough to keep track of or control, if possible, during the experiment.

Blocking used for nuisance factors that can be controlled

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.

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 of randomized block designs

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.

Randomized Block Designs (RBD)

Name of Design	Number of Factors k	Number of Runs n
2-factor RBD	2	L1 * L2
3-factor RBD	3	L1 * L2 * L3
4-factor RBD	4	L1 * L2 * L3 * L4
$\vdots$	$\vdots$	$\vdots$
k-factor RBD	k	L1 * L2 * $\cdots$ * Lk

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

$\vdots$

L_k = number of levels (settings) of factor k

Example of a Randomized Block Design

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:

X1	X2
1	1
1	2
1	3
2	1
2	2
2	3
3	1
3	2
3	3
4	1
4	2
4	3

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 for a Randomized Block Design

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 X₁ = i and X₂ = j

X₁ is the primary factor

X₂ 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 X₁)

B_j is the effect for being in block j (of factor X₂)

Estimates for a Randomized Block Design

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 X₁ = i.

Estimate for B_j : ${\overline {Y}}_{\cdot j}-{\overline {Y}}$ with ${\overline {Y}}_{\cdot j}$ = average of all Y for which X₂ = j.

Generalizations of randomized block designs

• 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
• Combinatorial design
• Generalized randomized block design
• Glossary of experimental design
• Optimal design

References

• Addelman, Sidney (Oct 1969). "The Generalized Randomized Block Design". The American Statistician. 23 (4): 35–36. doi:10.2307/2681737. JSTOR 2681737.
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• Anscombe, F. J. (1948). "The Validity of Comparative Experiments". Journal of the Royal Statistical Society. Series A (General). 111 (3): 181–211. doi:10.2307/2984159. JSTOR 2984159. | mr = 30181
• Bailey, R. A (2008). Design of Comparative Experiments. Cambridge University Press. ISBN 978-0-521-68357-9. {{cite book}}: External link in |author= and |publisher= (help) Pre-publication chapters are available on-line.
• Bapat, R. B. (2000). Linear Algebra and Linear Models (Second ed.). Springer. ISBN 978-0-387-98871-9. {{cite book}}: External link in |author= and |publisher= (help)
• Caliński, Tadeusz; Kageyama, Sanpei (2000). Block designs: A Randomization approach, Volume I: Analysis. Lecture Notes in Statistics. Vol. 150. New York: Springer-Verlag. ISBN 0-387-98578-6. {{cite book}}: Unknown parameter |lastauthoramp= ignored (|name-list-style= suggested) (help)
• Caliński, Tadeusz; Kageyama, Sanpei (2003). Block designs: A Randomization approach, Volume II: Design. Lecture Notes in Statistics. Vol. 170. New York: Springer-Verlag. ISBN 0-387-95470-8. MR 1994124. {{cite book}}: Unknown parameter |lastauthoramp= ignored (|name-list-style= suggested) (help)
• Gates, Charles E. (Nov 1995). "What Really Is Experimental Error in Block Designs?". The American Statistician. 49 (4): 362–363. doi:10.2307/2684574. JSTOR 2684574.
• Kempthorne, Oscar (1979). The Design and Analysis of Experiments (Corrected reprint of (1952) Wiley ed.). Robert E. Krieger. ISBN 0-88275-105-0.
• Hinkelmann, Klaus and Kempthorne, Oscar (2008). Design and Analysis of Experiments. Vol. I and II (Second ed.). Wiley. ISBN 978-0-470-38551-7.{{cite book}}: CS1 maint: multiple names: authors list (link)
  • Hinkelmann, Klaus and Kempthorne, Oscar (2008). Design and Analysis of Experiments, Volume I: Introduction to Experimental Design (Second ed.). Wiley. ISBN 978-0-471-72756-9.{{cite book}}: CS1 maint: multiple names: authors list (link)
  • Hinkelmann, Klaus and Kempthorne, Oscar (2005). Design and Analysis of Experiments, Volume 2: Advanced Experimental Design (First ed.). Wiley. ISBN 978-0-471-55177-5.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Lentner, Marvin; Thomas Bishop (1993). "The Generalized RCB Design (Chapter 6.13)". Experimental design and analysis (Second ed.). P.O. Box 884, Blacksburg, VA 24063: Valley Book Company. pp. 225–226. ISBN 0-9616255-2-X.{{cite book}}: CS1 maint: location (link)
• Raghavarao, Damaraju (1988). Constructions and Combinatorial Problems in Design of Experiments (corrected reprint of the 1971 Wiley ed.). New York: Dover. ISBN 0-486-65685-3.
• Raghavarao, Damaraju and Padgett, L.V. (2005). Block Designs: Analysis, Combinatorics and Applications. World Scientific. ISBN 981-256-360-1.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Shah, Kirti R.; Sinha, Bikas K. (1989). Theory of Optimal Designs. Lecture Notes in Statistics. Vol. 54. Springer-Verlag. pp. 171+viii. ISBN 0-387-96991-8. {{cite book}}: External link in |series= (help); Unknown parameter |lastauthoramp= ignored (|name-list-style= suggested) (help)
• Street, Anne Penfold; Street, Deborah J. (1987). Combinatorics of Experimental Design. Oxford U. P. [Clarendon]. pp. 400+xiv. ISBN 0-19-853256-3. {{cite book}}: Unknown parameter |lastauthoramp= ignored (|name-list-style= suggested) (help)
• Wilk, M. B. (June 1955). "The Randomization Analysis of a Generalized Randomized Block Design". Biometrika. 42 (1–2): 70–79. doi:10.2307/2333423. JSTOR 2333423.
• Zyskind, George (Dec 1963). "Some Consequences of randomization in a Generalization of the Balanced Incomplete Block Design". The Annals of Mathematical Statistics. 34 (4): 1569–1581. doi:10.1214/aoms/1177703889. JSTOR 2238364.

External links

• Examples of all ANOVA and ANCOVA models with up to three treatment factors, including randomized block, split plot, repeated measures, and Latin squares, and their analysis in R (University of Southampton)
• Randomized Block Designs

 This article incorporates public domain material from the National Institute of Standards and Technology