WordNet

a three-dimensional shape with six square or rectangular sides (同)cube
(computer science) a sector or group of sectors that function as the smallest data unit permitted; "since blocks are often defined as a single sector, the terms `block and `sector are sometimes used interchangeably"
support, secure, or raise with a block; "block a plate for printing"; "block the wheels of a car"
a solid piece of something (usually having flat rectangular sides); "the pyramids were built with large stone blocks"
housing in a large building that is divided into separate units; "there is a block of classrooms in the west wing"
a rectangular area in a city surrounded by streets and usually containing several buildings; "he lives in the next block" (同)city block
an inability to remember or think of something you normally can do; often caused by emotional tension; "I knew his name perfectly well but I had a temporary block" (同)mental_block
a number or quantity of related things dealt with as a unit; "he reserved a large block of seats"; "he held a large block of the companys stock"
interrupt the normal function of by means of anesthesia; "block a nerve"; "block a muscle"
run on a block system; "block trains"
shape by using a block; "Block a hat"; "block a garment"
shape into a block or blocks; "block the graphs so one can see the results clearly"
stamp or emboss a title or design on a book with a block; "block the book cover"
plan something for a specific role or purpose or effect; "This room is not designed for work"
the act of working out the form of something (as by making a sketch or outline or plan); "he contributed to the design of a new instrument" (同)designing
a decorative or artistic work; "the coach had a design on the doors" (同)pattern, figure
a preliminary sketch indicating the plan for something; "the design of a building"
an arrangement scheme; "the awkward design of the keyboard made operation difficult"; "it was an excellent design for living"; "a plan for seating guests" (同)plan
make a design of; plan out in systematic, often graphic form; "design a better mousetrap"; "plan the new wing of the museum" (同)plan
conceive or fashion in the mind; invent; "She designed a good excuse for not attending classes that day"
create designs; "Dupont designs for the house of Chanel"
create the design for; create or execute in an artistic or highly skilled manner; "Chanel designed the famous suit"
intend or have as a purpose; "She designed to go far in the world of business"
concealing crafty designs for advancing your own interest; "a selfish and designing nation obsessed with the dark schemes of European intrigue"- W.Churchill; "a scheming wife"; "a scheming gold digger" (同)scheming
completely obstructed or closed off; "the storm was responsible for many blocked roads and bridges"; "the drain was plugged" (同)plugged
closed to traffic; "the repaving results in many blocked streets" (同)out of use
a group of countries in special alliance (同)axis

PrepTutorEJDIC

(木・石などの,通例平らな面のある)『かたまり』 / 《米》(四団を街路に囲まれた)『区画』,街区 / 街区の1辺の距離(一般に40メートル前後) / 《英》ブロック(アパート・事務所・商店などに区切られている大きな建物) / 『妨害物』,障害物(obstacle);(障害によって生じた)停滞,渋滞 / 台木,台(まないた・肉切り台・まき割り台・せり売り台・帽子の木型など);《the~》炭頭台 / (特にアメリカンフットボールで)ブロック(相手の選手の突進を阻止する) / 滑車(pulley) / (座席券・株券などの)一組《+『of』+『名』》 / 〈通路など〉'を'『ふさぐ』,〈通行・進行など〉'を'妨げる / 〈帽子など〉の型取りをする / (特にアメリカンフットボールで)〈相手選手の突進〉'を'ブロックする;(ボクシングで)〈相手のパンチ〉'を'ブロックする
{C}(形式・構造などの)略図,見取り図,設計図 / 〈C〉『図案』,意匠,模様 / 〈U〉図案法,意匠術 / 〈C〉(…の)『計画』,企画,目的《+『for』+『名』》 / 《複数形で》(…への)たくらみ,下心《+『against』(『on』,『upon』)+『名』》 / 〈構造など〉‘を'『設計する』 / …‘の'『下絵を描く』,図案を描く / …‘を'『計画する』,頭の中で考える(plan) / …‘を'『予定する』
(人が)たくらみのある,下心のある,陰険な,腹黒い / 設計,図案作製,(服飾の)デザイン
ブロック,圏(地理・共通の利害・政治・経済上の目的などのために団結した団体・国家などの結合体)

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2013/06/03 14:36:58」(JST)

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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.

1 Blocking to "remove" the effect of nuisance factors
2 Blocking used for nuisance factors that can be controlled
3 Definition of blocking factors
4 Block for a few of the most important nuisance factors
5 Table of randomized block designs
6 Example of a Randomized Block Design
- 6.1 Description of the experiment
- 6.2 Matrix Representation
7 Model for a Randomized Block Design
8 Estimates for a Randomized Block Design
9 Generalizations of randomized block designs
10 See also
11 External links
12 References

Blocking to "remove" the effect of nuisance factors [edit]

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 [edit]

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 [edit]

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 for a few of the most important nuisance factors [edit]

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.

Table of randomized block designs [edit]

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

k-factor RBD	k	L₁ * L₂ * * L_k

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

L_k = number of levels (settings) of factor k

Example of a Randomized Block Design [edit]

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 [edit]

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 [edit]

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 [edit]

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 [edit]

Estimate for μ : = the average of all the data

Estimate for T_i : with = average of all Y for which X₁ = i.

Estimate for B_j : with = average of all Y for which X₂ = j.

Generalizations of randomized block designs [edit]

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

External links [edit]

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
Randomized Block Designs

References [edit]

Addelman, Sidney (Oct. 1969). "The Generalized Randomized Block Design". The American Statistician 23 (4): 35–36. doi:10.2307/2681737. JSTOR 2681737.

Addelman, Sidney (Sep. 1970). "Variability of Treatments and Experimental Units in the Design and Analysis of Experiments". Journal of the American Statistical Association 65 (331): 1095–1108. doi:10.2307/2284277. JSTOR 2284277.

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. 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.

Caliński, Tadeusz and Kageyama, Sanpei (2000). Block designs: A Randomization approach, Volume I: Analysis. Lecture Notes in Statistics 150. New York: Springer-Verlag. ISBN 0-387-98578-6.

Caliński, Tadeusz and Kageyama, Sanpei (2003). Block designs: A Randomization approach, Volume II: Design. Lecture Notes in Statistics 170. New York: Springer-Verlag. ISBN 0-387-95470-8. MR 1994124.

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. I and II (Second ed.). Wiley. ISBN 978-0-470-38551-7.
- 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.
- 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.

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.

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.
Shah, Kirti R. and Sinha, Bikas K. (1989). Theory of Optimal Designs. Lecture Notes in Statistics 54. Springer-Verlag. pp. 171+viii. ISBN 0-387-96991-8.
Street, Anne Penfold and Street, Deborah J. (1987). Combinatorics of Experimental Design. Oxford U. P. [Clarendon]. pp. 400+xiv. ISBN 0-19-853256-3.

Wilk, M. B. (June 1955). "The Randomization Analysis of a Generalized Randomized Block Design". Biometrika 42 (1–2): 70–79. 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.

This article incorporates public domain material from websites or documents of the National Institute of Standards and Technology.

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