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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. 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.

**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

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:

- “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.

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

*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*L*= number of levels (settings) of factor_{k}*k*

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:

*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:

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

*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})

- Estimate for μ : = the average of all the data
- Estimate for
*T*_{i}: with = average of all*Y*for which*X*_{1}=*i*. - Estimate for
*B*_{j}: with = average of all*Y*for which*X*_{2}=*j*.

- 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

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*.

**Analysis of variance** (**ANOVA**) is a collection of statistical models and their associated estimation procedures used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the *t*-test beyond two means. A non-parametric alternative is PERMANOVA.

The **design of experiments** is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associated with experiments in which the design introduces conditions that directly affect the variation, but may also refer to the design of quasi-experiments, in which natural conditions that influence the variation are selected for observation.

**Analysis of covariance** (**ANCOVA**) is a general linear model which blends ANOVA and regression. ANCOVA evaluates whether the means of a dependent variable (DV) are equal across levels of a categorical independent variable (IV) often called a treatment, while statistically controlling for the effects of other continuous variables that are not of primary interest, known as covariates (CV) or nuisance variables. Mathematically, ANCOVA decomposes the variance in the DV into variance explained by the CV(s), variance explained by the categorical IV, and residual variance. Intuitively, ANCOVA can be thought of as 'adjusting' the DV by the group means of the CV(s).

**Dependent** and **Independent variables** are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand that they depend, by some law or rule, on the values of other variables. Independent variables, in turn, are not seen as depending on any other variable in the scope of the experiment in question. In this sense, some common independent variables are time, space, density, mass, fluid flow rate, and previous values of some observed value of interest to predict future values.

In statistics, an **interaction** may arise when considering the relationship among three or more variables, and describes a situation in which the effect of one causal variable on an outcome depends on the state of a second causal variable. Although commonly thought of in terms of causal relationships, the concept of an interaction can also describe non-causal associations. Interactions are often considered in the context of regression analyses or factorial experiments.

In statistics, a full **factorial experiment** is an experiment whose design consists of two or more factors, each with discrete possible values or "levels", and whose experimental units take on all possible combinations of these levels across all such factors. A full **factorial design** may also be called a **fully crossed design**. Such an experiment allows the investigator to study the effect of each factor on the response variable, as well as the effects of interactions between factors on the response variable.

This **glossary of statistics and probability** is a list of definitions of terms and concepts used in the mathematical sciences of statistics and probability, their sub-disciplines, and related fields. For additional related terms, see Glossary of mathematics.

In statistics, a **confounder** is a variable that influences both the dependent variable and independent variable, causing a spurious association. Confounding is a causal concept, and as such, cannot be described in terms of correlations or associations. The existence of confounders is an important quantitative explanation why correlation does not imply causation.

In science, **randomized experiments** are the experiments that allow the greatest reliability and validity of statistical estimates of treatment effects. Randomization-based inference is especially important in experimental design and in survey sampling.

In the design of experiments and analysis of variance, a **main effect** is the effect of an independent variable on a dependent variable averaged across the levels of any other independent variables. The term is frequently used in the context of factorial designs and regression models to distinguish main effects from interaction effects.

A **quasi-experiment** is an empirical interventional study used to estimate the causal impact of an intervention on target population without random assignment. Quasi-experimental research shares similarities with the traditional experimental design or randomized controlled trial, but it specifically lacks the element of random assignment to treatment or control. Instead, quasi-experimental designs typically allow the researcher to control the assignment to the treatment condition, but using some criterion other than random assignment.

In statistics, **one-way analysis of variance** is a technique that can be used to compare whether two sample's means are significantly different or not. This technique can be used only for numerical response data, the "Y", usually one variable, and numerical or (usually) categorical input data, the "X", always one variable, hence "one-way".

The **average treatment effect** (**ATE**) is a measure used to compare treatments in randomized experiments, evaluation of policy interventions, and medical trials. The ATE measures the difference in mean (average) outcomes between units assigned to the treatment and units assigned to the control. In a randomized trial, the average treatment effect can be estimated from a sample using a comparison in mean outcomes for treated and untreated units. However, the ATE is generally understood as a causal parameter that a researcher desires to know, defined without reference to the study design or estimation procedure. Both observational studies and experimental study designs with random assignment may enable one to estimate an ATE in a variety of ways.

**Repeated measures design** is a research design that involves multiple measures of the same variable taken on the same or matched subjects either under different conditions or over two or more time periods. For instance, repeated measurements are collected in a longitudinal study in which change over time is assessed.

In statistics, econometrics, political science, epidemiology, and related disciplines, a **regression discontinuity design (RDD)** is a quasi-experimental pretest-posttest design that aims to determine the causal effects of interventions by assigning a cutoff or threshold above or below which an intervention is assigned. By comparing observations lying closely on either side of the threshold, it is possible to estimate the average treatment effect in environments in which randomisation is unfeasible. However, it remains impossible to make true causal inference with this method alone, as it does not automatically reject causal effects by any potential confounding variable. First applied by Donald Thistlethwaite and Donald Campbell to the evaluation of scholarship programs, the RDD has become increasingly popular in recent years. Recent study comparisons of randomised controlled trials (RCTs) and RDDs have empirically demonstrated the internal validity of the design.

**Algebraic statistics** is the use of algebra to advance statistics. Algebra has been useful for experimental design, parameter estimation, and hypothesis testing.

In statistics, **restricted randomization** occurs in the design of experiments and in particular in the context of randomized experiments and randomized controlled trials. Restricted randomization allows intuitively poor allocations of treatments to experimental units to be avoided, while retaining the theoretical benefits of randomization. For example, in a clinical trial of a new proposed treatment of obesity compared to a control, an experimenter would want to avoid outcomes of the randomization in which the new treatment was allocated only to the heaviest patients.

In the design of experiments, **completely randomized designs** are for studying the effects of one primary factor without the need to take other nuisance variables into account. This article describes completely randomized designs that have one primary factor. The experiment compares the values of a response variable based on the different levels of that primary factor. For completely randomized designs, the levels of the primary factor are randomly assigned to the experimental units.

The following is a glossary of terms. It is not intended to be all-inclusive.

In randomized statistical experiments, **generalized randomized block designs** (**GRBDs**) are used to study the interaction between blocks and treatments. For a GRBD, each treatment is replicated at least two times in each block; this replication allows the estimation and testing of an interaction term in the linear model.

- ↑ Bernstein S.N. (1926) Sur l'extension du théorème limite du calcul des probabilités aux sommes de quantités dépendantes. Math. Annalen, v. 97, 1-59.
- ↑ Ibragimov I.A. and Linnik Yu.V. (1971) Independent and stationary sequences of random variables. Wolters-Noordhoff, Groningen.
- ↑ Leadbetter M.R., Lindgren G. and Rootzén H. (1983) Extremes and Related Properties of Random Sequences and Processes. New York: Springer Verlag.
- ↑ Novak S.Y. (2011) Extreme Value Methods with Applications to Finance. Chapman & Hall/CRC Press, London.

- This article incorporates public domain material from the National Institute of Standards and Technology website https://www.nist.gov .

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- Caliński T. & Kageyama S. (2000).
*Block designs: A Randomization approach, Volume*. Lecture Notes in Statistics. Vol. 150. New York: Springer-Verlag. ISBN 0-387-98578-6.**I**: Analysis - Caliński T. & Kageyama S. (2003).
*Block designs: A Randomization approach, Volume*. Lecture Notes in Statistics. Vol. 170. New York: Springer-Verlag. ISBN 0-387-95470-8. MR 1994124.**II**: Design - Gates, C.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}}`

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*Design and Analysis of Experiments, Volume 2: Advanced Experimental Design*(First ed.). Wiley. ISBN 978-0-471-55177-5.`{{cite book}}`

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- 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}}`

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- Street, Anne Penfold; Street, Deborah J. (1987).
*Combinatorics of Experimental Design*. Oxford U. P. [Clarendon]. ISBN 0-19-853256-3. - Wilk, M. B. (1955). "The Randomization Analysis of a Generalized Randomized Block Design".
*Biometrika*.**42**(1–2): 70–79. doi:10.2307/2333423. JSTOR 2333423. - Zyskind, George (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.

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