Generalized randomized block design

In 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 (without making parametric assumptions about a normal distribution for the error).^[1]

Like a randomized complete block design (RCBD), a GRBD has randomization. Within each block, treatments are randomly assigned to experimental units and this randomization is independent between blocks. In a (classic) RCBD, there is no replication of treatments within blocks.^[2] Without replication, the (classic) RCBD has no block-treatment interaction-term that may be estimated and tested (using the randomization distribution rather than using a normal distribution for the error).^[3]

Many authors do not distinguish between RCBDs and GRBDs.^[4] The GRBD has the advantage that replication allows block-treatment interaction to be studied.^[5] However, if block-treatment interaction known to be negligible, the experimental protocol may specify that the interaction terms be assumed to be zero and that their degrees of freedom be used for the error term: Many textbooks on the analysis of variance and on the analysis of experimental data ignore the possibility of testing for block-treatment interaction and instead present only the RCBD analysis with extra degrees of freedom for error.^[6]

Notes

^
- Lentner and Biship, page 223.
- Addelman, page 35.
- Hinkelmann and Kempthorne, page 314, for example; c.f. page 312.
^
- Addelman, page 35.
- Hinkelmann and Kempthorne, page 314.
- Lentner and Bishop, page 223.
^
- Addelman, page 35.
- Lentner and Bishop, page 223.
A more detailed treatment occurs in Chapter 9.7 in Hinkelmann and Kempthorne. (Hinkelmann and Kempthorne do discuss block-treatment interaction for more complicated blocking structures, like crossed-blocking factors in Chapter 9.6, and for forms of "non-additivity" that may be removed by transformations).
^
- Complaints about the neglect of GRBDs in the literature and ignorance among practitioners are stated by Addelman (page 35).
^
- Addelman, page 35.
- Lentner and Bishop, page 223.
^
- Addelman, page 35.

References

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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}}: External link in |publisher= (help)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}}: External link in |publisher= and |title= (help)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}}: External link in |publisher= and |title= (help)CS1 maint: multiple names: authors list (link)

Lentner, Marvin (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}}: Unknown parameter |coauthor= ignored (|author= suggested) (help)CS1 maint: location (link)

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[1] 
Lentner and Biship, page 223.

Addelman, page 35.

Hinkelmann and Kempthorne, page 314, for example; c.f. page 312.

[2] Lentner and Biship, page 223.

[3] Addelman, page 35.

[4] Hinkelmann and Kempthorne, page 314, for example; c.f. page 312.

[2] 
Addelman, page 35.

Hinkelmann and Kempthorne, page 314.

Lentner and Bishop, page 223.

[6] Addelman, page 35.

[7] Hinkelmann and Kempthorne, page 314.

[8] Lentner and Bishop, page 223.

[3] 
Addelman, page 35.

Lentner and Bishop, page 223.
A more detailed treatment occurs in Chapter 9.7 in Hinkelmann and Kempthorne. (Hinkelmann and Kempthorne do discuss block-treatment interaction for more complicated blocking structures, like crossed-blocking factors in Chapter 9.6, and for forms of "non-additivity" that may be removed by transformations).

[10] Addelman, page 35.

[11] Lentner and Bishop, page 223.

[4] 
Complaints about the neglect of GRBDs in the literature and ignorance among practitioners are stated by Addelman (page 35).

[13] Complaints about the neglect of GRBDs in the literature and ignorance among practitioners are stated by Addelman (page 35).

[5] 
Addelman, page 35.

Lentner and Bishop, page 223.

[15] Addelman, page 35.

[16] Lentner and Bishop, page 223.

[6] 
Addelman, page 35.

[18] Addelman, page 35.

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v t e Design of experiments
Scientific method	Scientific experiment Statistical design Control Internal and external validity Experimental unit Blinding Optimal design: Bayesian Random assignment Randomization Restricted randomization Replication versus subsampling Sample size
Treatment and blocking	Treatment Effect size Contrast Interaction Confounding Orthogonality Blocking Covariate Nuisance variable
Models and inference	Linear regression Ordinary least squares Bayesian Random effect Mixed model Hierarchical model: Bayesian Analysis of variance (Anova) Cochran's theorem Manova (multivariate) Ancova (covariance) Compare means Multiple comparison
Designs Completely randomized	Factorial Fractional factorial Plackett–Burman Taguchi Response surface methodology Polynomial and rational modeling Box–Behnken Central composite Block Generalized randomized block design (GRBD) Latin square Graeco-Latin square Orthogonal array Latin hypercube Repeated measures design Crossover study Randomized controlled trial Sequential analysis Sequential probability ratio test
Glossary Category Mathematics portal Statistical outline Statistical topics

See also

Notes

References