Generalized randomized block design
generalized randomized block design example, generalized randomized block design statisticsIn 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 without making parametric assumptions about a normal distribution for the error
Contents
 1 Univariate response
 11 GRBDs versus RCBDs: Replication and interaction
 12 Twoway linear model: Blocks and treatments
 121 GRBDs when blocktreatment interaction lacks interest
 2 Multivariate analysis
 3 Functional models for blocktreatment interactions: Testing known forms of interaction
 4 See also
 5 Notes
 6 References
Univariate response
GRBDs versus RCBDs: Replication and interaction
See also: Randomized block design, Replication statistics, and Interaction statisticsLike a randomized complete block design RCBD, a GRBD is randomized Within each block, treatments are randomly assigned to experimental units: this randomization is also independent between blocks In a classic RCBD, however, there is no replication of treatments within blocks
Twoway linear model: Blocks and treatments
The experimental design guides the formulation of an appropriate linear model Without replication, the classic RCBD has a twoway linearmodel with treatment and blockeffects but without a blocktreatment interaction Without replicates, this twoway linearmodel that may be estimated and tested without making parametric assumptions by using the randomization distribution, without using a normal distribution for the error In the RCBD, the blocktreatment interaction cannot be estimated using the randomization distribution; a fortiori there exists no "valid" ie randomizationbased test for the blocktreatment interaction in the analysis of variance anova of the RCBD
The distinction between RCBDs and GRBDs has been ignored by some authors, and the ignorance regarding the GRCBD has been criticized by statisticians like Oscar Kempthorne and Sidney Addelman The GRBD has the advantage that replication allows blocktreatment interaction to be studied
GRBDs when blocktreatment interaction lacks interest
However, if blocktreatment interaction is known to be negligible, then 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 GRBD designs for models without interaction terms offer more degrees of freedom for testing treatmenteffects than do RCBs with more blocks: An experimenter wanting to increase power may use a GRBD rather than RCB with additional blocks, when extra blockseffects would lack genuine interest
Multivariate analysis
The GRBD has a realnumber response For vector responses, multivariate analysis considers similar twoway models with main effects and with interactions or errors Without replicates, error terms are confounded with interaction, and only error is estimated With replicates, interaction can be tested with the multivariate analysis of variance and coefficients in the linear model can be estimated without bias and with minimum variance by using the leastsquares method
Functional models for blocktreatment interactions: Testing known forms of interaction
Nonreplicated experiments are used by knowledgeable experimentalists when replications have prohibitive costs When the blockdesign lacks replicates, interactions have been modeled For example, Tukey's Ftest for interaction nonadditivity has been motivated by the multiplicative model of Mandel 1961; this model assumes that all treatmentblock interactions are proportion to the product of the mean treatmenteffect and the mean blockeffect, where the proportionality constant is identical for all treatmentblock combinations Tukey's test is valid when Mandel's multiplicative model holds and when the errors independently follow a normal distribution
Tukey's Fstatistic for testing interaction has a distribution based on the randomized assignment of treatments to experimental units When Mandel's multiplicative model holds, the Fstatistics randomization distribution is closely approximated by the distribution of the Fstatistic assuming a normal distribution for the error, according to the 1975 paper of Robinson
The rejection of multiplicative interaction need not imply the rejection of nonmultiplicative interaction, because there are many forms of interaction
Generalizing earlier models for Tukey's test are the “bundleofstraight lines” model of Mandel 1959 and the functional model of Milliken and Graybill 1970, which assumes that the interaction is a known function of the block and treatment maineffects Other methods and heuristics for blocktreatment interaction in unreplicated studies are surveyed in the monograph Milliken & Johnson 1989
See also
 Block design
 Complete block design
 Incomplete block design
 Randomized block design
 Randomization
 Randomized experiment
Notes
 ^
 Wilk, page 79
 Lentner and Biship, page 223
 Addelman 1969 page 35
 Hinkelmann and Kempthorne, page 314, for example; cf page 312
 ^
 Wilk, page 79
 Addelman 1969 page 35
 Hinkelmann and Kempthorne, page 314
 Lentner and Bishop, page 223
 ^
 Wilk, page 79
 Addelman 1969 page 35
 Lentner and Bishop, page 223
 ^ Wilk, Addelman, Hinkelmann and Kempthorne
 ^
 Complaints about the neglect of GRBDs in the literature and ignorance among practitioners are stated by Addelman 1969 page 35
 ^
 Wilk, page 79
 Addelman 1969 page 35
 Lentner and Bishop, page 223
 ^
 Addelman 1970 page 1104
 ^ Johnson & Wichern 2002, p 312, “Multivariate twoway fixedeffects model with interaction”, in “66 Twoway multivariate analysis of variance”, p 307–317
 ^ Mardia, Kent & Bibby 1979, p 352, “Tests for interactions”, in 127 Twoway classification, p 350356
 ^ Hinklemann & Kempthorne 2008, p 305
 ^ Milliken & Johnson 1989, 16 Tukey's single degreeoffreedom test for nonadditivity, pp 78
 ^ Lentner & Bishop 1993, p 214, in 68 Nonadditivity of blocks and treatments, pp 213–216
 ^ Milliken & Johnson 1989, 18 Mandel's bundleofstraight lines model, pp 1729
References
 Addelman, Sidney Oct 1969 "The Generalized Randomized Block Design" The American Statistician 23 4: 35–36 doi:102307/2681737 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:102307/2284277 JSTOR 2284277
 Gates, Charles E Nov 1995 "What Really Is Experimental Error in Block Designs" The American Statistician 49 4: 362–363 doi:102307/2684574 JSTOR 2684574
 Hinkelmann, Klaus; Kempthorne, Oscar 2008 Design and Analysis of Experiments, Volume I: Introduction to Experimental Design Second ed Wiley ISBN 9780471727569 MR 2363107
 Johnson, Richard A; Wichern, Dean W 2002 "6 Comparison of several multivariate means" Applied multivariate statistical analysis Fifth ed Prentice Hall pp 272–353 ISBN 0131219731
 Lentner, Marvin; Bishop, Thomas 1993 "The Generalized RCB Design Chapter 613" Experimental design and analysis Second ed PO Box 884, Blacksburg, VA 24063: Valley Book Company pp 225–226 ISBN 096162552X
 Mardia, K V; Kent, J T; Bibby, J M 1979 "12 Multivariate analysis of variance" Multivariate analysis Academic Press ISBN 0124712509
 Milliken, George A; Johnson, Dallas E 1989 Nonreplicated experiments: Designed experiments Analysis of messy data 2 New York: Van Nostrand Reinhold
 Wilk, M B June 1955 "The Randomization Analysis of a Generalized Randomized Block Design" Biometrika 42 1–2: 70–79 doi:102307/2333423 JSTOR 2333423 MR 0068800
 Zyskind, George December 1963 "Some Consequences of Randomization in a Generalization of the Balanced Incomplete Block Design" The Annals of Mathematical Statistics 34 4: 1569–1581 doi:101214/aoms/1177703889 JSTOR 2238364 MR 0157448



Scientific method 

Treatment and blocking 

Models and inference 

Designs Completely randomized 


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