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Generalized randomized block design

generalized randomized block design example, generalized randomized block design statistics
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 without making parametric assumptions about a normal distribution for the error

Contents

  • 1 Univariate response
    • 11 GRBDs versus RCBDs: Replication and interaction
    • 12 Two-way linear model: Blocks and treatments
      • 121 GRBDs when block-treatment interaction lacks interest
  • 2 Multivariate analysis
  • 3 Functional models for block-treatment 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 statistics

Like 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

Two-way linear model: Blocks and treatments

The experimental design guides the formulation of an appropriate linear model Without replication, the classic RCBD has a two-way linear-model with treatment- and block-effects but without a block-treatment interaction Without replicates, this two-way linear-model 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 block-treatment interaction cannot be estimated using the randomization distribution; a fortiori there exists no "valid" ie randomization-based test for the block-treatment 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 block-treatment interaction to be studied

GRBDs when block-treatment interaction lacks interest

However, if block-treatment 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 treatment-effects than do RCBs with more blocks: An experimenter wanting to increase power may use a GRBD rather than RCB with additional blocks, when extra blocks-effects would lack genuine interest

Multivariate analysis

The GRBD has a real-number response For vector responses, multivariate analysis considers similar two-way 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 least-squares method

Functional models for block-treatment interactions: Testing known forms of interaction

Non-replicated experiments are used by knowledgeable experimentalists when replications have prohibitive costs When the block-design lacks replicates, interactions have been modeled For example, Tukey's F-test for interaction non-additivity has been motivated by the multiplicative model of Mandel 1961; this model assumes that all treatment-block interactions are proportion to the product of the mean treatment-effect and the mean block-effect, where the proportionality constant is identical for all treatment-block combinations Tukey's test is valid when Mandel's multiplicative model holds and when the errors independently follow a normal distribution

Tukey's F-statistic for testing interaction has a distribution based on the randomized assignment of treatments to experimental units When Mandel's multiplicative model holds, the F-statistics randomization distribution is closely approximated by the distribution of the F-statistic 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 non-multiplicative interaction, because there are many forms of interaction

Generalizing earlier models for Tukey's test are the “bundle-of-straight 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 main-effects Other methods and heuristics for block-treatment 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

  1. ^
    • Wilk, page 79
    • Lentner and Biship, page 223
    • Addelman 1969 page 35
    • Hinkelmann and Kempthorne, page 314, for example; cf page 312
  2. ^
    • Wilk, page 79
    • Addelman 1969 page 35
    • Hinkelmann and Kempthorne, page 314
    • Lentner and Bishop, page 223
  3. ^
    • Wilk, page 79
    • Addelman 1969 page 35
    • Lentner and Bishop, page 223
    A more detailed treatment occurs in Chapter 97 in Hinkelmann and Kempthorne Hinkelmann and Kempthorne do discuss block-treatment interaction for more complicated blocking structures, like crossed-blocking factors in Chapter 96, and for forms of "non-additivity" that may be removed by transformations
  4. ^ Wilk, Addelman, Hinkelmann and Kempthorne
  5. ^
    • Complaints about the neglect of GRBDs in the literature and ignorance among practitioners are stated by Addelman 1969 page 35
  6. ^
    • Wilk, page 79
    • Addelman 1969 page 35
    • Lentner and Bishop, page 223
  7. ^
    • Addelman 1970 page 1104
    If the scientists do not know that the block-treatment interaction is zero, Addelman requires that the generalized randomized block design be used, because otherwise the block-treatment interaction and the error are confounded In this situation, where scientists are uncertain whether the block-treatment interaction is zero, Hinkelmann and Kempthorne recommend that the generalized randomized block design be used "if at all possible" page 312
  8. ^ Johnson & Wichern 2002, p 312, “Multivariate two-way fixed-effects model with interaction”, in “66 Two-way multivariate analysis of variance”, p 307–317
  9. ^ Mardia, Kent & Bibby 1979, p 352, “Tests for interactions”, in 127 Two-way classification, p 350-356
  10. ^ Hinklemann & Kempthorne 2008, p 305
  11. ^ Milliken & Johnson 1989, 16 Tukey's single degree-of-freedom test for nonadditivity, pp 7-8
  12. ^ Lentner & Bishop 1993, p 214, in 68 Nonadditivity of blocks and treatments, pp 213–216
  13. ^ Milliken & Johnson 1989, 18 Mandel's bundle-of-straight lines model, pp 17-29

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 978-0-471-72756-9 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 0-13-121973-1 
  • 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 0-9616255-2-X 
  • Mardia, K V; Kent, J T; Bibby, J M 1979 "12 Multivariate analysis of variance" Multivariate analysis Academic Press ISBN 0-12-471250-9 
  • 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 

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