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Generalized randomized block design

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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 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.[6]

See also

Notes

  1. ^
    • Wilk, page 79.
    • Lentner and Biship, page 223.
    • Addelman (1969) page 35.
    • Hinkelmann and Kempthorne, page 314, for example; c.f. 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 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).
  4. ^
    • Complaints about the neglect of GRBDs in the literature and ignorance among practitioners are stated by Addelman (1969) page 35.
  5. ^
    • Wilk, page 79.
    • Addelman (1969) page 35.
    • Lentner and Bishop, page 223.
  6. ^
    • Addelman (1970) page 1104.
    If the scientists do not know that the 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, Hinkelmann and Kempthorne argue that the generalized randomized block design be used "if at all possible" (page 312).

References

  • 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)
  • 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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