Generalized randomized block design

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).^[1]

GRBDs versus RCBDs: Replication and interaction

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.^[2] Without replication, the (classic) RCBD's linear model lacks a block-treatment interaction-term that may be estimated (using the randomization distribution rather than using a normal distribution for the error).^[3] In the RCBD, the block-treatment interaction cannot be estimated using the randomization distribution; a fortiori there exists no "valid" (i.e. randomization-based) test for the block-treatment interaction in the analysis of variance (anova) of the RCBD.^[4]

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.^[5] The GRBD has the advantage that replication allows block-treatment interaction to be studied.^[6]

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.^[7] During exploratory data analysis or the (post-hoc) secondary analysis of data from a GRBD, statisticians may check for non-additivity using residual diagnostics.

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; 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. 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).

References

• Addelman, Sidney (Oct 1969). "The Generalized Randomized Block Design". The American Statistician 23 (4): 35–36. doi:10.2307/2681737. 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:10.2307/2284277. JSTOR 2284277.

• Gates, Charles E. (Nov 1995). "What Really Is Experimental Error in Block Designs?". The American Statistician 49 (4): 362–363. doi:10.2307/2684574. 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. External link in |publisher= (help)

• Lentner, Marvin; Bishop, Thomas (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.

• Wilk, M. B. (June 1955). "The Randomization Analysis of a Generalized Randomized Block Design". Biometrika 42 (1–2): 70–79. doi:10.2307/2333423. JSTOR 2333423. MR 68800.

• 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:10.1214/aoms/1177703889. JSTOR 2238364. MR 157448.