Generalized randomized block design

In

error).^[1]

Univariate response

GRBDs versus RCBDs: Replication and interaction

Like a

experimental units: this randomization is also independent between blocks. In a (classic) RCBD, however, there is no replication of treatments within blocks.^[2]

Two-way linear model: Blocks and treatments

The experimental design guides the formulation of an appropriate

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] 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,

least-squares method).^[8]^[9]

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.^[10]

The rejection of multiplicative interaction need not imply the rejection of non-multiplicative interaction, because there are many forms of interaction.^[11]^[12]

Generalizing earlier models for Tukey's test are the “bundle-of-straight lines” model of Mandel (1959)^[13] 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).

Notes

^
- Wilk, page 79.
- Lentner and Biship, page 223.
- Addelman (1969) page 35.
- Hinkelmann and Kempthorne, page 314, for example; c.f. 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.
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
).
^ 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.
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).
^ Johnson & Wichern (2002, p. 312, “Multivariate two-way fixed-effects model with interaction”, in “6.6 Two-way multivariate analysis of variance”, p. 307–317)
^ Mardia, Kent & Bibby (1979, p. 352, “Tests for interactions”, in 12.7 Two-way classification, p. 350-356)
^ Hinkelmann & Kempthorne (2008, p. 305)
^ Milliken & Johnson (1989, 1.6 Tukey's single degree-of-freedom test for nonadditivity, pp. 7-8)
^ Lentner & Bishop (1993, p. 214, in 6.8 Nonadditivity of blocks and treatments, pp. 213–216)
^ Milliken & Johnson (1989, 1.8 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.
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.
JSTOR 2284277
.

Gates, Charles E. (Nov 1995). "What Really Is Experimental Error in Block Designs?". The American Statistician. 49 (4): 362–363.
JSTOR 2684574
.

Hinkelmann, Klaus;
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 6.13)". Experimental design and analysis (Second ed.). Blacksburg, VA: Valley Book Company. pp. 225–226.
ISBN 0-9616255-2-X
.

ISBN 0-12-471250-9
.

Milliken, George A.; Johnson, Dallas E. (1989). Nonreplicated experiments: Designed experiments. Analysis of messy data. Vol. 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.
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.
MR 0157448
.

[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.

[3] Lentner and Biship, page 223.

[4] Addelman (1969) page 35.

[5] 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.

[7] Wilk, page 79.

[8] Addelman (1969) page 35.

[9] Hinkelmann and Kempthorne, page 314.

[10] 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
).

[12] Wilk, page 79.

[13] Addelman (1969) page 35.

[14] Lentner and Bishop, page 223.

[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.

[17] 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.

[19] Wilk, page 79.

[20] Addelman (1969) page 35.

[21] 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).

[23] Addelman (1970) page 1104.

[8] Johnson & Wichern (2002, p. 312, “Multivariate two-way fixed-effects model with interaction”, in “6.6 Two-way multivariate analysis of variance”, p. 307–317)

[9] Mardia, Kent & Bibby (1979, p. 352, “Tests for interactions”, in 12.7 Two-way classification, p. 350-356)

[10] Hinkelmann & Kempthorne (2008, p. 305)

[11] Milliken & Johnson (1989, 1.6 Tukey's single degree-of-freedom test for nonadditivity, pp. 7-8)

[12] Lentner & Bishop (1993, p. 214, in 6.8 Nonadditivity of blocks and treatments, pp. 213–216)

[13] Milliken & Johnson (1989, 1.8 Mandel's bundle-of-straight lines model, pp. 17-29)

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v t e Design of experiments
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