# 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.

## Notes

1. ^
• Wilk, page 79.
• Lentner and Biship, page 223.
• Hinkelmann and Kempthorne, page 314, for example; c.f. page 312.
2. ^
• Wilk, page 79.
• Hinkelmann and Kempthorne, page 314.
• Lentner and Bishop, page 223.
3. ^
• Wilk, page 79.
• 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.