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]}
Contents 
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 term for blocktreatment interactionterm that may be estimated (using the randomization distribution rather than using a normal distribution for the error).^{[3]} In the RCBD, the blocktreatment interaction cannot be estimated using the randomization distribution; a fortiori there exists no "valid" (i.e. randomizationbased) test for the blocktreatment 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 blocktreatment interaction to be studied.^{[6]}
However, if blocktreatment 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 (posthoc) secondary analysis of data from a GRBD, statisticians may check for nonadditivity using residual diagnostics.
