I understand that RCBD experiments usually only contain one observation per cell (block-treatment combination). Also, there should be no interaction between the treatments and blocks.
In GRBD, there is more than one observation per cell (block-treatment combination), and this allows for interaction between the blocks and treatments in the model.
My question is, why is this the case? Why does the presence of >1 observation per cell mean that there is now possibly interaction?
I am surprised this is not answered on site so far (at least I cannot find it by search). We suppose balanced designs. First the unreplicated case, then the analysis is by a two-way anova. Let the data be $Y_{ij}, i=1,2, \dotsc, r, j=1,2,\dotsc, c$, for a total of $rc$ observations. Define the means by dot notation, $\bar{Y}_{..}=\frac1{rc}\sum_{i,j} Y_{i,j}$, $\bar{Y}_{i.}=\frac1c\sum_j Y_{i,j}$, $\bar{Y}_{.j}=\frac1r\sum_i Y_{i,j}$ Then we can make the following decomposition of the data $$ Y_{i,j}- \bar{Y}_{..} = \left(\bar{Y}_{i.} - \bar{Y}_{..}\right) + \left(\bar{Y}_{.j} - \bar{Y}_{..}\right) + \left(Y_{i,j} -\bar{Y}_{i.}- \bar{Y}_{.j}+ \bar{Y}_{..}\right) $$ Using orthogonality of the design, squaring and summing now leads to a decomposition of the (corrected) sum of squares. Three terms, one for treatments, one for blocks, and then the last one---is it interaction or error? It cant be both!
Let us now add replication, so we need a third index, but it should be clear how the notation is generalized. Now we get a decomposition of four terms: $$ Y_{i,j,k} - \bar{Y}_{...} = \left(\bar{Y}_{i..} - \bar{Y}_{...}\right) + \left(\bar{Y}_{.j.} - \bar{Y}_{...}\right) + \underbrace{\left(Y_{i,j,.} -\bar{Y}_{i..}- \bar{Y}_{.j.} + \bar{Y}_{...}\right)}_{\text{interaction}} + \underbrace{\left(Y_{i,j,k}-\bar{Y}_{ij.} \right)}_{\text{pure error}} $$ Comparing the two decompositions, we see that the last term in the first decomposition has the form of the interaction part of the last, and there is no term for pure error. This is because we have exhausted the degrees of freedom. So without replication, we really need to assume there is no interaction between blocks and treatment. Then, the apparent interaction must be because of the error term, so we use the interaction sum of squares as error sum of squares.
