Now we will generalize what we have shown by example. We will look at \(2^k\) designs in \(2^p\) blocks of size \(2^{k-p}\). We do this by choosing *k* and if we want to confound the design in \(2^p\) blocks then we need to choose *p* effects to confound. Then, due to the intereactions among these effects, we get \(2^{p}-1\) effects confounded with blocks.

To illustrate this, if \(p= 2\) then we have \(2^{p} = \text{four blocks}\), and thus \(2^{p} - 1 = 3\) effects confounded, i.e., the 2 effects we chose plus the interaction between these two. In general, we choose *p* effects and in addition to the p effects we choose, \(2^{p} - p - 1\) other effects are automatically confounded. We will call these "generalized interactions" which are also confounded.

Earlier we looked at a couple of examples - for instance when \(k = 3\) and \(p = 2\). We chose ABC and AB. Then the \(ABC \times AB = A^{2} B^{2} C = C\) which was also confounded. This shows that the generalized interaction can be a main effect, i.e. the generalized interaction affect can be a lower order term. This is not a good outcome. A better outcome that we settled on was to pick two 2-way interactions, AB and AC, which gave us \(AB \times AC = A^{2}BC \text{which} = BC\), another 2-way interaction. In this case we have all three 2-way interactions confounded, but all the main effects were estimable.