In a split-plot design with Factor A assigned to whole plots and Factor B assigned to split-plots, it is possible that there are no blocks for factor 1. An example is the cigarette experiment described on page 760. Suppose there are \(ar\) blocks where \(a\) is the number of levels of factor \(A\). The levels of factor \(A\) are assigned to the whole plots in the completed randomized design and each level is replicated \(\ell\) times, and levels of factor \(B\) are assigned to the split-plots as general complete block design where the whole plots are blocks.
\[ \begin{gathered} Y_{i u j t}=\mu+\alpha_i+\epsilon_{i u}^W \\ +\beta_j+(\alpha \beta)_{i j}+\epsilon_{j t(i u)}^S \\ \epsilon_{i u}^W \sim N\left(0, \sigma_W^2\right), \quad \epsilon_{j t(i u)}^S \sim N\left(0, \sigma_S^2\right) \\ \epsilon_{i u}^W \text{ and } \epsilon_{j t(i u)}^S \text{ mutually independent}\\ i=1, \ldots, a ; \quad u=1, \ldots, \ell ; j=1, \ldots, b ; t=1, \ldots, m. \end{gathered} \]
Note The whole error term \(\epsilon_{i u(h)}^W\) denotes the random effects of the whole plot. In some books it is denoted by \(\text{WP}_{u(i)}\). The whole plots are now nested in \(A\). The SAS code is given below:
proc mixed data=yourdata;
class WP A B;
model time=A B A*B;
random WP(A);
run;If the treatments assigned to the split-plots represent combinations of two factors, say, \(C\) and \(D\), the above model is equivalent to the following model that includes the interaction terms.
proc mixed data=yourdata;
class WP A C D;
model time=A C D C*D A*C A*D A*C*D;
random WP(A);
run;We will then be able to test the interaction effects.