7.5 Blocking in a factorial design

If the \(ab\) treatment combinations cannot all be run under homogeneous conditions, each replicate can be run in a separate block, giving a randomised complete block design whose treatments are the factor combinations. The model is \[ y_{ijk} = \mu + \tau _i + \beta _j + (\tau \beta )_{ij} + \delta _k + \varepsilon _{ijk} , \qquad k = 1, \dots , n , \] with block effects \(\delta _k\). The block sum of squares \(\frac {1}{ab}\sum _ky_{\cdot \cdot k}^2 - y_{\cdot \cdot \cdot }^2/N\), on \(n - 1\) degrees of freedom, is removed from the error, which keeps \((ab - 1)(n - 1)\) degrees of freedom; the factorial effects are computed exactly as before. When even one replicate is too large for a block, the blocks must contain only part of a replicate, and some effect is necessarily confounded with blocks; that is the subject of a later section.

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