12.2 The split-plot design

The design arises when one factor is hard or expensive to change, or must be applied to large areas. Irrigation must be applied to large areas of a field; varieties can be sown in small strips. An oven’s temperature is slow to change, but several specimens can go into one oven run. Running a fully randomised factorial would mean resetting the hard factor for every run. The split plot accepts a cost instead: the whole-plot factor is compared on fewer, larger units, with a larger error, while the subplot factor and the interaction are compared within whole plots, with a smaller error.

The commonest layout has the whole plots in \(r\) complete blocks. Randomisation has two stages: the \(a\) whole-plot levels are randomised to the \(a\) whole plots in each block, and the \(b\) subplot levels are randomised afresh within each whole plot (Figure 15).

The model

Let \(y_{ijk}\) be the response in block \(i\), whole-plot level \(j\) and subplot level \(k\). The model is \begin {gather*} y_{ijk} = \mu + \rho _i + \tau _j + \delta _{ij} + \beta _k + (\tau \beta )_{jk} + \varepsilon _{ijk} , \\ i = 1, \dots , r , \quad j = 1, \dots , a , \quad k = 1, \dots , b . \end {gather*} Here \(\rho _i\) is the block effect, \(\tau _j\) and \(\beta _k\) the fixed effects of the two factors and \((\tau \beta )_{jk}\) their interaction, with the usual side conditions. The term \(\delta _{ij} \sim N(0, \sigma ^2_\delta )\) is the whole-plot error, the deviation of whole plot \((i, j)\) shared by all its subplots, and \(\varepsilon _{ijk} \sim N(0, \sigma ^2)\) the subplot error; all random terms are independent. The blocks may be fixed or random (\(\rho _i \sim N(0, \sigma ^2_\rho )\)); the tests for the treatment factors are the same either way.

The model contains two error terms because the experiment has two kinds of experimental unit. Two subplots in the same whole plot share \(\delta _{ij}\), so their responses are correlated, with correlation \(\sigma ^2_\delta /(\sigma ^2_\delta + \sigma ^2)\).

The analysis of variance

The sums of squares are those of a three-factor analysis (blocks, \(A\), \(B\)) with one observation per cell, regrouped. The whole-plot error is the block \(\times \) \(A\) interaction, and the subplot error is what is left: \begin {gather*} SS_{\mathrm {WP}} = \frac 1b\sum _{i,j}y_{ij\cdot }^2 - C , \qquad E_a = SS_{\mathrm {WP}} - SS_{\mathrm {Blocks}} - SS_A , \\ E_b = SST - SS_{\mathrm {WP}} - SS_B - SS_{AB} , \end {gather*} where \(SS_{\mathrm {WP}}\) is the sum of squares between the \(ra\) whole-plot totals.

The proof shows the whole design in two lines. The whole-plot factor is compared using whole-plot means, whose error includes \(\delta _{ij}\), while everything measured within a whole plot is free of \(\delta _{ij}\). Since usually \(\sigma ^2_\delta > 0\), the subplot comparisons are the more precise.

Standard errors of comparisons

Four kinds of comparison arise, with different variances. With \(\bar y_{\cdot jk}\) the mean of treatment combination \((j, k)\) over blocks:

The first three follow from the model as before; for instance in \(\bar y_{\cdot jk} - \bar y_{\cdot jk'}\) the \(\delta _{ij}\) cancel, leaving only subplot errors. In the fourth, \(\bar y_{\cdot jk} - \bar y_{\cdot j'k}\) contains \(\bar \delta _{\cdot j} - \bar \delta _{\cdot j'}\) as well as \(\bar \varepsilon _{\cdot jk} - \bar \varepsilon _{\cdot j'k}\), so its variance is \(2(\sigma ^2_\delta + \sigma ^2)/r\). No single mean square estimates \(\sigma ^2 + \sigma ^2_\delta \), but \[ \frac {(b - 1)MS_{E_b} + MS_{E_a}}b \] does, because \(\sigma ^2_\delta \) has the estimate \((MS_{E_a} - MS_{E_b})/b\). Its degrees of freedom come from Satterthwaite’s formula (Proposition 11.2.3).