5.2 The model and the analysis of variance
The model for the RCBD is \[ y_{ij} = \mu + \tau _i + \beta _j + \varepsilon _{ij} , \qquad i = 1, \dots , t , \quad j = 1, \dots , b , \] where \(\tau _i\) is the effect of treatment \(i\), \(\beta _j\) the effect of block \(j\), and the \(\varepsilon _{ij}\) are independent \(N(0, \sigma ^2)\). Treatment and block effects are taken as fixed deviations from the overall mean, \(\sum _i\tau _i = 0\) and \(\sum _j\beta _j = 0\). The model is additive: the effect of a treatment is the same in every block. With one observation per treatment in each block there are no degrees of freedom left to estimate an interaction, and additivity is an assumption to be checked.
Proposition 5.2.1. The least squares estimates are \[ \hat \mu = \bar y_{\cdot \cdot } , \qquad \hat \tau _i = \bar y_{i\cdot } - \bar y_{\cdot \cdot } , \qquad \hat \beta _j = \bar y_{\cdot j} - \bar y_{\cdot \cdot } , \] the fitted values are \(\hat y_{ij} = \bar y_{i\cdot } + \bar y_{\cdot j} - \bar y_{\cdot \cdot }\), and the residuals are \(e_{ij} = y_{ij} - \bar y_{i\cdot } - \bar y_{\cdot j} + \bar y_{\cdot \cdot }\).
Proof. The normal equations are those of the parameters \(\mu \), each \(\tau _i\) and each \(\beta _j\): \begin {align*} tb\hat \mu + b\sum _i\hat \tau _i + t\sum _j\hat \beta _j &= y_{\cdot \cdot } , \\ b\hat \mu + b\hat \tau _i + \sum _j\hat \beta _j &= y_{i\cdot } , \qquad i = 1, \dots , t , \\ t\hat \mu + \sum _i\hat \tau _i + t\hat \beta _j &= y_{\cdot j} , \qquad j = 1, \dots , b . \end {align*}
With the side conditions \(\sum \hat \tau _i = \sum \hat \beta _j = 0\) they reduce to \(tb\hat \mu = y_{\cdot \cdot }\), \(b(\hat \mu + \hat \tau _i) = y_{i\cdot }\) and \(t(\hat \mu + \hat \beta _j) = y_{\cdot j}\), whose solution is as stated, and it satisfies the side conditions. □
The simplicity of the solution is a consequence of balance: every treatment appears once in every block, so treatment comparisons are unaffected by block effects and vice versa. The deviation of each observation from the grand mean splits accordingly, \[ y_{ij} - \bar y_{\cdot \cdot } = (\bar y_{i\cdot } - \bar y_{\cdot \cdot }) + (\bar y_{\cdot j} - \bar y_{\cdot \cdot }) + (y_{ij} - \bar y_{i\cdot } - \bar y_{\cdot j} + \bar y_{\cdot \cdot }) . \]
Theorem 5.2.2. \[ SST = SS_{\mathrm {Treatments}} + SS_{\mathrm {Blocks}} + SSE , \] where \begin {align*} SST &= \sum _i\sum _j(y_{ij} - \bar y_{\cdot \cdot })^2 , \qquad SS_{\mathrm {Treatments}} = b\sum _i(\bar y_{i\cdot } - \bar y_{\cdot \cdot })^2 , \\ SS_{\mathrm {Blocks}} &= t\sum _j(\bar y_{\cdot j} - \bar y_{\cdot \cdot })^2 , \qquad SSE = \sum _i\sum _j(y_{ij} - \bar y_{i\cdot } - \bar y_{\cdot j} + \bar y_{\cdot \cdot })^2 , \end {align*}
with \(tb - 1 = (t - 1) + (b - 1) + (t - 1)(b - 1)\) degrees of freedom.
Proof. Square the decomposition and sum. Each of the three cross-product sums vanishes. For example, \(\sum _i\sum _j(\bar y_{i\cdot } - \bar y_{\cdot \cdot })(\bar y_{\cdot j} - \bar y_{\cdot \cdot }) = \sum _i(\bar y_{i\cdot } - \bar y_{\cdot \cdot })\cdot \sum _j(\bar y_{\cdot j} - \bar y_{\cdot \cdot }) = 0\), and the residual sums to zero over \(j\) for each \(i\) and over \(i\) for each \(j\), which disposes of its products with the treatment and block terms. □
Theorem 5.2.3. Under the additive model, \begin {gather*} E(MS_{\mathrm {Treatments}}) = \sigma ^2 + \frac {b\sum _i\tau _i^2}{t - 1} , \qquad E(MS_{\mathrm {Blocks}}) = \sigma ^2 + \frac {t\sum _j\beta _j^2}{b - 1} , \\ E(MSE) = \sigma ^2 , \end {gather*} where \(MSE = SSE/\bigl ((t - 1)(b - 1)\bigr )\). With normal errors the three sums of squares are independent, \(SSE/\sigma ^2 \sim \chi ^2_{(t-1)(b-1)}\), and under \(H_0 : \tau _1 = \dots = \tau _t = 0\), \[ F_0 = \frac {MS_{\mathrm {Treatments}}}{MSE} \sim F_{t-1,\,(t-1)(b-1)} . \]
Proof. Let \(H_0\), \(H_T\) and \(H_B\) be the projections that replace each observation by \(\bar y_{\cdot \cdot }\), by its treatment mean and by its block mean. Then \begin {gather*} SS_{\mathrm {Treatments}} = \mathbf Y^\top (H_T - H_0)\mathbf Y , \qquad SS_{\mathrm {Blocks}} = \mathbf Y^\top (H_B - H_0)\mathbf Y , \\ SSE = \mathbf Y^\top (I - H_T - H_B + H_0)\mathbf Y . \end {gather*} Because the design is balanced, \(H_TH_B = H_0\): averaging over a block and then over a treatment gives the grand mean. From this and \(H_TH_0 = H_BH_0 = H_0\) it follows that the three matrices are symmetric and idempotent, with traces \(t - 1\), \(b - 1\) and \(tb - t - b + 1 = (t - 1)(b - 1)\), and that the product of any two of them is zero. Theorem 3.1.4 then gives the independence and the distributions, and Lemma 3.1.2 gives the expectations: with \(\boldsymbol \mu \) the vector of means \(\mu + \tau _i + \beta _j\), the vector \((H_T - H_0)\boldsymbol \mu \) has entries \(\tau _i\), so \(\boldsymbol \mu ^\top (H_T - H_0)\boldsymbol \mu = b\sum \tau _i^2\); similarly for blocks; and \((I - H_T - H_B + H_0)\boldsymbol \mu = \mathbf 0\) for an additive mean. □
For hand calculation, \begin {gather*} SST = \sum _i\sum _jy_{ij}^2 - \frac {y_{\cdot \cdot }^2}{tb} , \qquad SS_{\mathrm {Treatments}} = \frac 1b\sum _iy_{i\cdot }^2 - \frac {y_{\cdot \cdot }^2}{tb} , \\ SS_{\mathrm {Blocks}} = \frac 1t\sum _jy_{\cdot j}^2 - \frac {y_{\cdot \cdot }^2}{tb} , \qquad SSE = SST - SS_{\mathrm {Treatments}} - SS_{\mathrm {Blocks}} . \end {gather*}
Example 5.2.4. For the fertiliser data, the correction factor is \(911^2/20 = 41496.05\), and \begin {align*} SST &= 41671 - 41496.05 = 174.95 , \\ SS_{\mathrm {Treatments}} &= \frac {216^2 + 226^2 + 244^2 + 225^2}{5} - 41496.05 = 82.55 , \\ SS_{\mathrm {Blocks}} &= \frac {187^2 + 176^2 + 195^2 + 172^2 + 181^2}{4} - 41496.05 = 82.70 , \\ SSE &= 174.95 - 82.55 - 82.70 = 9.70 . \end {align*}
| source | SS | df | MS | \(F_0\) |
| fertilisers | \(82.55\) | \(3\) | \(27.517\) | \(34.04\) |
| fields (blocks) | \(82.70\) | \(4\) | \(20.675\) | |
| error | \(9.70\) | \(12\) | \(0.808\) | |
| total | \(174.95\) | \(19\) |
Since \(F_{0.05,3,12} = 3.49\), the fertilisers differ, with \(p = 4 \times 10^{-6}\). Tukey’s honestly significant difference is \[ HSD = q_{0.05}(4, 12)\sqrt {0.808/5} = 4.199(0.402) = 1.69 , \] so \(C\) (mean \(48.8\)) gives a higher yield than every other fertiliser, \(B\) (\(45.2\)) and \(D\) (\(45.0\)) do not differ, and \(A\) (\(43.2\)) is below \(B\) and \(D\) by more than \(1.69\).
Had the fields been ignored and the data analysed as a completely randomised design, the field sum of squares would have joined the error: \(SSE = 82.70 + 9.70 = 92.40\) on \(16\) degrees of freedom, \(MSE = 5.775\), and \(F_0 = 27.517/5.775 = 4.76\) (\(p = 0.015\)). The conclusion survives, but the evidence is weaker by a factor of seven, and the Tukey difference would be \(\sqrt {5.775/0.808} \approx 2.7\) times as wide.
Remark (Testing blocks). The ratio \(MS_{\mathrm {Blocks}}/MSE = 25.6\) suggests that the fields differ, and it is useful to look at it to see whether blocking was worthwhile. It is not a valid \(F\) test in the same sense as the test for treatments: the treatments were randomised within blocks, but the blocks themselves were not randomised to anything. Block effects may also be regarded as random, if the blocks are a sample from a population of possible blocks. Either way, the test for treatments is unchanged: for \(H_0\) about treatments the analysis is the same whether blocks are fixed or random.
Contrasts and multiple comparisons are as in Section 4, with \(r\) replaced by \(b\) and \(N - t\) by \((t - 1)(b - 1)\): a contrast \(\sum c_i\mu _i\) is estimated by \(\sum c_i\bar y_{i\cdot }\) with variance \(\sigma ^2\sum c_i^2/b\), free of the block effects.
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