11.3 The two-factor random effects model

Suppose both factors of a two-factor experiment are random: \(a\) levels of \(A\) and \(b\) levels of \(B\), each sampled from a population, with \(n\) replicates of every combination. The model is \[ y_{ijk} = \mu + \tau _i + \beta _j + (\tau \beta )_{ij} + \varepsilon _{ijk} , \] where \(\tau _i \sim N(0, \sigma ^2_\tau )\), \(\beta _j \sim N(0, \sigma ^2_\beta )\), \((\tau \beta )_{ij} \sim N(0, \sigma ^2_{\tau \beta })\) and \(\varepsilon _{ijk} \sim N(0, \sigma ^2)\), all independent. Then \(\Var (y_{ijk}) = \sigma ^2_\tau + \sigma ^2_\beta + \sigma ^2_{\tau \beta } + \sigma ^2\), and the hypotheses are \(\sigma ^2_\tau = 0\), \(\sigma ^2_\beta = 0\) and \(\sigma ^2_{\tau \beta } = 0\). The sums of squares are those of the fixed effects analysis. One lemma does most of the work.

Two cases are used repeatedly. If \(w_1, \dots , w_m\) are independent with variance \(v\), centring is a projection of rank \(m - 1\), so \(E\sum _i(w_i - \bar w)^2 = (m - 1)v\). For an \(a \times b\) array of independent \(w_{ij}\) of variance \(v\), the double centring \(w_{ij} - \bar w_{i\cdot } - \bar w_{\cdot j} + \bar w_{\cdot \cdot }\) is a projection of rank \((a - 1)(b - 1)\), and the map from \(w\) to the array of row-mean deviations \(\bar w_{i\cdot } - \bar w_{\cdot \cdot }\) (each repeated \(b\) times) is a projection of rank \(a - 1\), so \begin {gather*} E\sum _i\sum _j(w_{ij} - \bar w_{i\cdot } - \bar w_{\cdot j} + \bar w_{\cdot \cdot })^2 = (a - 1)(b - 1)v , \\ E\sum _i(\bar w_{i\cdot } - \bar w_{\cdot \cdot })^2 = \frac {(a - 1)v}b . \end {gather*}

The expected mean squares determine the tests. The interaction is tested against error, but the main effects are tested against the interaction: \begin {gather*} H_0 : \sigma ^2_{\tau \beta } = 0 : \quad F_0 = \frac {MS_{AB}}{MSE} , \\ H_0 : \sigma ^2_\tau = 0 : \quad F_0 = \frac {MS_A}{MS_{AB}} , \qquad H_0 : \sigma ^2_\beta = 0 : \quad F_0 = \frac {MS_B}{MS_{AB}} . \end {gather*} In each case the two mean squares have equal expectations under \(H_0\) and are independent scaled \(\chi ^2\) variables, so \(F_0\) has an \(F\) distribution. Using \(MSE\) as the denominator for the main effects, as in the fixed effects analysis, would be wrong whenever \(\sigma ^2_{\tau \beta } > 0\): the test would reject too often. The reason is visible in the model: with new levels of \(B\), the apparent effect of a level of \(A\) changes by an amount of variance \(\sigma ^2_{\tau \beta }/b\), and that is the noise against which a main effect of \(A\) must be judged. The variance components are estimated by \begin {gather*} \hat \sigma ^2 = MSE , \qquad \hat \sigma ^2_{\tau \beta } = \frac {MS_{AB} - MSE}n , \\ \hat \sigma ^2_\tau = \frac {MS_A - MS_{AB}}{bn} , \qquad \hat \sigma ^2_\beta = \frac {MS_B - MS_{AB}}{an} . \end {gather*}