11.2 The one-way random effects model

The model has a consequence that the fixed effects model does not: observations on the same level are correlated. Indeed \begin {gather*} \Var (y_{ij}) = \sigma ^2_\tau + \sigma ^2 , \qquad \Cov (y_{ij}, y_{ij'}) = \Var (\tau _i) = \sigma ^2_\tau \quad (j \ne j') , \\ \Cov (y_{ij}, y_{i'j'}) = 0 \quad (i \ne i') , \end {gather*} so two observations on the same level have correlation \[ \rho = \frac {\sigma ^2_\tau }{\sigma ^2_\tau + \sigma ^2} , \] the intraclass correlation. It is also the fraction of the variance of a single observation that is due to the factor.

The sums of squares are computed exactly as in the fixed effects case: \(SST = SS_{\mathrm {Treatments}} + SSE\), with \(N = tr\) observations. What changes is their expectations.

The test

The hypothesis that the factor contributes no variability is \(H_0 : \sigma ^2_\tau = 0\), against \(H_1 : \sigma ^2_\tau > 0\). By the theorem, with \(\theta = \sigma ^2_\tau /\sigma ^2\), \[ \frac {F_0}{1 + r\theta } \sim F_{t-1, N-t} , \qquad F_0 = \frac {MS_{\mathrm {Treatments}}}{MSE} . \] Under \(H_0\) this is the usual \(F\) statistic, so the test is computed exactly as for fixed effects: reject when \(F_0 > F_{\alpha , t-1, N-t}\). The power is different, and simpler: under the alternative the statistic is a multiple of a central \(F\), so \[ \text {power} = P\bigl (F_{t-1, N-t} > F_{\alpha , t-1, N-t}/(1 + r\theta )\bigr ) , \] with no noncentrality parameter needed.

Estimating the variance components

Equating the mean squares to their expectations gives the analysis of variance estimators \[ \hat \sigma ^2 = MSE , \qquad \hat \sigma ^2_\tau = \frac {MS_{\mathrm {Treatments}} - MSE}r . \] Both are unbiased, and for balanced data under normality they have minimum variance among all unbiased estimators. But \(\hat \sigma ^2_\tau \) is negative whenever \(F_0 < 1\), which by the pivot above has probability \[ P\bigl (F_{t-1, N-t} < 1/(1 + r\theta )\bigr ) , \] more than one half when \(\sigma ^2_\tau = 0\). A negative estimate is not an arithmetic error. It is evidence that \(\sigma ^2_\tau \) is small, and the usual practice is to report \(\hat \sigma ^2_\tau = 0\) (which makes the estimator biased) and to check that the model is right. The restricted maximum likelihood (REML) estimator, which is the standard method for unbalanced data, agrees with the analysis of variance estimator for balanced data whenever the latter is non-negative, and otherwise sets \(\hat \sigma ^2_\tau = 0\).

Confidence intervals

Three intervals are exact. Since \(SSE/\sigma ^2 \sim \chi ^2_{N-t}\), \[ \frac {SSE}{\chi ^2_{\alpha /2, N-t}} \le \sigma ^2 \le \frac {SSE}{\chi ^2_{1-\alpha /2, N-t}} , \] where \(\chi ^2_{\alpha , \nu }\) is the upper \(\alpha \) point. The pivot \(F_0/(1 + r\theta )\) gives an interval \([L, U]\) for the ratio \(\theta \): \[ L = \frac 1r\Bigl (\frac {F_0}{F_{\alpha /2, t-1, N-t}} - 1\Bigr ) , \qquad U = \frac 1r\Bigl (\frac {F_0}{F_{1-\alpha /2, t-1, N-t}} - 1\Bigr ) , \] and since \(\rho = \theta /(1 + \theta )\) is increasing in \(\theta \), \([L/(1 + L), U/(1 + U)]\) is an exact interval for the intraclass correlation. There is no exact interval for \(\sigma ^2_\tau \) itself, because its estimator is a difference of mean squares with different expectations. The standard approximation is the following.

Treating \(\hat \nu \hat \psi /\psi \) as \(\chi ^2_{\hat \nu }\) gives the approximate interval \(\hat \nu \hat \psi /\chi ^2_{\alpha /2, \hat \nu } \le \psi \le \hat \nu \hat \psi /\chi ^2_{1-\alpha /2, \hat \nu }\). The approximation is good when all \(c_i > 0\); with negative coefficients, as here, it is rougher, but it remains the usual choice.