Discussion
Consider a two-way classification model \(y_{ij} = \mu + \tau_{i} + \beta_{j} + \varepsilon_{ij}\) \(i = 1 \ldots v\), \(j = 1, \ldots, b\) with \(E(\varepsilon_{ij}) = 0\), \(\mathrm{Var}(\varepsilon_{ij}) = \sigma^{2}\), \(\varepsilon_{ij}\) uncorrelated where \(\tau_{i}\ \delta = 1 \ldots v\) are fixed effects and \(\beta_{j}\ j = 1, \ldots, b\) are random effects with mean 0 and variance \(\sigma_{\beta}^{2}\). Hence the hypothesis regarding the insignificance of random effects is :
Answer: B
The hypothesis is \(H_{0} : \sigma_{\beta}^{2} = 0\).
In a mixed model the block effects \(\beta_{j}\) are random variables drawn from a population with variance \(\sigma_{\beta}^{2}\). Interest is in that population, not in the particular blocks observed, so 'no block effect' means the random effects have no variability: \(\sigma_{\beta}^{2} = 0\).
\(\beta_{1} = \cdots = \beta_{b}\) is the hypothesis for fixed effects, \(\tau_{1} = \cdots = \tau_{v}\) concerns the fixed treatment effects, and \(\sigma^{2} = 0\) would mean no experimental error at all.
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