The following exercise is concerned with the distribution of the likelihoodratio statistic in a 'fixed-effects' model for a balanced design, where $\Sigma ={\sigma }^2{I}_n$, and either $\mu \in 1_n$ under the null hypothesis or $\mu \in \mathrm{span}\left(B\right)$ under the alternative. The meaning of the term 'residual' is unchanged, and the $F$-statistic in Exercise 18.2 is also unchanged.

Show that the residual $\log$ likelihood-ratio statistic for testing $\mu \in 1$ versus $\mu \in$ $\mathrm{span}\left(B\right)$ is

$$(n-1)\log (1+\frac{(m-1)F}{n-m})$$

By simulation or otherwise, show also that the null expected value exceeds that of ${\chi }_{m-1}^2$ by the approximate multiplicative factor

$$1+\frac{1}{2}(m+1)/(n-m).$$

This is a particular instance of the Bartlett correction factor.

Data From Exercises 18.2

Consider a balanced block design having \(m\) blocks each consisting of \(b\) observational units, and let \(B\) be the associated block factor as a Boolean matrix of order \(n=m b\). The three-parameter Gaussian model with moments

\[
\mu \in \mathbf{1}_{n}, \quad \Sigma=\sigma^{2}\left(I_{n}+\theta Bight),
\]

is parameterized by two scalars \(\mu, \sigma>0\) and one additional parameter. For the purposes of this exercise \(\theta>-1 / b\) is not necessarily positive, but \(\Sigma\) is positive definite. In addition, the residual refers to any linear transformation, such as \(Y_{i j}-\bar{Y}_{. .}\), whose kernel is \(\mathbf{1} \subset \mathbb{R}^{n}\).

Let \(Y_{i j}\) be the observation for unit \(j\) in block \(i\). Show that the within- and between- quadratic forms

\[
\mathrm{SS}_{W}=\sum_{i j}\left(Y_{i j}-\bar{Y}_{i .}ight)^{2}, \quad \mathrm{SS}_{B}=b \sum_{i}\left(\bar{Y}_{i .}-\bar{Y}_{. .}ight)^{2}
\]

are independent with distributions \(\sigma^{2} \chi_{n-m}^{2}\) and \(\sigma^{2}(1+b \theta) \chi_{m-1}^{2}\) respectively. Hence deduce that, if \(\theta=0\), the mean-square ratio

\[
F=\frac{\mathrm{SS}_{B} /(m-1)}{\mathrm{SS}_{W} /(n-m)}
\]

is distributed according to Fisher's \(F_{m-1, n-m}\) distribution.