3.5 The General Linear Hypothesis
Individual tests on single parameters have been treated. Most hypotheses of interest involve several parameters at once: that all treatments are alike, that a subset of regressors may be dropped, that two effects are equal. All of them are of one form, and one test covers all of them.
Definition 3.5.1 (General linear hypothesis). Let \(C\) be a \(q\times p\) matrix of full row rank whose rows are estimable functions. The general linear hypothesis is \[H_0:\ CB=\underline {d}\qquad \text {against}\qquad H_a:\ CB\neq \underline {d}.\]
Theorem 3.5.2 (Test of the general linear hypothesis). Under \(H_0\), with \(r=rank(X)\), \[F=\frac {\left (C\widehat {\underline {b}}-\underline {d}\right )^t \left [C\left (X^tX\right )^{-}C^t\right ]^{-1} \left (C\widehat {\underline {b}}-\underline {d}\right )\big /q}{MSE} \ \sim \ F_{q,\,n-r},\] and \(H_0\) is rejected at level \(\alpha \) when \(F>F_{q,\,n-r,\alpha }\).
Theorem 3.5.3 (Extra sum of squares form). The same test is obtained by fitting the model twice. Let \(SSE_{\text {full}}\) be the residual sum of squares of the unrestricted model and \(SSE_{\text {red}}\) that of the model fitted subject to \(CB=\underline {d}\). Then \[F=\frac {\left (SSE_{\text {red}}-SSE_{\text {full}}\right )\big /q} {SSE_{\text {full}}\big /(n-r)} .\]
Note. Theorem 3.5.3 is the form used in practice, and it is the one behind every “extra sum of squares” calculation in Section 4: the quantity written \(SSR\left (X_2,X_3\mid X_1,X_4\right )\) there is exactly \(SSE_{\text {red}}-SSE_{\text {full}}\) for the hypothesis \(\beta _2=\beta _3=0\). The two theorems are the same test seen from two directions — one computes the restriction, the other computes the cost of imposing it. The second needs no generalised inverse and no matrix \(C\), only two model fits, which is why software reports it that way.
Example 3.5.4. For the one-way layout, \(H_0:\tau _1=\tau _2=\cdots =\tau _k\) is written with \(q=k-1\) and \[C=\begin {pmatrix} 0 & 1 & -1 & 0 & \cdots & 0\\ 0 & 0 & 1 & -1 & \cdots & 0\\ \vdots & & & \ddots & & \vdots \\ 0 & 0 & \cdots & 0 & 1 & -1 \end {pmatrix}.\] Every row is a contrast, hence estimable, and the resulting \(F\) is the \(SSTr/SSE\) ratio of Section 1.2. The reduced model is \(Y_{ij}=\mu +\varepsilon _{ij}\), whose residual sum of squares is \(SST\); so \(SSE_{\text {red}}-SSE_{\text {full}}=SST-SSE=SSTr\), as Theorem 3.5.3 requires.
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.