8.3 Practice Problems
Problem 8.1. State the one-way MANOVA model, its assumptions, and the decomposition \(SSP_{\text {tot}} = SSP_{\text {tr}} + SSP_{\text {res}}\). What is the univariate statement of which this is the generalisation?
Problem 8.2. Wilks’ lambda is \(\Lambda = \left |SSP_{\text {res}}\right |/\left |SSP_{\text {res}}+SSP_{\text {tr}}\right |\). Explain why small values of \(\Lambda \) are evidence against equality of the mean vectors, and why the statistic uses determinants rather than traces. Where to start: the determinant is the generalised variance of Section 3.4; compare what a determinant and a trace each measure.
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Solution. \(SSP_{\text {res}}\) measures scatter within groups and the denominator measures total scatter. If the group means coincide, treatment contributes nothing and the two are close, so \(\Lambda \) is near one; the further apart the means, the larger the denominator and the smaller \(\Lambda \).
Determinants are used because the generalised variance measures the volume the scatter occupies, and volume responds to separation in any direction, including a direction that is a combination of the variables. A trace would add the marginal variances and could miss a separation that shows in no single variable - the same phenomenon as the significant \(T^{2}\) with no significant component in Section 6.3.
Problem 8.3. For \(g=2\) groups, show that Wilks’ lambda is a monotone function of the two-sample \(T^{2}\) of Theorem 6.5, so that the two tests agree.
Problem 8.4. In a two-way MANOVA the interaction is significant. Explain why the main effect tests should not then be interpreted in the usual way, and what should be reported instead.
Problem 8.5. An experiment has \(g=3\) treatments, \(b=2\) blocks, \(p=4\) responses and \(n=2\) replicates per cell. Write out the degrees of freedom for each line of the analysis, and check that the residual matrix is invertible. Where to start: the residual has \(gb(n-1)\) degrees of freedom, and needs at least \(p\) of them.
Problem 8.6. Explain why a MANOVA on \(p\) responses is preferable to \(p\) separate univariate analyses of variance, giving two distinct reasons.
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