Consider the latent roots of Table 4.13 in Example 4.14

(a) Using the chi-squared criterion (Eq. 4.6), test whether the covariance matrix is of diagonal form.

(b) Using Eq. (4.11) test whether the covariance matrix exhibits spherical form.

Example 4.14

The random (independent) normal data of Example 4.2 indicate complete sphere city of the latent roots when using the chi-squared test. The latent roots of the covariance matrix are given in Table 4.13 (Figs. 4.1 and 4.2). Plotting l_t, against i exhibits the characteristic exponential decline common for random normal data The plot can also he linearized by transforming to logarithms. Thus the linearity of In l_i Fig. 4.2) would seem to confirm the chi-squared test of complete sphere city. Linearity can be confirmed further by least squares regression or else more simply by a plot of In l_i-1 - In I_i (i= 2, 3, . , r) against rank number, such as in Figure 4.3, where approximate linearity is indicated by a random scatter of residuals. For random, independent (normal) data, linearity is simply a byproduct of

Example 4.2

The independence test is illustrated by Vieffa and Carlson (1981) using simulated data. Given a random sample from an independentp-variate normal distribution, it is always possible to obtain nonzero loadings since R (or Σ) can indicate an apparent departure from diagonality. The magnitudes of the loadings can at times be surprisingly high, and this may create false impressions concerning a PC analysis. The data and results are shown in Tables 4.1-4.4 where the correlation loadings are generally low except for several large values, which are certainly large enough to be retained by most informal rules of thumb. When latent roots are known the LR test is particularly easy to compute. Using Eq. (4.6) we have