Use the first eight principal components defined in Exercise 14.2 as independent variables and the sampling site means for BIOMASS as the dependent variable. Regress BIOMASS on the principal...

Use the first eight principal components defined in Exercise 14.2 as independent variables and the sampling site means for BIOMASS as the dependent variable. Regress BIOMASS on the principal components (plus an intercept) and compute the sum of squares attributable to each principal component. These sums of squares, multiplied by five to put them on a “per observation” basis, are an orthogonal partitioning of the “among site” sum of squares. Compute the analysis of variance for the original data to obtain the “among site” and “within site” sums of squares. Verify that the “among site” sums of squares computed by the two methods agree. Test the significance of each principal component using the “within site” mean square as the estimate of σ2. Which principal component dominates the regression and which variables does this result suggest might be most important? Which principal component is nearly orthogonal to BIOMASS and what does this imply, if anything, about some of the variables?

Exercise 14.2

Extract from the 9×15 matrix of centered and standardized variables the 14 independent variables to obtain Z. Do the principal component analysis on this matrix. Explain why only eight eigenvalues are nonzero. Describe the composition (in terms of the original variables) of the three principal components that account for the most dispersion. What proportion of the dispersion do they account for? Compare these principal components to those given for the case study using all observations.