10.5 Exercises

EXERCISE 10.1   In Example 10.4 we have computed $\widehat{\data{Q}}$ and $\widehat{\Psi}$ using the method of principal factors. We used a two-step iteration for $\widehat{\Psi}$. Perform the third iteration step and compare the results (i.e., use the given $\widehat{\data{Q}}$ as a pre-estimate to find the final $\Psi$).

EXERCISE 10.2   Using the bank data set, how many factors can you find with the Method of Principal Factors?

EXERCISE 10.3   Repeat Exercise 10.2 with the U.S. company data set!

EXERCISE 10.4   Generalize the two-dimensional rotation matrix in Section 10.2 to $n$-dimensional space.

EXERCISE 10.5   Compute the orthogonal factor model for

$$\Sigma=\left( \begin{array}{ccc} 1 & 0.9 & 0.7\\ 0.9 & 1 & 0.4\\ 0.7 & 0.4 & 1 \end{array} \right).$$

[Solution: $\psi_{11}=-0.575, q_{11}=1.255$]

EXERCISE 10.6   Perform a factor analysis on the type of families in the French food data set. Rotate the resulting factors in a way which provides the most reasonable interpretation. Compare your result with the varimax method.

EXERCISE 10.7   Perform a factor analysis on the variables $X_3$ to $X_9$ in the U.S. crime data set (Table B.10). Would it make sense to use all of the variables for the analysis?

EXERCISE 10.8   Analyze the athletic records data set (Table B.18). Can you recognize any patterns if you sort the countries according to the estimates of the factor scores?

EXERCISE 10.9   Perform a factor analysis on the U.S. health data set (Table B.16) and estimate the factor scores.

EXERCISE 10.10   Redo Exercise 10.9 using the U.S. crime data in Table B.10. Compare the estimated factor scores of the two data sets.

EXERCISE 10.11   Analyze the vocabulary data given in Table B.17.