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Math4424: Homework 5

9.1. Show that the covariance matrix   1.0 .63 .45 ρ = .63 1.0 .35  .45 .35 1.0 for the p = 3 standardized random variables Z1 , Z2 and Z3 can be generated by the m = 1 factor model Z1 = .9F1 + ε1 , Z2 = .7F1 + ε2 , Z3 = .5F1 + ε3 where Var(F1 ) = 1 and Cov(ε, F1 ) = 0, and   .19 0 0 Ψ = Cov(ε) =  0 .51 0  . 0 0 .75 That is, write ρ in the form ρ = LL′ + Ψ. 9.3. The eigenvalues and eigenvectors of the correlation matrix ρ in 9.1 above are λ1 = 1.96,

e′1 = [0.625, 0.593, 0.507],

λ1 = 0.68,

e′2 = [−0.219, −0.491, 0.843],

λ1 = 0.36,

e′3 = [0.749, −0.638, −0.177].

(a). Assuming an m = 1 factor model, calculate the loading matrix L and matrix of specific variances Ψ using the principal component solution method. Compare the 1

results with those in 9.1 above. (b). What proportion of the total population variance is explained by the first common factor? 9.10. The correlation matrix for chicken-bone measurements (see Example 9.14) is 

1.000 0.505  0.569  0.602  0.621 0.603

1.000 0.422 0.467 0.482 0.450

 1.000 0.926 1.000 0.877 0.874 1.000 0.878 0.894 0.937 1.000

   .   

The following estimated factor loadings were extracted by the maximum likelihood procedure:

Variable 1. Skull length 2. Skull breadth 3. Femur length 4. Tibia length 5. Humerus length 6. Ulna length

Estimated factor loadings F1 F2 0.602 0.200 0.467 0.154 0.926 0.143 1.000 0.000 0.874 0.476 0.894 0.327

Varimax rotated estimated factor loadings F1∗ F 2∗ 0.484 0.411 0.375 0.319 0.603 0.717 0.519 0.855 0.861 0.499 0.744 0.594

Using the unrotated estimated factor loadings, obtain the maximum likelihood estimates of the following. (a). The specific variances. (b). The communalities. (c). The proportion of variance explained by each factor. b zL b′ − Ψ b z. (d). The residual matrix R − L z

9.12. The covariance matrix for the logarithms of turtle measurements (see

Example 8.4) is   11.072  S = 10−3  8.019 6.417 8.160 6.005 6.773 2

The following maximum likelihood estimates of the factor loading for an m = 1 model were obtained: Variable 1. ln(length) 2. ln(width) 3. ln(height)

Estimated factor loading F1 0.1022 0.0752 0.0765

Using the estimated factor loadings, obtain the maximum likelihood estimates of each of the following. (a). Specific variances. (b). Communalities. (c). Proportion of variance explained by the factor. bL b ′ − Ψ. b (d). The residual matrix Sn − L

(Hint: Convert S to Sn .)

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