Show that the covariance matrix
\rho =[■(1.0&0.63&0.45@0.63&1.0&0.35@0.45&0.35&1.0)]
For the P=3 standardized random variables Z_1,Z_2 and Z_3 can be generated by m=1 factor model.
Z_1=0⋅9F_1+\epsi _1
Z_2=0.7F_(1^(+\epsi ) 2)
z_3=0⋅5F_1+\epsi _3
Where var(F_1 )=1 ,cov(\epsi ,F_1 )=0 and
\phi =cov(\epsi )=[■(0.19&0&0@0&0.51&0@0&0&0.75)]
That is write \rho in the form \rho =LL^'+\phi
Use the information in (1) above to:
(a)Calculate communalities h_i^2 ,ⅈ=1,2,3 and interpret these quantities.
(b)Calculate Corr(Z_(i ),F_1 ) for ⅈ=1,2,3⋅Which variable might carry the greatest weight in “naming” the common factor? Why?
(3) The eigenvalues and eigenvectors of the correlation matrix \rho in (1) are
\lambda _1=1⋅96 , e_1^T=[■(0.625&0.593&0.507)]
\lambda _2=0⋅68 , e_2^T=[■(-0.219&-0.491&0.843)]
\lambda _3=0⋅36 , e_3^T=[■(0.749&-0.638&-0.177)]
(a)Assuming an m=1factor model, calculate the loading matrix L,
And matrix of specific variances \phi using the principal
Components solution method. Compare the results with those in (1)
(b)What proportion of total population variance is explained by the
The first common factor?
(4) Given \rho and \phi in (1) and m=1 factor model, calculate the reduced correlation matrix \rho ̃=\rho -\phi and the principal factor solution for the loading matrix L.Is the result consistent with the information in (1)? Should it be?
(5) The covariance matrix for the logarithms of turtle measurements is
S=〖10〗^(-3)\times [■(11.072&&@8.019&6.417&@8.160&6.005&6.773)]
The following maximum likelihood estimates of factor loadings for an m=1
Model were obtained.
VARIABLE ESTIMATED FACTOR LOADING (F1)
1.ln(length) 0.1022
2.ln(width) 0.0752
3.ln(height) 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 variances explained by the factor.
(d)The residual matrix S_n-L ̂L ̂^T-\phi ̂.
Hint: convert S to〖 S〗_n.