\(\mathbf{S_n} = \frac{23}{24} \mathbf{S} = \begin{pmatrix} 0.01061 & 0.00768 & 0.00782 \\ 0.00768 & 0.00615 & 0.00575 \\ 0.00782 & 0.00575 & 0.00649 \\ \end{pmatrix}\)

\(\mathbf{S_n} \approx \hat{\mathbf{L}} \hat{\mathbf{L}}^T + \hat{\mathbf{\Psi}}\), where \(diag(\mathbf{S_n}) = diag(\hat{\mathbf{L}} \hat{\mathbf{L}}^T) + diag(\hat{\mathbf{\Psi}})\)

Hence, \(\hat{\mathbf{\Psi}} = diag(\hat{\mathbf{\Psi}}) = diag(\mathbf{S_n} - \hat{\mathbf{L}} \hat{\mathbf{L}}^T)\)

\(\hat{\mathbf{\Psi}} = \begin{pmatrix} 0.000166 & 0.000000 & 0.000000 \\ 0.000000 & 0.000495 & 0.000000 \\ 0.000000 & 0.000000 & 0.000639 \\ \end{pmatrix}\)

\(\sigma_{ii} = \ell_{i1}^2 + \ell_{i2}^2 + ~...~ + \ell_{im}^2 + \psi_i\)

\(h_i^2 = \ell_{i1}^2 + \ell_{i2}^2 + ~...~ + \ell_{im}^2 = \ell_{i1}^2\)

\(h_1^2 = 0.0104, ~ h_2^2 = 0.0057, ~ h_3^2 = 0.0059\)

\(\frac{s_{11} + s_{22} + ~...~ + s_{pp}}{h_i^2} = 0.9441\)

\(\mathbf{S_n} - \mathbf{\hat{L}}\mathbf{\hat{L}}^T - \mathbf{\hat{\Psi}} = \begin{pmatrix} 0.000000 & -0.000166 & -0.000164 \\ -0.000495 & 0.000000 & -0.000493 \\ -0.000637 & -0.000637 & 0.000000 \\ \end{pmatrix}\)

RC1 | RC2 | RC3 | |
---|---|---|---|

YrHgt | 0.566 | 0.734 | 0.87 |

FtFrBody | 31.179 |
83.935 |
24.022 |

PrctFFB | 0.601 | 1.488 | 2.742 |

Frame | 0.296 | 0.39 | 0.415 |

BkFat | 0.014 | -0.006 | -0.055 |

SaleHt | 1.021 | 0.934 | 0.82 |

SaleWt | 122.551 |
39.068 |
-17.48 |

ML1 | ML2 | ML3 | |
---|---|---|---|

YrHgt | -0.001 | 0.581 | 1.629 |

FtFrBody | 21.802 |
84.458 |
31.383 |

PrctFFB | -0.253 | 2.155 | 1.056 |

Frame | 0.019 | 0.307 | 0.817 |

BkFat | 0.033 | -0.015 | -0.027 |

SaleHt | 0.498 | 0.841 | 1.532 |

SaleWt | 119.136 |
33.925 |
38.8 |

RC1 | RC3 | RC2 | |
---|---|---|---|

YrHgt | 0.941 |
0.27 | -0.082 |

FtFrBody | 0.447 | 0.794 |
0.205 |

PrctFFB | 0.262 | 0.859 |
-0.295 |

Frame | 0.938 |
0.219 | -0.028 |

BkFat | -0.231 | -0.339 | 0.812 |

SaleHt | 0.833 |
0.419 | 0.109 |

SaleWt | 0.352 | 0.43 | 0.722 |

ML1 | ML2 | ML3 | |
---|---|---|---|

YrHgt | 0.941 |
0.286 | 0.164 |

FtFrBody | 0.414 | 0.505 |
0.553 |

PrctFFB | 0.231 | 0.947 |
0.212 |

Frame | 0.891 |
0.251 | 0.18 |

BkFat | -0.256 | -0.514 |
0.273 |

SaleHt | 0.755 |
0.269 | 0.434 |

SaleWt | 0.253 | -0.05 | 0.879 |

The results obtained using covariance matrices are hard to interpret here. Since there are three kinds of units used: **pound**, **inch**, and **self-defined scales(1-8)**, the factor loadings on some variables are very large and others small.

Interpretation of the factors is straightforward using results obtained from the covariance matrix.

By (c) and (d), 1. factor 1 has larger loadings on **YrHgt**, **Frame**, and **SaleHt**, which are all related to the **height**s of the bulls. 2. Factor 2 has large loadings on **FtFrBody** and **PrctFFB** and is negatively related to **BkFat** in both methods(PC & ML) of factor analysis. It might be called a **lean** factor. 3. In both PC & ML method of factor analysis, the loadings of factor 3 is large on **SaleWt**. The loading is also large on **BkFat** from the PC method, and the loading is medium on **FtFrBody** from the ML method. Factor 3 might be related to the **weight**s of the bulls.

The point, **51**, in the scatter plot on the left and the point, **16**, in the scatter plot on the right seem to be outliers.